Singular Traces: Volume 2 Trace Formulas [2nd corr. and exten. edition] 9783110700176, 9783110700008

Table of contents :
Preface
Notations
Contents
Introduction
Part I: Trace and integral formulas
1 Bounded operators and pseudodifferential operators
2 Trace formulas
3 Integration formulas
Part II: The principal symbol mapping in noncommutative geometry
4 Integration formula for the noncommutative plane
5 A C∗-algebraic approach to principal symbols and trace formulas
6 Quantum differentiability for the Euclidean plane
Part III: Further applications
7 Connes character formula
8 Density of states
Bibliography
Index

Citation preview

Steven Lord, Edward McDonald, Fedor Sukochev, Dmitriy Zanin Singular Traces

De Gruyter Studies in Mathematics

�

Edited by Carsten Carstensen, Berlin, Germany Gavril Farkas, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Waco, Texas, USA Niels Jacob, Swansea, United Kingdom Zenghu Li, Beijing, China Guozhen Lu, Storrs, USA Karl-Hermann Neeb, Erlangen, Germany René L. Schilling, Dresden, Germany Volkmar Welker, Marburg, Germany

Volume 46/2

Steven Lord, Edward McDonald, Fedor Sukochev, Dmitriy Zanin

Singular Traces �

Volume 2: Trace Formulas 2nd edition

Mathematics Subject Classification 2020 46L51, 47L20, 58B34, 47B06, 47B10, 47G10, 47G30, 58J42 Authors Dr. Steven Lord University of Oxford Environmental Change Institute Oxford OX1 3QY United Kingdom [email protected]

Prof. Fedor Sukochev University of New South Wales School of Mathematics & Statistics Sydney, NSW 2052 Australia [email protected]

Dr. Edward McDonald University of New South Wales School of Mathematics & Statistics Sydney, NSW 2052 Australia [email protected]

Dr. Dmitriy Zanin University of New South Wales School of Mathematics & Statistics Sydney, NSW 2052 Australia [email protected]

ISBN 978-3-11-070000-8 e-ISBN (PDF) 978-3-11-070017-6 e-ISBN (EPUB) 978-3-11-070024-4 ISSN 0179-0986 Library of Congress Control Number: 2022952234 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2023 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface This book is dedicated to Alain Connes. Almost all the trace formulas in the book originate from Alain’s inspired use of singular traces in geometry and physics. This new work expands on the applications of singular traces in the survey [194], which in turn is built upon the original book about singular traces [193]. Writing an accessible but largely self-contained text has split our exposition about singular traces into three volumes. Volume I describes the complete characterization of traces on a separable Hilbert space based on Kalton’s commutator approach and Pietsch’s dyadic decomposition approach. Volume II concentrates on applications of singular traces and trace formulas. Volume III describes the semifinite theory of singular traces and some applications. Our intention is that much of the material from [193] will be absorbed into Volume III. The authors thank Eva-Maria Hekkelman for close reading of the text and editing. End notes to each chapter give historical background and credit of results. We apologize in advance for possible omissions. Sydney, Australia, 30 November 2022

https://doi.org/10.1515/9783110700176-201

Steven Lord Ed McDonald Fedor Sukochev Dmitriy Zanin

Notations ℕ ℤ ℤ+ ℤd+ ℝ ℝ+ ℝd |⋅| ℂ MN (ℂ) Ek,l Tr(A) det(A) SO(d) SU(d) ⊕ ⊗ ⊗ ⊗min ⋆ ⊖ ≺ ≺≺ ≺≺log s → a+ s → a− aα ↑ a (aα ↓ a) Γ log+ (x) f+ (x) sgn(f ) Ḃ r (ℝ) p,q

r Bp,q (ℝ) H ξ, η ⟨⋅, ⋅⟩ ‖⋅‖ ‖ ⋅ ‖∞ ℒ(H) {en }∞ n=0 ξ ⊗η 1 diag(x) [A, B] A∗ |A|

set of natural numbers set of integers set of nonnegative integers positive quadrant of ℤd field of real numbers set of positive real numbers Euclidean space of dimension d Euclidean norm on ℝd field of complex numbers algebra of square N × N complex matrices matrix units such that (Ek,l )i,j = δk,i δj,k , 1 ≤ i, j ≤ N trace of a matrix A determinant of a matrix A special orthogonal group in dimension d special unitary group in dimension d direct sum tensor product, usually the algebraic tensor product spatial tensor product of von Neumann algebras minimal tensor product convolution orthocomplement majorization submajorization logarithmic submajorization limit approaching from s > a limit approaching from s < a increasing (decreasing) net with respect to a partial order gamma function max{log |x|, 0} max{f (x), 0} sgn ∘ f where sgn is the sign function homogeneous Besov space

inhomogeneous Besov space separable complex Hilbert space elements of an abstract separable complex Hilbert space inner product (complex linear in the first variable) norm (usually the vector norm on a Hilbert space) operator (uniform) norm or supremum norm on a function space algebra of bounded operators on H orthonormal basis of H one-dimensional operator on H defined by (ξ ⊗ η)χ := ⟨χ, ξ⟩η identity map on H diagonal operator on H associated with x ∈ l∞ commutator of operators A and B adjoint of the operator A absolute value of the operator A

https://doi.org/10.1515/9783110700176-202

VIII � Notations

A≥0 A+ (A− ) ℜA (ℑA) A⊕n dom(A) ker(A) λ(A) μ(A) c0 c l∞ l1 l2 lp lp,∞ lp,q p∗ 𝒞00 (H) 𝒞0 (H) ℒ1 ℒ2 ℒp ℒp,q ℒ1,∞ ℒp,∞ (ℒp,∞ )0 ‖ ⋅ ‖p ‖ ⋅ ‖p,q Tr(A) Trω (A) φθ (A) C(X) C0 (X) Cb (X) (Ω, Σ, ν) L∞ (Ω) L0 (Ω) L1 (Ω) L2 (Ω) Lp (Ω) Lp,q (Ω) Ẇ k (Ω) p Wpk (Ω)

S(Ω)

E(Ω) E (p) (Ω)

operator A is positive positive (negative) part of a self-adjoint operator A real (imaginary) part of the operator A direct sum of n copies of the operator A domain of the operator A kernel of the operator A an eigenvalue sequence of the operator A singular value sequence, or singular value function, of A space of sequences converging to zero space of convergent sequences space of bounded sequences space of summable sequences Hilbert space of square-summable sequences space of p-summable sequences weak-lp sequence space Lorentz sequence space Hölder conjugate of p ideal of finite-rank operators on H ideal of compact operators on H ideal of trace class operators ideal of Hilbert–Schmidt operators Schatten ideal of compact operators ideal of compact operators corresponding to the lp,q Lorentz sequence space ideal of weak trace class operators weak-lp ideal of compact operators separable part of ℒp,∞ (quasi)norm, shorthand for ‖ ⋅ ‖lp , ‖ ⋅ ‖Lp , or ‖ ⋅ ‖ℒp (quasi)norm, shorthand for ‖ ⋅ ‖lp,q , ‖ ⋅ ‖Lp,q , or ‖ ⋅ ‖ℒp,q (standard) trace of the trace class operator A Dixmier trace of a compact operator A for an extended limit ω positive trace of a compact operator A for a Banach limit θ space of continuous functions on a compact Hausdorff topological space X space of continuous functions vanishing at infinity on a locally compact Hausdorff topological space X space of bounded continuous functions on a locally compact Hausdorff topological space X σ-finite measure space algebra of (equivalence classes of) bounded functions on Ω set of measurable functions on Ω space of (equivalence classes of) integrable functions on Ω Hilbert space of (equivalence classes of) square-integrable functions on Ω space of (equivalence classes of) p-integrable functions on Ω space of (equivalence classes of) functions on Ω with bounded p, q-quasinorm. homogeneous Sobolev space on Ω

inhomogeneous Sobolev space on Ω space of measurable functions on Ω such that its singular value function μ(t, f ) is finite for each t > 0 Banach subspace of S(Ω) p-convexification of E(Ω)

Notations �

χA Lp Mf supp(x) 𝒜′ τ Proj(𝒜) ∨ (∧) μ𝒜,τ (t, A) 𝒮(𝒜, τ) ℒp (𝒜, τ) ℒp,q (𝒜, τ) ℰ(𝒜, τ) EA nA K E ‖ ⋅ ‖E E0 ℰ ‖ ⋅ ‖ℰ ℰ0 ‖ ⋅ ‖E(L2 ) d

(E(L2 ))(ℝ ) φ φ ∘ diag ⟨t⟩, ⟨A⟩ σt lim a. c.- lim ω θ O(⋅) o(⋅) D ,D Tϕ 0 1 (X) If BS ‖ ⋅ ‖BS f [1] ℱ kA pA 𝕊d−1 𝕋d γi D Mk

IX

characteristic function of a ν-measurable set A ⊆ Ω Lebesgue Lp -space on (0, ∞) multiplication operator on L2 (X) for f ∈ L∞ (X) support of a sequence x, a complex-valued function x, or a distribution x commutant of a semifinite von Neumann algebra 𝒜 faithful normal semifinite trace on 𝒜 lattice of projections in a von Neumann algebra 𝒜 supremum (infimum) operation (in a lattice) generalized singular value function of A set of τ-measurable operators noncommutative Lp -space noncommutative Lorentz space quasi-Banach ideal in 𝒮(𝒜, τ) spectral measure of a self-adjoint operator A ∈ ℒ(H) spectral distribution function of an operator A ∈ ℒ(H) concavity modulus of a quasinorm quasi-Banach symmetric sequence or function space symmetric quasinorm on E separable part of E quasi-Banach ideal of compact operators symmetric norm on ℰ separable part of ℰ with E a quasi-Banach sequence space embedding continuously in l2 , K = [0, 1]d a fixed unit cube, and f ∈ L0 (ℝd ) locally square-integrable, ‖f ‖E(L2 ) = ‖{‖f ‖L2 (m+K ) }m∈ℤd ‖E(ℤd ) the set of measurable functions f on ℝd for which ‖f ‖E(L2 ) is finite trace in ℒ1,∞ symmetric functional on l1,∞ associated with a trace φ 1

1

abbreviation of (1 + |t|2 ) 2 for real t or (1 + |A|2 ) 2 for an operator A, respectively dilation action on function spaces on ℝd ordinary limit on c the value of a Banach limit on an almost convergent sequence in l∞ extended limit on l∞ Banach limit on l∞ big O notation for asymptotics little o notation for asymptotics double operator integral weak operator integral of an operator-valued function f the Birman–Solomyak class of functions the norm on the Birman–Solomyak class of functions the divided difference of f Fourier transform kernel of a pseudodifferential or Hilbert–Schmidt operator A symbol of a Hilbert–Schmidt operator A unit sphere in ℝd flat torus of dimension d Clifford matrices Dirac operator pointwise product operator for the coordinate tk , t ∈ ℝd

X � Notations

Dk Δ Δℱ ∇ ∇ℱ g(∇)

−i𝜕k , the Fourier dual of Mk Laplacian operator on ℝd or another (noncommutative) space Fourier dual of Δ differential operator (D1 , . . . , Dd ) on ℝd operator (M1 , . . . , Md ) on ℝd or Fourier dual of ∇ convolution operator ℱ ∗ Mg ℱ

𝒮 ′ (ℝd ) Hr (ℝd ) Ψm T† Sm σ Op(σ) Op0 (σ) Ψm cl σm Ψ−∞ S −∞ ResW Lmod;1 (ℝd ) Lmod;2 (ℝd ) ‖ ⋅ ‖mod S mod ℝdθ L∞ (ℝdθ ) τθ Lp (ℝdθ )

space of tempered distributions (the dual space of 𝒮(ℝd )) Sobolev Hilbert space of order r ∈ ℝ set of uniform pseudodifferential operators of order m formal adjoint of a pseudodifferential operator T set of uniform symbols of order m symbol of a pseudodifferential operator uniform pseudodifferential operator defined by a symbol σ that is a function on ℝd × ℝd uniform pseudodifferential operator defined by a symbol σ that is a function on ℝd ×𝕊d−1 set of classical uniform pseudodifferential operators of order m principal symbol of a classical uniform pseudodifferential operator set of smoothing operators set of symbols of smoothing operators noncommutative residue on classical pseudodifferential operators algebra of modulated functions Banach space of functions whose absolute value squared is a modulated function norm on the space of modulated operators Banach space of symbols of Laplacian modulated operators symbolic notation for the noncommutative plane von Neumann algebra of the noncommutative plane trace on L∞ (ℝdθ ) Lp -space on the noncommutative plane

Δg C m (Ω) ‖ ⋅ ‖C m (Ω) C ∞ (Ω) Cc∞ (ℝd ) 𝒮(ℝd ) pα,β

E(ℝdθ ) C0 (ℝdθ ) 𝒮(ℝdθ ) 𝒮 ′ (ℝdθ ) Lθ Wpk (ℝdθ ) ⋆θ ⋅θ Sp(θ, d) L∞ (𝕋dθ ) 𝜕sα |α| α! sym q

Laplace–Beltrami operator associated with a bounded Riemannian metric g on ℝd set of functions on Ω with m continuous derivatives (k) ‖f ‖C m (Ω) = ∑m k=0 ‖f ‖∞ set of smooth functions on Ω Fréchet space of smooth functions on ℝd with compact support Fréchet space of Schwartz functions on ℝd Schwartz function seminorms

symmetric ideal in L∞ (ℝdθ ) corresponding to the symmetric sequence space E algebra in L∞ (ℝdθ ) equivalent to the continuous functions vanishing at infinity Schwartz function space on the noncommutative plane distributions on the noncommutative plane Weyl transform Sobolev space on the noncommutative plane twisted convolution on 𝒮(ℝd ) Moyal product on 𝒮(ℝd ) symplectic group on ℝdθ von Neumann algebra of the noncommutative torus α α 𝜕s11 ⋅ ⋅ ⋅ 𝜕sdd for multiindex α α1 + ⋅ ⋅ ⋅ + αd for multiindex α α1 ! ⋅ ⋅ ⋅ αd ! for multiindex α (abstract) symbol map Calkin quotient map

Notations �

Π(𝒜1 , 𝒜2 ) -df Ch C n (𝒜) Cn (𝒜) HHn (𝒜) HHn (𝒜)

XI

C ∗ -algebra of bounded operators on a Hilbert space H generated by representations of C ∗ -algebras 𝒜1 and 𝒜2 quantized differential of f Connes–Chern character space of Hochschild chains on the algebra 𝒜 space of Hochschild cochains on the algebra 𝒜 Hochschild homology of the algebra 𝒜 Hochschild cohomology of the algebra 𝒜

Contents Preface � V Notations � VII Introduction � XVII

Part I: Trace and integral formulas 1 1.1 1.1.1 1.1.2 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.4 1.4.1 1.4.2 1.4.3 1.5 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.6 1.6.1 1.6.2

Bounded operators and pseudodifferential operators � 3 Bounded operators on Hilbert space and traces � 3 Singular values and ideals � 4 Traces on ℒ1,∞ � 7 Submajorization and interpolation � 12 Fully symmetric ideals � 12 Lorentz function spaces � 15 Noncommutative Lorentz spaces � 20 Operator inequalities � 24 Submajorization and majorization inequalities � 24 Hölder inequalities and commutators in Lorentz ideals � 31 Double operator integrals � 41 Integration of operator-valued functions in the weak operator topology � 41 Definition of a double operator integral � 47 Double operator integrals and estimates of commutators � 50 Compactness estimates of product-convolution operators � 54 Product-convolution operators on the Euclidean plane � 54 Hilbert–Schmidt operators on the Euclidean plane and the trace formula � 57 Compactness of product-convolution operators for square-integrable functions � 60 Abstract product-convolution estimates � 63 Estimates for product-convolution operators on the Euclidean plane � 74 Estimates for product-convolution operators on tori � 78 Pseudodifferential operators � 81 The Schwartz space and distributions � 82 Symbols and operators � 83

XIV � Contents 1.6.3 1.6.4 1.6.5 1.6.6 1.6.7 1.7

Sobolev spaces and mapping properties of pseudodifferential operators � 87 Inverses and complex powers of pseudodifferential operators � 92 Compactness estimates for pseudodifferential operators � 107 Asymptotic expansion of symbols � 110 The noncommutative residue � 115 Notes � 118

2 2.1 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.5 2.5.1 2.5.2 2.6

Trace formulas � 129 Introduction � 129 The Banach algebra of modulated functions � 132 Closure of the algebra of modulated functions � 133 Square modulated functions and behavior of the Fourier transform � 134 Laplacian modulated operators � 136 Symbols and characterization of Laplacian modulated operators � 136 Examples of Laplacian modulated operators � 141 Pseudodifferential operators are Laplacian modulated � 143 Eigenvalues of Laplacian modulated operators � 145 Trace formulas for Laplacian modulated operators � 152 Computing traces from symbols � 152 Connes’ trace theorem � 159 Notes � 162

3 3.1 3.2

Integration formulas � 167 Introduction � 167 Integration formula for elliptic differential operators and the curved plane � 170 Integration formula for elliptic differential operators � 171 Integration theorem on a curved plane � 177 Integration of square-integrable functions on 𝕋d � 180 Integration of compactly supported square-integrable functions on ℝd � 185 Integration formula on the noncommutative torus � 190 Definition of the noncommutative torus � 190 Differential calculus of the noncommutative torus � 191 Product-convolution operators for the noncommutative torus � 192 Integration of square-integrable functions on the noncommutative torus � 196 Notes � 197

3.2.1 3.2.2 3.3 3.4 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6

Contents

� XV

Part II: The principal symbol mapping in noncommutative geometry 4 4.1 4.2 4.2.1 4.2.2 4.2.3 4.2.4 4.3 4.4

4.5 4.6 4.7

Integration formula for the noncommutative plane � 209 Introduction � 209 Definition of the noncommutative plane � 214 Canonical commutation relations � 214 Distributions and the Schwartz space of the noncommutative plane � 222 Differentiation for the noncommutative plane � 231 Additional symmetries of the noncommutative plane � 239 Product-convolution estimates for the noncommutative plane � 241 Smooth product-convolution estimates for the noncommutative plane � 251 Integral formula for derivations of operators acting on invariant subspaces � 252 Product-convolution estimates for partial derivations on the noncommutative plane � 257 Integration formula up to a constant � 260 Measurability in the noncommutative plane � 266 Notes � 276

5 5.1 5.2 5.2.1 5.2.2 5.2.3 5.3 5.3.1 5.3.2 5.3.3 5.4 5.4.1 5.4.2 5.5 5.5.1 5.5.2 5.6 5.6.1 5.6.2 5.6.3 5.6.4

A C ∗ -algebraic approach to principal symbols and trace formulas � 278 Introduction � 278 Principal symbol mapping in the C ∗ -algebraic context � 281 C ∗ -norms on tensor products of C ∗ -algebras � 282 Principal symbol mapping using the Calkin quotient map � 283 An abstract trace formula � 286 Trace formula for zero-order pseudodifferential operators � 289 Symbol map on product-convolution operators on ℝd � 290 Extension of Connes’ trace theorem � 297 Failure of a symbol map for nonhomogeneous operators � 299 Principal symbol and trace formula for noncommutative tori � 302 Principal symbol map � 302 Trace formula � 310 Principal symbol and trace formula for the noncommutative plane � 314 Principal symbol map � 314 Trace formula � 321 Principal symbol and trace formula for SU(2) � 330 Differential calculus of SU(2) � 330 Principal symbol map � 334 Description of the symbol algebra for SU(2) � 338 Trace formula � 344

4.4.1 4.4.2

XVI � Contents 5.7 6 6.1 6.2 6.3 6.3.1 6.3.2 6.4 6.5 6.6 6.7

Notes � 347 Quantum differentiability for the Euclidean plane � 349 Introduction � 349 Compactness estimates for the quantized differential of a differentiable function � 354 Approximation of quantized differentials by principal terms � 361 Approximation of differentials � 361 Approximation of densities � 367 Integration of quantum densities � 371 Approximation of homogeneous Sobolev functions by smooth functions � 377 Trace formula for quantum densities on the Euclidean plane � 381 Notes � 386

Part III: Further applications 7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.3 7.4 7.5 7.6

Connes character formula � 391 Introduction � 391 Spectral triples and the Connes–Chern character � 391 Hochschild homology and cohomology � 394 Statement of the Connes character formula � 395 Exploiting Hochschild cohomology � 397 Commutator estimates � 406 Asymptotics for the heat semigroup � 412 Proof of the Connes character formula � 418 Notes � 425

8 8.1 8.2 8.3 8.4 8.5 8.6

Density of states � 426 Introduction � 426 Compactness estimate for the density of states � 433 The density of states trace formula � 442 Invariance under perturbation � 448 Density of states for homogeneous potentials � 451 Notes � 454

Bibliography � 457 Index � 469

Introduction In the late 1960s Jacques Dixmier constructed a trace on the ideal of bounded weak trace class operators from a separable infinite dimensional Hilbert space to itself. Alain Connes, one of the founders of noncommutative geometry, called it the Dixmier trace. The Dixmier trace vanishes on trace class operators and is therefore not an extension of the matrix trace of finite-rank operators. Dixmier’s construction was the start of the study of singular traces. From 1988, Alain Connes developed Dixmier’s trace and used it centrally in his noncommutative geometry. Connes demonstrated its role in a unique and remarkable theory of noncommutative integration based on differential geometry. Connes applied the Dixmier trace to extensions of classical Yang–Mills and Polyakov actions, extended the noncommutative residue of Adler, Manin, Wodzicki and Guillemin, showed an equivalent expression for the Hochschild class of the Chern character in terms of the Dixmier trace, recovered geometric measures on quasi-Fuschian groups using his quantized calculus, and introduced the fundamental relation between the noncommutative integral defined by the Dixmier trace and Voiculescu’s obstruction to the Berg–Weyl–von Neumann discretization theorem for tuples. The original presentation of Dixmier’s trace by Connes from 1989, or the monograph “Noncommutative Geometry” from 1994, is still referred to by most users in noncommutative geometry. However, due to Albrecht Pietsch’s work on dyadic decomposition and traces and Nigel Kalton’s work on symmetric functionals and the spectral characterization of commutators, with contributions of others, there is a complete description of traces on two-sided ideals of compact operators. Volume I of Singular Traces describes the characterization of traces on the ideal of weak trace class operators, including Dixmier traces, and formulas for the calculation of traces such as the heat trace and zeta-function residue formulas. This work (Volume II) provides applications of the results from Volume I. The applications are mainly associated with trace formulas from Alain Connes’ noncommutative geometry. Chapter 1 in Part I provides background on compactness of product-convolution operators on ℝd , pseudodifferential operators on ℝd , and introduces double operator integrals. Double operator integrals are a central tool for extending statements about pseudodifferential operators without the availability of the pseudodifferential calculus. Chapters 2 and 3 in Part I prove that traces on weak trace class operators extend the noncommutative residue of Adler, Manin, Wodzicki and Guillemin on the Euclidean plane ℝd and, in combination with the Laplace operator, extend Lebesgue integration on ℝd . Part II considers noncommutative versions of the residue and integral formulas on ℝd . The role of commutative algebras of functions on ℝd is replaced by a noncommutative algebra of operators acting on the Hilbert space L2 (ℝd ) called the noncommutative plane, or variants thereof for tori, and spheres in the form of SU(2). Traces of weak trace https://doi.org/10.1515/9783110700176-203

XVIII � Introduction class operators are associated with noncommutative versions of integration of principal symbols in phase space (Liouville measure), and the recovery of semifinite normal traces on algebras of noncommuting operators (integrals in the sense of Irving Segal and the standard theory of von Neumann algebras). Part III shows two applications to algebraic geometry and physics. It proves the Connes character formula in noncommutative geometry for any singular trace, and examines the density of states formula in mathematical physics for a Schrödinger operator with a real essentially bounded potential. Each chapter of the book has its own introduction and notes. Instead of repeating all results here, we give an overview of some of the main results of interest for the application of traces on weak trace class operators. Trace formulas for the Euclidean plane ℝd The trace formulas in the book are based on trace formulas of integral operators on ℝd . Let A ∈ ℒ2 (L2 (ℝd )) be a Hilbert–Schmidt operator on the Hilbert space L2 (ℝd ) of square-integrable functions on ℝd . As an integral operator, A has a symbol pA ∈ L2 (ℝd × ℝd ) such that d

(Au)(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ u)(ξ)dξ,

u ∈ L2 (ℝd ).

ℝd

Here ℱ denotes the Fourier transform d

(ℱ u)(ξ) := (2π)− 2 ∫ e−i⟨s,ξ⟩ u(s) ds,

ξ ∈ ℝd , u ∈ L2 (ℝd ).

ℝd

Volume I introduced traces on ideals of the ∗-algebra ℒ(L2 (ℝd )) of all bounded operators on the Hilbert space L2 (ℝd ). Definition. Let 𝒥 be a two-sided ideal of ℒ(L2 (ℝd )). A trace φ : 𝒥 → ℂ is a linear functional φ on 𝒥 that is unitarily invariant, φ(A) = φ(U ∗ AU),

A ∈ 𝒥 , U ∗ = U −1 ∈ ℒ(L2 (ℝd )).

A trace φ : 𝒥 → ℂ is positive if φ(A) ≥ 0 when A ≥ 0. The two-sided ideal ℒ2 (L2 (ℝd )) of Hilbert–Schmidt operators has only the zero functional as a trace. Two ideals composed of Hilbert–Schmidt operators are of principal interest for trace formulas. Definition. If A ∈ ℒ2 (L2 (ℝd )) is a Hilbert–Schmidt operator, then the singular value sequence μ(A) ∈ l2 is the sequence of eigenvalues of |A| listed with multiplicity in decreasing order. Here l2 denotes the space of square summable sequences.

Introduction

� XIX

The two-sided ideal ℒ1 (ℝd ) of trace class operators of the ∗-algebra ℒ(L2 (ℝd )) is defined by d

d

ℒ1 (ℝ ) := {A ∈ ℒ2 (L2 (ℝ )) : μ(A) ∈ l1 }.

Here l1 denotes the space of summable sequences. The two-sided ideal ℒ1,∞ (ℝd ) of weak trace class operators of the ∗-algebra ℒ(L2 (ℝd )) is defined by d

d

ℒ1,∞ (ℝ ) := {A ∈ ℒ2 (L2 (ℝ )) : μ(A) ∈ l1,∞ }.

Here l1,∞ denotes the space of weakly summable sequences, that is, the sequences whose singular value sequence is dominated by a multiple of (1 + n)−1 , n ≥ 0. A trace φ : ℒ1,∞ → ℂ is normalized if φ(A) = 1 for any positive weak trace class operator A such that μ(n, A) = (1 + n)−1 , n ≥ 0. Volume I proved that the space of positive traces on ℒ1 is one-dimensional and spanned by the trace Tr on trace class operators. Definition. An extended limit θ on the space of bounded sequences l∞ is a positive linear functional on l∞ such that θ(a) = limn→∞ a(n) for any convergent sequence a ∈ l∞ . A Banach limit is an extended limit such that θ(a) = θ(Sa) for any sequence a ∈ l∞ , where (Sa)(n) = a(n + 1), n ≥ 0, is the shift of a sequence a ∈ l∞ . Volume I proved that the space of continuous traces on ℒ1,∞ is infinite dimensional and bijective with the linear space spanned by the set of Banach limits on l∞ . Positive normalized traces on ℒ1,∞ correspond bijectively to Banach limits on l∞ , and the set of Dixmier traces on ℒ1,∞ corresponds to the set of factorizable Banach limits on l∞ . The next theorem collates Theorem 1.5.5, Theorem 2.1.1, and Theorem 2.1.2 from the text. They show how to compute the trace of the Hilbert–Schmidt operator A using the symbol pA . Theorem (Trace formulas on ℝd ). (a) Integrable symbol. Let A ∈ ℒ1 (L2 (ℝd )) be a trace class Hilbert–Schmidt operator with symbol pA ∈ L2 (ℝd × ℝd ). If pA ∈ L1 (ℝd × ℝd ) then Tr(A) = (2π)−d ⋅ ∫ ∫ pA (t, ξ)dtdξ. ℝd ℝd

(b) Modulated symbol. Let A ∈ ℒ2 (L2 (ℝd )) be a Hilbert–Schmidt operator with symbol pA ∈ L2 (ℝd × ℝd ). If pA has compact support in the first variable and

XX � Introduction 󵄨 󵄨2 sup s ⋅ ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ < ∞, s>0

1 d |ξ|>s d ℝ

then A ∈ ℒ1,∞ (L2 (ℝd )) and (i) Dixmier traces. Trω (A) = (2π)−d ⋅ ω(

1 log(2 + n)

∫ pA (t, ξ)dtdξ)

∫

1 d |ξ| 0, we have t

t

∫ μ(s, B)ds ≤ ∫ μ(s, A)ds. 0

0

A Banach ideal ℰ of ℒ(H) with norm ‖ ⋅ ‖ℰ is fully symmetric if B ≺≺ A, where B ∈ ℒ(H) and A ∈ ℰ , implies that B ∈ ℰ and ‖B‖ℰ ≤ ‖A‖ℰ . This section proves the association between submajorization and interpolation spaces between ℒ1 and ℒ(H). Theorem 1.2.2. Let T : ℒ(H) → ℒ(H) be a linear map. Then the following statements are equivalent: (a) 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩1 ≤ ‖A‖1 ,

A ∈ ℒ1 ,

and 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩∞ ≤ ‖A‖∞ ,

A ∈ ℒ(H).

1.2 Submajorization and interpolation

(b)

T(A) ≺≺ A,

�

13

A ∈ ℒ(H).

(c) For every fully symmetric Banach ideal ℰ of ℒ(H) with norm ‖ ⋅ ‖ℰ , T :ℰ →ℰ and 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩ℰ ≤ ‖A‖ℰ ,

A ∈ ℰ.

Proof. From the definition of a fully symmetric Banach ideal it follows that condition (b) implies condition (c). Condition (c) implies condition (a) since ℒ(H) and ℒ1 are fully symmetric Banach ideals in their respective norms. We prove that (a) implies (b). Assume that (a) holds and let A ∈ ℒ(H). Let A = A0 + A1 with A0 ∈ ℒ1 and A1 ∈ ℒ(H). For n ≥ 0, n+1

n+1

∫ μ(t, T(A)) dt = ∫ μ(t, T(A0 ) + T(A1 )) dt 0

0

n+1

n+1

≤ ∫ μ(t, T(A0 )) dt + ∫ μ(t, T(A1 )) dt. 0

0

The latter equality follows from Theorem 2.3.5 in Volume I. Then n+1

∞

n+1

∫ μ(t, T(A)) dt ≤ ∫ μ(t, T(A0 )) dt + ∫ μ(0, T(A1 )) dt 0

0

0

󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩T(A0 )󵄩󵄩󵄩1 + (n + 1)󵄩󵄩󵄩T(A1 )󵄩󵄩󵄩∞ .

Using (a), we get n+1

∫ μ(t, T(A)) dt ≤ ‖A0 ‖1 + (n + 1)‖A1 ‖∞ . 0

We now choose particular operators A0 and A1 . Let A = U|A| be a polar decomposition of A. Fix n > 0 and set A0 := U(|A| − μ(n + 1, A))+ ,

A1 := U min{|A|, μ(n + 1, A)}.

Then A = A0 + A1 , and μ(A0 ) = (μ(A) − μ(n + 1, A))+ ,

μ(A1 ) = min{μ(A), μ(n + 1, A)}.

14 � 1 Bounded operators and pseudodifferential operators Thus n+1

n+1

‖A0 ‖1 + (n + 1)‖A1 ‖∞ = ∫ μ(t, A) − μ(n + 1, A)dt + ∫ μ(n + 1, A)dt 0

0

n+1

= ∫ μ(t, A)dt. 0

Since n ≥ 0 is arbitrary, the statement follows. An exact interpolation space ℱ between ℒ1 and ℒ(H) is a Banach space ℱ such that ℒ1 ⊂ ℱ ⊂ ℒ(H)

with continuous embeddings, and if 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩1 ≤ ‖A‖1 ,

A ∈ ℒ1 ,

and 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩∞ ≤ ‖A‖∞ ,

A ∈ ℒ(H),

for a linear operator T : ℒ(H) → ℒ(H), then T :ℱ →ℱ and 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩ℱ ≤ ‖A‖ℱ . It follows from Theorem 1.2.2 that the exact interpolation spaces between ℒ1 and ℒ(H) and the fully symmetric ideals of ℒ(H) are the same. Corollary 1.2.3. Let H be a separable Hilbert space. The following are equivalent: (a) ℱ is an exact interpolation space for (ℒ1 , ℒ(H)). (b) ℱ is a fully symmetric ideal of ℒ(H). Proof. From Theorem 1.2.2 it follows that statement (b) implies statement (a). By the Calkin correspondence and Theorem II.4.1 of [180], a fully symmetric ideal ℱ embeds continuously between ℒ1 and ℒ(H). Then Theorem 1.2.2 implies that every linear operator T : ℒ(H) → ℒ(H) that is continuous on ℒ1 and ℒ(H) with operator norm 1 is also continuous on ℱ with operator norm 1. From Theorem 1.2.2 it also follows that statement (a) implies statement (b). Assume that ℱ is an exact interpolation space. Let B, C ∈ ℒ(H). The linear map

1.2 Submajorization and interpolation

TB,C (A) := BAC,

�

15

A ∈ ℒ(H),

satisfies 󵄩󵄩 󵄩 󵄩󵄩TB,C (A)󵄩󵄩󵄩∞ ≤ (‖B‖∞ ‖C‖∞ ) ⋅ ‖A‖∞ ,

A ∈ ℒ(H),

and 󵄩󵄩 󵄩 󵄩󵄩TB,C (A)󵄩󵄩󵄩1 ≤ (‖B‖∞ ‖C‖∞ ) ⋅ ‖A‖1 ,

A ∈ ℒ1 .

Hence, by the condition that ℱ is an exact interpolation space, BAC ∈ ℱ when A ∈ ℱ , and ‖BAC‖ℱ ≤ ‖B‖∞ ‖C‖∞ ‖A‖ℱ . Hence ℱ is a Banach ideal with symmetric norm. Now assume that B ∈ ℒ(H), A ∈ ℱ , and B ≺≺ A. By Theorem 2.2 of [116] there exists a linear map T : ℒ(H) → ℒ(H) that is continuous on ℒ(H) and ℒ1 with operator norm 1 such that T(A) = B. By the condition that ℱ is an interpolation space, B = T(A) ∈ ℱ when A ∈ ℱ , and 󵄩 󵄩 ‖B‖ℱ = 󵄩󵄩󵄩T(A)󵄩󵄩󵄩ℱ ≤ ‖A‖ℱ . Thus ℱ is a fully symmetric ideal of ℒ(H).

1.2.2 Lorentz function spaces Similarly to Definition 1.1.2, we can define the decreasing rearrangement of a measurable function f on a σ-finite measure space (Ω, Σ, ν). Let L0 (Ω) denote the set of measurable functions on Ω. Definition 1.2.4. The singular value function μ(f ) of f ∈ L0 (Ω), also called the decreasing rearrangement of f , is defined by 󵄩 󵄩 μ(t, f ) := inf {󵄩󵄩󵄩f ⋅ (1 − χF )󵄩󵄩󵄩∞ : ν(F) ≤ t}, F∈Σ

t > 0,

where the infimum, if it exists, is taken over the σ-algebra Σ. The space of functions f ∈ L0 (Ω) such that μ(t, f ) is finite for each t > 0 is denoted by S(Ω).

16 � 1 Bounded operators and pseudodifferential operators Note that the singular value function μ(f ) depends on the measure ν. With the decreasing rearrangement of a function f ∈ S(Ω), we can define the Lorentz function spaces Lp,q (Ω), p, q ∈ (0, ∞). Example 1.2.5 (Lorentz function spaces). For p, q ∈ (0, ∞), denote by Lp,q (Ω) the Lorentz function space defined as the subspace of f ∈ S(Ω) such that ∞

‖f ‖p,q := ( ∫ s

q −1 p

1 q

q

μ(s, f ) ds) < ∞.

0

This is modified when p or q is infinite as 1 p

∞

‖f ‖p,∞ := sup t μ(t, f ), t>0

1 q

‖f ‖∞,q := ( ∫ s−1 μ(s, f )q ds) . 0

The function spaces Lp,q (Ω) are quasinormed spaces given the above quasinorms. The spaces Lp (Ω) are the Lorentz spaces Lp,p (Ω), p ∈ (0, ∞]. The context should make clear whether ‖⋅‖p refers to the Lorentz norm of a function, an operator, or a sequence. When it is necessary to make the distinction clear, we will use the notation ‖ ⋅ ‖Lp,q to refer to the Lorentz norm of a function. The Hölder inequality for Lorentz function spaces can be found in [215]. Theorem 1.2.6. Let 0 < p1 , p2 < ∞ and 0 < q1 , q2 ≤ ∞, and let 1 1 1 = + , p p1 p2

1 1 1 = + . q q1 q2

If f ∈ Lp1 ,q1 (Ω) and g ∈ Lp2 ,q2 (Ω), then fg ∈ Lp,q (Ω), and ‖fg‖p,q ≤ cp1 ,p2 ,q1 ,q2 ⋅ ‖f ‖p1 ,q1 ‖g‖p2 ,q2 for a constant cp1 ,p2 ,q1 ,q2 > 0. Fully symmetric function spaces are defined using decreasing rearrangements and submajorization. Definition 1.2.7. If f , g ∈ S(Ω), then g is submajorized by f (denoted g ≺≺ f ) if t

t

∫ μ(s, g)ds ≤ ∫ μ(s, f )ds, 0

0

t > 0.

1.2 Submajorization and interpolation

� 17

A Banach subspace E(Ω) of S(Ω) with norm ‖ ⋅ ‖E(Ω) is called fully symmetric if g ≺≺ f , where g ∈ S(Ω) and f ∈ E(Ω), implies that g ∈ E(Ω) and ‖g‖E(Ω) ≤ ‖f ‖E(Ω) . Theorem 3.1.2 of [193] states that a fully symmetric function space E(Ω) is obtained from a fully symmetric function space E ⊂ S(0, ∞) by 󵄩 󵄩 ‖f ‖E(Ω) = 󵄩󵄩󵄩μ(f )󵄩󵄩󵄩E .

E(Ω) = {f ∈ S(Ω) : μ(f ) ∈ E},

Here the measure on the semiaxis (0, ∞) is the Lebesgue measure. The measure on the Euclidean space ℝd , or any Borel subset of ℝd , will be the Lebesgue measure unless stated otherwise. The decreasing rearrangement μ : S(Ω) → S(0, ∞) is given in Definition 1.2.4. If E is a fully symmetric space of functions on the semiaxis (0, ∞), then it follows from Theorem 3.1.2 of [193] and the Calkin correspondence that there is a fully symmetric ideal ℰ in ℒ(H) such that 󵄩 󵄩 ‖A‖ℰ = 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩E .

ℰ = {A ∈ ℒ(H) : μ(A) ∈ E},

Here the singular value function of an operator A ∈ ℒ(H) is given by Definition 1.1.2. Definition 1.2.8. Let E ⊂ S(0, ∞) be a fully symmetric function space, and let p ≥ 1. Define 󵄩 󵄩1/p ‖f ‖E(p) (Ω) := 󵄩󵄩󵄩μ(f )p 󵄩󵄩󵄩E

E (p) (Ω) := {f ∈ S(Ω) : μ(f )p ∈ E}, and ℰ

(p)

:= {A ∈ ℒ(H) : μ(A)p ∈ E},

󵄩 󵄩1/p ‖A‖ℰ (p) := 󵄩󵄩󵄩μ(A)p 󵄩󵄩󵄩E .

From the preceding remarks, E (p) (Ω) and ℰ (p) are the p-convexifications of the Banach spaces E(Ω) and ℰ , respectively. If A ∈ ℒ(H) and f ∈ S(Ω), then we write A ≺≺ f if μ(A) ≺≺ μ(f ),

18 � 1 Bounded operators and pseudodifferential operators where the singular value function of an operator A on the left side is given by Definition 1.1.2, and the singular value function of a measurable function f on Ω with finite distribution function on the right side is given by Definition 1.2.4. The following theorem extends Theorem 1.2.2. When p = 1, the statements in Theorem 1.2.9 become equivalent. Theorem 1.2.9. Let p ≥ 1, and let T : (Lp + L∞ )(Ω) → ℒ(H) be a linear map. Then (a) ⇒ (b) ⇒ (c) for the following statements: (a) 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℒp ≤ ‖f ‖Lp (Ω) ,

f ∈ Lp (Ω),

and 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℒ(H) ≤ ‖f ‖L∞ (Ω) , (b)

󵄨󵄨 󵄨p p−1 p 󵄨󵄨T(f )󵄨󵄨󵄨 ≺≺ 2 ⋅ |f | ,

f ∈ L∞ (Ω).

f ∈ (Lp + L∞ )(Ω).

(c) For every fully symmetric Banach space E of functions on (0, ∞), T : E (p) (Ω) → ℰ (p) and p−1 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℰ (p) ≤ 2 p ⋅ ‖f ‖E(p) (Ω) ,

f ∈ E (p) (Ω),

where ℰ (p) is the ideal of ℒ(H) corresponding to the p-convexification E (p) of E. Proof. From the definition of ℰ it follows that condition (b) implies condition (c). Assume that (b) holds, and let f ∈ E (p) (Ω). Then μ(f )p ∈ E. Since p

μ(T(f )) ≺≺ 2p−1 ⋅ μ(f )p by assumption and E is fully symmetric, we have p

μ(T(f )) ∈ E and p󵄩 󵄩󵄩 p−1 󵄩 p󵄩 󵄩󵄩μ(T(f )) 󵄩󵄩󵄩E ≤ 2 ⋅ 󵄩󵄩󵄩μ(f ) 󵄩󵄩󵄩E .

Then

1.2 Submajorization and interpolation 1

�

19

1

p−1 p−1 p󵄩 p 󵄩󵄩 󵄩 󵄩 󵄩 p󵄩 p 󵄩󵄩T(f )󵄩󵄩󵄩ℰ (p) = 󵄩󵄩󵄩μ(T(f )) 󵄩󵄩󵄩E ≤ 2 p ⋅ 󵄩󵄩󵄩μ(f ) 󵄩󵄩󵄩E = 2 p ⋅ ‖f ‖E(p) (Ω) .

We prove that (a) implies (b). Assume that (a) holds, and let f ∈ (Lp + L∞ )(Ω). Let f = f0 + f1 with f0 ∈ Lp (Ω) and f1 ∈ L∞ (Ω). Fix t > 0. Using the linearity of T, we have t

t

p

p

∫ μ(s, T(f )) ds = ∫ μ(s, T(f0 ) + T(f1 )) ds. 0

0

From the Fan inequalities (see Corollary 2.2.9 in Volume I or Section 1.3.1) we have t

t

p

p

∫ μ(s, T(f0 ) + T(f1 )) ds ≤ ∫(μ(s, T(f0 )) + μ(0, T(f1 ))) ds 0

0

t

t

p

p

≤ 2p−1 (∫ μ(s, T(f0 )) ds + ∫ μ(0, T(f1 )) ds) 0

0

󵄩 󵄩p 󵄩 󵄩p ≤ 2p−1 (󵄩󵄩󵄩T(f0 )󵄩󵄩󵄩p + t 󵄩󵄩󵄩T(f1 )󵄩󵄩󵄩∞ ). Using the assumption in part (a) that ‖T(f0 )‖p ≤ ‖f0 ‖p and ‖T(f1 )‖∞ ≤ ‖f1 ‖∞ , we obtain t

p

∫ μ(s, T(f )) ds ≤ 2p−1 (‖f0 ‖pp + t‖f1 ‖p∞ ). 0

Choose f0 := sgn(f )(|f | − μ(t, f ))+ ,

f1 := sgn(f ) min{|f |, μ(t, f )}.

Then f = f0 + f1 , and μ(f0 ) = (μ(f ) − μ(t, f ))+ ,

μ(f1 ) = min{μ(f ), μ(t, f )}.

Thus t

p

t

‖f0 ‖pp + t‖f1 ‖p∞ = ∫(μ(s, f ) − μ(t, f )) ds + ∫ μ(t, f )p ds. 0

0

From the pointwise inequality p

(μ(s, f ) − μ(t, f )) + μ(t, f )p ≤ μ(s, f )p , it follows that

0 < s < t,

(1.2)

20 � 1 Bounded operators and pseudodifferential operators

‖f0 ‖pp

+

t‖f1 ‖p∞

t

≤ ∫ μ(s, f )p ds.

(1.3)

0

Since t > 0 is arbitrary, the statement follows from (1.2) and (1.3). Let p ≥ 1. An interpolation pair (𝒥 , X) between the couples (ℒp , ℒ(H)) and (Lp (Ω), L∞ (Ω)) is a pair of Banach spaces (𝒥 , X) such that ℒp ⊂ 𝒥 ⊂ ℒ(H),

Lp (Ω) ∩ L∞ (Ω) ⊂ X ⊂ Lp (Ω) + L∞ (Ω)

with continuous embeddings, and if 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℒp ≤ ‖f ‖Lp (Ω) ,

f ∈ Lp (Ω),

and 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩ℒ(H) ≤ ‖f ‖L∞ (Ω) ,

f ∈ L∞ (Ω),

for a linear operator T : Lp (Ω) + L∞ (Ω) → ℒ(H), then T :X →𝒥 and 󵄩󵄩 󵄩 󵄩󵄩T(f )󵄩󵄩󵄩𝒥 ≤ cp,𝒥 ,X ⋅ ‖f ‖X for some constant cp,𝒥 ,X > 0. By Theorem 1.2.9 the pair (ℰ (p) , E (p) (Ω)) associated with the p-convexification of a fully symmetric function space E = E(0, ∞) forms an interpolation pair for the couples (ℒp , ℒ(H)) and (Lp (Ω), L∞ (Ω)).

1.2.3 Noncommutative Lorentz spaces Let 𝒜 be a von Neumann algebra in ℒ(H) for separable Hilbert space H. Let τ be a faithful normal semifinite trace on 𝒜. The set of projections Proj(𝒜) := {p ∈ 𝒜 : p∗ = p = p2 } is a lattice with supremum and infimum of p, q ∈ Proj(𝒜) given by p ∨ q := inf{r ∈ Proj(𝒜) : r ≥ p, r ≥ q}, p ∧ q := sup{r ∈ Proj(𝒜) : r ≤ p, r ≤ q}.

1.2 Submajorization and interpolation

� 21

By Vigier’s theorem (Theorem 2.1.1 in [193]) Proj(𝒜) is a complete sublattice of Proj(ℒ(H)) [265, p. 41, 69] with ⋁ pi ∈ Proj(𝒜) i∈I

and ⋀ pi ∈ Proj(𝒜), i∈I

for any net of projections {pi }i∈I ⊂ Proj(𝒜). Any net of pairwise orthogonal projections {pk }∞ k=0 in ℒ(H) is countable ([273, Proposition 3.19]), and the faithful normal semifinite trace is countably additive for pairwise orthogonal projections, ∞

∞

k=0

k=0

τ( ⋁ pk ) = ∑ τ(pk ). The next example shows that the lattice of projections Proj(𝒜) generalizes the structure of a σ-algebra and that τ generalizes a σ-finite measure. Example 1.2.10. Let (Ω, Σ, ν) be a σ-finite measure space. Let L2 (Ω) denote the Hilbert space of square-integrable functions on Ω, and let L∞ (Ω) denote the von Neumann algebra of essentially bounded functions acting as product operators on L2 (Ω). (a) Every set F ∈ Σ defines a projection χF ∈ L∞ (Ω). For every projection p ∈ L∞ (Ω), there exists F ∈ Σ such that p = χF . (b) If p1 = χF1 and p2 = χF2 , then p1 ∨ p2 = χF1 ∪F2 ,

p1 ∧ p2 = χF1 ∩F2 .

(c) The integral τν (f ) := ∫ fdν,

f ∈ L∞ (Ω),

Ω

is a faithful normal semifinite trace on L∞ (Ω) such that τν (χF ) = ν(F),

F ∈ Σ.

To define noncommutative Lorentz spaces in the same fashion as Examples 1.1.6 and 1.2.5, we require the equivalent of the set S(Ω) of measurable functions that admit a decreasing rearrangement (Definition 1.2.4). Let ′

𝒜 := {B ∈ ℒ(H) : AB = BA, ∀A ∈ 𝒜}

denote the commutant of 𝒜.

22 � 1 Bounded operators and pseudodifferential operators Definition 1.2.11. Let 𝒜 be a von Neumann algebra in ℒ(H) for separable Hilbert space H. Let τ be a faithful normal semifinite trace on 𝒜. A closed and densely defined operator A : dom(A) → H is said to be (a) affiliated with 𝒜 if it commutes with every unitary operator from the commutant 𝒜′ of 𝒜, (b) τ-measurable if it is affiliated with 𝒜 and if, for every ε > 0, there exists a projection p ∈ Proj(𝒜) such that (1 − p)H ⊂ dom(A) and τ(p) ≤ ε. The set of all τ-measurable operators is denoted by 𝒮 (𝒜, τ). Given the structure of the lattice of projections Proj(𝒜) and a faithful normal semifinite trace τ, we can generalize the singular value functions from Definitions 1.1.2 and 1.2.4. Definition 1.2.12. Let 𝒜 be a von Neumann algebra in ℒ(H) for a separable Hilbert space H. Let τ be a faithful normal semifinite trace on 𝒜. For every A ∈ 𝒮 (𝒜, τ), the generalized singular value function μ𝒜,τ (A) of A is defined by 󵄩 󵄩 μ𝒜,τ (t, A) := inf{󵄩󵄩󵄩A(1 − p)󵄩󵄩󵄩∞ : p ∈ Proj(𝒜), τ(p) ≤ t},

t > 0.

Remark 1.2.13. (a) If p ∈ Proj(𝒜) is a projection such that τ(p) ≤ t as in Definition 1.2.11(b), then A(1 − p) : H → H is a closed operator. By the closed graph theorem, A(1 − p) is a bounded operator. Hence the operator norm ‖A(1 − p)‖∞ is finite, and the infimum in the definition of μ𝒜,τ (A) exists. Thus, when A ∈ 𝒮 (𝒜, τ), the function μ𝒜,τ (A) is everywhere defined for t > 0. (b) The singular value function μ𝒜,τ (t, A) is bounded as t → 0+ if and only if A ∈ 𝒜. In this case, we have μ𝒜,τ (0, A) = ‖A‖∞ . Following Section 1.2.2, let E be a fully symmetric space of functions in S(0, ∞). Then ℰ (𝒜, τ) := {𝒮 (𝒜, τ) : μ(A) ∈ E},

󵄩 󵄩 ‖A‖ℰ(𝒜,τ) := 󵄩󵄩󵄩μ𝒜,τ (A)󵄩󵄩󵄩E

is a Banach bimodule of 𝒜 within 𝒮 (𝒜, τ) that is fully symmetric in the sense that μ𝒜,τ (B) ≺≺ μ𝒜,τ (A) when B ∈ 𝒮 (𝒜, τ) and A ∈ ℰ (𝒜, τ) implies that B ∈ ℰ (𝒜, τ) and ‖B‖ℰ(𝒜,τ) ≤ ‖A‖ℰ(𝒜,τ) (see Chapter 3 of [193]). Example 1.2.14 (Noncommutative Lorentz space). Let 0 < p, q < ∞. Define ∞

‖A‖p,q := ( ∫ s 0

q −1 p

1 q

μ𝒜,τ (s, A)q ds) = ‖μ𝒜,τ ‖p,q .

1.2 Submajorization and interpolation

�

23

This is modified when p or q is infinite as ∞

1 p

‖A‖p,∞ := sup t μ𝒜,τ (t, A), t>0

1 q

‖A‖∞,q := ( ∫ s−1 μ𝒜,τ (s, A)q ds) . 0

Let 0 < p, q ≤ ∞. Define ℒp,q (𝒜, τ) := {A ∈ 𝒮 (𝒜, τ) : ‖A‖p,q < ∞}.

Denote ℒp (𝒜, τ) := ℒp,p (𝒜, τ), and note from Remark 1.2.13 that ℒ∞ (𝒜, τ) = 𝒜. Note that the faithful normal semifinite trace τ has an extension from its finite domain to ℒ1 (𝒜, τ) by defining ∞

τ(A) := ∫ μ𝒜,τ (s, A)ds,

A ∈ ℒ1 (𝒜, τ).

0

It follows that 1

‖A‖p = τ(|A|p ) p ,

A ∈ ℒp (𝒜, τ).

If E is a fully symmetric space of functions on (0, ∞), then the noncommutative operator space ℰ (ℒ(H), Tr) is a fully symmetric ideal as introduced in Section 1.2.1. Example 1.2.15. (a) Let H be a separable Hilbert space. For the von Neumann algebra ℒ(H) with faithful normal semifinite trace Tr, 𝒮 (ℒ(H), Tr) is the set ℒ(H). The singular value function from Definition 1.2.12 on 𝒮 (ℒ(H), Tr) and the singular value function from Definition 1.1.2 on A ∈ ℒ(H) coincide, which we denote by μℒ(H),Tr (t, A),

t > 0,

when required. Let 0 < p, q ≤ ∞. The Lorentz ideals ℒp,q in Example 1.1.6 are equivalent to ℒp,q (ℒ(H), Tr). (b) Let (Ω, Σ, ν) be a σ-finite measure space with faithful normal semifinite trace τν on L∞ (Ω) as in Example 1.2.10. For the von Neumann algebra L∞ (Ω) acting on the Hilbert space L2 (Ω), 𝒮 (L∞ (Ω), τν ) is the set S(Ω) from Definition 1.2.4. The decreasing rearrangement of a function f ∈ S(Ω) from Definition 1.2.4 and the singular value function from Definition 1.2.12 on 𝒮 (L∞ (Ω), τν ) coincide, which we denote by μL∞ (Ω),τν (t, f ),

t > 0,

when required. Let 0 < p, q ≤ ∞. The Lorentz function spaces Lp,q (Ω) in Example 1.2.5 are equivalent to ℒp,q (L∞ (Ω), τν ).

24 � 1 Bounded operators and pseudodifferential operators The following theorem generalizes both Theorems 1.2.2 and 1.2.9. When p = 1, the statements in Theorem 1.2.16 become equivalent. The proof is omitted since it identically repeats the proofs of Theorems 1.2.2 and 1.2.9. Theorem 1.2.16. Let 𝒜1 and 𝒜2 be von Neumann algebras with faithful normal semifinite traces τ1 and τ2 , and let p ≥ 1. If T : (ℒp + ℒ∞ )(𝒜1 , τ1 ) → (ℒp + ℒ∞ )(𝒜2 , τ2 ) is a linear map, then (a) ⇒ (b) ⇒ (c) for the following statements: (a) 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩ℒ

p (𝒜2 ,τ2 )

A ∈ ℒp (𝒜1 , τ1 ),

≤ ‖A‖ℒp (𝒜1 ,τ1 ) ,

and 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩𝒜2 ≤ ‖A‖𝒜1 , (b)

p

μ𝒜2 ,τ2 (T(A)) ≺≺ 2p−1 μ𝒜1 ,τ1 (A)p ,

A ∈ 𝒜1 . A ∈ (ℒp + ℒ∞ )(𝒜1 , τ1 ).

(c) For every fully symmetric Banach space E of functions on (0, ∞), T : ℰ (p) (𝒜1 , τ1 ) → ℰ (p) (𝒜2 , τ2 ), and p−1 󵄩󵄩 󵄩 󵄩󵄩T(A)󵄩󵄩󵄩ℰ (p) (𝒜2 ,τ2 ) ≤ 2 p ⋅ ‖A‖ℰ (p) (𝒜1 ,τ1 ) ,

A ∈ ℰ (p) (𝒜1 , τ1 ),

where ℰ (p) (𝒜1 , τ1 ) and ℰ (p) (𝒜2 , τ2 ) are the bimodules of 𝒜1 and 𝒜2 , respectively, corresponding to E (p) .

1.3 Operator inequalities This section provides some operator inequalities based on submajorization and logarithmic submajorization. Operator inequalities for the Lorentz ideals ℒp,q from Example 1.1.6 are also presented. These inequalities will be used in later chapters.

1.3.1 Submajorization and majorization inequalities Let H be a separable Hilbert space. In Section 1.2.1, B ∈ ℒ(H) was submajorized by A ∈ ℒ(H) (denoted B ≺≺ A) if for all n ≥ 0,

1.3 Operator inequalities n

n

k=0

k=0

�

25

∑ μ(k, B) ≤ ∑ μ(k, A).

Here μ(A) and μ(B) are the singular value functions of the operators A and B, respectively. The summation criteria is equivalent to the integral form of submajorization given in Section 1.2.1, since the singular value function of a bounded operator from Definition 1.1.2 is a step function. We recall from Chapter 2 of Volume I the notion of logarithmic submajorization. Given two bounded operators A, B ∈ ℒ(H), we say that B is logarithmically submajorized by A (denoted B ≺≺log A) if for all n ≥ 0, n

n

k=0

k=0

∏ μ(k, B) ≤ ∏ μ(k, A).

Inequalities for powers of positive operators The following statement concerning submajorization of the absolute value of products of positive operators is proved on p. 478 in [178]. Theorem 1.3.1 (Araki–Lieb–Thirring inequality). If A, B ∈ ℒ(H) are positive and if r ≥ 1, then |AB|r ≺≺log Ar Br . Since any quasi-Banach ideal of ℒ(H) is closed under logarithmic submajorization (Proposition 2.4.18 in Volume I), the following is an immediate consequence of Theorem 1.3.1. Corollary 1.3.2. Let 𝒥 be a quasi-Banach ideal with quasinorm ‖ ⋅ ‖𝒥 . If r ≥ 1 and if A, B ∈ ℒ(H) are positive and such that Ar Br ∈ 𝒥 , then |AB|r ∈ 𝒥 . Moreover, there is a constant c𝒥 > 0 such that 󵄩 󵄩 ‖|AB|r ‖𝒥 ≤ c𝒥 ⋅ 󵄩󵄩󵄩Ar Br 󵄩󵄩󵄩𝒥 . The following submajorization estimate is needed in the proof of Lemma 1.3.18. The historical development of this lemma is discussed in the end notes. Theorem 1.3.3 (Birman–Koplienko–Ricard–Solomyak inequality). If 0 ≤ A, B ∈ ℒ(H) and if θ ∈ (0, 1) and p > 0, then there is a constant cp,θ > 0 such that 󵄨󵄨 θ θ 󵄨p pθ 󵄨󵄨A − B 󵄨󵄨󵄨 ≺≺ cp,θ ⋅ |A − B| . Theorem 1.3.3 has a similar statement for commutators. Theorem 1.3.4 (Birman–Koplienko–Ricard–Solomyak inequality for commutators). If A, B ∈ ℒ(H), A ≥ 0, and if θ ∈ (0, 1) and p > 0, then there is a constant cp,θ > 0 such that

26 � 1 Bounded operators and pseudodifferential operators p(1−θ) 󵄨 󵄨󵄨 θ 󵄨󵄨p 󵄨pθ 󵄨󵄨[A , B]󵄨󵄨 ≺≺ cp,θ ⋅ ‖B‖ℒ(H) 󵄨󵄨󵄨[A, B]󵄨󵄨󵄨 .

If A, B ∈ ℒ(H) are positive compact operators, then μ(n, A1/2 BA1/2 ) = λ(n, AB),

n ≥ 0,

where λ(AB) is the eigenvalue sequence of AB. Weyl’s lemma (Lemma 1.1.20 in Volume I), which states that an eigenvalue sequence is submajorized by the singular value sequence, then proves that A1/2 BA1/2 ≺≺ AB. The following theorem extends this submajorization inequality and is used in Chapter 4. Theorem 1.3.5. If A, B ∈ ℒ(H) are positive and if θ ∈ (0, 1), then Aθ BA1−θ ≺≺ AB. Direct sums and majorization Let (Ω, Σ, ν) be a σ-finite measure space. Definition 1.3.6. If 0 ≤ f , g ∈ L1 (Ω) are positive functions, then g is majorized by f (denoted g ≺ f ) if g is submajorized by f (g ≺≺ f as defined in Section 1.2.2) and ∞

∞

∫ μ(s, g)ds = ∫ μ(s, f )ds. 0

0

Here the singular value function μ(f ) of a measurable function f on Ω with finite distribution function is given by Definition 1.2.4. The following lemma is Lemma 2 in [201]. Lemma 1.3.7. Let 0 ≤ f , g ∈ L1 ([a, b]) be such that t

t

∫ g(s)ds ≤ ∫ f (s)ds, a

a < t < b,

a

and b

b

∫ g(s)ds = ∫ f (s)ds. a

a

Let g be decreasing on [a, b]. If ϕ ∈ C 1 (0, ∞) is a convex function, then

1.3 Operator inequalities b

�

27

b

∫ ϕ(g(s))ds ≤ ∫ ϕ(f (s))ds. a

a

If ϕ ∈ C 1 (0, ∞) is concave, the inequality is reversed. Proof. Suppose ϕ ∈ C 1 (0, ∞). Set t

A(t) := ∫(g(s) − f (s))ds,

a < t < b.

a

By assumption, A(t) ≤ 0 and A(b) = A(a) = 0. Then, since ϕ is convex, b

b

a

a

∫ ϕ(g(s)) − ϕ(f (s))ds ≤ ∫ ϕ′ (g(s))(g(s) − f (s))ds. Using integration by parts, we have b

′

∫ ϕ (g(s))(g(s) − f (s))ds = ϕ a

′

(g(t))A(t)|ba

b

− ∫ A(s)d(ϕ′ (g(s))). a

The first part is zero by assumption. The second part is negative since A is negative on [a, b] and ϕ′ (g(t)) is decreasing on t ∈ [a, b] by the assumption that ϕ is convex and g is decreasing. Hence b

∫ ϕ(g(s)) − ϕ(f (s))ds ≤ 0. a

When ϕ is concave, the statement is proved similarly. Submajorization will be used in later chapters for compactness estimates of productconvolution operators and pseudodifferential operators for the Lorentz ideals ℒp and ℒp,∞ when p > 2. For 0 < p < 2, majorization and direct sums will be the main tool to prove compactness estimates. Lemma 1.3.8. Let (Ω, Σ, ν) be a σ-finite measure space, and let 0 < p < 2. Let 0 ≤ f , g ∈ L2 (Ω) be such that g 2 ≺ f 2 . (a) If g ∈ Lp (Ω), then f ∈ Lp (Ω), and ‖f ‖p ≤ ‖g‖p . (b) If g ∈ Lp,∞ (Ω), then f ∈ Lp,∞ (Ω), and 1

‖f ‖p,∞ ≤ 2 p √

p ⋅ ‖g‖p,∞ . 2−p

28 � 1 Bounded operators and pseudodifferential operators n

n

∞

∞

Proof. (a) Let n ∈ ℕ. Since ∫0 μ(s, g)2 ds ≤ ∫0 μ(s, f )2 ds and ∫0 μ(s, g)2 ds = ∫0 μ(s, f )2 ds, there exists an ∈ (0, n] such that an

n

2

∫ μ(s, f ) ds = ∫ μ(s, g)2 ds. 0

0

Set fn := μ(f )χ(0,an ) ,

gn := μ(g)χ(0,n) .

We show that gn2 ≺ fn2 . Let 0 < t ≤ an . Since an ≤ n, we have t

t

t

t

0

0

0

0

t

n

∫ gn2 (s)ds = ∫ g 2 (s)ds ≤ ∫ f 2 (s)ds = ∫ fn2 (s)ds. If t > an , then ∫ gn2 (s)ds 0

an

2

t

≤ ∫ g (s)ds = ∫ f (s)ds = ∫ fn2 (s)ds. 0

2

0

0

In addition, the choice of an ensures that ∞

∞

0

0

∫ gn2 (s)ds = ∫ fn2 (s)ds. Hence fn and gn are decreasing functions supported on the interval [0, n] such that gn2 ≺ fn2 . Using Lemma 1.3.7 for the concave function ϕ(s) = sp/2 , s ∈ [0, n], w ehave n

n

0

0

‖fn ‖pp = ∫ ϕ(fn2 (s))ds ≤ ∫ ϕ(gn2 (s))ds = ‖gn ‖pp . Passing to the limit as n → ∞, we obtain ‖f ‖p ≤ ‖g‖p , as required. (b) By assumption, t

t

∫ μ (s, g)ds ≤ ∫ μ2 (s, f )ds, 0

and

2

0

t > 0,

1.3 Operator inequalities ∞

∞

0

0

�

29

∫ μ2 (s, g)ds = ∫ μ2 (s, f )ds. Therefore, for t > 0, ∞

2

∞

t

∫ μ (s, f )ds = ∫ μ (s, f )ds − ∫ μ2 (s, f )ds t

2

0 ∞

0

t

2

∞

≤ ∫ μ (s, g)ds − ∫ μ (s, g)ds = ∫ μ2 (s, g)ds. 0

2

t

0

Hence 2

2t

∞

tμ (2t, f ) ≤ ∫ μ (s, f )ds ≤ ∫ μ2 (s, f )ds t

2

t ∞

∞

≤ ∫ μ2 (s, g)ds ≤ ‖g‖2p,∞ ∫ s t

− p2

ds =

t

p − 2 +1 ⋅ t p ‖g‖2p,∞ . 2−p

Multiplying by t −1 and taking the square root, we obtain 1

μ(2t, f ) ≤ 2 p √

p −1 ⋅ (2t) p ‖g‖p,∞ . 2−p

Taking the supremum over t > 0, we complete the proof. Let H be a separable Hilbert space. If A, B ∈ ℒ(H), then the direct sum A⊕B is conventionally defined as the linear operator on the Hilbert space H ⊕H given by (A⊕B)(η⊕ξ) = Aη ⊕ Bξ, η, ξ ∈ H. Dealing with a direct sum of Hilbert spaces will be inconvenient, and therefore we adopt the following equivalent notion of a direct sum using commuting projections in the Hilbert space H. Definition 1.3.9. Let Ak ∈ ℒ(H), k ≥ 0, satisfy supk≥0 ‖Ak ‖∞ < ∞. If pk ∈ ℒ(H), k ≥ 0, are pairwise orthogonal projections, and if Sk ∈ pk ℒ(H)pk are such that μ(Sk ) = μ(Ak ), k ≥ 0, and the sequence of partial sums ∑k≤n Sk strongly converges as n → ∞ to a bounded operator ∑k≥0 Sk ∈ ℒ(H), then we write ⨁ Ak := ∑ Sk . k≥0

k≥0

Here the singular value function of an operator A is given by Definition 1.1.2. In the definition of ⨁k≥0 Ak , there is some freedom in choosing the decomposition of H and the operators Sk . The direct sum will be used for estimates of the singular value function μ(⨁k≥0 Ak ) that are independent of the specific choices of Sk .

30 � 1 Bounded operators and pseudodifferential operators Definition 1.3.10. An operator 0 ≤ B ∈ ℒ(H) is majorized by 0 ≤ A ∈ ℒ1 if μ(B) ≺ μ(A). We will denote this by B ≺ A as well. Equivalently, B ≺≺ A, and ∞

∞

k=0

k=0

∑ μ(k, B) = ∑ μ(k, A).

Let Ak , Bk ∈ ℒ(H), k ≥ 0, with supk≥0 ‖Ak ‖∞ < ∞ and supk≥0 ‖Bk ‖∞ < ∞. Observe that if Bk ≺ Ak , k ≥ 0, then ⨁ Bk ≺ ⨁ Ak . k≥0

k≥0

(1.4)

We say that Ak ∈ ℒ(H), k ≥ 0, are disjoint from the left if A∗k Al = 0 when k ≠ l. Similarly, Ak ∈ ℒ(H), k ≥ 0, are disjoint from the right if Ak A∗l = 0 when k ≠ l. The following lemma allows us to compare the direct sum of disjoint operators with series of the disjoint operators in terms of majorization. Lemma 1.3.11. (a) Let Ak ∈ ℒ(H), k ≥ 0, be such that ∑k≥0 ‖Ak ‖22 < ∞. If {Ak }k≥0 are disjoint from the right or from the left, then μ2 (⨁ Ak ) ≺ μ2 ( ∑ Ak ). k≥0

k≥0

(b) Let Ak , Bk ∈ ℒ(H), k, l ≥ 0, be such that ∑k,l≥0 ‖Ak Bl ‖22 < ∞. If {Ak }k≥0 are disjoint from the left and {Bl }l≥0 are disjoint from the right, then μ2 (⨁ Ak Bl ) ≺ μ2 ( ∑ Ak Bl ). k,l≥0

k,l≥0

Proof. (a) It is sufficient to prove the statement under the assumption that Ak are disjoint from the left, since the statement for operators disjoint from the right can be proved similarly by taking adjoints. We first note that since Ak are disjoint from the left, it follows that 󵄩󵄩 m 󵄩󵄩2 m m m 󵄩󵄩 󵄩 󵄩󵄩 ∑ Ak 󵄩󵄩󵄩 = Tr( ∑ A∗ Al ) = ∑ Tr(A∗ Ak ) = ∑ ‖Ak ‖2 k k 2 󵄩󵄩 󵄩󵄩 󵄩󵄩k=n 󵄩󵄩2 k,l=n k=n k=n whenever m > n, n, m ∈ ℕ. Since the series ∑k≥0 ‖Ak ‖22 converges, we obtain that the series ∑k≥0 Ak converges in ℒ2 and, moreover, 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨 ∗ ∗ 󵄨󵄨 ∑ Ak 󵄨󵄨󵄨 = ∑ Ak Al = ∑ Ak Ak . 󵄨󵄨k≥0 󵄨󵄨 k,l≥0 k≥0

1.3 Operator inequalities

�

31

By Theorem 2.3.7 in Volume I we have that μ2 (⨁ Ak ) = μ(⨁ |Ak |2 ) ≺≺ μ( ∑ A∗k Ak ) = μ2 ( ∑ Ak ), k≥0

k≥0

k≥0

k≥0

which proves submajorization. Since the series ∑k≥0 Ak converges in ℒ2 , we have ∞

󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 ∞ 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ∫ μ (s, ⨁ Ak )ds = 󵄩󵄩⨁ Ak 󵄩󵄩 = 󵄩󵄩 ∑ Ak 󵄩󵄩󵄩 = ∫ μ2 (s, ∑ Ak )ds, 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 k≥0 k≥0 k≥0 k≥0 2

0

0

which concludes the proof. (b) By assumption we have that the operators Ak Bl are disjoint from the left and from the right. Hence, repeating the same argument as in part (a), we have that the series ∑k,l≥0 Ak Bl converges in ℒ2 . Since for any fixed k, the operators Ak Bl are disjoint from the right, part (a) implies that μ2 (⨁ Ak Bl ) ≺ μ2 (∑ Ak Bl ), l≥0

l≥0

k ∈ ℕ.

Therefore by (1.4) we have that μ2 (⨁ Ak Bl ) = μ2 (⨁ ⨁ Ak Bl ) ≺ μ2 (⨁(∑ Ak Bl )). k,l≥0

k≥0 l≥0

k≥0

l≥0

Using part (a) again for operator ∑l≥0 Ak Bl , k ∈ ℕ, with Ak Bl disjoint from the left, we obtain that μ2 (⨁ Ak Bl ) ≺ μ2 ( ∑ (∑ Ak Bl )) = μ2 ( ∑ Ak Bl ). k,l≥0

k≥0 l≥0

k,l≥0

Lemma 1.3.11 combined with Lemma 1.3.8 allows us to compare disjoint series in ℒp and ℒp,∞ , 0 < p < 2, by comparing direct sums. 1.3.2 Hölder inequalities and commutators in Lorentz ideals We recall here some Hölder-type inequalities and commutator inequalities for the Lorentz ideals ℒp,q from Example 1.1.6. Hölder inequalities in Lorentz ideals The following estimates follow directly from Theorem 1.2.6 and the inequalities of Fan [130, 137]: if A, B ∈ ℒ(H), then

32 � 1 Bounded operators and pseudodifferential operators μ(t + s, A + B) ≤ μ(t, A) + μ(s, B),

s, t > 0,

and μ(t + s, AB) ≤ μ(t, A)μ(s, B),

s, t > 0.

A proof of the Fan inequalities is given in Corollary 2.2.9 in Volume I. Theorem 1.3.12. (a) Let 0 < p1 , p2 < ∞ and 0 < q1 , q2 ≤ ∞. Let 1 1 1 = + , p p1 p2

1 1 1 = + . q q1 q2

If A ∈ ℒp1 ,q1 and B ∈ ℒp2 ,q2 , then AB ∈ ℒp,q , and ‖AB‖p,q ≤ cp1 ,p2 ,q1 ,q2 ⋅ ‖A‖p1 ,q1 ‖B‖p2 ,q2

(1.5)

for a constant cp1 ,p2 ,q1 ,q2 > 0. (b) Let 0 < r, p, q < ∞. Let 1 1 1 ≤ + . r p q In the particular case of the Schatten ideals, if A ∈ ℒp and B ∈ ℒq , then AB ∈ ℒr , and ‖AB‖r ≤ ‖A‖p ‖B‖q .

(1.6)

(c) Let 0 < r, p, q < ∞. Let 1 1 1 ≤ + . r p q In the particular case of the weak Schatten ideals, if A ∈ ℒp,∞ and B ∈ ℒq,∞ , then AB ∈ ℒr,∞ , and ‖AB‖r,∞ ≤ cp,q ⋅ ‖A‖p,∞ ‖B‖q,∞

(1.7)

for a constant cp,q > 0. The Lorentz ideals ℒp,q are ordered lexicographically in (p, q) in the sense that ℒp0 ,q0 ⊂ ℒp1 ,q1 ,

p0 < p1 or p0 = p1 , q0 < q1 .

(1.8)

1.3 Operator inequalities

� 33

If A, B ∈ ℒ(H) for a separable Hilbert space H, then the direct product A ⊗ B is a bounded linear operator on the Hilbert space H ⊗ H defined densely by (A ⊗ B)(η ⊗ ξ) := Aη ⊗ Bξ,

η, ξ ∈ H.

Recall from Definition 1.1.8 that, given a trace φ : ℒ1,∞ (H) → ℂ, the trace φ : ℒ1,∞ (H ⊗ H) → ℂ refers to the trace in bijective correspondence to φ on ℒ1,∞ (H) using any unitary operator U : H → H ⊗ H. The following lemma is a form of Fubini’s theorem. Lemma 1.3.13. If A ∈ ℒ1,∞ (H) and B ∈ ℒ1 (H), then A ⊗ B ∈ ℒ1,∞ (H ⊗ H) and ‖A ⊗ B‖1,∞ ≤ ‖A‖1,∞ ‖B‖1 . For every continuous trace φ on ℒ1,∞ (H), we have φ(A ⊗ B) = φ(A) ⋅ Tr(B). Proof. For brevity, we denote all singular value functions (see Definition 1.2.12) by μ when the von Neumann algebra and trace are clear from the context. We have μ(t, A) ≤ t −1 ‖A‖1,∞ ,

t > 0.

Thus μ(A ⊗ B) = μ(μ(A) ⊗ μ(B)) ≤ ‖A‖1,∞ ⋅ μ(z ⊗ μ(B)), where z(t) = t −1 , t > 0. A standard calculation, which can be seen in the proof of Theorem 2.f.2 in [187], shows that μ(z ⊗ f ) = ‖f ‖1 ⋅ z. It follows that μ(A ⊗ B) ≤ ‖A‖1,∞ ‖B‖1 ⋅ z since ‖μ(B)‖1 = ‖B‖1 . Consequently, for all s ∈ [0, 1), μ(n, A ⊗ B) = μ(n + s, A ⊗ B) ≤ ‖A‖1,∞ ‖B‖1 ⋅ z(n + s),

n ∈ ℤ+ .

Hence, taking s → 1− , μ(n, A ⊗ B) ≤ ‖A‖1,∞ ‖B‖1 ⋅ (n + 1)−1 ,

n ∈ ℤ+ .

Multiplying by n + 1 and taking the supremum over n ∈ ℤ+ , we infer the first statement.

34 � 1 Bounded operators and pseudodifferential operators Let now φ be a continuous trace, let 0 ≤ A ∈ ℒ1,∞ , and let 0 ≤ B be a finite-rank operator. We write B=

rank(B)−1

∑

k=0

μ(k, B)pk ,

where {pk }rank(B) are pairwise orthogonal rank 1 projections. We have k=0 φ(A ⊗ B) =

rank(B)−1

∑

k=0

μ(k, B)φ(A ⊗ pk ).

Since A ≥ 0, A ⊗ pk ≥ 0 and μ(A ⊗ pk ) = μ(A) for every 0 ≤ k < rank(B), it follows that φ(A ⊗ pk ) = φ(A),

0 ≤ k < rank(B).

Hence φ(A ⊗ B) =

rank(B)−1

∑

k=0

μ(k, B)φ(A) = φ(A) ⋅ Tr(B).

This proves the second statement for the case where A ≥ 0 and where B ≥ 0 has finite rank. By linearity the second statement holds for every A ∈ ℒ1,∞ and for every finiterank B. By the continuity (i. e., by the first statement) the second assertion holds in full generality. Differences and commutators in Lorentz spaces The next theorem concerns Lipschitz continuity of the absolute value function on self-adjoint operators, and commutator estimates, for the ideals ℒp and ℒp,∞ when 1 < p < ∞. Theorem 1.3.14. Let A = A∗ , B = B∗ ∈ ℒ(H), and 1 < p < ∞. (a) If A − B ∈ ℒp , then |A| − |B| ∈ ℒp , and there is a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩|A| − |B|󵄩󵄩󵄩p ≤ cp ⋅ ‖A − B‖p . (b) If A − B ∈ ℒp,∞ , then |A| − |B| ∈ ℒp,∞ , and there is a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩|A| − |B|󵄩󵄩󵄩p,∞ ≤ cp ⋅ ‖A − B‖p,∞ . (c) If [A, B] ∈ ℒp , then [A, |B|] ∈ ℒp , and there is a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩[A, |B|]󵄩󵄩󵄩p ≤ cp ⋅ 󵄩󵄩󵄩[A, B]󵄩󵄩󵄩p .

1.3 Operator inequalities

� 35

(d) If [A, B] ∈ ℒp,∞ , then [A, |B|] ∈ ℒp,∞ , and there is a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩[A, |B|]󵄩󵄩󵄩p,∞ ≤ cp ⋅ 󵄩󵄩󵄩[A, B]󵄩󵄩󵄩p,∞ . Recall that the weak ideals ℒp,∞ from Example 1.1.6 are not separable. When differences and commutators belong to the separable part (ℒp,∞ )0 of the ideal ℒp,∞ , so do the differences and commutators involving the absolute value. Corollary 1.3.15. Let A = A∗ , B = B∗ ∈ ℒ(H), and 1 < p < ∞. (a) If A − B ∈ (ℒp,∞ )0 , then |A| − |B| ∈ (ℒp,∞ )0 . (b) If [A, B] ∈ (ℒp,∞ )0 , then [A, |B|] ∈ (ℒp,∞ )0 . Proof. Let V = A − B ∈ (ℒp,∞ )0 . Choose a sequence {Vn }n≥0 of finite-rank operators such that Vn → V in ℒp,∞ . Set An = B + Vn and note that An − B ∈ ℒp . By Theorem 1.3.14(a) we have |An | − |B| ∈ ℒp ⊂ (ℒp,∞ )0 . Note that An → A in ℒp,∞ . By Theorem 1.3.14(b) we have |An | → |A| in ℒp,∞ . Thus |An | − |B| → |A| − |B| in ℒp,∞ . This proves the first statement. To see the second statement, we may assume without loss of generality that ‖A‖∞ < 1. Choose ϕ ∈ Cc∞ (ℝ) such that ϕ(t) = √1 − t 2 on [−‖A‖∞ , ‖A‖∞ ]. We write √1 − A2 = ϕ(A) = (2π)−1 ∫ ∫ ϕ(v)e−isv eisA dsdv. ℝℝ

Therefore [√1 − A2 , B] = [ϕ(A), B] = (2π)−1 ∫ ∫ ϕ(v)e−isv [eisA , B]dsdv. ℝℝ

Since [eisA , B] ∈ (ℒp,∞ )0 ,

󵄩󵄩 isA 󵄩󵄩 󵄩 󵄩 󵄩󵄩[e , B]󵄩󵄩p,∞ ≤ |s|󵄩󵄩󵄩[A, B]󵄩󵄩󵄩p,∞ ,

it follows that the latter integral is a Bochner integral in (ℒp,∞ )0 . In particular, [√1 − A2 , B] ∈ (ℒp,∞ )0 . Set U = A + i√1 − A2 and C = U −1 BU. Since [U, B] ∈ (ℒp,∞ )0 , it follows that C − B ∈ (ℒp,∞ )0 . By the first statement we have U −1 |B|U − |B| = |C| − |B| ∈ (ℒp,∞ )0 . Henc, [U, |B|] ∈ (ℒp,∞ )0 . Taking the imaginary part, we complete the proof of the second statement. The following lemma is used in Chapter 6.

36 � 1 Bounded operators and pseudodifferential operators Lemma 1.3.16. Let A, B ∈ ℒ(H) and 1 < p < ∞. If [A, B] ∈ (ℒp,∞ )0 and [A, B∗ ] ∈ (ℒp,∞ )0 , then [A, |B|] ∈ (ℒp,∞ )0 . Proof. Assume that A = A∗ ∈ ℒ(H) and let B ∈ ℒ(H) be arbitrary. Consider the operators C and D on the Hilbert space H ⊗ ℂ2 given by the formulas C := (

A 0

0 ), A

0 D := ( ∗ B

B ). 0

We have [C, D] = (

0 [A, B∗ ]

[A, B] ) ∈ (ℒp,∞ )0 . 0

Since D is self-adjoint, it follows from Corollary 1.3.15 that [C, |D|] ∈ (ℒp,∞ )0 . However, |D| = (

|B∗ | 0

0 ). |B|

Thus [C, |D|] = (

[A, |B∗ |] 0

0 ) [A, |B|]

belongs to the ideal (ℒp,∞ )0 (H ⊗ ℂ2 ). It follows that [A, |B|] ∈ (ℒp,∞ )0 (H). This concludes the proof when A = A∗ . Suppose now that A ∈ ℒ(H) is arbitrary. By hypothesis, [A, B] ∈ (ℒp,∞ )0 and [A, B∗ ] ∈ (ℒp,∞ )0 . Then [A∗ , B] = −[A, B∗ ]∗ ∈ (ℒp,∞ )0 and [A∗ , B∗ ] = −[A, B]∗ ∈ (ℒp,∞ )0 . Hence the self-adjoint operators 1 A1 := (A + A∗ ) 2 and A2 :=

1 (A − A∗ ) 2i

satisfy [Ai , B] ∈ (ℒp,∞ )0 and [Ai , B∗ ] ∈ (ℒp,∞ )0 for i = 1, 2. We then have that [Ai , |B|] ∈ (ℒp,∞ )0 , i = 1, 2, and [A, |B|] = [A1 + A2 , |B|] = [A1 , |B|] + [A2 , |B|] ∈ (ℒp,∞ )0 . This concludes the proof. The next corollary is required in the proof of Lemma 1.3.18.

1.3 Operator inequalities

�

37

Corollary 1.3.17. If 0 ≤ A, B ∈ ℒp,∞ and 1 < p < ∞, then there is a constant cp > 0 such that p−1 󵄩󵄩 p p󵄩 p−⌊p⌋ ⌊p⌋ 󵄩󵄩A − B 󵄩󵄩󵄩1,∞ ≤ cp ⋅ (‖A − B‖p,∞ ⋅ max{‖A‖p,∞ , ‖B‖p,∞ } + ‖A − B‖p,∞ ⋅ ‖B‖p,∞ ).

Here ⌊p⌋ denotes the greatest integer less than or equal to p. Proof. Set m := ⌊p⌋ and θ := p − ⌊p⌋ ∈ [0, 1). In what follows, we assume that θ > 0 (otherwise, the proof becomes easier as the second term disappears). We have Ap − Bp = Aθ (Am − Bm ) + (Aθ − Bθ )Bm . By the triangle inequality and the Hölder inequalities in Theorem 1.3.12 using the relationship 1 p θ

+

1

p m

=

θ+m = 1, p

we have 󵄩󵄩 p 󵄩 θ󵄩 󵄩 m 󵄩 m󵄩 󵄩 θ p󵄩 ′ m󵄩 θ󵄩 󵄩󵄩A − B 󵄩󵄩󵄩1,∞ ≤ cp ⋅ (󵄩󵄩󵄩A 󵄩󵄩󵄩 p ,∞ 󵄩󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ + 󵄩󵄩󵄩B 󵄩󵄩󵄩∞ 󵄩󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ ) θ m θ for a constant cp′ > 0, or 󵄩󵄩 p p󵄩 ′ θ 󵄩 m m󵄩 m󵄩 θ θ󵄩 󵄩󵄩A − B 󵄩󵄩󵄩1,∞ ≤ cp ⋅ (‖A‖p,∞ 󵄩󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ + ‖B‖∞ 󵄩󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ ). m θ Clearly, m−1

Am − Bm = ∑ Ak (A − B)Bm−1−k . k=0

By the triangle inequality and the Hölder inequalities in Theorem 1.3.12 using 1

p k

+

1 + p

1

p m−1−k

=

k+1+m−1−k 1 = p, p m

we have m−1

󵄩󵄩 m m󵄩 ′′ k m−1−k 󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ ≤ cp ∑ ‖A‖p,∞ ⋅ ‖A − B‖p,∞ ⋅ ‖B‖p,∞ m k=0

for a constant cp′′ > 0. By Theorem 1.3.3 we have 󵄩󵄩 θ θ󵄩 ′′′ θ 󵄩󵄩A − B 󵄩󵄩󵄩 p ,∞ ≤ cp ⋅ ‖A − B‖p,∞ θ for a constant cp′′′ > 0. Combining these estimates, we complete the proof.

38 � 1 Bounded operators and pseudodifferential operators The following lemma is Proposition 10, part (3), on p. 320 in [72]. The estimate is required in Chapter 6. Lemma 1.3.18. Let A, B ∈ ℒp,∞ and 1 < p < ∞ be such that A − B ∈ (ℒp,∞ )0 . Then |A|p − |B|p ∈ (ℒ1,∞ )0 . Proof. Step 1. First, we prove the statement for the case A ≥ B ≥ 0. Set V := A − B ∈ (ℒp,∞ )0 . We have V = ∑ μ(k, V )pk , k≥0

where {pk }k≥0 is a sequence of pairwise orthogonal rank 1 projections. Set n

An := B + ∑ μ(k, V )pk . k=0

Since An − B has finite rank, it follows that An − B ∈ ℒr,∞ for every r > 0. Let m := ⌊p⌋ ≥ 1 and θ := p − ⌊p⌋ ∈ [0, 1). By Theorem 1.3.3 we have Aθn − Bθ ∈ ℒ r ,∞ for every r > 0. In θ

particular, Aθn − Bθ ∈ ℒ1 . Thus

m−1

Apn − Bp = (Aθn − Bθ ) ⋅ Bm + Aθn ⋅ ∑ Akn (A − B)Bm−1−k ∈ ℒ1 .

(1.9)

k=0

Since V ∈ (ℒp,∞ )0 , the difference ∞

An − A = ∑ μ(k, V )pk k=n+1

p

vanishes in the ℒp,∞ -quasinorm as n → ∞. By Corollary 1.3.17 it follows that An → Ap in p p ℒ1,∞ . Thus An − Bp → Ap − Bp in ℒ1,∞ . Since An − Bp ∈ ℒ1 for n ≥ 0 by (1.9), it follows that Ap − Bp belongs to the closure of ℒ1 in ℒ1,∞ and hence belongs to (ℒ1,∞ )0 . This proves the statement for the case A ≥ B ≥ 0. Step 2. Next, we prove the statement for the case of self-adjoint bounded operators A = A∗ and B = B∗ . Assume that A, B ≥ 0. Let V := A − B ∈ (ℒp,∞ )0 . It is immediate that V+ , V− ∈ (ℒp,∞ )0 . Set C := B + V+ . Clearly, C − A, C − B ∈ (ℒp,∞ )0 , C ≥ A ≥ 0, and C ≥ B ≥ 0. By Step 1 we have that C p − Ap ∈ (ℒ1,∞ )0 and C p − Bp ∈ (ℒ1,∞ )0 . Hence Ap − Bp = (C p − Bp ) − (C p − Ap ) ∈ (ℒ1,∞ )0 . This proves the statement for the case A, B ≥ 0.

1.3 Operator inequalities

� 39

Now assume that A = A∗ , B = B∗ ∈ ℒ(H). By Corollary 1.3.15 we have that |A| − |B| ∈ (ℒp,∞ )0 . Applying the preceding paragraph to the positive operators |A|, |B| ∈ ℒ(H), we have |A|p − |B|p ∈ (ℒ1,∞ )0 . This proves the statement for the case A = A∗ and B = B∗ . Step 3. We now prove the statement for general A, B ∈ ℒ(H). Define self-adjoint operators X and Y on the Hilbert space H ⊗ ℂ2 by X := (

0 A∗

A ), 0

Y := (

0 ), |A|

Y =(

0 B∗

B ) 0

with |X| = (

|A∗ | 0

|B∗ | 0

0 ). |B|

Then X, Y ∈ (ℒp,∞ )(H ⊗ ℂ2 ), and X −Y =(

0 (A − B)∗

A−B ) 0

belongs to the ideal (ℒp,∞ )0 (H ⊗ ℂ2 ). From Step 2, |X|p − |Y |p ∈ (ℒ1,∞ )0 (H ⊗ ℂ2 ), where |A∗ |p − |B∗ |p |X|p − |Y |p = ( 0

0 ). |A|p − |B|p

Thus |A|p − |B|p ∈ (ℒ1,∞ )0 (H). This proves the statement in full generality. The next lemma was first stated as Lemma 3.β.11 in [72] without proof. The end notes to Chapter 1 discuss a proof. 1

Theorem 1.3.19. Let 0 ≤ A ∈ ℒ(H), and let 0 ≤ B ∈ ℒp,∞ , 1 < p < ∞. If [A 2 , B] ∈ (ℒp,∞ )0 , then 1

1

p

Bp Ap − (A 2 BA 2 ) ∈ (ℒ1,∞ )0 . Theorem 9.1.5(b) of Volume I has the consequence that if 0 ≤ B ∈ ℒ1,∞ and A ∈ ℒ(H) and if the limit lim (p − 1) Tr(Bp A)

p→1+

exists, then for all extended limits ω on l∞ , we have Trω (AB) = lim+ (p − 1) Tr(Bp A), p→1

40 � 1 Bounded operators and pseudodifferential operators where Trω is the Dixmier associated with the extended limit ω on l∞ as described in Section 1.1.2. The next theorem is a variant of this result when the positive operator B is no longer a compact operator. 1

Theorem 1.3.20. Let 0 ≤ A, B ∈ ℒ(H) be such that AB ∈ ℒ1,∞ , [B, A 2 ] ∈ ℒ1 , and 󵄩󵄩 p 21 󵄩󵄩 −2 󵄩󵄩B A 󵄩󵄩1 = o((p − 1) ),

p → 1+ .

For every extended limit ω on l∞ , we have Trω (AB) = lim+ (p − 1) Tr(Bp A),

(1.10)

p→1

where Trω is the Dixmier associated with the extended limit ω on l∞ , provided that the limit in the right-hand side exists. Proof. By Hölder’s inequality we have 1 󵄩󵄩 p p 󵄩 p 1 󵄩 󵄩 p− 1 󵄩 p 󵄩 󵄩󵄩B A − B A󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩B A 2 󵄩󵄩󵄩1 󵄩󵄩󵄩A 2 − A 2 󵄩󵄩󵄩∞ , 1

where Bp A 2 ∈ ℒ1 for p > 1 by assumption. From the spectral theorem, 1 󵄩󵄩 p− 21 󵄩 − A 2 󵄩󵄩󵄩∞ = 󵄩󵄩A

1 󵄨 1 󵄨 sup 󵄨󵄨󵄨t p− 2 − t 2 󵄨󵄨󵄨 ≤ cA ⋅ (p − 1)

t∈[0,‖A‖∞ ]

for some constant cA > 0. From the assumption 󵄩󵄩 p 21 󵄩󵄩 −2 󵄩󵄩B A 󵄩󵄩1 = o((p − 1) ),

p → 1+ ,

we have 󵄩󵄩 p p p 󵄩 −2 −1 󵄩󵄩B A − B A󵄩󵄩󵄩1 ≤ o((p − 1) ) ⋅ cA ⋅ (p − 1) = o((p − 1) ),

p → 1+ .

Hence lim (p − 1) Tr(Bp Ap ) = lim+ (p − 1) Tr(Bp A)

p→1+

p→1

1

1

assuming that the limit in the right-hand side exists. Let Y := A 2 BA 2 . Then Y − AB = A1/2 [A1/2 , B] ∈ ℒ1 by assumption and Y ∈ ℒ1,∞ . Theorem 4.5 in [226] states that lim (p − 1) Tr(Y p ) = lim+ (p − 1) Tr(Bp Ap ).

p→1+

p→1

From Theorem 9.1.5(b) of Volume I we have Trω (Y ) = lim+ (p − 1) Tr(Y p ) p→1

1.4 Double operator integrals

�

41

for every extended limit ω on l∞ . Finally, Trω (AB) = Trω (Y ) for every extended limit ω on l∞ since a Dixmier trace vanishes on ℒ1 .

1.4 Double operator integrals Let D0 : dom(D0 ) → H and D1 : dom(D1 ) → H be two unbounded self-adjoint operators on a separable Hilbert space H. According to the spectral theorem [231, Theorem VIII.6], D0 and D1 both have spectral resolutions of the form Dj = ∫ t dEj (t),

j = 0, 1,

ℝ

where E0 and E1 are projection-valued measures on ℝ. The theory of double operator integrals associates a linear operator with formal expressions such as ∬ ϕ(t, s) dE0 (t)XdE1 (s), ℝ×ℝ

where ϕ is a measurable function on ℝ2 , and X is a bounded operator on H. We consider the functions ϕ that admit a separation of variables in the sense that there are a σ-finite measure space (Ω, Σ, ν) and measurable functions α, β on ℝ × Ω such that ϕ(t, s) = ∫ α(t, ω)β(s, ω) dν(ω),

t, s ∈ ℝ,

Ω

and 󵄨 󵄨 󵄨 󵄨 ∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω) < ∞. Ω

t∈ℝ

s∈ℝ

In this case, ∬ℝ×ℝ ϕ(t, s) dE0 (t)XdE1 (s) can be defined as a linear operator on H using the weak operator topology on ℒ(H). Estimates of commutators derived from double operator integrals will be used in later chapters. 1.4.1 Integration of operator-valued functions in the weak operator topology Let (Ω, Σ, ν) be a measure space, where the signed measure ν is not necessarily positive. The theory of weak operator topology integration concerns integrals of functions f : Ω → ℒ(H).

42 � 1 Bounded operators and pseudodifferential operators Definition 1.4.1. A function f : Ω → ℒ(H) is said to be weak operator topology measurable if for all ξ, η ∈ H, the scalar-valued function ω 󳨃→ ⟨f (ω)ξ, η⟩,

ω ∈ Ω,

is measurable. Similarly, f is said to be weak operator topology integrable if for all ξ and η, the above map is integrable. The following is adapted from [107, Chapter II, Section 3, Lemma 1]. Lemma 1.4.2. Let f : Ω → ℒ(H) be weak operator topology measurable. Then the norm function 󵄩 󵄩 ω 󳨃→ 󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ ,

ω ∈ Ω,

is measurable. If 󵄩 󵄩 ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ d|ν|(ω) < ∞,

Ω

then there exists a unique If ∈ ℒ(H) such that ⟨If ξ, η⟩ = ∫⟨f (ω)ξ, η⟩ dν(ω),

ξ, η ∈ H,

Ω

and 󵄩 󵄩 ‖If ‖∞ ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ d|ν|(ω). Ω

Proof. Since the underlying Hilbert space H is separable, we may select a countable dense subset A of the unit ball of H. Then 󵄩󵄩 󵄩 󵄩󵄩f (ω)󵄩󵄩󵄩∞ = sup ⟨f (ω)ξ, η⟩, ξ,η∈A

ω ∈ Ω.

Since the functions ω 󳨃→ ⟨f (ω)ξ, η⟩ are measurable for each pair ξ, η ∈ A, and A is countable, then the supremum ω 󳨃→ ‖f (ω)‖∞ is measurable. Since 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨⟨f (ω)ξ, η⟩󵄨󵄨󵄨 ≤ ‖ξ‖‖η‖󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ ,

ω ∈ Ω, ξ, η ∈ H,

the assumption that ω 󳨃→ ‖f (ω)‖∞ is a |ν|-integrable function on Ω implies that f is integrable in the weak operator topology.

1.4 Double operator integrals

� 43

Since f is integrable in the weak operator topology, for every ξ, η ∈ H, we may define ξ,η

If

:= ∫⟨f (ω)ξ, η⟩ dν(ω),

ξ, η ∈ H.

Ω

This is a sesquilinear functional of ξ, η ∈ H and obeys the bound 󵄨󵄨 ξ,η 󵄨󵄨 󵄩 󵄩 󵄨󵄨If 󵄨󵄨 ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ d|ν|(ω) ⋅ ‖ξ‖‖η‖,

ξ, η ∈ H.

Ω

By the Riesz representation theorem [231, p. 44] there exists a unique operator If ∈ ℒ(H) ξ,η

such that ⟨If ξ, η⟩ = If , ξ, η ∈ H, with norm bounded by the integral of ‖f (ω)‖∞ .

For a weak operator topology measurable function f : Ω → ℒ(H) satisfying the conditions of Lemma 1.4.2, we define the bounded operator If ∈ ℒ(H) by If = ∫ f (ω) dν(ω). Ω

This operator is the weak operator topology integral. Note that if A and B are bounded linear operators and f : Ω → ℒ(H) is integrable in the weak operator topology, then ω → Af (ω)B, ω ∈ Ω, is integrable in the weak operator topology, and A(∫ f (ω) dν(ω))B = ∫ Af (ω)B dμ(ω). Ω

(1.11)

Ω

Indeed, for all ξ, η ∈ H, we have ⟨AIf Bξ, η⟩ = ⟨If (Bξ), A∗ η⟩ = ∫⟨f (ω)Bξ, A∗ η⟩ dω = ⟨IAfB ξ, η⟩. Ω

The next lemma gives conditions under which the weak operator topology integral belongs to the ideal ℒ1 . Lemma 1.4.3. Let f : Ω → ℒ(H) be a weak operator topology measurable function, and assume that f (ω) ∈ ℒ1 for almost all ω ∈ Ω. Then the functions 󵄩 󵄩 ω 󳨃→ 󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 ,

ω 󳨃→ Tr(f (ω)),

are measurable. If we further have 󵄩 󵄩 ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 d|ν|(ω) < ∞,

Ω

ω ∈ Ω,

44 � 1 Bounded operators and pseudodifferential operators then ∫Ω f (ω) dν(ω) ∈ ℒ1 , and we have the norm bound 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩∫ f (ω) dν(ω)󵄩󵄩󵄩 ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 d|ν|(ω) 󵄩󵄩 󵄩󵄩1 Ω

Ω

and the trace formula Tr(∫ f (ω) dν(ω)) = ∫ Tr(f (ω)) dν(ω). Ω

Ω

Proof. If H is finite-dimensional with orthonormal basis {ek }N−1 k=0 , then f may be identified with the n × n-matrix-valued function N−1

ω 󳨃→ ∑ ⟨f (ω)ek , el ⟩(ek ⊗ el ). k,l=0

The entries ⟨f (ω)ek , el ⟩, 1 ≤ k, l ≤ N − 1, are measurable by assumption, and the trace class norm ‖f (ω)‖1 is a continuous function of the matrix components of f . Hence in the finite-dimensional case, ‖f (ω)‖1 is measurable. Now we consider the infinite-dimensional case. If P is a finite-rank orthogonal projection, then the function 󵄩 󵄩 ω 󳨃→ 󵄩󵄩󵄩Pf (ω)P󵄩󵄩󵄩1 ,

ω ∈ Ω,

is measurable. Indeed, ω 󳨃→ Pf (ω)P may be identified with a weak operator topology measurable function with values in the finite-dimensional space ℒ(PH). By the separability of the Hilbert space H there exists a sequence of projections ∞ {Pn }n=0 , each of finite rank, such that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩f (ω)󵄩󵄩󵄩1 = sup󵄩󵄩󵄩Pn f (ω)Pn 󵄩󵄩󵄩1 , n≥0

ω ∈ Ω.

It follows that ω 󳨃→ ‖f (ω)‖1 is the supremum of a countable sequence of measurable functions and hence is measurable. Note also that 󵄩 󵄩 󵄩 󵄩 ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩∞ d|ν|(ω) ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 d|ν|(ω) < ∞.

Ω

Ω

Hence the weak operator topology measurable function f : Ω → ℒ(H) satisfies the conditions of Lemma 1.4.2, and the weak operator topology integral exists. Taking a polar decomposition of the bounded operator ∫Ω f (ω) dν(ω), there exists a partial isometry U such that 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨∫ f (ω) dν(ω)󵄨󵄨󵄨 = U ∫ f (ω) dν(ω). 󵄨󵄨 󵄨󵄨 Ω

Ω

1.4 Double operator integrals

� 45

Applying (1.11) to the bounded operators Pn U and Pn , it follows that 󵄨󵄨 󵄨󵄨 󵄨 󵄨 Pn 󵄨󵄨󵄨∫ f (ω) dν(ω)󵄨󵄨󵄨Pn = ∫ Pn Uf (ω)Pn dν(ω), 󵄨󵄨 󵄨󵄨 Ω

Ω

where the latter integral is finite-dimensional. Hence 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄩 󵄩 Tr(Pn 󵄨󵄨󵄨∫ f (ω) dν(ω)󵄨󵄨󵄨Pn ) = ∫ Tr(Pn Uf (ω)Pn ) dν(ω) ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 dν(ω). 󵄨󵄨 󵄨󵄨 Ω

Ω

Ω

Since the operator trace Tr is normal, we have 󵄩󵄩 󵄩󵄩 󵄨󵄨 󵄨󵄨 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩∫ f (ω) dν(ω)󵄩󵄩󵄩 = Tr(󵄨󵄨󵄨∫ f (ω) dν(ω)󵄨󵄨󵄨) 󵄩󵄩 󵄩󵄩1 󵄨󵄨 󵄨󵄨 Ω

Ω

󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄩 󵄩 = lim Tr(Pn 󵄨󵄨󵄨∫ f (ω) dν(ω)󵄨󵄨󵄨Pn ) ≤ ∫󵄩󵄩󵄩f (ω)󵄩󵄩󵄩1 dν(ω). n→∞ 󵄨󵄨 󵄨󵄨 Ω

Ω

This proves the norm bound. Next, we prove the trace formula. Let {en }n=0∞ be an orthonormal basis for H. For each N ≥ 0, the function N

ω 󳨃→ ∑ ⟨f (ω)ek , ek ⟩, k=0

ω ∈ Ω,

is a finite sum of measurable functions on Ω. Then the function N

ω 󳨃→ Tr(f (ω)) = lim ∑ ⟨f (ω)ek , ek ⟩, N→∞

k=0

ω ∈ Ω,

is measurable since it is the pointwise limit of measurable functions. For all ω ∈ Ω and N ≥ 0, we have 󵄨󵄨 N 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ ⟨f (ω)en , en ⟩󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩f (ω)󵄩󵄩󵄩 . 󵄨󵄨 󵄨󵄨 󵄩 󵄩1 󵄨󵄨n=0 󵄨󵄨 Since the function ω 󳨃→ ‖f (ω)‖1 is integrable, the dominated convergence theorem implies that N

Tr(∫ f (ω) dν(ω)) = lim ∑ ∫⟨f (ω)en , en ⟩ dν(ω) Ω

N→∞

n=0 Ω N

= ∫ lim ∑ ⟨f (ω)en , en ⟩ dν(ω) Ω

N→∞

n=0

46 � 1 Bounded operators and pseudodifferential operators = ∫ Tr(f (ω)) dν(ω). Ω

Double operator integrals are further defined in Section 1.4.2 using the weak operator topology integral. The next lemma is used in Theorem 1.4.9 to obtain submajorization bounds on the double operator integral transformer of a bounded operator. Lemma 1.4.4. Let ν be a Borel probability measure on ℝ with compact support. Let t 󳨃→ Vt , t ∈ ℝ, be a strongly continuous family of unitary operators on a separable Hilbert space ℋ. Let A be a bounded operator ℋ. Then the function t 󳨃→ Vt−1 AVt from ℝ to ℒ(H) is integrable in the weak operator topology, and (a) ‖ ∫ℝ Vt−1 AVt dν(t)‖∞ ≤ ‖A‖∞ for A ∈ ℒ(H). (b) ‖ ∫ℝ Vt−1 AVt dν(t)‖1 ≤ ‖A‖1 for A ∈ ℒ1 . (c) ∫ℝ Vt−1 AVt dν(t) ≺≺ A. Proof. The fact that the function t 󳨃→ Vt−1 AVt , t ∈ ℝ, is measurable in the weak operator topology follows from the assumption that t 󳨃→ ⟨Vt−1 AVt ξ, η⟩, t ∈ ℝ, is continuous. Since 󵄩󵄩 −1 󵄩 󵄩󵄩Vt AVt 󵄩󵄩󵄩∞ ≤ ‖A‖∞ ,

t ∈ ℝ,

we have 󵄩 󵄩 ∫󵄩󵄩󵄩Vt−1 AVt 󵄩󵄩󵄩∞ dν(t) ≤ ‖A‖∞ ⋅ ν(ℝ) = ‖A‖∞ . ℝ

Similarly, if A ∈ ℒ1 , then 󵄩󵄩 −1 󵄩 󵄩󵄩Vt AVt 󵄩󵄩󵄩1 ≤ ‖A‖1 ,

t ∈ ℝ,

and 󵄩 󵄩 ∫󵄩󵄩󵄩Vt−1 AVt 󵄩󵄩󵄩1 dν(t) ≤ ‖A‖1 ⋅ ν(ℝ) = ‖A‖1 . ℝ

From Lemma 1.4.2 and Lemma 1.4.3, respectively, it follows that the weak operator integral T(A) := ∫ Vt−1 AVt dν(t) ℝ

is a bounded linear operator and estimates (a) and (b) are satisfied. To prove part (c), note that T : ℒ(H) → ℒ(H) is a linear map. Parts (a) and (b) prove that T is a contraction on ℒ1 and ℒ(H). Part (c) follows from the equivalent conditions in Theorem 1.2.2.

1.4 Double operator integrals

�

47

1.4.2 Definition of a double operator integral We will define double operator integrals on ℒ(H) for symbol functions belonging to the Birman–Solomyak class BS. Definition 1.4.5. Denote by BS the space of all bounded Borel-measurable functions ϕ on ℝ2 such that there exist a σ-finite measure space (Ω, Σ, ν) and measurable functions α, β on ℝ × Ω such that ϕ(t, s) = ∫ α(t, ω)β(s, ω) dν(ω),

t, s ∈ ℝ,

Ω

and 󵄨 󵄨 󵄨 󵄨 ∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω) < ∞. Ω

t∈ℝ

s∈ℝ

We define the norm ‖ϕ‖BS as the infimum of the above quantity over all representations of ϕ as above. The class BS is closed under pointwise sum and product. Indeed, if ϕ1 , ϕ2 ∈ BS admit representations ϕj (t, s) = ∫ αj (t, ω)βj (s, ω) dν(ω),

j = 1, 2,

Ωj

then ϕ1 + ϕ2 can be represented as an integral over the disjoint union Ω1 ⊔ Ω2 , and ϕ1 ϕ2 admits a representation over the product Ω1 × Ω2 . We have ‖ϕ1 + ϕ2 ‖BS ≤ ‖ϕ1 ‖BS + ‖ϕ2 ‖BS ,

‖ϕ1 ϕ2 ‖BS ≤ ‖ϕ1 ‖BS ‖ϕ2 ‖BS .

Given ϕ ∈ BS with representation ϕ(t, s) = ∫ α(t, ω)β(s, ω) dν(ω),

s, t ∈ ℝ,

(1.12)

Ω

X ∈ ℒ(H), and self-adjoint linear operators D0 and D1 on H, we can define the bounded linear operator ∫ α(D0 , ω)Xβ(D1 , ω) dν(ω) ∈ ℒ(H) Ω

using the weak operator topology integral. The existence of the weak operator topology integral follows from Lemma 1.4.2. For η, ξ ∈ H, we have

48 � 1 Bounded operators and pseudodifferential operators ⟨η, α(D0 , ω)Xβ(D1 , ω)ξ⟩ = ∑ ⟨η, α(D0 , ω)Xen ⟩⟨en , β(D1 , ω)ξ⟩ n≥0

for any orthonormal basis {en }∞ n=0 of the separable Hilbert space H. Hence the function ω 󳨃→ ⟨η, α(D0 , ω)Xβ(D1 , ω)ξ⟩ is a measurable function because it is the limit of a countable sequence of products of measurable functions, and 󵄩 󵄩 󵄨 󵄨 󵄨 󵄨 ∫󵄩󵄩󵄩α(D0 , ω)Xβ(D1 , ω)󵄩󵄩󵄩∞ d|ν|(ω) ≤ ‖X‖∞ ∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω) < ∞.

Ω

Ω

t∈ℝ

s∈ℝ

The next lemma shows that the operator ∫ α(D0 , ω)Xβ(D1 , ω) dν(ω) ∈ ℒ(H) Ω

does not depend on the choice of representation of ϕ in (1.12). Lemma 1.4.6 ([118, Lemma 4.1]). Let α, β be measurable functions on ℝ×Ω, where (Ω, Σ, ν) is a σ-finite measure space, such that 󵄨 󵄨 󵄨 󵄨 ∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω) < ∞. Ω

t∈ℝ

s∈ℝ

If ∫ α(t, ω)β(s, ω) dν(ω) = 0, Ω

then for all bounded linear operators X and self-adjoint operators D0 , D1 , we have ∫ α(D0 , ω)Xβ(D1 , ω) dν(ω) = 0. Ω

It follows from the lemma that if ϕ ∈ BS, then the operator ∫ α(D0 , ω)Xβ(D1 , Ω) dν(ω),

X ∈ ℒ(H),

Ω

depends only on ϕ, D0 , and D1 , and is independent of the decomposition α and β used on the measure space (Ω, Σ, ν). It is also independent of the measure space (Ω, Σ, ν) by using the disjoint union: if ϕ has a decomposition using measurable functions α and β on the measure space (Ω, Σ, ν) and a decomposition using α′ and β′ on the measure space

1.4 Double operator integrals

� 49

(Ω′ , Σ′ , ν′ ), then both decompositions can be lifted to the disjoint union of the measure spaces. Definition 1.4.7. Let ϕ ∈ BS be a function on ℝ2 as in Definition 1.4.5. For self-adjoint linear operators D0 and D1 on H, define the bounded operator D ,D1

Tϕ 0 D ,D1

The operator Tϕ 0 D ,D Tϕ 0 1

(X) := ∫ α(D0 , ω)Xβ(D1 , ω) dν(ω),

X ∈ ℒ(H).

Ω

(X) is called a double operator integral, and the linear operator

: ℒ(H) → ℒ(H) is called a transformer.

Double operator integration is a representation of the algebra of functions BS into the algebra of continuous operators from ℒ(H) to ℒ(H). Theorem 1.4.8. If ϕ1 , ϕ2 ∈ BS, then D ,D

D ,D1

Tϕ 0+ϕ1 (X) = Tϕ 0 1

2

1

D ,D1

(X) + Tϕ 0 2

(X),

D ,D

D ,D1

Tϕ 0ϕ 1 (X) = Tϕ 0 1 2

1

D ,D1

(Tϕ 0 2

(X))

for all X ∈ ℒ(H). D ,D

The next theorem proves the continuity of the transformer Tϕ 0 1 : ℰ → ℰ on a fully symmetric Banach ideal ℰ of ℒ(H). Fully symmetric ideals were introduced in Section 1.2.1. Theorem 1.4.9. Let ϕ ∈ BS be a function on ℝ2 as in Definition 1.4.5. For self-adjoint linear operators D0 and D1 on H, we have D ,D1

Tϕ 0

(X) ≺≺ ‖ϕ‖BS ⋅ X,

X ∈ ℒ(H).

Equivalently, for every fully symmetric Banach ideal ℰ of ℒ(H), we have D ,D1

Tϕ 0

:ℰ →ℰ

and 󵄩󵄩 D0 ,D1 󵄩󵄩 󵄩󵄩Tϕ (X)󵄩󵄩ℰ ≤ ‖ϕ‖BS ⋅ ‖X‖ℰ ,

X ∈ ℒ(H).

Proof. If X ∈ ℒ(H), then Lemma 1.4.2 implies that 󵄩󵄩 D0 ,D1 󵄩󵄩 󵄩 󵄩 󵄩󵄩Tϕ (X)󵄩󵄩∞ ≤ ∫󵄩󵄩󵄩α(D0 , ω)Xβ(D1 , ω)󵄩󵄩󵄩∞ d|ν|(ω) Ω

󵄨 󵄨 󵄨 󵄨 ≤ (∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω)) ⋅ ‖X‖∞ . Ω

t∈ℝ

s∈ℝ

50 � 1 Bounded operators and pseudodifferential operators Taking the infimum over all representations of ϕ yields 󵄩󵄩 D0 ,D1 󵄩󵄩 󵄩󵄩Tϕ (X)󵄩󵄩∞ ≤ ‖ϕ‖BS ⋅ ‖X‖∞ ,

X ∈ ℒ(H).

Similarly, using Lemma 1.4.3 in place of Lemma 1.4.2 yields 󵄩󵄩 D0 ,D1 󵄩󵄩 󵄩󵄩Tϕ (X)󵄩󵄩1 ≤ ‖ϕ‖BS ⋅ ‖X‖1 ,

X ∈ ℒ1 .

Hence D ,D1

Tϕ 0

: ℒ(H) → ℒ(H)

is a linear map, which is a contraction on ℒ1 and ℒ(H). The statements follow from the equivalent conditions in Theorem 1.2.2.

1.4.3 Double operator integrals and estimates of commutators Let D0 and D1 be self-adjoint linear operators on H. Let ϕ ∈ BS be a function on ℝ2 as in D ,D Definition 1.4.5. Let Tϕ 0 1 : ℒ(H) → ℒ(H) denote the transformer as in Definition 1.4.7. If g is a bounded Borel function on ℝ, then it follows from the bounded Borel functional calculus that D ,D1

g(D0 )Tϕ 0

D ,D1

(X) = Tϕ 0

D ,D Tϕ 0 1 (X)g(D1 )

=

(g(D0 )X),

X ∈ ℒ(H),

(1.13)

D ,D Tϕ 0 1 (Xg(D1 )),

X ∈ ℒ(H).

(1.14)

If ϕ is a function of only the first variable, then it also follows that D ,D1

Tϕ 0

(X) = ϕ(D0 )X,

X ∈ ℒ(H),

and if ϕ is only a function of the second variable, then it follows that D ,D1

Tϕ 0

(X) = Xϕ(D1 ),

X ∈ ℒ(H).

Definition 1.4.10. Let f ∈ C 1 (ℝ). The divided difference function f [1] : ℝ2 → ℂ is defined by f [1] (t, s) := {

f (t)−f (s) , t−s ′

f (t),

t ≠ s, t = s.

The next theorem indicates the role of double operator integrals in Lipschitz and commutator estimates of operators.

1.4 Double operator integrals

�

51

Theorem 1.4.11. Let f ∈ C 1 (ℝ). If f [1] ∈ BS and A, B ∈ ℒ(H) are self-adjoint, then TfA,B [1] (AX − XB) = f (A)X − Xf (B),

X ∈ ℒ(H),

(1.15)

and 󵄩 󵄩 f (A)X − Xf (B) ≺≺ 󵄩󵄩󵄩f [1] 󵄩󵄩󵄩BS ⋅ (AX − XB),

X ∈ ℒ(H).

If ℰ is a fully symmetric Banach ideal of ℒ(H) and A, B ∈ ℒ(H) are self-adjoint, then 󵄩󵄩 󵄩 󵄩 [1] 󵄩 󵄩󵄩f (A)X − Xf (B)󵄩󵄩󵄩ℰ ≤ 󵄩󵄩󵄩f 󵄩󵄩󵄩BS ⋅ ‖AX − XB‖ℰ ,

X ∈ ℰ.

Proof. If f [1] ∈ BS and A = A∗ , B = B∗ , X ∈ ℒ(H), then f [1] has a representation with functions α and β as in Definition 1.4.5, and TfA,B [1] (AX − XB) = ∫ Aα(A, ω)Xβ(B, ω)dν(ω) − ∫ α(A, ω)Xβ(B, ω)Bdν(ω) Ω

Ω

= TϕA,B (X), where

ϕ(s, t) = sf [1] (s, t) − f [1] (s, t)t = f (s) − f (t),

s, t ∈ ℝ.

Setting ϕ1 (s, t) = f (s) and ϕ2 (s, t) = f (t), from Theorem 1.4.8 we have A,B A,B TfA,B [1] (AX − XB) = Tϕ (X) − Tϕ (X) = f (A)X − Xf (B). 1

2

The submajorization inequality and norm estimates for fully symmetric ideals now follow from Theorem 1.4.9. Theorem 1.4.11 and identity (1.15) will be used to estimate the norms of differences f (A) − f (B) and commutators [f (A), X] for bounded operators A, B, X on H. The divided difference can be defined for a more general class than the function with continuous first derivative. A difficulty in using Theorem 1.4.11 is determining which functions f have the property f [1] ∈ BS, but the following class of functions will be general enough for the application of Theorem 1.4.11 in later chapters. Sobolev Hilbert spaces will be defined in Section 1.6, in Definition 1.6.8, in terms of the Bessel potential and tempered distributions on ℝd . For the present section, we define 󵄨 󵄨2 H 1 (ℝ) := {f ∈ L2 (ℝ) : ∫(1 + ξ 2 )󵄨󵄨󵄨(ℱ f )(ξ)󵄨󵄨󵄨 dξ < ∞} ℝ

with

52 � 1 Bounded operators and pseudodifferential operators 1

2 󵄨 󵄨2 ‖f ‖H 1 := (∫(1 + ξ 2 )󵄨󵄨󵄨(ℱ f )(ξ)󵄨󵄨󵄨 dξ) ,

f ∈ H 1 (ℝ).

ℝ

Here ℱ : L2 (ℝ) → L2 (ℝ) is the Fourier transform, 1

(ℱ f )(t) = (2π)− 2 ∫ e−itξ f (ξ) dξ,

t ∈ ℝ, f ∈ L2 (ℝ).

ℝ

Lemma 1.4.12. Let f ∈ C 1 (ℝ) with derivative f ′ ∈ H 1 (ℝ). Then 󵄩󵄩 [1] 󵄩󵄩 󵄩 ′󵄩 󵄩󵄩f 󵄩󵄩BS ≤ c ⋅ 󵄩󵄩󵄩f 󵄩󵄩󵄩H 1 for a constant c > 0 that does not depend on f . Proof. By the Cauchy–Schwarz inequality we have ∞

∞

−∞

−∞

1 2

∞

1 2

−1 󵄨 󵄨 󵄨 󵄨2 ∫ 󵄨󵄨󵄨(ℱ f ′ )(ξ)󵄨󵄨󵄨 dξ ≤ ( ∫ (1 + ξ 2 ) dξ) ( ∫ (1 + ξ 2 )󵄨󵄨󵄨(ℱ f ′ )(ξ)󵄨󵄨󵄨 dξ) , −∞

that is, 󵄩󵄩 ′ 󵄩󵄩 󵄩 ′󵄩 󵄩󵄩ℱ f 󵄩󵄩L1 (ℝ) ≤ c1 ⋅ 󵄩󵄩󵄩f 󵄩󵄩󵄩H 1 (ℝ) for a constant c1 > 0 independent of f . For all t, s ∈ ℝ, we have the formula 1

f [1] (t, s) = ∫ f ′ (s(1 − θ) + tθ) dθ. 0

Since the Fourier transform of f ′ is integrable, the Fourier inversion theorem implies that 1

1 ∞

f [1] (t, s) = (2π)− 2 ∫ ∫ (ℱ f ′ )(ξ)eiξs(1−θ) eiξtθ dξdθ. 0 −∞

With Ω = ℝ × [0, 1] and dν(θ, ξ) = (ℱ f ′ )(ξ) dξdθ, this is a decomposition such that f [1] ∈ BS. Moreover, 󵄩󵄩 [1] 󵄩󵄩 󵄩 ′󵄩 −1 󵄩 ′ 󵄩 −1 󵄩󵄩f 󵄩󵄩BS ≤ (2π) 2 󵄩󵄩󵄩(ℱ f )󵄩󵄩󵄩L (ℝ) ≤ ((2π) 2 c1 ) ⋅ 󵄩󵄩󵄩f 󵄩󵄩󵄩H 1 (ℝ) . 1

From Theorem 1.4.11 and Lemma 1.4.12 we have the following Lipschitz and commutator estimates, which will be used in later chapters.

1.4 Double operator integrals

� 53

Corollary 1.4.13. Let f ∈ C 1 (ℝ) with derivative f ′ ∈ H 1 (ℝ). There exists a constant c > 0 such that for bounded self-adjoint operators A, B ∈ ℒ(H) and X ∈ ℒ(H), we have 󵄩 󵄩 f (A)X − Xf (B) ≺≺ c ⋅ 󵄩󵄩󵄩f ′ 󵄩󵄩󵄩H 1 ⋅ (AX − XB). For a fully symmetric Banach ideal ℰ of ℒ(H), we have 󵄩󵄩 󵄩 󵄩 ′󵄩 󵄩󵄩f (A)X − Xf (B)󵄩󵄩󵄩ℰ ≤ c ⋅ 󵄩󵄩󵄩f 󵄩󵄩󵄩H 1 ‖AX − XB‖ℰ . The following is a change-of-variables formula for double operator integrals. Lemma 1.4.14. Let f,g : ℝ → ℝ be bounded Borel-measurable functions. For ϕ ∈ BS, the function ϕf ,g given by ϕf ,g (t, s) = ϕ(f (t), g(s)),

t, s ∈ ℝ,

belongs to BS, and for all self-adjoint operators D0 and D1 , D ,D1

Tϕ 0

f ,g

f (D0 ),g(D1 )

(X) = Tϕ

(X),

X ∈ ℒ(H).

Proof. This is almost an immediate consequence of the definition. Suppose ϕ ∈ BS is written as ϕ(t, s) = ∫ α(t, ω)β(s, ω)dν(ω),

s, t ∈ ℝ.

Ω

Then ϕf ,g (t, s) = ∫ α(f (t), ω)β(g(s), ω)dν(ω),

s, t ∈ ℝ,

Ω

is a decomposition of the measurable function ϕf ,g on ℝ2 with 󵄨 󵄨 󵄨 󵄨 ∫ sup󵄨󵄨󵄨α(f (t), ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(g(s), ω)󵄨󵄨󵄨 d|ν|(ω) Ω

t∈ℝ

s∈ℝ

󵄨 󵄨 󵄨 󵄨 ≤ ∫ sup󵄨󵄨󵄨α(t, ω)󵄨󵄨󵄨 sup󵄨󵄨󵄨β(s, ω)󵄨󵄨󵄨 d|ν|(ω) < ∞. Ω

t∈ℝ

s∈ℝ

Hence ϕf ,g ∈ BS. We have D ,D1

Tϕ 0

f ,g

f (D0 ),g(D1 )

(X) = ∫ α(f (D0 ), ω)Xβ(g(D1 ), ω) dν(ω) = Tϕ Ω

(X).

54 � 1 Bounded operators and pseudodifferential operators Note that for ν-almost all ω, α(f (D0 ), ω), β(g(D1 ), ω) ∈ ℒ(H) by the bounded Borel functional calculus, since α(f (t), ω), t ∈ ℝ, and β(g(s), ω), s ∈ ℝ, are bounded Borel functions on ℝ for ν-almost all ω.

1.5 Compactness estimates of product-convolution operators In this section, we examine the compactness of product-convolution operators on the Euclidean plane and the torus. Estimates of the decay of the singular values of compact product-convolution operators are proved using submajorization and majorization. A submajorization theorem is proved for the noncommutative Lorentz spaces introduced in Section 1.2.3 and will be used for noncommutative tori and noncommutative Euclidean spaces in later chapters.

1.5.1 Product-convolution operators on the Euclidean plane By the Hausdorff–Young inequality, for 1 ≤ p ≤ 2, the Fourier transform d

(ℱ u)(t) := (2π)− 2 ∫ e−i⟨t,ξ⟩ u(ξ) dξ,

u ∈ Lp (ℝd ),

(1.16)

u ∈ Lp (ℝd ),

(1.17)

ℝd

is a continuous linear operator d

d

ℱ : Lp (ℝ ) → Lp∗ (ℝ ),

where 2 ≤ p∗ ≤ ∞ is the Hölder conjugate of p, 1 1 + = 1. p∗ p The dual map d

(ℱ ∗ u)(t) := (2π)− 2 ∫ ei⟨t,ξ⟩ u(ξ) dξ, ℝd

is a continuous linear operator d

∗

d

ℱ : Lp (ℝ ) → Lp∗ (ℝ )

such that ⟨ℱ ∗ u, v⟩ = ⟨u, ℱ v⟩ = ∫ u(t)(ℱ v)(t)dt, ℝd

u, v ∈ Lp (ℝd ).

1.5 Compactness estimates of product-convolution operators

� 55

For p = 2, the Fourier transform d

d

ℱ : L2 (ℝ ) → L2 (ℝ )

is a unitary operator on the Hilbert space L2 (ℝd ) by the Plancherel theorem, ‖ℱ u‖2 = ‖u‖2 , and (ℱ ∗ u)(t) = (ℱ −1 u)(t),

t ∈ ℝd ,

u ∈ L2 (ℝd ).

We define convolution operators using the fact that the Fourier transform maps products to convolutions. Definition 1.5.1. Let g ∈ Lp (ℝd ) for 2 ≤ p ≤ ∞. The convolution operator g(∇) : L2 (ℝd ) → L 2p (ℝd ) p−2

is a continuous linear operator defined by d

(g(∇)u)(t) := (2π)− 2 ∫ ei⟨t,ξ⟩ g(ξ)(ℱ u)(ξ) dξ,

u ∈ L2 (ℝd ).

ℝd

Equivalently, g(∇) = ℱ ∗ Mg ℱ , where Mg : L2 (ℝd ) → L 2p (ℝd ) p+2

is the product operator (Mg u)(t) := g(t)u(t),

t ∈ ℝd ,

u ∈ L2 (ℝd ).

We caution the reader that this notation is not consistent with writing ∇ for the gradient operator (𝜕1 , . . . , 𝜕d ), where 𝜕j , j = 1, . . . , d, are the usual partial derivatives on ℝd . Instead, ∇ denotes the operator (−i𝜕1 , . . . , −i𝜕d ). The operator g(∇) : L2 (ℝd ) → L 2p (ℝd ) p−2

is continuous, which follows by an application of the Hölder and Hausdorff–Young inequalities.

56 � 1 Bounded operators and pseudodifferential operators Definition 1.5.2. Let f , g ∈ Lp (ℝd ) for 2 ≤ p ≤ ∞. The product-convolution operator Mf g(∇) : L2 (ℝd ) → L2 (ℝd ) is a bounded linear operator defined by d

(Mf g(∇)u)(t) := (2π)− 2 f (t) ∫ ei⟨t,ξ⟩ g(ξ)(ℱ u)(ξ) dξ,

u ∈ L2 (ℝd ).

ℝd

Equivalently, Mf g(∇) = Mf ℱ ∗ Mg ℱ , where Mf : L 2p (ℝd ) → L2 (ℝd ) p−2

is the product operator t ∈ ℝd ,

(Mf u)(t) := f (t)u(t),

u ∈ L 2p (ℝd ). p−2

For f , g ∈ Lp (ℝd ), the product-convolution operator is bounded, and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)u󵄩󵄩󵄩2 ≤ ‖f ‖p ‖g‖p ‖u‖2 ,

u ∈ L2 (ℝd ),

(1.18)

which follows by the Hausdorff–Young and Hölder inequalities. Using the Marcinkiewicz interpolation theorem, Calderón proved a sharper Hausdorff–Young inequality for 1 < p < 2 and 1 ≤ q ≤ ∞, d

d

ℱ : Lp,q (ℝ ) → Lp∗ ,q (ℝ ),

and ∗

d

d

ℱ : Lp,q (ℝ ) → Lp∗ ,q (ℝ )

where p∗ is the Hölder conjugate of p. Combined with the Hölder inequality for Lorentz spaces (Theorem 1.2.6), when p > 2 and g ∈ Lp,∞ (ℝd ), the convolution operator defined by g(∇) : ℱ ∗ Mg ℱ : L2 (ℝd ) → L 2p ,2 (ℝd ) ⊂ L 2p (ℝd ) p−2

is everywhere defined and continuous.

p−2

(1.19)

1.5 Compactness estimates of product-convolution operators �

57

Definition 1.5.3. Let f ∈ Lp (ℝd ) and g ∈ Lp,∞ (ℝd ) for 2 < p ≤ ∞. The productconvolution operator Mf g(∇) : L2 (ℝd ) → L2 (ℝd ) is a bounded linear operator defined by d

(Mf g(∇)u)(t) := (2π)− 2 f (t) ∫ ei⟨t,ξ⟩ g(ξ)(ℱ u)(ξ) dξ,

u ∈ L2 (ℝd ).

ℝd

Equivalently, Mf g(∇) = Mf ℱ ∗ Mg ℱ . For f ∈ Lp (ℝd ) and g ∈ Lp,∞ (ℝd ), the product-convolution operator is bounded and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)u󵄩󵄩󵄩2 ≤ cp ⋅ ‖f ‖p ‖g‖p,∞ ‖u‖2 ,

u ∈ L2 (ℝd ),

(1.20)

for a constant cp > 0, which follows from (1.19) and the Hölder inequality on Lorentz spaces.

1.5.2 Hilbert–Schmidt operators on the Euclidean plane and the trace formula The following characterization of Hilbert–Schmidt operators on the Hilbert space L2 (ℝd ) follows from the Schmidt decomposition of a compact operator (Theorem 1.1.14 of Volume I) and the Plancherel theorem. Theorem 1.5.4. If A : L2 (ℝd ) → L2 (ℝd ) is a bounded linear operator, then A ∈ ℒ2 (L2 (ℝd )) if and only if there is a square-integrable function kA ∈ L2 (ℝd × ℝd ) such that (Au)(t) = ∫ kA (t, s)u(s)ds,

t ∈ ℝd ,

u ∈ L2 (ℝd ).

(1.21)

ℝd

If A ∈ ℒ2 (L2 (ℝd )), then ‖A‖ℒ2 = ‖kA ‖L2 (ℝd ×ℝd ) . Equivalently, A ∈ ℒ2 (L2 (ℝd )) if and only if there is a square-integrable function pA ∈ L2 (ℝd × ℝd ) such that d

(Au)(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ u)(ξ)dξ, ℝd

t ∈ ℝd ,

u ∈ L2 (ℝd ).

(1.22)

58 � 1 Bounded operators and pseudodifferential operators If A ∈ ℒ2 (L2 (ℝd )), then d

‖A‖ℒ2 = (2π)− 2 ‖pA ‖L2 (ℝd ×ℝd ) . If A ∈ ℒ2 (L2 (ℝd )), then the function kA ∈ L2 (ℝd × ℝd ) is called the kernel of the operator A, and the function pA ∈ L2 (ℝd × ℝd ) is called the symbol of A. Theorem 1.5.4 is well known for the representation of a Hilbert–Schmidt operator as an integral operator with square-integrable kernel. The characterization of a Hilbert– Schmidt operator in terms of its symbol follows from pA (t, ξ) = ∫ ei⟨s,ξ⟩ kA (t, t + s)ds,

t, ξ ∈ ℝd ,

(1.23)

ℝd

and kA (t, s) = (2π)−d ∫ ei⟨t−s,ξ⟩ pA (t, ξ)dξ,

t, s ∈ ℝd .

(1.24)

ℝd

A trace class operator A ∈ ℒ1 (L2 (ℝd )) is Hilbert–Schmidt, and hence it has a squareintegrable symbol pA ∈ L2 (ℝd × ℝd ). If the symbol pA is integrable, then the trace of A can be calculated from the symbol pA . Most of the applications in this book are directly, or indirectly, an extension of the following trace formula for trace class operators. Theorem 1.5.5. If A ∈ ℒ1 (L2 (ℝd )) and pA ∈ L1 (ℝd × ℝd ), then Tr(A) = (2π)−d

∫ pA (t, ξ)dtdξ. ℝd ×ℝd

Proof. We use the Hermite basis in L2 (ℝd ) defined by setting d

ψn (ξ) := ∏ ψnk (ξk ), k=1

ξ = (ξ1 , . . . , ξd ) ∈ ℝd ,

n = (n1 , . . . , nd ) ∈ ℤd+ ,

where ψm , m ∈ ℤ+ , are the Hermite functions. Since A ∈ ℒ1 , it follows that Tr(A) = ∑ ⟨Aψn , ψn ⟩. n∈ℤd+

By the Abel summation method we have Tr(A) = lim− ∑ λ|n|1 ⟨Aψn , ψn ⟩, λ→1

where |n|1 = ∑dk=1 nk for n = (n1 , . . . , nd ).

n∈ℤd+

1.5 Compactness estimates of product-convolution operators

� 59

Since A ∈ ℒ1 is Hilbert–Schmidt, from (1.22) we have d

(Aψn )(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ ψn )(ξ)dξ. ℝd

The Hermite functions are the eigenfunctions of the Fourier transform with ℱ ψn = (−i)

|n|1

ψn ,

n ∈ ℤd+ .

Therefore d

⟨Aψn , ψn ⟩ = (−i)|n|1 (2π)− 2

∫ ei⟨t,ξ⟩ pA (t, ξ)ψn (t)ψn (ξ)dtdξ. ℝd ×ℝd

Thus d

∫ ei⟨t,ξ⟩ pA (t, ξ)ψn (t)ψn (ξ)dtdξ.

Tr(A) = (2π)− 2 lim− ∑ (−iλ)|n|1 λ→1

n∈ℤd+

ℝd ×ℝd

Cramer’s inequality shows the uniform boundedness of Hermite functions (see, e. g., formula (22.14.17) in [1]), ‖ψn ‖∞ ≤ 1,

n ∈ ℤd+ .

Hence, for every λ ∈ (0, 1), the series F(λ, t, ξ) := ∑ (−iλ)|n|1 ψn (t)ψn (ξ) n∈ℤd+

converges uniformly in t and ξ. Since pA ∈ L1 (ℝd × ℝd ), it follows that d

Tr(A) = (2π)− 2 lim− ∫ ei⟨t,ξ⟩ pA (t, ξ)F(λ, t, ξ)dtdξ. λ→1

ℝd ×ℝd

We clearly have d

F(λ, t, ξ) = ∏ ∑ (−iλ)nk ψnk (tk )ψnk (ξk ). k=1 nk ≥0

From Mehler’s formula, valid for z ∈ ℂ with |z| < 1 and x, y ∈ ℝ, ∑ zm ψm (x)ψm (y) = (π(1 − z2 ))

m≥0

we have the identity

− 21

exp(

4zxy − (x 2 + y2 )(1 + z2 ) ), 2(1 − z2 )

(1.25)

60 � 1 Bounded operators and pseudodifferential operators

F(λ, t, ξ) = (π(1 + λ2 ))

− d2

exp(−

4iλ⟨t, ξ⟩ + (|ξ|2 + |t|2 )(1 − λ2 ) ). 2(1 + λ2 )

Hence d

F(λ, t, ξ) → (2π)− 2 exp(−i⟨t, ξ⟩),

λ → 1− ,

t, ξ ∈ ℝd ,

and 󵄨󵄨 󵄨 󵄨󵄨F(λ, t, ξ)󵄨󵄨󵄨 ≤ 1,

λ ∈ (0, 1),

t, ξ ∈ ℝd .

Since pA ∈ L1 (ℝd × ℝd ), the statement now follows from (1.25) and the dominated convergence theorem.

1.5.3 Compactness of product-convolution operators for square-integrable functions When f , g ∈ L2 (ℝd ), the product-convolution operator Mf g(∇) : L2 (ℝd ) → L2 (ℝd ) from Definition 1.5.2 is an integral operator of the form d

(Mf g(∇)u)(t) = (2π)− 2 ∫ ei⟨t,s⟩ f (t)g(s)(ℱ u)(s) ds,

t ∈ ℝd ,

u ∈ L2 (ℝd ),

(1.26)

ℝd

with symbol (f ⊗ g)(t, s) = f (t)g(s),

s, t ∈ ℝd .

Since f , g ∈ L2 (ℝd ), we have f ⊗ g ∈ L2 (ℝd × ℝd ), and the next theorem follows immediately from Theorem 1.5.4. Theorem 1.5.6. If f , g ∈ L2 (ℝd ), then Mf g(∇) ∈ ℒ2 (L2 (ℝd )) and 󵄩󵄩 󵄩 −d 󵄩󵄩Mf g(∇)󵄩󵄩󵄩ℒ2 = (2π) 2 ‖f ⊗ g‖L2 (ℝd ×ℝd ) . When f , g ∈ L2 (ℝd ) have compact supports, the estimate for the decay of the singular values of the product-convolution operator can be improved.

1.5 Compactness estimates of product-convolution operators �

61

Theorem 1.5.7. If f , g ∈ L2 (ℝd ) and f , g have support in [0, 1]d , then Mf g(∇) ∈ ℒp (L2 (ℝd )) for every 0 < p ≤ 2, and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)󵄩󵄩󵄩ℒp ≤ cp ⋅ ‖f ⊗ g‖L2 (ℝd ×ℝd ) for a constant cp > 0 depending only on d and p. To prove Theorem 1.5.7, we need the following lemma. Lemma 1.5.8. Let h ∈ L1 (ℝd ) be supported on [0, 1]d , and let ϕ ∈ C ∞ (ℝd ) be supported on [−3, 3]d . For the function ψ = ϕ ⋅ ℱ −1 h and every 0 < p ≤ 2, we have p 󵄨 󵄨p ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 ≤ cϕ,p,d ⋅ ‖h‖1

k∈ℤd

for a constant cϕ,p,d > 0. Proof. Since h ∈ L1 (ℝd ) is compactly supported, it follows that ℱ −1 h ∈ C ∞ (ℝd ). By the assumptions the function ψ = ϕ ⋅ ℱ −1 h ∈ C ∞ ([−π, π]d ) and vanishes near the boundary together with all its derivatives. Fix an even integer m > dp . We have 󵄨 󵄨p 󵄨 󵄨p 󵄨 󵄨 p ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 = ∑ |k|−mp (|k|m 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨) ≤ ( ∑ |k|−mp ) ⋅ (sup |k|m 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨) .

k =0 ̸

k =0 ̸

k =0 ̸

k =0 ̸

Since ψ ∈ C ∞ ([−π, π]d ) and m is even, it follows that |k|m (ℱ ψ)(k) is the kth Fourier m coefficient of (−Δ) 2 ψ. Hence m m 󵄨 󵄨 󵄩 󵄩 sup |k|m 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩(−Δ) 2 ψ󵄩󵄩󵄩∞ ≤ d 2 ‖ψ‖C m ([−π,π]d ) .

k =0 ̸

By the definition of ψ and the Leibniz rule we have m 󵄨 󵄨 󵄩 󵄩 sup |k|m 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 ≤ d 2 2m ‖ϕ‖C m (ℝd ) 󵄩󵄩󵄩ℱ −1 h󵄩󵄩󵄩C m (ℝd ) .

k =0 ̸

l

To estimate the norm ‖ℱ −1 h‖C m (ℝd ) , for every l ∈ ℤd+ , we set hl (t) = ∏dj=1 tj j , t ∈ ℝd . We have 󵄩󵄩 −1 󵄩󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩ℱ h󵄩󵄩C m (ℝd ) = max 󵄩󵄩󵄩hl (∇)ℱ h󵄩󵄩󵄩∞ = max 󵄩󵄩󵄩ℱ (hl h)󵄩󵄩󵄩∞ ‖l‖ ≤m ‖l‖ ≤m 1

≤ max ‖hl h‖1 ≤ ‖h‖1 , ‖l‖1 ≤m

1

62 � 1 Bounded operators and pseudodifferential operators where the last inequality follows from the fact that h is supported on [0, 1]d . Set p

m

cϕ,p,d := ( inf d 2 2m ‖ϕ‖C m (ℝd ) ) . m>d/p

Thus p p 󵄨 󵄨p ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 ≤ cϕ,p,d ⋅ ( ∑ |k|−mp ) ⋅ ‖h‖1 = cϕ,p,d ⋅ ‖h‖1 ,

k =0 ̸

k =0 ̸

which concludes the proof. Proof of Theorem 1.5.7. We can write (see, e. g., [229, Theorem IX.29]) d

(Mf |g|2 (∇)Mf ξ)(s) = (2π)− 2 f (s) ∫ f (t)ξ(t)(ℱ −1 |g|2 )(s − t)dt,

ξ ∈ L2 (ℝd ).

(1.27)

[0,1]d

Let ϕ be a Schwartz function such that ϕ|[0,1]d = 1 and set ψ := ϕ ⋅ ℱ −1 |g|2 as in Lemma 1.5.8. Since |g|2 ∈ L1 (ℝd ), it follows that ψ ∈ C ∞ ([−π, π]d ) and vanishes near the boundary together with all its derivatives. Hence, using Fourier series, we have, for almost every v ∈ [−π, π]d , d

ψ(v) = (2π)− 2 ∑ (ℱ ψ)(k)ei⟨k,v⟩ , k∈ℤd

󵄨 󵄨 ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 < ∞.

k∈ℤd

(1.28)

Since ϕ equals 1 for u ∈ [−1, 1]d , it follows that d

(ℱ −1 |g|2 )(v) = (2π)− 2 ∑ (ℱ ψ)(k)ei⟨k,v⟩ ,

v ∈ [−1, 1]d .

k∈ℤd

We set fk (t) := f (t)ei⟨k,t⟩ , k ∈ ℤ, and define the rank-one operator Ak on L2 (ℝd ) by setting Ak ξ := ⟨ξ, fk ⟩fk ,

ξ ∈ L2 (ℝd ).

By (1.27) we obtain that (Mf |g|2 (∇)Mf ξ)(s) = (2π)−d f (s) ∫ f (t)ξ(t) ∑ (ℱ ψ)(k)ei⟨k,s−t⟩ dt. [0,1]d

k∈ℤd

By (1.28) and the dominated convergence theorem we infer that (Mf |g|2 (∇)Mf ξ)(s) = (2π)−d ∑ (ℱ ψ)(k)f (s)ei⟨k,s⟩ ∫ f (t)ξ(t)e−i⟨k,t⟩ dt k∈ℤd

[0,1]d

1.5 Compactness estimates of product-convolution operators

� 63

= (2π)−d ∑ (ℱ ψ)(k)(Ak ξ)(s). k∈ℤd

The quasinorm ‖ ⋅ ‖ p is p2 -subadditive, and therefore 2

pd p/2 󵄩󵄩 󵄩p 󵄩 󵄩p/2 󵄨 󵄨p 2 − 󵄩󵄩Mf g(∇)󵄩󵄩󵄩p = 󵄩󵄩󵄩Mf |g| (∇)Mf 󵄩󵄩󵄩p/2 ≤ (2π) 2 ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 2 ‖Ak ‖p/2

= (2π)

pd −2

p 2

k∈ℤd

p 󵄨 󵄨 ‖f ‖2 ∑ 󵄨󵄨󵄨(ℱ ψ)(k)󵄨󵄨󵄨 . k∈ℤd

(1.29)

By Lemma 1.5.8 (with h = |g|2 ) we obtain that p 󵄩 2 󵄩p/2 p p 󵄩󵄩 󵄩p 󵄩󵄩Mf g(∇)󵄩󵄩󵄩p ≤ cp ⋅ ‖f ‖2 󵄩󵄩󵄩g 󵄩󵄩󵄩1 = cp ⋅ ‖f ‖2 ‖g‖2

for a constant cp > 0, which concludes the proof. Theorems 1.5.6 and 1.5.7 form the basis for compactness estimates of productconvolution operators. 1.5.4 Abstract product-convolution estimates Let H1 and H2 be separable Hilbert spaces. Let 𝒜1 and 𝒜2 be von Neumann algebras in ℒ(H1 ) and ℒ(H2 ), respectively, and let τ1 and τ2 be faithful normal semifinite traces on 𝒜1 and 𝒜2 . Let 𝒜1 ⊗̄ 𝒜2 denote the spatial tensor product (the weak operator topology closure of the algebraic tensor product) in ℒ(H1 ⊗H2 ) with faithful semifinite trace τ1 ⊗τ2 . Suppose π1 and π2 are representations of, respectively, 𝒜1 and 𝒜2 in ℒ(H) for a separable Hilbert space H. Let A ∈ 𝒜1 and B ∈ 𝒜2 . We have 󵄩󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒ(H) ≤ ‖A ⊗ B‖L∞ (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) . To obtain abstract versions of the results on the decay of singular values of productconvolution operators, we assume that the following estimate holds for the representations π1 and π2 . It is the analogue of Theorem 1.5.6 when π1 is the representation of the von Neumann algebra L∞ (ℝd ) by product operators and π2 is the representation of the von Neumann algebra L∞ (ℝd ) by convolution operators. Assumption 1.5.9. Assume that for all A ∈ 𝒜1 ∩ ℒ2 (𝒜1 , τ1 ) and B ∈ 𝒜2 ∩ ℒ2 (𝒜2 , τ2 ), we have π1 (A)π2 (B) ∈ ℒ2 (H) and 󵄩󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒ (H) ≤ ‖A ⊗ B‖L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) . 2

In the following theorem the singular value function of π1 (A)π2 (B) ∈ ℒ(H) is given by Definition 1.1.2, and the singular value function of A ⊗ B ∈ 𝒜1 ⊗̄ 𝒜2 is from Defini-

64 � 1 Bounded operators and pseudodifferential operators tion 1.2.12. The submajorization estimate allows us to obtain estimates similar to Theorem 1.2.16 in Section 1.2, even though the contraction properties of the bilinear product A ⊗ B 󳨃→ π1 (A)π2 (B) do not necessarily extend to a linear contraction on 𝒜1 ⊗̄ 𝒜2 when taking linear combinations. Theorem 1.5.10. Let 𝒜1 and 𝒜2 be von Neumann algebras with faithful normal semifinite traces τ1 and τ2 as above. Suppose that 𝒜1 and 𝒜2 have representations π1 and π2 in ℒ(H) for a separable Hilbert space H that satisfy Assumption 1.5.9. If A ∈ 𝒜1 and B ∈ 𝒜2 , then 2

2 μℒ(H),Tr (π1 (A)π2 (B)) ≺≺ 160 ⋅ μ𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 (A ⊗ B) .

We will prove Theorem 1.5.10 by a sequence of lemmas. The first two lemmas are technical estimates for sequences of products of pairwise orthogonal operators. A sequence of self-adjoint bounded operators {Ak }k∈ℤ is pairwise orthogonal if Aj Ak = Ak Aj = 0,

k ≠ j, k, j ∈ ℤ.

Lemma 1.5.11. Let {pk }k∈ℤ and {qk }k∈ℤ be sequences of pairwise orthogonal projections in 𝒜1 and 𝒜2 , respectively. For every finite matrix {ckl }k,l∈ℤ , we have 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 . 󵄩󵄩 ∑ ckl π1 (pk )π2 (ql )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩 ∑ ckl pk ⊗ ql 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩ℒ2 󵄩󵄩 󵄩󵄩ℒ2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) k,l∈ℤ k,l∈ℤ Proof. It is immediate that 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 󵄩2 2󵄩 󵄩󵄩 ∑ ckl π1 (pk )π2 (ql )󵄩󵄩󵄩 = ∑ |ckl | 󵄩󵄩󵄩π1 (pk )π2 (ql )󵄩󵄩󵄩2 . 󵄩󵄩 󵄩󵄩2 k,l∈ℤ k,l∈ℤ By Assumption 1.5.9 we have 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 󵄩 󵄩 2 2 󵄩󵄩 ∑ ckl π1 (pk )π2 (ql )󵄩󵄩󵄩 ≤ ∑ |ckl | ‖pk ⊗ ql ‖2 = 󵄩󵄩󵄩 ∑ ckl pk ⊗ ql 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 k,l∈ℤ k,l∈ℤ k,l∈ℤ Lemma 1.5.12. Let {Ak }k∈ℤ and {Bk }k∈ℤ be sequences of pairwise orthogonal self-adjoint contractions. Then 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ Ak Bk 󵄩󵄩󵄩 ≤ 1. 󵄩󵄩 󵄩󵄩∞ k∈ℤ

1.5 Compactness estimates of product-convolution operators

� 65

Proof. Clearly, 󵄨󵄨 󵄨󵄨2 󵄨󵄨 󵄨 2 󵄨󵄨 ∑ Ak Bk 󵄨󵄨󵄨 = ∑ Bk1 Ak1 Ak2 Bk2 = ∑ Bk1 Ak1 Bk1 . 󵄨󵄨 󵄨󵄨 k∈ℤ k1 ,k2 ∈ℤ k1 ∈ℤ The summands are pairwise orthogonal. Thus 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 2 2 2 2 󵄩󵄩 ∑ Bk1 Ak1 Bk1 󵄩󵄩󵄩 = max󵄩󵄩󵄩Bk1 Ak1 Bk1 󵄩󵄩󵄩∞ = max ‖Bk1 ‖∞ ‖Ak1 ‖∞ ≤ 1. 󵄩󵄩 󵄩󵄩∞ k1 ∈ℤ k1 ∈ℤ k1 ∈ℤ Finally, 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 2 󵄩󵄩 ∑ Ak Bk 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ Bk1 Ak1 Bk1 󵄩󵄩󵄩 ≤ 1. 󵄩󵄩 󵄩󵄩∞ 󵄩󵄩 󵄩󵄩∞ k∈ℤ k1 ∈ℤ The next lemma proves Theorem 1.5.10 when the spectra of the operators A and B are finite and dyadic. Lemma 1.5.13. Let 0 ≤ A ∈ 𝒜1 and 0 ≤ B ∈ 𝒜2 have finite spectra consisting of powers of 2 and 0. We have 2

2 μℒ(H),Tr (π1 (A)π2 (B)) ≺≺ 10 ⋅ μ𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 (A ⊗ B) .

Proof. For brevity, we denote all singular value functions by μ when the von Neumann algebra and trace are clear from the context. By assumption the spectrum of A ⊗ B is a discrete set with finite number of elements consisting of powers of 2 and 0. Hence μ(A ⊗ B) takes values in powers of 2 and 0. Fix t > 0. We have either μ(t, A ⊗ B) = 0 or μ(t, A ⊗ B) = 2n for some n ∈ ℤ. If μ(t, A ⊗ B) = 0, then t

t

0

0

󵄩 󵄩2 ∫ μ2 (s, π1 (A)π2 (B))ds ≤ 󵄩󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩2 ≤ ‖A ⊗ B‖22 = ∫ μ2 (s, A ⊗ B)ds. If μ(t, A ⊗ B) = 2n , then denote pk := χ{2k } (A) and qk := χ{2k } (B) for k ∈ ℤ. We have π1 (A)π2 (B) = ∑ 2k+l π1 (pk )π2 (ql ) = X + Y , k,l∈ℤ

X := ∑ 2k+l π1 (pk )π2 (ql ), k+l>n

Y := ∑ 2k+l π1 (pk )π2 (ql ). k+l≤n

Here the sums are finite, so we do not need to worry about convergence. By Theorem 2.3.5 from Volume I,

66 � 1 Bounded operators and pseudodifferential operators t

t

2

t

0

2

t

∫ μ (s, π1 (A)π2 (B))ds = ∫ μ (s, X + Y )ds ≤ 2 ∫ μ (s, X)ds + 2 ∫ μ2 (s, Y )ds. 0

2

0

0

Thus t

∫ μ2 (s, π1 (A)π2 (B))ds ≤ 2‖X‖22 + 2t‖Y ‖2∞ .

(1.30)

0

By Lemma 1.5.11 we have 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖X‖2 ≤ 󵄩󵄩󵄩 ∑ 2k+l pk ⊗ ql 󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩2 k+l>n Next, ∑ 2k+l pk ⊗ ql = ( ∑ 2k+l pk ⊗ ql ) ⋅ ( ∑ pk ⊗ ql ) = (A ⊗ B)χ(2n ,∞) (A ⊗ B).

k+l>n

k,l∈ℤ

k+l>n

Since μ(t, A ⊗ B) = 2n , it follows that (τ1 ⊗ τ2 )(χ(2n ,∞) (A ⊗ B)) ≤ t. Thus t

󵄩 󵄩2 ‖X‖22 ≤ 󵄩󵄩󵄩(A ⊗ B)χ(2n ,∞) (A ⊗ B)󵄩󵄩󵄩2 ≤ ∫ μ2 (s, A ⊗ B)ds.

(1.31)

0

Next, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖Y ‖∞ ≤ ∑ 2m 󵄩󵄩󵄩 ∑ π1 (pk )π2 (ql )󵄩󵄩󵄩 . 󵄩󵄩 󵄩󵄩∞ m≤n k+l=m Applying Lemma 1.5.12 with Ak := π1 (pk ) and Bk := π2 (qm−k ), we obtain 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩󵄩 ∑ π1 (pk )π2 (ql )󵄩󵄩󵄩 ≤ 1, 󵄩󵄩 󵄩󵄩∞ k+l=m

m ∈ ℤ.

Thus ‖Y ‖∞ ≤ ∑ 2m = 2n+1 = 2μ(t, A ⊗ B). m≤n

(1.32)

1.5 Compactness estimates of product-convolution operators �

67

Combining (1.30), (1.31), and (1.32), we obtain t

t

0

0

∫ μ2 (s, π1 (A)π2 (B))ds ≤ 2 ∫ μ2 (s, A ⊗ B)ds + 8tμ2 (t, A ⊗ B). This completes the proof since μ2 is decreasing and t

tμ2 (t, A ⊗ B) ≤ ∫ μ2 (s, A ⊗ B)ds. 0

Proof of Theorem 1.5.10. For brevity, we denote all singular value functions by μ when the von Neumann algebra and trace are clear from the context. Suppose 0 ≤ A ∈ 𝒜1 and 0 ≤ B ∈ 𝒜2 have spectra consisting of powers of 2 and 0. Set An := Aχ[2−n ,2n ] (A),

Bn := Bχ[2−n ,2n ] (B),

n ∈ ℤ+ .

The operators An and Bn satisfy the assumptions in Lemma 1.5.13. By Lemma 1.5.13 we have μ2 (π1 (An )π2 (Bn )) ≺≺ 10μ2 (An ⊗ Bn ) ≤ 10μ2 (A ⊗ B). The latter equality follows since An ≤ A, Bn ≤ B, and μ(An ⊗ Bn ) = μ(μ(An ) ⊗ μ(Bn )) ≤ μ(μ(A) ⊗ μ(B)) = μ(A ⊗ B). Clearly, π1 (An ) → π1 (A), π2 (Bn ) → π2 (B) as n → ∞ in the strong operator topology. Thus π1 (An )π2 (Bn ) → π1 (A)π2 (B),

n → ∞,

in the strong operator topology. Consequently (using Lemma 2.2.13 in Volume I), μ2 (π1 (A)π2 (B)) ≺≺ 10μ2 (A ⊗ B). Suppose 0 ≤ A ∈ 𝒜1 and 0 ≤ B ∈ 𝒜2 have arbitrary spectra. Set A := EC and B := DF where C := ∑ 2k χ[2k ,2k+1 ) (A), k∈ℤ

D := ∑ 2k χ[2k ,2k+1 ) (B),

E := ∑ 2−k Aχ[2k ,2k+1 ) (A), k∈ℤ

Since ‖E‖∞ ≤ 2 and ‖F‖∞ ≤ 2, it follows that

k∈ℤ

F := ∑ 2−k Bχ[2k ,2k+1 ) (B). k∈ℤ

68 � 1 Bounded operators and pseudodifferential operators μ(π1 (A)π2 (B)) = μ(π1 (E)π1 (C)π2 (D)π2 (F)) ≤ 4μ(π1 (C)π2 (D)). The operators C and D have spectra consisting of powers of 2 and 0. By the preceding paragraph we have μ2 (π1 (C)π2 (D)) ≺≺ 10μ2 (C ⊗ D). Thus μ2 (π1 (A)π2 (B)) ≺≺ 16 ⋅ 10μ2 (C ⊗ D) ≤ 160μ2 (A ⊗ B). The last inequality follows since C ≤ A and D ≤ B. Suppose A ∈ 𝒜1 and B ∈ 𝒜2 are arbitrary. By the preceding paragraph we have 󵄨 󵄨 󵄨 󵄨 μ2 (π1 (A)π2 (B)) = μ2 (π1 (|A|)π2 (󵄨󵄨󵄨B∗ 󵄨󵄨󵄨)) ≺≺ 160μ2 (|A| ⊗ 󵄨󵄨󵄨B∗ 󵄨󵄨󵄨) = 160μ2 (A ⊗ B).

Compactness estimates of products Submajorization estimates provide norm estimates for the fully symmetric ideals of ℒ(H) introduced in Section 1.2 and the Lorentz ideals introduced in Section 1.1. Remark 1.5.14. Note for p ≥ 2 that A ∈ ℒp (𝒜1 , τ1 ) and B ∈ ℒp (𝒜2 , τ2 ) is equivalent to A ⊗ B ∈ ℒp (𝒜1 ⊗̄ 𝒜2 , τ1 ⊗ τ2 ) with ‖A ⊗ B‖ℒp (𝒜1 ⊗𝒜2 ,τ1 ⊗τ2 ) = ‖A‖ℒp (𝒜1 ,τ1 ) ‖B‖ℒp (𝒜2 ,τ2 ) . For p > 2, A ∈ ℒp (𝒜1 , τ1 ) and B ∈ ℒp,∞ (𝒜2 , τ2 ) imply that A ⊗ B ∈ ℒp,∞ (𝒜1 ⊗ 𝒜2 , τ1 ⊗ τ2 ) with ‖A ⊗ B‖ℒp,∞ (𝒜1 ⊗𝒜2 ,τ1 ⊗τ2 ) ≤ ‖A‖ℒp (𝒜1 ,τ1 ) ‖B‖ℒp,∞ (𝒜2 ,τ2 ) (see Proposition 3.8 in [195]). The following result is a corollary to Theorem 1.5.10 observed from the implication (b) ⇒ (c) in Theorem 1.2.16 and Remark 1.5.14. Corollary 1.5.15. Let 𝒜1 and 𝒜2 be von Neumann algebras with faithful normal semifinite traces τ1 and τ2 as above. Suppose that 𝒜1 and 𝒜2 have representations π1 and π2 in ℒ(H) for a separable Hilbert space H that satisfy Assumption 1.5.9. Let E be a fully symmetric Banach space of functions on (0, ∞). (a) If A ∈ 𝒜1 , B ∈ 𝒜2 , and A ⊗ B ∈ ℰ (2) (𝒜1 ⊗ A2 , τ1 ⊗ τ2 ), then π1 (A)π2 (B) ∈ ℰ (2) , and

1.5 Compactness estimates of product-convolution operators

� 69

󵄩󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) ≤ 13 ⋅ ‖A ⊗ B‖ℰ (2) (𝒜1 ⊗A ̄ 2 ,τ1 ⊗τ2 ) , where ℰ (2) is the ideal of ℒ(H) corresponding to the 2-convexification E (2) of E, and ̄ 2 , τ1 ⊗ τ2 ) is the bimodule of 𝒜1 ⊗̄ 𝒜2 corresponding to E (2) . ℰ (2) (𝒜1 ⊗A (b) If A ∈ 𝒜1 ∩ ℒp (𝒜1 , τ1 ) and B ∈ 𝒜2 ∩ ℒp (𝒜2 , τ2 ) for p ≥ 2, then π1 (A)π2 (B) ∈ ℒp , and 󵄩󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒp ≤ 13 ⋅ ‖A‖ℒp (𝒜1 ,τ1 ) ‖B‖ℒp (𝒜2 ,τ2 ) . (c) If A ∈ 𝒜1 ∩ ℒp (𝒜1 , τ1 ) and B ∈ 𝒜2 ∩ ℒp,∞ (𝒜2 , τ2 ) for p > 2, then π1 (A)π2 (B) ∈ ℒp,∞ , and p 󵄩󵄩 󵄩 ⋅ ‖A‖ℒp (𝒜1 ,τ1 ) ‖B‖ℒp,∞ (𝒜2 ,τ2 ) . 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒp,∞ ≤ 18√ p−2 Proof. For brevity, we denote all singular value functions by μ when the von Neumann algebra and trace are clear from the context. The factors in (a) and (b) are obtained from Theorem 1.5.10 by noting that √160 ≤ 13. We now explain the factor in (c). Let E be the fully symmetric function space L p ,∞ (0, ∞) with norm 2

‖f ‖E = sup t

2 −1 p

t>0

t

∫ μ(s, f )ds,

f ∈ E.

0

Then E (2) = Lp,∞ (0, ∞), but the 2-convexified norm is not the Lorentz quasinorm introduced in Example 1.2.5. The 2-convexified norm is an equivalent norm as follows. Note that if f ∈ E (2) , then 2

2

‖|f |2 ‖ p ,∞ = sup t p μ(t, f )2 ≤ sup t p 2

t>0

≤2

−1

t>0

1− p2

2

sup(2t) p t>0

−1

2t

2t

∫ μ(s, f )2 ds t

∫ μ(s, f )2 ds = 2

1− p2

⋅ ‖|f |2 ‖E .

0

Conversely, if f ∈ Lp,∞ (0, ∞), then 2

‖|f | ‖E = sup t t>0

2 −1 p

t

2

2

∫ μ(s, f ) ds ≤ ‖|f | ‖ 0

p ,∞ 2

⋅ sup t t>0

2 −1 p

t

∫s 0

− p2

ds

70 � 1 Bounded operators and pseudodifferential operators

≤

p ⋅ ‖|f |2 ‖ p ,∞ . 2 p−2

By Theorem 1.5.10, 󵄩󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) ≤ √160 ⋅ ‖A ⊗ B‖ℰ (2) (𝒜1 ⊗A ̄ 2 ,τ1 ⊗τ2 ) .

(1.33)

Substituting the Lorentz quasinorm into the left-hand side of inequality (1.33), we have 1 1 − 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒp,∞ ≤ 2 2 p 󵄩󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) .

Substituting the Lorentz quasinorm into the right-hand side of inequality (1.33), we have ‖A ⊗ B‖ℰ (2) (𝒜1 ⊗A2 ,τ1 ⊗τ2 ) ≤ √

p ⋅ ‖A‖ℒp (𝒜1 ,τ1 ) ‖B‖ℒp,∞ (𝒜2 ,τ2 ), p−2

since (see Proposition 3.8 in [195]) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ‖A ⊗ B‖ℰ (2) (𝒜1 ⊗A2 ,τ1 ⊗τ2 ) = 󵄩󵄩󵄩μ(A) ⊗ μ(B)󵄩󵄩󵄩E(2) ≤ 󵄩󵄩󵄩μ(A)󵄩󵄩󵄩p 󵄩󵄩󵄩μ(B)󵄩󵄩󵄩E(2) . 1

To explain the constant in (c), note that √160 ⋅ 2 2

− p1

≤ 18.

To prove an equivalent to Corollary 1.5.15 for noncommutative Lorentz spaces where 0 < p < 2, the following theorem has a condition on a quasi-Banach sequence space contained in l1 , which is a reverse of the fully symmetric property for quasiBanach sequence spaces containing l1 . The condition employs majorization instead of submajorization. Theorem 1.5.17 also requires an assumption that is an analogue of Theorem 1.5.7 when π1 is the representation of the von Neumann algebra L∞ (ℝd ) by product operators and π2 is the representation of the von Neumann algebra L∞ (ℝd ) by convolution operators. Assumption 1.5.16. There exist sequences {pk }k∈ℕ and {qk }k∈ℕ of pairwise orthogonal projections in 𝒜1 and 𝒜2 , respectively, such that ∞

∞

j=1

k=1

∑ π1 (pj ) = ∑ π2 (qk ) = 1, and, for all A ∈ 𝒜1 and B ∈ 𝒜2 , π1 (pj A)π2 (Bqk ) ∈ ℒp (H),

0 < p ≤ 2, j, k ∈ ℕ,

with 󵄩󵄩 󵄩 󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩ℒp (H) ≤ cp ⋅ ‖pj A ⊗ Bqk ‖L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) , for a constant cp > 0.

0 < p ≤ 2, j, k ∈ ℕ,

1.5 Compactness estimates of product-convolution operators

� 71

Assumption 1.5.16 implies Assumption 1.5.9 up to the rescaling one of the representations by a constant. If A ∈ 𝒜1 ∩ ℒ2 (𝒜1 , τ1 ) and B ∈ 𝒜2 ∩ ℒ2 (𝒜2 , τ2 ), and Assumption 1.5.16 holds, then, by the same argument in the first display in the proof of Lemma 1.3.11, ∞

󵄩󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒ (H) = ∑ 󵄩󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩ℒ (H) 2

2

j,k=1

∞

2 2 ≤ c22 ⋅ ∑ ‖pj A ⊗ Bqk ‖2L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) = c2 ⋅ ‖A ⊗ B‖L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) . j,k=1

Theorem 1.5.17. Let 𝒜1 and 𝒜2 be von Neumann algebras with faithful normal semifinite traces τ1 and τ2 as above. Suppose that 𝒜1 and 𝒜2 have representations π1 and π2 in ℒ(H) for a separable Hilbert space H that satisfy Assumption 1.5.16. Let E ⊂ l1 be a quasi-Banach sequence space such that μ(x) ≺ μ(y), x ∈ E, implies that y ∈ E and ‖y‖E ≤ cE′ ⋅ ‖x‖E for a constant cE′ > 0. Let A ∈ 𝒜1 , B ∈ 𝒜2 . If (2) {‖pj A ⊗ Bqk ‖L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) }j,k∈ℕ ∈ E (ℕ × ℕ),

then π1 (A)π2 (B) ∈ ℰ (2) , and 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) ≤ cE ⋅ 󵄩󵄩󵄩{‖pj A ⊗ Bqk ‖L2 (𝒜1 ⊗𝒜 ̄ 2 ,τ1 ⊗τ2 ) }j,k∈ℕ 󵄩 󵄩E(2) (ℕ×ℕ) , where ℰ (2) is the ideal of ℒ(H) corresponding to the 2-convexification E (2) of E, and cE > 0 is a constant. It follows from Lemma 1.3.8 in Section 1.3.1 that the quasi-Banach sequence spaces l p and l p ,∞ satisfy the conditions of Theorem 1.5.17 for 0 < p < 2. 2

2

Corollary 1.5.18. Suppose that 𝒜1 and 𝒜2 have representations π1 and π2 in ℒ(H) for a separable Hilbert space H that satisfy Assumption 1.5.16. Let A ∈ 𝒜1 , B ∈ 𝒜2 , and 0 < p < 2. (a) If {‖pj A‖L2 (𝒜1 ,τ1 ) }j∈ℕ ∈ lp ,

{‖Bqk ‖L2 (𝒜2 ,τ2 ) }k∈ℕ ∈ lp ,

then π1 (A)π2 (B) ∈ ℒp , and

72 � 1 Bounded operators and pseudodifferential operators 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒp ≤ cp ⋅ 󵄩󵄩󵄩{‖pj A‖L2 (𝒜1 ,τ1 ) }j∈ℕ 󵄩󵄩󵄩lp 󵄩󵄩󵄩{‖Bqk ‖L2 (𝒜2 ,τ2 ) }k∈ℕ 󵄩󵄩󵄩lp for some constant cp > 0. (b) If {‖pj A‖L2 (𝒜1 ,τ1 ) }j∈ℕ ∈ lp ,

{‖Bqk ‖L2 (𝒜2 ,τ2 ) }k∈ℕ ∈ lp,∞ ,

then π1 (A)π2 (B) ∈ ℒp,∞ , and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℒp,∞ ≤ cp ⋅ 󵄩󵄩󵄩{‖pj A‖L2 (𝒜1 ,τ1 ) }j∈ℕ 󵄩󵄩󵄩lp 󵄩󵄩󵄩{‖Bqk ‖L2 (𝒜2 ,τ2 ) }k∈ℕ 󵄩󵄩󵄩lp,∞ for some constant cp > 0. To prove Theorem 1.5.17, we need the following lemma. The modulus of concavity of a quasinorm was introduced in Definition 2.4.7 in Volume I. Lemma 1.5.19. Let E be a quasi-Banach sequence space with modulus of concavity K. For 1

every p > 0 such that 2 p

−1

> K, the operator T : E(ℕ2 ) → E(ℕ3 ) defined by

T : x 󳨃→ x ⊗ {k

− p1

}k∈ℕ ,

x ∈ E(ℕ2 ),

is bounded. 1

Proof. By the Aoki–Rolewicz theorem, for q > 0 satisfying 2 q alent quasinorm, if necessary), we have q

q

q

‖x + y‖E ≤ ‖x‖E + ‖y‖E ,

−1

= K (passing to an equiv-

x, y ∈ E(ℕ2 ).

Hence, for p < q, we have ∞

q − 1 󵄩q − q q 󵄩󵄩 󵄩󵄩x ⊗ {(k + 1) p }󵄩󵄩󵄩E ≤ ∑ n p ‖x‖E = c ⋅ ‖x‖E

n=1

for the constant c = ∑∞ n=1 n

q

−p

.

Proof of Theorem 1.5.17. For brevity, we denote all singular value functions by μ when the von Neumann algebra and trace are clear from the context. The operators π1 (pj A), j ≥ 1, are disjoint from the left, and π2 (Bqk ), k ≥ 1, are disjoint from the right. By assumption and from E (2) ⊂ l2 we have

1.5 Compactness estimates of product-convolution operators � ∞

∞

j,k=1

j,k=1

73

󵄩 󵄩2 ∑ 󵄩󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩2 ≤ c2 ⋅ ∑ ‖pj A ⊗ Bqk ‖22 󵄩 󵄩2 ≤ c2 cE ⋅ 󵄩󵄩󵄩{‖pj A ⊗ Bqk ‖2 }j,k∈ℕ 󵄩󵄩󵄩E(2) < ∞

for constants c2 , cE > 0. Hence the assumptions of Lemma 1.3.11 are satisfied, and therefore μ2 (⨁ π1 (pj A)π2 (Bqk )) ≺ μ2 ( ∑ π1 (pj A)π2 (Bqk )) = μ2 (π1 (A)π2 (B)). j,k≥1

j,k≥1

By the assumptions on the quasi-Banach sequence space E, 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ′ 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) ≤ cE ⋅ 󵄩󵄩󵄩⨁ π1 (pj A)π2 (Bqk )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩ℰ (2) j,k≥1 for a constant cE′ > 0. To complete the proof, we now calculate 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩⨁ μ(π1 (pj A)π2 (Bqk ))󵄩󵄩󵄩 . 󵄩󵄩⨁ π1 (pj A)π2 (Bqk )󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩ℰ (2) 󵄩󵄩 󵄩󵄩E(2) j,k≥1 j,k≥1 1

Choose p > 0 such that 2 p tion

−1

> K, where K is the modulus of concavity for E. By assump-

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩ℒp,∞ ≤ 󵄩󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩ℒp ≤ cp ⋅ ‖pj A ⊗ Bqk ‖2 for a constant cp > 0. Hence μ(π1 (pj A)π2 (Bqk )) ≤ cp ‖pj A ⊗ Bqk ‖2 ⋅ zp , where zp = {m

− p1

}m∈ℕ .

Thus μ(⨁ μ(π1 (pj A)π2 (Bqk ))) ≤ cp ⋅ μ(⨁ ‖pj A ⊗ Bqk ‖2 ⋅ zp ). j,k≥1

j,k≥1

Using the standard equality μ(⨁ cm ⋅ f ) = μ({cm }m≥0 ⊗ f ), m≥0

we write

74 � 1 Bounded operators and pseudodifferential operators

μ(⨁ μ(π1 (pj A)π2 (Bqk ))) ≤ cp ⋅ μ({‖pj A ⊗ Bqk ‖2 }j,k≥1 ⊗ zp ). j,k≥1

Thus 󵄩󵄩 󵄩 󵄩 ′ 󵄩 󵄩󵄩π1 (A)π2 (B)󵄩󵄩󵄩ℰ (2) ≤ cp cE ⋅ 󵄩󵄩󵄩{‖pj A ⊗ Bqk ‖2 }j,k≥1 ⊗ zp 󵄩󵄩󵄩ℰ (2) . The statement now follows from Lemma 1.5.19.

1.5.5 Estimates for product-convolution operators on the Euclidean plane Theorems 1.5.20 and 1.5.22 provide estimates for the decay of singular values for productconvolution operators on the Euclidean plane. In the Euclidean case, Theorems 1.5.10 and 1.5.17 provide the conditions required for the abstract results of the preceding section. Part (b) of the following theorem was originally proved by Michael Cwikel [93]. Theorem 1.5.20. Let p > 2. (a) If f ∈ Lp (ℝd ) and g ∈ Lp (ℝd ), then Mf g(∇) ∈ ℒp (L2 (ℝd )), and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)󵄩󵄩󵄩p ≤ 13 ⋅ ‖f ‖p ‖g‖p , where Mf g(∇) is the product-convolution operator in Definition 1.5.2. (b) If f ∈ Lp (ℝd ) and g ∈ Lp,∞ (ℝd ), then Mf g(∇) ∈ ℒp,∞ (L2 (ℝd )), and p 󵄩󵄩 󵄩 ⋅ ‖f ‖p ‖g‖p,∞ , 󵄩󵄩Mf g(∇)󵄩󵄩󵄩p,∞ ≤ 18√ p−2 where Mf g(∇) is the product-convolution operator in Definition 1.5.3. Proof. For brevity, we denote all singular value functions by μ when the von Neumann algebra and trace are clear from the context. Suppose f ∈ Lp (ℝd ) and g ∈ Lp,∞ (ℝd ). Let n, m ≥ 0. Set fn = pn f , where pn = χ{s:|f (s)|≤n} and gm = qm g with qm = χ{s:|g(s)|≤m} . Then fn , gm ∈ L∞ (ℝd ). Let π1 be the representation of the von Neumann algebra L∞ (ℝd ) by product operators, and let π2 be the representation of the von Neumann algebra L∞ (ℝd ) by convolution operators. Then

1.5 Compactness estimates of product-convolution operators �

75

π1 (fn )π2 (gm ) = π1 (pn )Mf g(∇)π2 (qm ), and π1 (fn )π2 (gm ) → Mf g(∇),

n, m → ∞,

in the strong operator topology since Mf g(∇) is a bounded operator. We have, using Lemma 2.2.13 in Volume I, t

t

2

2

∫ μ(s, Mf g(∇)) ds = lim ∫ μ(s, π1 (fn )π2 (gm )) ds. m,n→∞

0

0

Assumption 1.5.9 is satisfied because of Theorem 1.5.6. It follows from Theorem 1.5.10 that t

2

t

∫ μ(s, π1 (fn )π2 (gm )) ds ≤ 160 ⋅ ∫ μ(s, fn ⊗ gm )2 ds. 0

0

Since |fn | ≤ |f | and |gm | ≤ |g|, m, n ≥ 0, we have μ(fn ⊗ gm )2 ≤ μ(f ⊗ g)2 . Combining the estimates, we get 2

μ(Mf g(∇)) ≺≺ 160 ⋅ μ(f ⊗ g)2 . Both statements now follow from the same observations and calculations as in Corollary 1.5.15. Remark 1.5.21. Part (b) of Theorem 1.5.20 is false for p = 2. There exist f ∈ L2 (ℝ) and g ∈ L2,∞ (ℝ) such that Mf g(∇) does not belong to ℒ4 (L2 (ℝ)); see [186, Theorem 5.1]. Let K = [0, 1]d be a fixed unit cube. Let f ∈ L0 (ℝd ) be locally square integrable. Let E ⊂ l2 be a quasi-Banach sequence space that embeds continuously in l2 and denote 󵄩 󵄩 ‖f ‖E(L2 ) = 󵄩󵄩󵄩{‖f ‖L2 (m+K) }m∈ℤd 󵄩󵄩󵄩E(ℤd ) and (E(L2 ))(ℝd ) := {f ∈ L0 (ℝd ) : ‖f ‖E(L2 ) < ∞}. If f , g ∈ (E(L2 ))(ℝd ), then f , g ∈ L2 (ℝd ), and the linear operator Mf g(∇) : L2 (ℝd ) → L2 (ℝd )

76 � 1 Bounded operators and pseudodifferential operators from Definition 1.5.2 is everywhere defined and bounded. The following theorem was originally proved by Birman and Solomyak [33]. Theorem 1.5.22. Let 0 < p < 2. (a) If f ∈ (lp (L2 ))(ℝd ) and g ∈ (lp (L2 ))(ℝd ), then Mf g(∇) ∈ ℒp (L2 (ℝd )), and there exists a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)󵄩󵄩󵄩ℒp ≤ cp ⋅ ‖f ‖lp (L2 ) ‖g‖lp (L2 ) . (b) If f ∈ (lp (L2 ))(ℝd ) and g ∈ (lp,∞ (L2 ))(ℝd ), then Mf g(∇) ∈ ℒp,∞ (L2 (ℝd )), and there exists a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇)󵄩󵄩󵄩ℒp,∞ ≤ cp ⋅ ‖f ‖lp (L2 ) ‖g‖lp,∞ (L2 ) . Proof. Suppose f ∈ (lp (L2 ))(ℝd ) and g ∈ (lp,∞ (L2 ))(ℝd ) are bounded. Let π1 be the representation of the von Neumann algebra L∞ (ℝd ) by product operators, and let π2 be the representation of the von Neumann algebra L∞ (ℝd ) by convolution operators. Set A := f , B := g, and pj = qj := χK+m(j) , j ∈ ℕ, for an enumeration m : ℕ → ℤd . From Theorem 1.5.7 we have 󵄩󵄩 󵄩 󵄩󵄩π1 (pj A)π2 (Bqk )󵄩󵄩󵄩ℒs ≤ cs ⋅ ‖pj A ⊗ Bqk ‖2 ,

0 < s ≤ 2, j, k ∈ ℕ,

for a constant cs > 0. Hence Assumption 1.5.16 is satisfied. For bounded f and g, both statements follow from Corollary 1.5.18 since 󵄩󵄩 󵄩 󵄩󵄩{‖pj A‖2 }j∈ℕ 󵄩󵄩󵄩lp = ‖f ‖lp (L2 ) ,

f ∈ (lp (L2 ))(ℝ𝕕 ),

and 󵄩󵄩 󵄩 󵄩󵄩{‖Bqk ‖2 }k∈ℕ 󵄩󵄩󵄩lp,∞ = ‖g‖lp,∞ (L2 ) ,

g ∈ (lp (L2 ))(ℝ𝕕 ).

Completing the proof for arbitrary f ∈ (lp (L2 ))(ℝd ) and g ∈ (lp,∞ (L2 ))(ℝd ) using the bounded approximations fn = fχ|f |≤n and gn = gχ|g|≤n , n ≥ 0, is similar to the proof in Theorem 1.5.20. Observe that since L2 (K) ⊂ L1 (K), the space (l1 (L2 ))(ℝd ) is continuously embedded in L1 (ℝd ) ∩ L2 (ℝd ), and hence the integration map

1.5 Compactness estimates of product-convolution operators

� 77

f ∈ (l1 (L2 ))(ℝd ),

f 󳨃→ ∫ f (t) dt, ℝd

is continuous on (l1 (L2 ))(ℝd ). From this continuity and Theorem 1.5.22 we have the following trace formula. Corollary 1.5.23. For all f ∈ (l1 (L2 ))(ℝd ) and g ∈ (l1 (L2 ))(ℝd ), we have Tr(Mf g(∇)) = (2π)−d ∫ f (t) dt ∫ g(t) dt. ℝd

ℝd

Proof. The integral operator Mf g(∇) is trace class by Theorem 1.5.22. The symbol of Mf g(∇) is f ⊗ g. We have f ⊗ g ∈ L1 (ℝd × ℝd ) by the assumption on f and g. The statement follows from Theorem 1.5.5. Let Δ be the Laplacian operator on ℝd . Denote by d

(1 − Δ)− 2

the bounded operator g(∇) : L2 (ℝd ) → (L∞ ∩ L2 )(ℝd ) from Definition 1.5.1 where − d2

g(t) = (1 + |t|2 )

t ∈ ℝd .

,

The square-integrable function g satisfies ‖g‖L2 (m+K) ≤ c ⋅ (1 + |m|2 )

− d2

,

m ∈ ℤd ,

for a constant c > 0. Hence g ∈ (l1,∞ (L2 ))(ℝd ). Remark 1.5.24. If f ∈ (l1 (L2 ))(ℝd ), then the product-convolution operator d

Mf (1 − Δ)− 2

belongs to the weak trace class ideal ℒ1,∞ by Theorem 1.5.22. The linear functional d

f 󳨃→ φ(Mf (1 − Δ)− 2 ),

f ∈ (l1 (L2 ))(ℝd ), d

obtained by applying a trace φ on ℒ1,∞ to the operator Mf (1 − Δ)− 2 , is examined in Chapter 3.

78 � 1 Bounded operators and pseudodifferential operators 1.5.6 Estimates for product-convolution operators on tori Supplementing the discussion of estimates of the decay of singular values for productconvolution operators on Euclidean space ℝd , we include some brief details about the corresponding results for tori 𝕋d , where 𝕋 is the unit circle in the complex plane, 𝕋 := {z ∈ ℂ : |z| = 1}, with its canonical group multiplication. We write Lp (𝕋d ), and so forth, for the Lp -spaces of the d-torus equipped with the normalized Haar measure. For n ∈ ℤd , we denote by zn the trigonometric basis function n

n

zn := z1 1 ⋅ ⋅ ⋅ zdd ,

z = (z1 , . . . , zd ) ∈ 𝕋d ,

n = (n1 , . . . , nd ) ∈ ℤd .

The family {zn }n∈ℤd forms an orthonormal basis for L2 (𝕋d ). In the discrete case the Hausdorff–Young inequality implies that d

(ℱ u)(n) := (2π)− 2 ∫ z−n u(z)dz,

n ∈ ℤd ,

u ∈ L2 (𝕋d ),

𝕋d

is a linear continuous operator d

d

ℱ : Lp (𝕋 ) → lp∗ (ℤ )

for the Hölder conjugate p∗ of 1 ≤ p ≤ 2. In the discrete case, d

(ℱ ∗ x)(z) := (2π)− 2 ∑ x(n)zn , n∈ℤd

z ∈ 𝕋d ,

x ∈ lp (ℤd ),

is the dual map ∗

d

d

ℱ : lp (ℤ ) → Lp∗ (𝕋 ).

When p = 2, the Fourier transform is a unitary operator d

d

ℱ : L2 (𝕋 ) → l2 (ℤ )

with inverse ℱ −1 = ℱ ∗ . Similarly to the ℝd case, for p ≥ 2 and g ∈ lp (ℤd ), the Hausdorff–Young and Hölder inequalities can be used to define the continuous linear operator g(∇𝕋d ) : L2 (𝕋d ) → L 2p (𝕋d ) p−2

by

1.5 Compactness estimates of product-convolution operators �

d

(g(∇𝕋d )u)(z) := (2π)− 2 ∑ g(n)(ℱ u)(n)zn ,

z ∈ 𝕋d ,

n∈ℤd

79

u ∈ L2 (𝕋d ).

Definition 1.5.25. Let f ∈ Lp (𝕋d ) and g ∈ lp (ℤd ) for 2 ≤ p ≤ ∞. The bounded linear operator Mf g(∇𝕋d ) : L2 (𝕋d ) → L2 (𝕋d ) is defined via the Fourier transform ℱ as d

(Mf g(∇𝕋d )u)(z) := (2π)− 2 f (t) ∑ g(n)(ℱ u)(n)zn , n∈ℤd

z ∈ 𝕋d ,

u ∈ L2 (𝕋d ).

Equivalently, Mf g(∇𝕋d ) = Mf ℱ ∗ Mg ℱ , where Mg : l2 (ℤd ) → l 2p (ℤd ) p+2

is the multiplication operator (Mg x)(n) := g(n)x(n),

n ∈ ℤd ,

x ∈ l2 (ℤd ),

and Mf : L 2p (𝕋d ) → L2 (𝕋d ) p−2

is the multiplication operator (Mf u)(z) := f (z)u(z),

z ∈ 𝕋d ,

u ∈ L 2p (𝕋d ). p−2

The operator Mf g(∇𝕋d ) is Hilbert–Schmidt when f ∈ L2 (𝕋d ) and g ∈ l2 (ℤd ). Theorem 1.5.26. Let f ∈ L2 (𝕋d ) and g ∈ lp (ℤd ) for 0 < p ≤ 2. Then Mf g(∇𝕋d ) ∈ ℒp , and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp ≤ ‖f ‖L2 (𝕋d ) ‖g‖lp (ℤd ) .

80 � 1 Bounded operators and pseudodifferential operators Proof. Note that 󵄩󵄩 󵄩p 󵄩 n 󵄩p 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp ≤ ∑ 󵄩󵄩󵄩Mf g(∇𝕋d )z 󵄩󵄩󵄩L (𝕋d ) ; 2 n∈ℤd

see, for example, [137, p. 95]. Clearly, 󵄩󵄩 󵄩 󵄨 󵄨 󵄩 n󵄩 n󵄩 n󵄩 󵄩󵄩Mf g(∇𝕋d )z 󵄩󵄩󵄩L2 (𝕋d ) = 󵄩󵄩󵄩Mf g(n)z 󵄩󵄩󵄩L2 (𝕋d ) = 󵄨󵄨󵄨g(n)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩fz 󵄩󵄩󵄩L2 (𝕋d ) 󵄨 󵄨 = 󵄨󵄨󵄨g(n)󵄨󵄨󵄨 ⋅ ‖f ‖L2 (𝕋d ) , n ∈ ℤd . The combination of these equalities yields the statement. The observation in Theorem 1.5.26 is sufficient to prove compactness estimates for product-convolution operators on tori. Theorem 1.5.27. (a) Let p ≥ 2. If f ∈ Lp (𝕋d ) and g ∈ lp (ℤd ), then Mf g(∇𝕋d ) ∈ ℒp (L2 (𝕋d )), and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp ≤ 13 ⋅ ‖f ‖Lp (𝕋d ) ‖g‖lp (ℤd ) . (b) Let p > 2. If f ∈ Lp (𝕋d ) and g ∈ lp,∞ (ℤd ), then Mf g(∇𝕋d ) ∈ ℒp,∞ (L2 (𝕋d )), and p 󵄩󵄩 󵄩 ⋅ ‖f ‖Lp (𝕋d ) ‖g‖lp,∞ (ℤd ) . 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp,∞ ≤ 18√ p−2 (c) Let 0 < p ≤ 2. If f ∈ L2 (𝕋d ) and g ∈ lp (ℤd ), then Mf g(∇𝕋d ) ∈ ℒp (L2 (𝕋d )), and 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp ≤ ‖f ‖L2 (𝕋d ) ‖g‖lp (ℤd ) . (d) Let 0 < p < 2. If f ∈ L2 (𝕋d ) and g ∈ lp,∞ (ℤd ), then Mf g(∇𝕋d ) ∈ ℒp,∞ (L2 (𝕋d )),

1.6 Pseudodifferential operators

�

81

and there exists a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒp,∞ ≤ cp ⋅ ‖f ‖L2 (𝕋d ) ‖g‖lp,∞ (ℤd ) . Proof. Let π1 be the representation of L∞ (𝕋d ) on L2 (𝕋d ) by product operators, and let π2 be the representation of l∞ (ℤd ) on L2 (𝕋d ) by convolution operators. Parts (a) and (b) follow for f bounded from Corollary 1.5.15 since Assumption 1.5.9 is satisfied because of Theorem 1.5.26. An approximation argument using fn = fχ|f |≤n , n ≥ 0, for arbitrary f ∈ L2 (𝕋d ) is similar to the argument in the proof of Theorem 1.5.20 and is omitted. Part (c) is the statement of Theorem 1.5.26. To prove Part (d), let pj , j ≥ 1, be the sequence of projections in L∞ (𝕋d ) given by p1 = 1, pj = 0, j ≥ 1. Let qk , k ≥ 1, be the sequence of projections in l∞ (ℤd ) given by the elementary vector with 1 in the n(k)th position, where n : ℕ → ℤd is an enumeration. It follows from Theorem 1.5.26 that 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩π1 (pj f )π2 (gqk )󵄩󵄩󵄩ℒs ≤ ‖pj f ‖L2 󵄨󵄨󵄨g(n(k))󵄨󵄨󵄨 = ‖pj f ⊗ gqk ‖L2 (𝕋d )⊗l2 (ℤd ) ,

k ∈ ℕ.

Hence Assumption 1.5.16 is satisfied. Part (d) for f bounded is then obtained from Corollary 1.5.18, since 󵄩󵄩 󵄩 󵄩󵄩{‖pj f ‖L2 (𝕋d ) }j≥1 󵄩󵄩󵄩lp (ℕ) = ‖f ‖L2 (𝕋d ) , and 󵄩󵄩 󵄩 󵄩󵄩{‖gqk ‖l2 (ℤd ) }k≥1 󵄩󵄩󵄩l

p,∞ (ℕ)

= ‖g‖lp,∞ (ℤd ) .

An approximation argument using fn = fχ|f |≤n , n ≥ 0, for arbitrary f ∈ L2 (𝕋d ) is similar to that in the proof of Theorem 1.5.20.

1.6 Pseudodifferential operators This section introduces the pseudodifferential calculus on ℝd . Standard results in pseudodifferential operator theory are given without proofs. Detailed references are indicated in the section end notes. Let ℝd denote the Euclidean space of dimension d ≥ 1. A multiindex is an element β = (β1 , . . . , βd ) ∈ ℤd+ , and we use the notation |β| := β1 + ⋅ ⋅ ⋅ + βd

82 � 1 Bounded operators and pseudodifferential operators and β! := β1 ! ⋅ ⋅ ⋅ βd !. For partial derivatives, we use the notation β

𝜕t :=

β

𝜕|β|

β

𝜕t11 . . . 𝜕tdd

t = (t1 , . . . , td ) ∈ ℝd .

,

The subscript t will be dropped when it is clear from the context which variables the partial derivatives are acting upon. We denote the inner product on ℝd by ⟨t, s⟩, t, s ∈ ℝd , and the associated vector norm by |t| = √⟨t, t⟩, t ∈ ℝd . This should cause no confusion with the order | ⋅ | of a multiindex. 1.6.1 The Schwartz space and distributions The Schwartz class 𝒮 (ℝd ) is the linear space of all smooth functions f : ℝd → ℂ such that for all multiindices α, β ∈ ℤd+ , |α| 󵄨 β 󵄨 pα,β (f ) := sup (1 + |t|) 󵄨󵄨󵄨𝜕t f (t)󵄨󵄨󵄨 < ∞. d t∈ℝ

The Schwartz class 𝒮 (ℝd ) is a Fréchet space when equipped with the family of seminorms {pα,β }α,β∈ℤd+ . We denote by (⋅, ⋅) the bilinear operation on 𝒮 (ℝd ) defined by (f , g) := ∫ f (t)g(t) dt. ℝd

The space 𝒮 ′ (ℝd ) of tempered distributions on ℝd is defined to be the topological dual of 𝒮 (ℝd ). We denote the dual pairing as (ω, f ) := ω(f ) for ω ∈ 𝒮 ′ (ℝd ) and f ∈ 𝒮 (ℝd ). Concretely, 𝒮 ′ (ℝd ) is the space of linear functionals ω : 𝒮 (ℝd ) → ℂ such that there exist Nω > 0 and a constant Cω such that 󵄨󵄨 󵄨 󵄨󵄨(ω, f )󵄨󵄨󵄨 ≤ Cω ⋅

∑ |α|,|β|≤Nω

pα,β (f ),

f ∈ 𝒮 (ℝd ).

The dual 𝒮 ′ (ℝd ) is equipped with the weak∗ -topology. A net {ωi }i∈𝕀 of distributions converges to ω ∈ 𝒮 ′ (ℝd ) if for every f ∈ 𝒮 (ℝd ), we have (ωi , f ) → (ω, f ). We typically identify 𝒮 (ℝd ) with its canonical embedding into 𝒮 ′ (ℝd ) given by the injection

1.6 Pseudodifferential operators

� 83

f ∈ 𝒮 (ℝd ) 󳨃→ (g 󳨃→ (f , g)) ∈ 𝒮 ′ (ℝd ). According to this embedding, 𝒮 (ℝd ) is a dense subset of 𝒮 ′ (ℝd ), and for every ω ∈ 𝒮 ′ (ℝd ), there exists a sequence of Schwartz class functions {ωn }n∈ℕ such that ωn → ω in the weak∗ -sense. The Lebesgue spaces Lp (ℝd ), 1 ≤ p ≤ ∞, continuously embed in 𝒮 ′ (ℝd ) in the same way and conventionally are identified with their embeddings. If F is a smooth function on ℝd with polynomial growth at infinity, and all derivatives of F have polynomial growth, then f ∈ 𝒮 (ℝd ) implies that the pointwise product Ff ∈ 𝒮 (ℝd ). This definition is extended to 𝒮 ′ (ℝd ) by the identity (Fω, f ) := (ω, Ff ),

f ∈ 𝒮 (ℝd ),

ω ∈ 𝒮 ′ (ℝd ).

Given a multiindex α ∈ ℤd+ , the derivative 𝜕α ω of ω ∈ 𝒮 ′ (ℝd ) is defined by (𝜕α ω, f ) := (−1)|α| (ω, 𝜕α f ). Recall from (1.16) that we denote the Fourier transform of f ∈ 𝒮 (ℝd ) as d

(ℱ f )(t) = (2π)− 2 ∫ e−i⟨t,ξ⟩ f (ξ) dξ,

t ∈ ℝd .

ℝd

Since ℱ : 𝒮 (ℝd ) → 𝒮 (ℝd ), the Fourier transform is extended to 𝒮 ′ (ℝd ) by the identity (ℱ ω, f ) := (ω, ℱ f ),

f ∈ 𝒮 (ℝd ),

ω ∈ 𝒮 ′ (ℝd ).

1.6.2 Symbols and operators The class of ordinary pseudodifferential operators Ψm (ℝd ) of order m is defined in terms of the Hörmander symbol class S m (ℝd × ℝd ) as follows. Symbols Definition 1.6.1. A smooth function σ : ℝd × ℝd → ℂ is said to be a symbol of order m ∈ ℝ if for all multiindices α, β ∈ ℤd+ , |β|−m 󵄨 α β 󵄨󵄨𝜕 𝜕 σ(t, ξ)󵄨󵄨󵄨 󵄨 t ξ 󵄨

‖σ‖m,α,β := sup (1 + |ξ|) t,ξ∈ℝd

< ∞.

The set of symbols of order m is denoted S m (ℝd × ℝd ). Denote S −∞ (ℝd × ℝd ) := ⋂m∈ℝ S m (ℝd × ℝd ). For every m ∈ ℝ, the symbol space S m (ℝd × ℝd ) is a Fréchet space equipped with the family of norms {‖ ⋅ ‖m,α,β }α,β∈ℤd+ .

84 � 1 Bounded operators and pseudodifferential operators Given m ∈ ℝ, let σ ∈ S m (ℝd × ℝd ) and u ∈ 𝒮 (ℝd ). The function Op(σ)u defined by d

(Op(σ)u)(t) := (2π)− 2 ∫ ei⟨t,ξ⟩ σ(t, ξ)(ℱ u)(ξ) dξ,

t ∈ ℝd ,

(1.34)

ℝd

again belongs to the Schwartz class 𝒮 (ℝd ) and defines a continuous endomorphism Op(σ) : 𝒮 (ℝd ) → 𝒮 (ℝd ). In fact, the mapping S m (ℝd × ℝd ) × 𝒮 (ℝd ) → 𝒮 (ℝd ) given by (σ, f ) 󳨃→ Op(σ)f is jointly continuous from S m (ℝd × ℝd ) × 𝒮 (ℝd ) to 𝒮 (ℝd ) in the sense that for any Schwartz seminorm pα,β , there exist N ≥ 0 and CN > 0 such that pα,β (Op(σ)f ) ≤ CN

∑ |γ|+|δ|≤N

‖σ‖m,γ,δ + pγ,δ (f ).

There exists a unique continuous extension of Op(σ) to a linear endomorphism Op(σ) : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝd ). This extension may be described equivalently in either of the following ways: (i) For σ ∈ S m (ℝd × ℝd ) and f ∈ 𝒮 (ℝd ), define Tf ∈ 𝒮 (ℝd ) by the iterated integral (Tf )(s) := (2π)−d ∫ ( ∫ ei⟨s−t,ξ⟩ σ(t, ξ)f (t) dt) dξ,

s ∈ ℝd .

ℝd ℝd

Then for ω ∈ 𝒮 ′ (ℝd ), we may define Op(σ)ω ∈ 𝒮 ′ (ℝd ) in terms of T by (Op(σ)ω, f ) = (ω, Tf ),

f ∈ 𝒮 (ℝd ).

(ii) Equivalently, if {ωn }n∈ℕ is any sequence in 𝒮 (ℝd ) converging to ω in the topology of 𝒮 ′ (ℝd ), then the sequence {Op(σ)ωn }n∈ℕ converges to a unique distribution, and we define (Op(σ)ω, f ) = lim (Op(σ)ωn , f ), n→∞

f ∈ 𝒮 (ℝd ).

These definitions are consistent. Indeed, it suffices only to observe that (Op(σ)g, f ) = (g, Tf ) for all f , g ∈ 𝒮 (ℝd ). Hence if {ωn }n∈ℕ converges to ω in 𝒮 ′ (ℝd ), then we have lim (Op(σ)ωn , f ) = lim (ωn , Tf ) = (ω, Tf ).

n→∞

n→∞

The fact that the extension of Op(σ) to 𝒮 ′ (ℝd ) is continuous in the topology of 𝒮 ′ (ℝd ) follows straightforwardly from the second definition.

1.6 Pseudodifferential operators

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Pseudodifferential operators Having extended Op(σ) to 𝒮 ′ (ℝd ), we now define the set of all pseudodifferential operators on ℝd . Definition 1.6.2. A continuous linear operator T : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝd ) is said to be a pseudodifferential operator of order m ∈ ℝ if there exists σ ∈ S m (ℝd × ℝd ) such that Tu = Op(σ)u for all u ∈ 𝒮 (ℝd ). The linear space of all pseudodifferential operators of order m is denoted Ψm (ℝd ). We adopt the notations Ψ∞ (ℝd ) := ⋃ Ψm (ℝd ), m∈ℝ

Ψ−∞ (ℝd ) := ⋂ Ψm (ℝd ). m∈ℝ

An operator T ∈ Ψ−∞ (ℝd ) is called a smoothing operator. Observe that if T ∈ Ψm (ℝd ) coincides with Op(σ) on 𝒮 (ℝd ), then (Tei⟨⋅,ξ⟩ )(t) = σ(t, ξ)ei⟨t,ξ⟩ ,

t, ξ ∈ ℝd .

It follows that if T ∈ Ψm (ℝd ), then there is a unique σ ∈ S m (ℝd × ℝd ) such that Tu = Op(σ)u for all u ∈ 𝒮 (ℝd ). Henceforth σ is referred to as the symbol of the operator T, and we will identify Op(σ) with its unique continuous extension to 𝒮 ′ (ℝd ). Example 1.6.3. If f is a smooth function on ℝd with all derivatives bounded, then the product operator (Mf ω, u) := (ω, fu),

ω ∈ 𝒮 ′ (ℝd ),

u ∈ 𝒮 (ℝd ),

is a pseudodifferential operator of order 0 with symbol σ(t, ξ) = f (t), t, ξ ∈ ℝd . Similarly, if {aα }|α|≤m is a family of smooth functions on ℝd with all derivatives bounded, then the differential operator P = ∑ Maα 𝜕tα |α|≤m

is a pseudodifferential operator of order m with symbol σ(t, ξ) = ∑ aα (t)(iξ)α , |α|≤m

t, ξ ∈ ℝd .

The pseudodifferential calculus A pair of classical theorems state that the composition of pseudodifferential operators is a pseudodifferential operator and that a pseudodifferential operator has a pseudodifferential adjoint; see [264, Section VI.3] and [242, Section II.2.5].

86 � 1 Bounded operators and pseudodifferential operators Proposition 1.6.4. The linear space Ψ∞ (ℝd ) is an algebra in the sense that if T ∈ Ψm1 (ℝd ) and S ∈ Ψm2 (ℝd ), then the composition TS is a pseudodifferential operator of order at most m1 + m2 . If σ1 ∈ S m1 (ℝd × ℝd ) and σ2 ∈ S m2 (ℝd × ℝd ), then Op(σ1 ) Op(σ2 ) − Op(σ1 σ2 ) ∈ Ψm1 +m2 −1 (ℝd ). Moreover, the symbol of the above difference depends continuously on σ1 and σ2 in the following sense: if Op(σ3 ) = Op(σ1 ) Op(σ2 ) − Op(σ1 σ2 ), then for all α, β ∈ ℤd+ , there exist C, N > 0 such that ‖σ3 ‖m1 +m2 −1,α,β ≤ C ⋅

∑ |α1 |+|α2 |+|β1 |+|β2 |≤N

‖σ1 ‖m1 ,α1 ,β1 ‖σ2 ‖m2 ,α2 ,β2 .

Corollary 1.6.5. Let T ∈ Ψm1 (ℝd ) and S ∈ Ψm2 (ℝd ). We have [T, S] = TS − ST ∈ Ψm1 +m2 −1 (ℝd ). Proof. Let σ1 and σ2 be the symbols of T and S, respectively. By Proposition 1.6.4 we have [Op(σ1 ), Op(σ2 )] − (Op(σ1 σ2 ) − Op(σ2 σ1 )) ∈ Ψm1 +m2 −1 (ℝd ). As σ1 σ2 = σ2 σ1 , this completes the proof. Recall that the bilinear functional (⋅, ⋅) is defined on the Schwartz class by (f , g) := ∫ f (t)g(t) dt,

f , g ∈ 𝒮 (ℝd ).

ℝd

This is to be distinguished from the sesquilinear inner product ⟨⋅, ⋅⟩ of L2 (ℝd ), which is instead given by ⟨f , g⟩ = (f , g) = ∫ f (t)g(t) dt. ℝd

Proposition 1.6.6. Let T ∈ Ψm (ℝd ). There exists a unique T † ∈ Ψm (ℝd ), the formal adjoint of T, that satisfies ⟨Tf , g⟩ = ⟨f , T † g⟩, If T = Op(σ), then T † = Op(σ † ), where

f , g ∈ 𝒮 (ℝd ).

1.6 Pseudodifferential operators

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87

σ † − σ ∈ S m−1 (ℝd × ℝd ). Equivalently, Op(σ)† − Op(σ) ∈ Ψm−1 (ℝd ). If τ ∈ S m−1 (ℝd × ℝd ) is the symbol of the above operator, that is, Op(τ) = Op(σ)† − Op(σ), then for all α, β ∈ ℤd+ , there exist constants C, N > 0 such that ‖τ‖m−1,α,β ≤ C ⋅

∑ |γ|+|δ|≤N

‖σ‖m,γ,δ .

Remark 1.6.7. The composition Op(σ1 ) Op(σ2 ) can be approximated “up to order −N” for every N due to repeated application of Proposition 1.6.4, Op(σ1 ) Op(σ2 ) − ∑

|α|≤N

(−i)|α| Op(𝜕ξα σ1 𝜕tα σ2 ) ∈ Ψm1 +m2 −N−1 (ℝd ). α!

Here the partial derivative (𝜕tα σ2 )(t, ξ) of the function σ2 on ℝd × ℝd denotes taking the partial derivative on the first copy of ℝd , and the partial derivative (𝜕ξα σ1 )(t, ξ) of

the function σ1 on ℝd × ℝd denotes taking the partial derivative on the second copy of ℝd . There also exists an expression for Op(σ)† to arbitrary order, which states that for every N ≥ 0, we have Op(σ)† − ∑

|α|≤N

(−i)|α| Op(𝜕ξα 𝜕tα σ) ∈ Ψm−N−1 (ℝd ). α!

However, we only require the N = 0 cases of these statements, as stated in Propositions 1.6.4 and 1.6.6.

1.6.3 Sobolev spaces and mapping properties of pseudodifferential operators Let r ∈ ℝ, and denote by jr the function r

jr (t, ξ) := (1 + |ξ|2 ) 2 , It is readily verified that jr ∈ S r (ℝd × ℝd ).

t, ξ ∈ ℝd .

88 � 1 Bounded operators and pseudodifferential operators Sobolev spaces and Bessel potentials The pseudodifferential operator Op(jr ) is called a Bessel potential of order r and is denoted r

(1 − Δ) 2 := Op(jr ). The Bessel potential Sobolev space H r (ℝd ) for r ∈ ℝ may be defined as follows. Definition 1.6.8. For r ∈ ℝ, we define the Sobolev space H r (ℝd ) as the linear subspace of ω ∈ 𝒮 ′ (ℝd ) such that r

(1 − Δ) 2 ω ∈ L2 (ℝd ). The space H r (ℝd ) is equipped with the norm r 󵄩 󵄩 ‖ω‖H r := 󵄩󵄩󵄩(1 − Δ) 2 ω󵄩󵄩󵄩L (ℝd ) . 2

For ω1 ∈ H r (ℝd ) and ω2 ∈ H −r (ℝd ), we denote r

r

(ω1 , ω2 ) := ∫ ((1 − Δ) 2 ω1 )(t)((1 − Δ)− 2 ω2 (t)) dt. ℝd

According to this bilinear pairing, we follow the convention that H −r (ℝd ) is identified with the dual of H r (ℝd ). It is a standard fact that H r (ℝd ) is a Hilbert space for every r ∈ ℝ. Remark 1.6.9. The notation (⋅, ⋅) for the bilinear pairing of H r (ℝd ) with H −r (ℝd ) is consistent with the identical notation used for the pairing of 𝒮 ′ (ℝd ) and 𝒮 (ℝd ). Denote temporarily r

r

(ω1 , ω2 )r := ∫ ((1 − Δ) 2 ω1 )(t)((1 − Δ)− 2 ω2 (t)) dt,

ω1 ∈ H r (ℝd ), ω2 ∈ H −r (ℝd ).

ℝd

Recall that we identify 𝒮 (ℝd ) with its canonical embedding into 𝒮 ′ (ℝd ). Since 𝒮 (ℝd ) is r invariant under (1 − Δ) 2 for every r ∈ ℝ, it follows that d

r

d

𝒮 (ℝ ) ⊂ H (ℝ )

for every r ∈ ℝ. For f ∈ 𝒮 (ℝd ) and ω ∈ H r (ℝd ), we have r

r

r

r

(ω, f )r := ∫ ((1 − Δ) 2 ω)(t)((1 − Δ)− 2 f (t)) dt = ((1 − Δ) 2 ω, (1 − Δ)− 2 f ). ℝd r

By definition the action of (1 − Δ) 2 on the distribution ω is defined by

1.6 Pseudodifferential operators

r

r

((1 − Δ) 2 ω, f ) = (ω, (1 − Δ) 2 f ),

� 89

f ∈ 𝒮 (ℝd ).

r

Since (1 − Δ)− 2 f ∈ 𝒮 (ℝd ), it follows that r

r

(ω, f )r = ((1 − Δ) 2 ω, (1 − Δ)− 2 f ) = (ω, f ). Hence the bilinear functional H r (ℝd ) × H −r (ℝd ) → ℂ defined by (⋅, ⋅)r coincides with the distributional pairing of 𝒮 ′ (ℝd ) with 𝒮 (ℝd ) whenever the second argument belongs to 𝒮 (ℝd ). that

The set of Hilbert spaces {H r (ℝd )}r∈ℝd is a complex interpolation scale in the sense (H r0 (ℝd ), H r1 (ℝd ))[θ] = H rθ (ℝd ),

0 < θ < 1, rθ = r0 (1 − θ) + r1 θ.

Here (⋅, ⋅)[θ] is the functor of complex interpolation [26, Chapter 4]. It follows that if r0 < r1 , and T ∈ ℒ(H r0 (ℝd ), H r0 (ℝd )) restricts to a bounded linear operator T|H r1 ∈ ℒ(H r1 (ℝd ), H r1 (ℝd )), then T also acts boundedly on H rθ (ℝd ) for every 0 < θ < 1 with norm bound θ ‖T‖H rθ →H rθ ≤ ‖T‖1−θ H r0 →H r0 ‖T‖H r1 →H r1 .

(1.35)

Regularity of pseudodifferential operators The following theorem describes the mapping properties of pseudodifferential operators restricted to Sobolev spaces. Theorem 1.6.10. For every r, m ∈ ℝ, an operator T ∈ Ψm (ℝd ) restricts to a bounded linear operator from H r+m (ℝd ) into H r (ℝd ). Moreover, the norm of Op(σ) depends continuously on σ in the sense that for every r ∈ ℝ, there exist C, N > 0 (which depend only on m, r, and d) such that 󵄩󵄩 󵄩 󵄩󵄩Op(σ)󵄩󵄩󵄩H r+m →H r ≤ C ⋅

∑ |α|+|β|≤N

‖σ‖m,α,β .

We give a short proof of Theorem 1.6.10 using the symbolic calculus for Ψ∞ (ℝd ). The following lemma is a form of Schur’s test for the boundedness of integral operators. Lemma 1.6.11. Let T : (L1 + L∞ )(ℝd ) → (L1 + L∞ )(ℝd ) be a linear operator defined by (Tu)(t) = ∫ K(t, s)u(s) ds,

u ∈ (L1 + L∞ )(ℝd ),

ℝd

for almost every t ∈ ℝd , where K is a bounded function such that

90 � 1 Bounded operators and pseudodifferential operators 󵄨 󵄨 A := sup ∫ 󵄨󵄨󵄨K(t, s)󵄨󵄨󵄨 ds < ∞ d t∈ℝ

ℝd

and 󵄨 󵄨 B := sup ∫ 󵄨󵄨󵄨K(t, s)󵄨󵄨󵄨 dt < ∞. d s∈ℝ

ℝd

Then Tu ≺≺ max{A, B} ⋅ u,

u ∈ (L1 + L∞ )(ℝd ),

where ≺≺ denotes submajorization of functions as in Definition 1.2.7. Proof. Note that by the assumption on the bounded function K, ‖Tu‖∞ ≤ A‖u‖∞ ,

u ∈ L∞ (ℝd ),

and ‖Tu‖1 ≤ B‖u‖1 ,

u ∈ L1 (ℝd ).

Hence T is well-defined, and T/max{A, B} is an absolute contraction. The statement follows since Su ≺≺ u for every absolute contraction S : (L1 + L∞ )(ℝd ) → (L1 + L∞ )(ℝd ) by Theorem 1.2.16 (a) 󳨐⇒ (b) (see also [47, Theorem 2]). Now we can begin the proof of Theorem 1.6.10. Proof of Theorem 1.6.10. The boundedness of T from H r+m (ℝd ) to H r (ℝd ) is equivalent to the boundedness on L2 (ℝd ) of the operator r

(1 − Δ) 2 T(1 − Δ)−

r+m 2

(1.36)

.

According to Proposition 1.6.4, operator (1.36) belongs to Ψ0 (ℝd ). It therefore suffices to prove that every T ∈ Ψ0 (ℝd ) restricts to a bounded linear operator on L2 (ℝd ). We prove this in several steps. Step 1. Assume that T ∈ Ψ−d−1 (ℝd ). It follows that for all α, β ∈ ℤd+ , there exists a constant Cα,β such that the symbol σ of T satisfies −d−1−|β| 󵄨󵄨 α β 󵄨 , 󵄨󵄨𝜕t 𝜕ξ σ(t, ξ)󵄨󵄨󵄨 ≤ Cα,β ⋅ (1 + |ξ|)

t, ξ ∈ ℝd .

1.6 Pseudodifferential operators

�

91

For u ∈ 𝒮 (ℝd ), we have (Tu)(t) = (2π)−d ∬ ei⟨t−s,ξ⟩ σ(t, ξ)u(s) dsdξ,

t ∈ ℝd .

ℝd ×ℝd

This is an absolutely convergent integral, since the assumption σ ∈ S −d−1 (ℝd × ℝd ) ensures that σ(t, ξ) is integrable in the second variable. We apply integration by parts N times to get (Tu)(t) = (2π)−d ∬ ei⟨t−s,ξ⟩ (1 + |t − s|2 )

−N

((1 − Δξ )N σ(t, ξ))u(s) dsdξ,

t ∈ ℝd .

ℝd ×ℝd

Hence T is given by the integral kernel K(t, s) = (2π)−d (1 + |t − s|2 )

−N

∫ ei⟨t−s,ξ⟩ (1 − Δξ )N σ(t, ξ) dξ,

t, s ∈ ℝd .

ℝd

Taking N > d2 , we see that K obeys the requirements of Lemma 1.6.11. Hence T is bounded from L2 (ℝd ) to L2 (ℝd ). Step 2. Now assume that T ∈ Ψm (ℝd ) for some m < 0. Combining Propositions 1.6.4 and 1.6.6, we see that the operator T † T has order 2m. If u is a Schwartz class function, then by the Cauchy–Schwarz inequality we have 󵄩 󵄩 ‖Tu‖2L (ℝd ) = ⟨Tu, Tu⟩ = ⟨T † Tu, u⟩ ≤ 󵄩󵄩󵄩T † Tu󵄩󵄩󵄩L (ℝd ) ‖u‖L2 (ℝd ) . 2

2

Hence if T † T is bounded, it follows that T is bounded on L2 (ℝd ). By Step 1 it follows that operators of order − d+1 are bounded, hence operators − d+1 are bounded, and so on. 2 4 Hence all operators of strictly negative order are bounded. Step 3. Now assume that T ∈ Ψ0 (ℝd ). If σ is the symbol of T, then since σ ∈ S 0 (ℝd × ℝ ), there exists a constant M > 0 such that d

󵄨 󵄨 1 sup 󵄨󵄨󵄨σ(t, ξ)󵄨󵄨󵄨 ≤ M. 2 d t,ξ∈ℝ Let 󵄨 󵄨2 1/2 τ(t, ξ) := (M 2 − 󵄨󵄨󵄨σ(t, ξ)󵄨󵄨󵄨 ) ,

t, ξ ∈ ℝd .

Since the function f (t) = (M 2 − |t|2 )1/2 is smooth on the image of σ, we have that τ ∈ S 0 (ℝd × ℝd ). Combining Theorems 1.6.4 and 1.6.6, we get that R := Op(τ)† Op(τ) − (M 2 − T † T) ∈ Ψ−1 (ℝd ).

92 � 1 Bounded operators and pseudodifferential operators By Step 2, R is bounded on L2 (ℝd ). For u ∈ 𝒮 (ℝd ), we have ⟨Ru, u⟩ = ⟨Op(τ)† Op(τ)u, u⟩ − M 2 ‖u‖2L (ℝd ) + ⟨T † Tu, u⟩. 2

Rearranging and applying the definition of the formal adjoint, it follows that 󵄩 󵄩2 ‖Tu‖2L (ℝd ) = M 2 ‖u‖2L (ℝd ) − 󵄩󵄩󵄩Op(τ)u󵄩󵄩󵄩L (ℝd ) + ⟨Ru, u⟩. 2

2

2

Hence by the Cauchy–Schwarz inequality and the boundedness of R, for all u ∈ 𝒮 (ℝd ), we have ‖Tu‖2L (ℝd ) ≤ M 2 ‖u‖2L (ℝd ) + ‖R‖L2 (ℝd )→L2 (ℝd ) ‖u‖2L (ℝd ) . 2

2

2

Since 𝒮 (ℝd ) is dense in L2 (ℝd ), the boundedness of T on L2 (ℝd ) follows. From the preceding arguments, it is clear that ‖Op(σ)‖H 0 →H 0 depends continuously on σ in the required sense. 1.6.4 Inverses and complex powers of pseudodifferential operators In this section, we define complex powers of elliptic pseudodifferential operators and a functional calculus on their principal symbol. We also prove that the restriction of a complex power of a positive elliptic pseudodifferential operator T of order m to the Sobolev space H m (ℝd ) coincides with the complex power defined by the functional calculus when T restricts to a positive unbounded operator T : H m (ℝd ) → L2 (ℝd ) on the Hilbert space L2 (ℝd ). Beals’ characterization of pseudodifferential operators In Section 1.6.3, we have shown that T ∈ Ψm (ℝd ) defines a bounded linear operator from H r+m (ℝd ) into H r (ℝd ) for every r ∈ ℝ. It is often helpful to characterize the class Ψm (ℝd ) in terms of its mapping properties on Sobolev spaces. Let T : 𝒮 (ℝd ) → 𝒮 ′ (ℝd ) be a linear continuous map. For j = 1, . . . , d, denote by Mj (T) : 𝒮 (ℝd ) → 𝒮 ′ (ℝd ) the linear map given by (Mj (T)u)(t) := (tj Tu)(t) − (Ttj u)(t),

u ∈ 𝒮 (ℝd ),

t ∈ ℝd ,

u ∈ 𝒮 (ℝd ),

t ∈ ℝd .

and denote by 𝜕j (T) the linear map (𝜕j (T)u)(t) := (𝜕j Tu)(t) − (T𝜕j u)(t), Similarly, for α, β ∈ ℤd+ , denote α

α

𝜕α (T) := 𝜕1 1 ⋅ ⋅ ⋅ 𝜕d d (T),

β

β

M β (T) := M1 1 ⋅ ⋅ ⋅ Md d (T).

1.6 Pseudodifferential operators

� 93

The following result is due to Beals [21]. Proofs may also be found in [109] and [296]. Theorem 1.6.12 (Beals’ theorem). A linear operator T : 𝒮 (ℝd ) → 𝒮 ′ (ℝd ) is the restriction of a pseudodifferential operator of order m ∈ ℝ if and only if for every α, β ∈ ℤd+ , the operator M α 𝜕β (T) acts boundedly from H m−|β| (ℝd ) to H 0 (ℝd ). Moreover, the symbol σ of T = Op(σ) is bounded by the norms of T in the following sense: there exists a constant Cd > 0 such that for all α, β ∈ ℤd+ , ‖σ‖m,α,β ≤ Cd cot

∑ |γ|+|δ|≤2d+1

󵄩󵄩 γ+β δ+α 󵄩󵄩 󵄩󵄩M 𝜕 (T)󵄩󵄩H m−|β| →H 0 .

Remark 1.6.13. The space Ψm (ℝd ) is a Fréchet space when equipped with the family of seminorms 󵄩 󵄩 qα,β (T) := 󵄩󵄩󵄩M α 𝜕β (T)󵄩󵄩󵄩H m−|α| →H m ,

α, β ∈ ℤd+ .

Beals’ theorem implies that the mapping Op : S m (ℝd × ℝd ) → Ψm (ℝd ) is a topological linear isomorphism. The “only if” component of Beals’ theorem is a straightforward consequence of Theorem 1.6.10 and the observation that β

M β (Op(σ)) = Op(i|β| 𝜕ξ σ),

𝜕α (Op(σ)) = Op(𝜕tα σ).

On the other hand, the converse implication that an operator T obeying the conditions of the theorem is pseudodifferential is more difficult to prove. Beals’ theorem is useful for a number of purposes. For us, it will be used to ensure that Ψ∞ (ℝd ) is closed under inversion and that a complex power of a positive elliptic pseudodifferential operator is again a pseudodifferential operator. Inverses of pseudodifferential operators If T is a closed invertible linear operator on a Hilbert space that is (the restriction of) a pseudodifferential operator, then the inverse is (the restriction of) a pseudodifferential operator. Theorem 1.6.14. Let T ∈ Ψ0 (ℝd ), and assume that T has a bounded inverse T −1 on L2 (ℝd ). Then, for every r > 0, T −1 |H r is an inverse for T on H r (ℝd ), and for r < 0, T −1 extends continuously to an inverse for T|H r .

94 � 1 Bounded operators and pseudodifferential operators Proof. We prove that if v ∈ H r (ℝd ), then T −1 v ∈ H r (ℝd ), initially when r ∈ ℤ+ , by induction on r. The case r = 0 is true by assumption. Assume now that T −1 (H r (ℝd )) ⊆ H r (ℝd ) for some r ∈ ℤ+ . If v ∈ H r+1 (ℝd ), then in particular v ∈ H r (ℝd ), and hence T −1 v ∈ H r (ℝd ). Then 1

1

1

1

1

(1 − Δ) 2 T −1 v = T −1 (1 − Δ) 2 v + [(1 − Δ) 2 , T −1 ]v = T −1 (1 − Δ) 2 v − T −1 [(1 − Δ) 2 , T]T −1 v. 1

1

We have (1 − Δ) 2 v ∈ H r (ℝd ), and therefore T −1 (1 − Δ) 2 v ∈ H r (ℝd ). Since T ∈ Ψ0 (ℝd ), it 1 follows from Corollary 1.6.5 that [(1 − Δ) 2 , T] ∈ Ψ0 (ℝd ). Hence 1

T −1 [(1 − Δ) 2 , T]T −1 v ∈ H r (ℝd ). So we have 1

(1 − Δ) 2 T −1 v ∈ H r (ℝd ), that is, T −1 v ∈ H r+1 (ℝd ). This completes the proof for r ∈ ℤ+ . Similarly, we deduce the case r ∈ ℤ− via induction. The general case r ∈ ℝ follows from the complex interpolation formula (1.35), which in this case yields 󵄩󵄩 −1 󵄩󵄩 󵄩 −1 󵄩1−θ 󵄩 −1 󵄩θ 󵄩󵄩T 󵄩󵄩H r+θ →H r+θ ≤ 󵄩󵄩󵄩T 󵄩󵄩󵄩H r →H r 󵄩󵄩󵄩T 󵄩󵄩󵄩H r+1 →H r+1 ,

θ ∈ (0, 1).

Corollary 1.6.15. Let m ∈ ℝ and T ∈ Ψm (ℝd ). If, for some r ∈ ℝ, the operator T restricts to an isomorphism T : H m+r (ℝd ) → H r (ℝd ), then T : H m+r (ℝd ) → H r (ℝd ) is an isomorphism for every r ∈ ℝ. Proof. Let r

S = (1 − Δ) 2 T(1 − Δ)−

m+r 2

∈ Ψ0 (ℝd ).

m+r

r

Since (1−Δ)− 2 : H 0 (ℝd ) → H m+r (ℝd ), T : H m+r (ℝd ) → H r (ℝd ), and (1−Δ) 2 : H r (ℝd ) → H 0 (ℝd ) are isomorphisms, it follows that S : H 0 (ℝd ) → H 0 (ℝd ) is an isomorphism. By Theorem 1.6.14, S : H s (ℝd ) → H s (ℝd ) is an isomorphism for every s ∈ ℝ. Clearly, T = (1 − Δ) r

m+r 2

r

S(1 − Δ)− 2 . m+r

Since (1 − Δ)− 2 : H s−r (ℝd ) → H s (ℝd ), S : H s (ℝd ) → H s (ℝd ), and (1 − Δ) 2 : H s (ℝd ) → H s−m−r (ℝd ) are isomorphisms, it follows that their composition T : H s−r (ℝd ) → H s−m−r (ℝd ) is an isomorphism for every s ∈ ℝ.

1.6 Pseudodifferential operators

� 95

The next theorem states that if T is a pseudodifferential operator of order m that is a topological isomorphism of Sobolev spaces, then there exists a pseudodifferential operator T −1 of order −m that is the inverse of T in the pseudodifferential calculus. Theorem 1.6.16. Let m ∈ ℝ. If, for some s ∈ ℝ, T ∈ Ψm (ℝd ) restricts to a topological isomorphism T|H s+m (ℝd ) : H s+m (ℝd ) → H s (ℝd ), then the inverse T −1 : H s (ℝd ) → H s+m (ℝd ) is the restriction of a pseudodifferential operator belonging to Ψ−m (ℝd ). Proof. In view of Theorem 1.6.12, it suffices to check that M β 𝜕α (T −1 ) ∈ ℒ(H s−|β| , H s+m ),

α, β ∈ ℤd+ ,

(1.37)

for all s ∈ ℝ. We prove this by induction on |α| + |β|. The base of induction (i. e., the case α = β = 0) is established in Corollary 1.6.15. If (1.37) holds for all |α| + |β| ≤ l, then M β 𝜕α (Mj (T −1 )) = −M β 𝜕α (T −1 Mj (T)T −1 ). By the Leibniz rule this expands to a sum of operators of the form M β1 𝜕α1 (T −1 ) ⋅ M β2 +ej 𝜕α2 (T) ⋅ M β3 𝜕α3 (T −1 ),

(1.38)

where α = α1 + α2 + α3 , β = β1 + β2 + β3 , and ej is the standard basis element of ℤd+ with 1 in the jth position and 0 in the other positions. By induction the mapping M β3 𝜕α3 (T −1 ) : H s−|β|−1 (ℝd ) → H s−|β|−1+|β3 |+m (ℝd ) = H s−|β1 |−|β2 |+m−1 (ℝd ) is bounded. Since T ∈ Ψm (ℝd ), it follows that M β2 +ej 𝜕α2 (T) : H s−|β1 |−|β2 |+m−1 (ℝd ) → H s−|β1 | (ℝd ) is a bounded mapping. By induction the mapping M β1 𝜕α1 (T −1 ) : H s−|β1 | (ℝd ) → H s+m (ℝd ) is bounded. Therefore the composition M β1 𝜕α1 (T −1 ) ⋅ M β2 +ej 𝜕α2 (T) ⋅ M β3 𝜕α3 (T −1 ) : H s−|β|−1 (ℝd ) → H s+m (ℝd ) is also bounded.

96 � 1 Bounded operators and pseudodifferential operators Thus we have a bounded linear map M β 𝜕α (Mj (T −1 )) : H s−|β|−1 (ℝd ) → H s+m (ℝd ). This proves that (1.37) holds for β + ej and α when |α| + |β| = l. We can reason identically to show that it also holds for β and α + ej when |α| + |β| = l. This proves that if (1.37) holds for |α| + |β| = l, then it holds for |α| + |β| = l + 1, which completes the induction. Complex powers of pseudodifferential operators If T ≥ 1 is a self-adjoint linear operator on L2 (ℝd ), then we can form the semigroup {T z }z∈ℂ of complex powers, defined by the self-adjoint functional calculus. When ℜ(z) < 0, T z can be constructed from resolvents and the integral Tz =

1 ∫ λz (λ − T)−1 dλ, 2πi Γ

where Γ is a contour that winds anticlockwise around a portion of the positive real axis, and the integral converges in the Bochner sense in the space ℒ(L2 (ℝd )). Specifically, we take 1 Γ = { + |t| − it : −∞ < t < ∞}. 2

(1.39)

To see that the integral converges, note that by the spectral theorem we have the upper bound 󵄩󵄩 −1 󵄩 −1 󵄩󵄩(λ − T) 󵄩󵄩󵄩∞ ≤ C|λ| ,

λ ∈ Γ.

(1.40)

The resolvent identity states that (λ − T)−1 − (μ − T)−1 = (μ − λ)(λ − T)−1 (μ − T)−1 ,

λ, μ ∈ Γ,

which implies that 󵄩󵄩 −1 −1 󵄩 2 −1 −1 󵄩󵄩(λ − T) − (μ − T) 󵄩󵄩󵄩∞ ≤ C |μ − λ||μ| |λ| ,

λ, μ ∈ Γ.

Hence the function λ 󳨃→ (λ − T)−1 is continuous in the operator norm, which implies that the integral with contour in (1.39) converges in the ℒ(L2 (ℝd ))-valued Bochner sense. If T ∈ Ψm (ℝd ), m ≥ 0, and T ≥ 1, then Theorem 1.6.16 implies that for all λ ∈ Γ, the resolvent (λ − T)−1 extends to a pseudodifferential operator of order −m. A more precise statement on the mapping property of the resolvent is as follows.

1.6 Pseudodifferential operators

�

97

Lemma 1.6.17. Let T ∈ Ψm (ℝd ) be a pseudodifferential operator of order m ≥ 0. Assume that T ≥ 1 in the sense that ⟨Tu, u⟩ ≥ ⟨u, u⟩ for all u ∈ H m (ℝd ). Let Γ be the contour (1.39). We have (i) for all α, β ∈ ℤd+ , 󵄩󵄩 β α −1 󵄩 󵄩󵄩M 𝜕 ((λ − T) )󵄩󵄩󵄩H r−m−|β| →H r ≤ Cα,β ,

λ ∈ Γ,

for some constant Cα,β > 0. (ii) for all α, β ∈ ℤd+ , 󵄩󵄩 β α −1 󵄩 −1 󵄩󵄩M 𝜕 ((λ − T) )󵄩󵄩󵄩H −|β| →H 0 ≤ Cα,β ⋅ |λ| ,

λ ∈ Γ,

for some constant Cα,β > 0. Proof. We initially prove (i) in the case that α = β = 0 and r = 0. Note that λ(λ − T)−1 = 1 + (λ − T)−1 T.

(1.41)

Hence 󵄩󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩H −m →H 0 ≤ 󵄩󵄩󵄩(λ − T) T 󵄩󵄩󵄩H 0 →H 0 󵄩󵄩󵄩T 󵄩󵄩󵄩H −m →H 0 󵄩 󵄩 󵄩 󵄩 ≤ (|λ|󵄩󵄩󵄩(λ − T)−1 󵄩󵄩󵄩H 0 →H 0 + 1)󵄩󵄩󵄩T −1 󵄩󵄩󵄩H −m →H 0 . Due to (1.40), the term |λ|‖(λ − T)−1 ‖H 0 →H 0 is uniformly bounded for λ ∈ Γ. Hence 󵄩 󵄩 sup󵄩󵄩󵄩(λ − T)−1 󵄩󵄩󵄩H −m →H 0 < ∞. λ∈Γ

This is (i) when α = β = 0 and r = 0. The statement may be extended to general r ∈ ℝ by an argument identical to Theorem 1.6.14 and Corollary 1.6.15, noting that the relevant bounds are uniform in λ ∈ Γ. Having established (i) for all r ∈ ℝ and α = β = 0, we extend the result to general α and β by induction on |α| + |β| similarly to the proof of Theorem 1.6.16. Indeed, if (i) holds for some α, β ∈ ℤd+ , then it holds for α and β + ej , 1 ≤ j ≤ d, since M β+ej 𝜕α ((λ − T)−1 ) is a linear combination of operators of the form M β1 𝜕α1 ((λ − T)−1 ) ⋅ M β2 +ej 𝜕α2 (λ − T) ⋅ M β3 𝜕α3 ((λ − T)−1 ). Since M β2 +ej (λ − T) = M β2 +ej (T), it follows from the inductive hypothesis that (i) holds for α, β + ej . Similarly, we may reason that it holds for α + ej and β. This completes the argument for (i). To prove the second assertion, we again apply (1.41), which shows that it suffices to have

98 � 1 Bounded operators and pseudodifferential operators 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (T(λ − T)−1 )󵄩󵄩󵄩H −|β| →H 0 < ∞. λ∈Γ

The above statement is proved similarly to (i) by induction on α and β with the α = β = 0 case being (1.40). We will now proceed to proving that if T ∈ Ψm (ℝd ) is positive in the correct sense, then the complex powers defined by the functional calculus on the restriction of T to a potentially unbounded operator on the Hilbert space L2 (ℝd ) can be extended to pseudodifferential operators T z ∈ Ψℜ(z)m (ℝd ). The first step to proving this is verifying that T z is indeed a pseudodifferential operator, which is accomplished by the following lemma. Lemma 1.6.18. Let m ≥ 0. Let T ∈ Ψm (ℝd ) be a pseudodifferential operator of order m such that T restricts to a self-adjoint operator T : H m (ℝd ) → L2 (ℝd ) with domain H m (ℝd ) and T ≥ 1 in the sense that ⟨Tu, u⟩ ≥ ⟨u, u⟩ for all u ∈ H m (ℝd ). Then the complex powers {T z }z∈ℂ , defined by the self-adjoint functional calculus, coincide with the restrictions of pseudodifferential operators to L2 (ℝd ). Proof. It is an immediate consequence of Proposition 1.6.4 that T n ∈ Ψmn (ℝd ) for n ∈ ℤ+ . Due to the operator identity T z = T n T z−n ,

n ∈ ℤ+ ,

it suffices to take ℜ(z) < −1. We represent T z by the integral Tz =

1 ∫ λz (λ − T)−1 dλ. 2πi Γ

By Theorem 1.6.16 the resolvent (λ − T)−1 belongs to Ψ−m (ℝd ) for every λ ∈ Γ. By Lemma 1.6.17 and the Leibniz rule we have 󵄩 󵄩 sup 󵄩󵄩󵄩M β 𝜕α ((λ − T)−1 (μ − T)−1 )󵄩󵄩󵄩H −m−|β| →H 0 < ∞.

λ,μ∈Γ

Hence by the resolvent identity it follows that for all λ, μ ∈ Γ, 󵄩󵄩 β α −1 β α −1 󵄩 󵄩󵄩M 𝜕 ((λ − T) ) − M 𝜕 ((μ − T) )󵄩󵄩󵄩H −m−|β| →H 0 ≤ Cα,β |λ − μ|. Thus, for all α, β ∈ ℤd+ the integral

(1.42)

1.6 Pseudodifferential operators

� 99

1 ∫ λz M β 𝜕α ((λ − T)−1 ) dλ 2πi Γ

converges as a Bochner integral in the space ℒ∞ (H −m−|β| , H 0 ). Since ℜ(z) < −1, the integral converges absolutely. This integral coincides with M β 𝜕α (T z ), because for all ϕ, ψ ∈ 𝒮 (ℝd ), (ϕ, M β 𝜕α (T z )ψ) =

1 ∫ λz (ϕ, M β 𝜕α ((λ − T)−1 )ψ) dλ. 2πi Γ

Thus M β 𝜕α (T z ) ∈ ℒ∞ (H −m−|β| , H 0 ),

α, β ∈ ℤd+ .

Hence, by Theorem 1.6.12, T z ∈ Ψ−m (ℝd ). Having proved that T z extends to a pseudodifferential operator in Lemma 1.6.18, we denote by T z also the pseudodifferential operator and call it the complex power of the pseudodifferential operator T. The following theorem proves that the pseudodifferential operator T z has order ℜ(z)m. Theorem 1.6.19. Let m ≥ 0. Let T ∈ Ψm (ℝd ) be a pseudodifferential operator of order m such that T restricts to a self-adjoint operator T : H m (ℝd ) → L2 (ℝd ) with domain H m (ℝd ) and T ≥ 1 in the sense that ⟨Tu, u⟩ ≥ ⟨u, u⟩ for all u ∈ H m (ℝd ). Then T z ∈ Ψmℜ(z) (ℝd ). Proof. As an immediate consequence of Proposition 1.6.4, we have that T n ∈ Ψmn (ℝd ) for n ∈ ℤ+ . Due to the operator identity T z = T n T z−n ,

n ∈ ℤ+ ,

it suffices to take ℜ(z) ∈ (−1, 0). For λ ∈ Γ, let r(λ, ⋅, ⋅) ∈ S −m (ℝd ×ℝd ) be the symbol function of the resolvent (λ−T)−1 . By Lemma 1.6.17 and Theorem 1.6.12 we have that m+|β| 󵄨 α β 󵄨󵄨𝜕 𝜕 r(λ, t, ξ)󵄨󵄨󵄨 󵄨 t ξ 󵄨

sup (1 + |ξ|)

t,ξ∈ℝd

≤ Cα,β ,

λ ∈ Γ,

and |β| 󵄨 β 󵄨 sup (1 + |ξ|) 󵄨󵄨󵄨𝜕tα 𝜕ξ r(λ, t, ξ)󵄨󵄨󵄨 ≤ Cα,β |λ|−1 , d

t,ξ∈ℝ

λ ∈ Γ.

100 � 1 Bounded operators and pseudodifferential operators Combining these estimates, we obtain m −1 −|β| 󵄨󵄨 α β 󵄨 󵄨󵄨𝜕t 𝜕ξ r(λ, t, ξ)󵄨󵄨󵄨 ≤ Cα,β ((1 + |ξ|) + |λ|) (1 + |ξ|) .

(1.43)

By Lemma 1.6.18 the operator T z is pseudodifferential (in particular, it has a smooth symbol). Define the function σ(z, ⋅, ⋅) by σ(z, t, ξ) :=

1 ∫ λz r(λ, t, ξ) dλ. 2πi Γ

This is the symbol of T z , as can be seen by examining the action of T z on Schwartz class functions. For all α, β ∈ ℤd+ , estimate (1.43) and the triangle inequality imply that m −1 −|β| 󵄨󵄨 α β 󵄨 ℜ(z) 󵄨󵄨𝜕t 𝜕ξ σ(z, t, ξ)󵄨󵄨󵄨 ≤ Cα,β ∫ |λ| ((1 + |ξ|) + |λ|) |dλ| ⋅ (1 + |ξ|) . Γ

Since ℜ(z) ∈ (−1, 0), it follows that m

−1

∫ |λ|ℜ(z) ((1 + |ξ|) + |λ|) |dλ| ≤ Cm,z (1 + |ξ|)

mℜ(z)

.

Γ

Thus mℜ(z)−|β| 󵄨󵄨 α β 󵄨 . 󵄨󵄨𝜕t 𝜕ξ σ(z, t, ξ)󵄨󵄨󵄨 ≤ Cα,β,m,z (1 + |ξ|)

Since σ(z, ⋅, ⋅) is the symbol of T z , this means precisely that T z ∈ Ψmℜ(z) (ℝd ). Principal symbols of complex powers To calculate the traces of complex powers when ℜ(z) < 0, we require a first-order expansion of the power T z in terms of a principal symbol of T. The notion of principal symbol will be discussed in more detail in Section 1.6.7. Approximately, a pseudodifferential operator T ∈ Ψm (ℝd ) has a principal symbol σ0 ∈ S m (ℝd × ℝd ) if T − Op(σ0 ) ∈ Ψm−1 (ℝd ). If this is the case, then Proposition 1.6.4 implies that for all n ∈ ℤ+ , T n − Op(σ0n ) ∈ Ψnm−1 (ℝd ). This continues to hold when n is noninteger. We prove this as a consequence of the following theorem, which can be considered a refinement of Lemma 1.6.17. Theorem 1.6.20. Let T = Op(σ) be a pseudodifferential operator obeying the conditions of Theorem 1.6.19, and let σ0 ∈ S m (ℝd × ℝd ) be positive and such that

101

1.6 Pseudodifferential operators �

σ − σ0 ∈ S m−1 (ℝd × ℝd ). Let Γ be the contour in (1.39). For all α, β ∈ ℤd+ , there exists a constant Cα,β such that (i) 󵄩󵄩 β α −1 −1 󵄩 󵄩󵄩M 𝜕 ((λ − T) − Op((λ − σ0 ) ))󵄩󵄩󵄩H −m−1−|β| →H 0 ≤ Cα,β , (ii)

󵄩󵄩 β α −1 −1 󵄩 −1 󵄩󵄩M 𝜕 ((λ − T) − Op((λ − σ0 ) ))󵄩󵄩󵄩H −1−|β| →H 0 ≤ Cα,β ⋅ |λ| ,

λ ∈ Γ; λ ∈ Γ.

Remark 1.6.21. Before proving Theorem 1.6.20, it is helpful to make explicit the following fact. If {A(λ)}λ∈Γ ⊂ Ψm1 (ℝd ) is a parameterized family of pseudodifferential operators such that for all y α, β ∈ ℤd+ and r ∈ ℝ, 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (A(λ))󵄩󵄩󵄩H r−|β|+m1 →H r < ∞ λ∈Γ

and {B(λ)}λ∈Γ ∈ Ψm1 (ℝd ) is a similar family such that 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (B(λ))󵄩󵄩󵄩H r−|β|+m2 →H r < ∞, λ∈Γ

then 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (BA(λ))󵄩󵄩󵄩H r−|β|+m1 +m2 →H r < ∞. λ∈Γ

This is a consequence of the Leibniz rule for the derivations Mj and 𝜕j but can also be proved as a consequence of Theorems 1.6.4 and 1.6.12. Proof of Theorem 1.6.20. Observe that for every λ ∈ Γ, the function (λ − σ0 )−1 belongs to S −m (ℝd × ℝd ), and for every α, β ∈ ℤd+ , we have 󵄩 󵄩 sup󵄩󵄩󵄩(λ − σ0 )−1 󵄩󵄩󵄩−m,α,β < ∞. λ∈Γ

Applying the norm estimate in Proposition 1.6.4 and Theorem 1.6.10, it follows that for all r ∈ ℝ and α, β ∈ ℤd+ , we have 󵄩󵄩 󵄩󵄩 σ 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (Op(σ0 ) Op((λ − σ0 )−1 ) − Op( 0 ))󵄩󵄩󵄩 < ∞. 󵄩 󵄩󵄩H r−1−|β| →H r λ − σ λ∈Γ 󵄩 0 Due to the algebraic identity Op(σ0 ) Op((λ − σ0 )−1 ) − Op( we have

σ0 ) = 1 − Op(λ − σ0 ) Op((λ − σ0 )−1 ), λ − σ0

(1.44)

102 � 1 Bounded operators and pseudodifferential operators 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (Op(λ − σ0 ) Op((λ − σ0 )−1 ) − 1)󵄩󵄩󵄩H r−1−|β| →H r < ∞. λ∈Γ

By assumption we have Op(λ − σ0 ) − (λ − T) ∈ Ψm−1 (ℝd ), and hence for all r ∈ ℝ and α, β ∈ ℤd+ , 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α ((λ − T) Op((λ − σ0 )−1 ) − 1)󵄩󵄩󵄩H r−1−|β| →H r < ∞. λ∈Γ

Now we write the difference Op((λ − σ0 )−1 ) − (λ − T)−1 as Op((λ − σ0 )−1 ) − (λ − T)−1 = (λ − T)−1 ((λ − T) Op((λ − σ0 )−1 ) − 1). Applying the Leibniz rule and Theorem 1.6.17(i) gives us 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α ((λ − T)−1 − Op((λ − σ0 )−1 ))󵄩󵄩󵄩H −m−1−|β|+r →H r < ∞ λ∈Γ

(1.45)

for every r ∈ ℝ. With r = 0, this completes the proof of (i). Since T acts boundedly from H r+m to H r for every r ∈ ℝ, it follows from (1.45) and Remark 1.6.21 that for all α, β ∈ ℤd+ , we have 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (T(λ − T)−1 − T Op((λ − σ0 )−1 ))󵄩󵄩󵄩H −1−|β|+r →H r < ∞. λ∈Γ

(1.46)

Now we write T Op((λ − σ0 )−1 ) as T Op((λ − σ0 )−1 ) − Op(σ0 (λ − σ0 )−1 ) = (T − Op(σ0 )) Op((λ − σ0 )−1 ) + (Op(σ0 ) Op((λ − σ0 )−1 ) − Op(σ0 (λ − σ0 )−1 )). Since T − Op(σ0 ) ∈ Ψm−1 (ℝd ) and 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α Op((λ − σ0 )−1 )󵄩󵄩󵄩H −m−|β|+r →H r < ∞ λ∈Γ

for all α, β ∈ ℤd+ and r ∈ ℝ, it follows from (1.44) that 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (T Op((λ − σ0 )−1 ) − Op(σ0 (λ − σ0 )−1 ))󵄩󵄩󵄩H −1−|β|+r →H r < ∞. λ∈Γ

See again Remark 1.6.21. Combining this with (1.46) yields 󵄩 󵄩 sup󵄩󵄩󵄩M β 𝜕α (T(λ − T)−1 − Op(σ0 (λ − σ0 )−1 ))󵄩󵄩󵄩H −|β|−1+r →H r . λ∈Γ

Since σ T − Op( 0 ) = λ((λ − T)−1 − Op((λ − σ0 )−1 )), λ−T λ − σ0

1.6 Pseudodifferential operators

� 103

it follows that for all α, β ∈ ℤd+ and r ∈ ℝ, we have 󵄩 󵄩 sup |λ|󵄩󵄩󵄩M β 𝜕α ((λ − T)−1 − (λ − σ0 )−1 )󵄩󵄩󵄩H −1−|β|+r →H r < ∞. λ∈Γ

Taking r = 0, this completes the proof of statement (ii). With sharp estimates for the symbol of the resolvent at hand, we may prove the following statement concerning the symbols of complex powers. Theorem 1.6.22. Let T = Op(σ) be a pseudodifferential operator obeying the conditions of Theorem 1.6.19, and let σ0 ∈ S m (ℝd × ℝd ) be positive and such that σ − σ0 ∈ S m−1 (ℝd × ℝd ). Then T z − Op(σ0z ) ∈ Ψmℜ(z)−1 (ℝd ) for all z ∈ ℂ. Proof. For λ ∈ Γ, where Γ is once again defined by (1.39), let (t, ξ) 󳨃→ r(λ, t, ξ) be the symbol function of (λ − T)−1 as in the proof of Theorem 1.6.19. Via Theorem 1.6.12, the two inequalities proved in Theorem 1.6.20 imply that for all α, β ∈ ℤd+ , there exists Cα,β such that we have the following estimates for the function r(λ, t, ξ) − (λ − σ0 )−1 : −1 󵄨 −m−|β|−1 󵄨󵄨 α β , 󵄨󵄨𝜕t 𝜕ξ [r(λ, t, ξ) − (λ − σ0 (t, ξ)) ]󵄨󵄨󵄨 ≤ Cα,β (1 + |ξ|) −1 󵄨 −|β|−1 󵄨󵄨 α β −1 . 󵄨󵄨𝜕t 𝜕ξ [r(λ, t, ξ) − (λ − σ0 (t, ξ)) ]󵄨󵄨󵄨 ≤ Cα,β |λ| (1 + |ξ|)

Here Cα,β is independent of λ. Combining these inequalities, we get the following strengthening of (1.6.17): −1 󵄨 m −1 −|β|−1 󵄨󵄨 α β . 󵄨󵄨𝜕t 𝜕ξ [r(λ, t, ξ) − (λ − σ0 (t, ξ)) ]󵄨󵄨󵄨 ≤ 2Cα,β ((1 + |ξ|) + |λ|) (1 + |ξ|)

(1.47)

In light of Theorem 1.6.19, for ℜ(z) < −1, we have T z − Op(σ0z ) = (2πi)−1 ∫ λz ((λ − T)−1 − Op((λ − σ0 )−1 )) dλ, Γ

where the integral converges in the ℒ(H 0 )-valued Bochner sense. Examining the behavior of this operator on the Schwartz class, we get that the operator T z − Op(σ0z ) has the symbol function σ(z, t, ξ) − σ0 (t, ξ)z = (2πi)−1 ∫ λz [r(λ, t, ξ) − (λ − σ0 (t, ξ)) ] dλ, −1

Γ

104 � 1 Bounded operators and pseudodifferential operators where σ(z, ⋅, ⋅) is the symbol of T z . For all α, β ∈ ℤd+ , we apply (1.47) to the integrand, which yields ∞

m −1 −1−|β| 󵄨󵄨 α β z 󵄨 ℜ(z) ds. 󵄨󵄨𝜕t 𝜕ξ [σ(z, t, ξ) − σ0 (t, ξ) ]󵄨󵄨󵄨 ≤ Cα,β ∫ s ((1 + |ξ|) + s) (1 + |ξ|) 1

The latter integral is bounded by a multiple of (1 + |ξ|)mℜ(z)−1−|β| , which implies that σ(z, t, ξ) − σ0 (t, ξ)z ∈ S mℜ(z)−1 (ℝd × ℝd ). Equivalently, T z − Op(σ0z ) ∈ Ψmℜ(z)−1 (ℝd ). This completes the proof in the case ℜ(z) < −1. For general z, we write T z = T n T z−n , where n is large enough so that ℜ(z) − n < −1. By the result just proved and Proposition 1.6.4 we have T n T z−n − T n Op(σ0z−n ) ∈ Ψmn+m(ℜ(z)−n)−1 (ℝd ) = Ψℜ(z)m−1 (ℝd ). Again by Proposition 1.6.4 we have T n − Op(σ0n ) ∈ Ψmn−1 , and therefore T z − Op(σ0n ) Op(σ0z−n ) ∈ Ψℜ(z)m−1 (ℝd ). Finally, Proposition 1.6.4 implies that Op(σ0n ) Op(σ0z−n ) − Op(σ0z ) ∈ Ψℜ(z)m−1 (ℝd ), which completes the proof. Elliptic pseudodifferential operators The condition that T ∈ Ψm (ℝd ) is self-adjoint on L2 (ℝd ) is in principle difficult to check since the self-adjointness requires that the domain of the adjoint operator T ∗ coincides with that of T. Recall that if T : dom(T) → L2 (ℝd ) is a linear operator defined on a subspace dom(T) ⊂ L2 (ℝd ), then dom(T ∗ ) = {u ∈ L2 (ℝd ) : there exists v ∈ L2 (ℝd )

such that for all f ∈ dom(T), ⟨v, f ⟩ = ⟨u, Tf ⟩}.

1.6 Pseudodifferential operators

� 105

However, the proof of self-adjointness is relatively straightforward for elliptic operators. Definition 1.6.23. A symbol σ ∈ S m (ℝd × ℝd ) is said to be elliptic if σ − σ0 ∈ S m−1 (ℝd × ℝd ) for a symbol σ0 ∈ S m (ℝd × ℝd ) such that 󵄨 󵄨 c|ξ|m ≤ 󵄨󵄨󵄨σ0 (t, ξ)󵄨󵄨󵄨 ≤ C|ξ|m ,

|ξ| ≥ 1,

t ∈ ℝd ,

for positive constants c, C > 0 independent of t and ξ. An operator T ∈ Ψm (ℝd ) is said to be elliptic if its symbol is elliptic. Lemma 1.6.24 (Elliptic regularity). Let T ∈ Ψm (ℝd ) be elliptic. Then for all r, r0 ∈ ℝ, u ∈ H r0 (ℝd ),

Tu ∈ H r (ℝd ) 󳨐⇒ u ∈ H r+m (ℝd ).

Proof. Let T = Op(σ), where σ is elliptic. It follows from the definition of ellipticity (see, e. g., Proposition 5.1′ on p. 40 in [256]) that there exists a smooth compactly supported function ϕ such that the function τ(t, ξ) = (1 − ϕ(ξ))σ0 (t, ξ)−1 ,

t, ξ ∈ ℝd ,

is a symbol of order −m. By Proposition 1.6.4 the operator R = 1 − Op(τ) Op(σ) belongs to Ψ−1 (ℝd ). Let u ∈ H r0 (ℝd ) be such that Tu = v ∈ H r (ℝd ). Since 1 − R = Op(τ)T, we have (1 − R)u = Op(τ) Op(σ)u = Op(τ)v. Let N > m. Since (1 + R + R2 + ⋅ ⋅ ⋅ + RN )(1 − R) = 1 − RN+1 , it follows that (1 − RN+1 )u = (1 + R + R2 + ⋅ ⋅ ⋅ + RN ) Op(τ)v, that is,

106 � 1 Bounded operators and pseudodifferential operators u = (1 + R + R2 + ⋅ ⋅ ⋅ + RN ) Op(τ)v + RN+1 u. Given that R ∈ Ψ−1 (ℝd ) and Op(τ) ∈ Ψ−m (ℝd ), Theorem 1.6.10 implies that u ∈ H r+m (ℝd ) + H r0 +N+1 (ℝd ). Choosing N large enough such that r0 + N + 1 > r + m, it follows that u ∈ H r+m (ℝd ). The next theorem proves that the restriction T : H m (ℝd ) → L2 (ℝd ) of an elliptic pseudodifferential operator T ∈ Ψm (ℝd ), m ≥ 0, is a self-adjoint operator if T is symmetric on 𝒮 (ℝd ). Theorem 1.6.25. Let m ≥ 0. Let T ∈ Ψm (ℝd ) be such that for all f , g ∈ 𝒮 (ℝd ), ⟨Tf , g⟩ = ⟨f , Tg⟩. If T is elliptic, then T restricts to a self-adjoint linear operator T : H m (ℝd ) → L2 (ℝd ) with domain H m (ℝd ). Proof. Let T0 denote the restriction of T to H m (ℝd ). Theorem 1.6.10 implies that T0 defines a linear operator T0 : H m (ℝd ) → L2 (ℝd ). We will prove that dom(T0∗ ) = H m (ℝd ). Since 𝒮 (ℝd ) ⊂ H m (ℝd ), if u ∈ dom(T0∗ ), then there exists v ∈ L2 (ℝd ) such that for all f ∈ 𝒮 (ℝd ), ⟨v, f ⟩ = ⟨u, T0 f ⟩ = ⟨u, Tf ⟩. By the definition of the formal adjoint T † in Proposition 1.6.6 we have ⟨u, Tf ⟩ = ⟨T † u, f ⟩,

f ∈ 𝒮 (ℝd ).

Here, T † u is a priori merely a distribution, but since f is arbitrary, the above statement tells us that T † u and v are identical distributions. Thus T † u = v ∈ L2 (ℝd ). We have assumed that ⟨Tf , g⟩ = ⟨f , Tg⟩ for all f , g ∈ 𝒮 (ℝd ). This can be restated as

1.6 Pseudodifferential operators �

Tf = T † f ,

107

f ∈ 𝒮 (ℝd ).

It follows that T coincides with T † on 𝒮 ′ (ℝd ), and therefore Tu = T † u ∈ L2 (ℝd ) = H 0 (ℝd ). Since T is assumed to be elliptic, from Lemma 1.6.24 it follows that u ∈ H m (ℝd ). Therefore dom(T0∗ ) ⊆ H m (ℝd ). Conversely, if u ∈ H m (ℝd ), then Tu = T † u ∈ H 0 (ℝd ) = L2 (ℝd ). Hence, taking v = Tu in the definition of dom(T0∗ ) gives that u ∈ dom(T0∗ ). Thus H m (ℝd ) ⊆ dom(T0∗ ). This completes the proof that dom(T0∗ ) = H m (ℝd ) = dom(T0 ). To see that T is self-adjoint, it suffices to observe that T0 u = Tu = v =: T0∗ u. 1.6.5 Compactness estimates for pseudodifferential operators Due to Theorem 1.6.10, a pseudodifferential operator T of negative order restricts to a bounded linear operator L2 (ℝd ) → L2 (ℝd ). In general, the restriction of negative order 1 operators is not compact. For example, the operator (1 − Δ)− 2 acting on L2 (ℝd ) has the spectrum consisting of the interval [0, 1]. 1 However, if f is a smooth compactly supported function, then Mf (1 − Δ)− 2 is a com1

pact operator. This follows from taking p > d + 1 in Theorem 1.5.20, since Mf (1 − Δ)− 2 restricts to a product-convolution operator on L2 (ℝd ). Generalizing this, the notion of a compactly supported pseudodifferential operator provides a sufficiently reduced class so that operators of order m < 0 extend to compact linear operators and those of order m < −d extend to trace class operators. Definition 1.6.26. A pseudodifferential operator T ∈ Ψ∞ (ℝd ) is said to be (a) compactly supported if there exist ϕ, ψ ∈ Cc∞ (ℝd ) such that Mϕ TMψ = T, (b) compactly supported on the left if there exists ϕ ∈ Cc∞ (ℝd ) such that Mϕ T = T, (c) compactly supported on the right if there exists ϕ ∈ Cc∞ (ℝd ) such that TMϕ = T.

108 � 1 Bounded operators and pseudodifferential operators Remark 1.6.27. It is clear that if T = Op(σ), then Mϕ T = Op(ϕσ), and hence T is compactly supported on the left if and only if the symbol σ of T has compact support in the first variable. Compact support and compact support on the right do not have straightforward characterizations in terms of the symbol, but their meanings are clear when looking at integral kernels. If σ ∈ S m (ℝd ×ℝd ) and m < −d, then Op(σ) may be described as an integral operator Op(σ)u(t) = ∫ K(t, s)u(s) ds,

t ∈ ℝd ,

ℝd

where the kernel K is given by the absolutely convergent integral K(t, s) = (2π)−d ∫ ei⟨t−s,ξ⟩ σ(t, ξ) dξ. ℝd

The pseudodifferential operator Op(σ) is compactly supported if and only if K is a compactly supported function on ℝd ×ℝd , and Op(σ) is compactly supported on the left or the right if and only if K is compactly supported in its first or second variable, respectively. The linear space of all compactly supported pseudodifferential operators forms an algebra with adjoint, but if T is compactly supported on the right, then the same may not be true for the adjoint T † . Instead, the adjoint of an operator compactly supported on the right is compactly supported on the left. However, every pseudodifferential operator compactly supported on one side is “almost” compactly supported as the following proposition demonstrates. Proposition 1.6.28. Let T, S ∈ Ψ∞ (ℝd ). (a) If T and S are compactly supported, then so are T † , TS, and ST. (b) T is compactly supported if and only if both T and T † are compactly supported on the left. (c) T is compactly supported if and only if both T and T † are compactly supported on the right. (d) If T is compactly supported on the left, then so is TS. (e) If T is compactly supported on the right, then so is ST. (f) If T is compactly supported on the left or the right, then there exists a compactly supported pseudodifferential operator T ′ such that T − T ′ ∈ Ψ−∞ (ℝd ). Proof. We only provide an argument for the final statement as the others are immediate. Suppose that T is compactly supported on the left. Let Mψ T = T for some ψ ∈ Cc∞ (ℝd ). Choose ϕ ∈ Cc∞ (ℝd ) such that ϕψ = ψ. Set T ′ = TMϕ = Mϕ TMϕ . For every n ∈ ℕ, choose a sequence {ϕk }nk=1 ⊂ Cc∞ (ℝd ) such that ϕk+1 ϕk = ϕk , 1 ≤ k < n, ϕ1 = ψ, and ϕn = ϕ. We have

1.6 Pseudodifferential operators

� 109

[Mϕn , [Mϕn−1 , . . . , [Mϕ1 , T]]] = T − T ′ . Let T have order m. By Corollary 1.6.5, T − T ′ has order m − n. Since n is arbitrarily large, the statement follows. A virtually identical argument applies if T is compactly supported on the right. Lemma 1.6.29. If T ∈ Ψ−m (ℝd ), −m < 0, is compactly supported on the left or the right, then T ∈ ℒ d ,∞ . In particular, if −m < −d, then T is trace class. m

m

Proof. Since T ∈ Ψm (ℝd ), it follows from Theorem 1.6.10 that the operator S = (1 − Δ) 2 T is bounded. If T is compactly supported on the left, then m

T = Mϕ T = Mϕ (1 − Δ)− 2 ⋅ S. m

The factor Mϕ (1 − Δ)− 2 belongs to ℒ d ,∞ by Theorem 1.5.22 when d < 2m and by Theom rem 1.5.20 when d > 2m. When d = 2m, then by Theorem 1.5.22 󵄨󵄨 󵄨2 −m −m 󵄨󵄨(1 − Δ) 2 Mϕ 󵄨󵄨󵄨 = Mϕ (1 − Δ) ⋅ Mϕ ∈ ℒ1,∞ . m

Hence Mϕ (1 − Δ)− 2 ∈ ℒ2,∞ . Since the operator S is bounded, T ∈ ℒ d ,∞ . m

If T is compactly supported on the right, then a similar argument applies with S ′ = m T(1 − Δ) 2 in place of S. Since compactly supported smoothing operators restrict to trace class operators on L2 (ℝd ) by Lemma 1.6.29, we obtain the following: Corollary 1.6.30. Let T ∈ Ψm (ℝd ) be compactly supported on the left or on the right. Then there exists a compactly supported pseudodifferential operator T ′ of order m such that T − T ′ is trace class. Proof. Let T ′ be as in Proposition 1.6.28(f). Then T −T ′ is a compactly supported smoothing operator. In particular, the operator T − T ′ is trace class. Another consequence of Lemma 1.6.29 is the following trace formula for pseudodifferential operators that are compactly supported on the left. Corollary 1.6.31. Let T = Op(σ) ∈ Ψ−m (ℝd ), −m < −d, be compactly supported on the left. Then σ ∈ L1 (ℝd × ℝd ), and Tr(T) = (2π)−d

∫ σ(t, ξ) dtdξ. ℝd ×ℝd

Proof. By assumption there exists a compactly supported bounded positive function χ ∈ L∞ (ℝd ) such that −m 󵄨󵄨 󵄨 󵄨󵄨σ(t, ξ)󵄨󵄨󵄨 ≤ Cχ(t)(1 + |ξ|) =: σ0 (t, ξ),

(t, ξ) ∈ ℝd × ℝd .

110 � 1 Bounded operators and pseudodifferential operators Since σ0 ∈ L1 (ℝd × ℝd ), it follows that σ ∈ L1 (ℝd × ℝd ). The statement follows from Theorem 1.5.5. Sometimes, an operator that is not compactly supported is still compact. We mention in particular the following useful estimate. Lemma 1.6.32. Let T ∈ Ψm (ℝd ) be compactly supported on the left or on the right, and let S ∈ Ψ−r (ℝd ) for some r > m − 1. Then [S, T] ∈ ℒ

d ,∞ r−m+1

.

Proof. We begin by proving that if S ∈ Ψα (ℝd ) for some α ∈ ℝ and R ∈ Ψ−∞ (ℝd ), then for all compactly supported smooth functions ϕ, we have SMϕ R ∈ ⋂ ℒp . p>0

This is an easy consequence of Lemma 1.6.29 when α ≤ 0. To see the general case, suppose that the statement holds for some α ∈ ℝ and let S ∈ Ψα+1 (ℝd ). Taking ψ ∈ Cc∞ (ℝd ) with ψϕ = ϕ, we have SMϕ R = [S, Mϕ ]Mψ R + Mϕ SMψ R. Lemma 1.6.29 shows that Mϕ SMψ R ∈ ⋂p>0 ℒp . By Corollary 1.6.5, [S, Mϕ ] ∈ Ψα (ℝd ), and hence [S, Mϕ ]Mψ R ∈ ⋂p>0 ℒp by assumption. Hence the statement holds for α + 1 if it holds for α, which proves the general case. Now let T ∈ Ψm (ℝd ) and S ∈ Ψ−r (ℝd ) be as in the statement of the lemma. Assume for definiteness that T is compactly supported on the left; the other case is proven similarly. By Proposition 1.6.28(f) there exists a compactly supported T ′ ∈ Ψm (ℝd ) such that R := T − T ′ ∈ Ψ−∞ (ℝd ). Choosing ϕ, ψ ∈ Cc∞ (ℝd ) such that T = Mϕ T = Mϕ T ′ Mψ + Mϕ R, we have [S, T] = [S, Mϕ T ′ Mψ ] + [S, Mϕ R] = [S, Mϕ T ′ ]Mψ + Mϕ T ′ [S, Mψ ] + SMϕ R − Mϕ RS. Each of the above four terms belongs to ℒ and the statement just proved.

d ,∞ r−m+1

due to Corollary 1.6.5, Lemma 1.6.29,

1.6.6 Asymptotic expansion of symbols The class of classical pseudodifferential operators of order m is defined using asymptotic expansion of symbols.

1.6 Pseudodifferential operators �

111

mj d d Definition 1.6.33. Let {σj }∞ j=0 be a sequence of symbol functions σj ∈ S (ℝ × ℝ ), j ≥ 0, ∞ of orders {mj }j=0 such that mj → −∞ as j → ∞.

A symbol σ ∈ S ∞ (ℝd × ℝd ) is said to have an asymptotic expansion ∞

σ ∼ ∑ σj j=0

if for every N > 0, there exists k > 0 such that k

σ − ∑ σj ∈ S −N (ℝd × ℝd ). j=0

Similarly, if {Tj }∞ j=0 is a sequence of pseudodifferential operators of orders mj such that mj → −∞ as j → ∞, then an operator T ∈ Ψ∞ (ℝd ) is said to have an asymptotic expansion ∞

T ∼ ∑ Tj j=0

if for every N > 0, there exists k > 0 such that k

T − ∑ Tj ∈ Ψ−N (ℝd ). j=0

It is easy to see that asymptotic expansions are compatible with algebraic operations on operators. For example, if T ∈ Ψm1 (ℝd ) and S ∈ Ψm2 (ℝd ) have asymptotic expansions ∞

∞

T ∼ ∑ Tj ,

S ∼ ∑ Sj ,

j=0

j=0

then ∞

TS ∼ ∑ Tj Sk , j,k=0

where the summation is ordered so that the orders of the terms are decreasing. Note that if T, T ′ ∈ Ψm (ℝd ) have the same asymptotic expansions ∞

T ∼ ∑ Tj , j=0

then for every k ≥ 0, we have

∞

T ′ ∼ ∑ Tj , j=0

112 � 1 Bounded operators and pseudodifferential operators k

k

j=0

j=0

T − T ′ = (T − ∑ Tj ) − (T ′ − ∑ Tj ) ∈ Ψmk+1 (ℝd ) for some mk+1 ≤ 0. Since k is arbitrary and mk+1 → −∞, it follows that T − T ′ ∈ Ψ−∞ (ℝd ), that is, asymptotic expansions determine an operator up to a smoothing term. Similarly, if two symbols σ, σ ′ ∈ S m (ℝd × ℝd ) have the same asymptotic expansion, then σ − σ ′ ∈ S −∞ (ℝd × ℝd ). A symbol function σ ∈ S m (ℝd × ℝd ) is said to be m-homogeneous except near zero if σ(t, αξ) = αm σ(t, ξ),

t ∈ ℝd ,

|ξ| ≥ 1,

α > 1.

Note that an m-homogeneous symbol is determined, up to a symbol of order −∞, by its restriction to the unit sphere. Definition 1.6.34. A pseudodifferential operator T ∈ Ψm (ℝd ), m ∈ ℝ, and its symbol σ ∈ S m (ℝd × ℝd ) are called classical if σ has an asymptotic expansion ∞

σ ∼ ∑ σm−j j=0

where each σm−j ∈ S m−j (ℝd × ℝd ) is a homogeneous function of order m − j except near zero. The first term σm ∈ S m (ℝd × ℝd ) in the asymptotic expansion of σ is called the principal symbol of T. The classical pseudodifferential operators of order m are denoted d Ψm cl (ℝ ). All partial differential operators are classical pseudodifferential operators. Example 1.6.35. A linear partial differential operator P = ∑ aα (s)𝜕tα |α|≤m

with smooth and bounded coefficients, with all derivatives bounded, is a classical pseudodifferential operator of order m. The symbol σ has the finite asymptotic expansion m

σ(t, ξ) = ∑ σm−j (t, ξ), j=0

t, ξ ∈ ℝd ,

where σm−j is the homogeneous polynomial σm−j (t, ξ) =

∑ aα (t)(iξ)α ,

|α|=m−j

t, ξ ∈ ℝd .

1.6 Pseudodifferential operators

� 113

In particular, P has the principal symbol σm (t, ξ) = ∑ aα (t)(iξ)α ,

t, ξ ∈ ℝd .

|α|=m

As observed above, an operator is determined up to a smoothing term by an asymptotic expansion; however, in principle, it can happen that an operator has many different representations by an asymptotic series. In the case of classical operators, the homogeneous components σm−j of the symbol σ are determined uniquely up to smoothing terms. In particular, by Lemma 1.6.36 the principal symbol of a classical pseudodifferential operator is unique up to a smoothing term. Lemma 1.6.36. Suppose ∞

σ ∼ ∑ σm−j j=0

and ∞

′ σ ∼ ∑ σm−j j=0

are asymptotic expansions of a classical symbol σ ∈ S m (ℝd × ℝd ). Then, for each integer j = 0, 1, 2, . . . , ′ σm−j (t, ξ) = σm−j (t, ξ),

|ξ| ≥ 1,

t ∈ ℝd .

Proof. Set ′ ′ ρ0 := σm − σm = (σ − σm ) − (σ − σm ) ∈ S m−1 (ℝd × ℝd ). ′ Given that σm and σm are m-homogeneous except near zero, it follows that ρ0 is also m-homogeneous except near zero. We will demonstrate that ρ0 (t, ξ) vanishes for all |ξ| ≥ 1. If ξ ∈ ℝd , |ξ| ≥ 1, and t ∈ ℝd is such that ρ0 (t, ξ) ≠ 0, then the m-homogeneity of ρ0 implies that for all s > 0,

sm =

ρ0 (t, sξ) . ρ0 (t, ξ)

Given that ρ0 ∈ S m−1 (ℝd × ℝd ), there exists a constant C > 0 such that m−1 󵄨󵄨 󵄨 󵄨󵄨ρ0 (t, sξ)󵄨󵄨󵄨 ≤ C(1 + s|ξ|) ,

Therefore, for all s > 0, we have

t ∈ ℝd ,

s > 0.

114 � 1 Bounded operators and pseudodifferential operators (1 + s|ξ|)m−1 󵄨󵄨 󵄨 . 󵄨󵄨ρ0 (t, ξ)󵄨󵄨󵄨 ≤ C ⋅ sm Letting s → ∞, m−1

1 1 󵄨󵄨 󵄨 󵄨󵄨ρ0 (t, ξ)󵄨󵄨󵄨 ≤ C ⋅ lim ( + |ξ|) s→∞ s s

.

Hence ρ0 (t, ξ) = 0 for all |ξ| ≥ 1. Replacing σ by σ − σm , we obtain ∞

∞

j=1

j=1

′ σ − σm ∼ ∑ σm−j ∼ ∑ σm−j .

′ By the preceding argument we have σm−1 (t, ξ) = σm−1 (t, ξ) for all |ξ| ≥ 1, and similarly ′ σm−j (t, ξ) = σm−j (t, ξ) for all j ≥ 0 and |ξ| ≥ 1.

Ellipticity for classical pseudodifferential operators has a simple form. d Remark 1.6.37. A classical pseudodifferential operator T ∈ Ψm cl (ℝ ) is elliptic according to Definition 1.6.23 if and only if the principal symbol σm of T satisfies

󵄨 󵄨 c|ξ|m ≤ 󵄨󵄨󵄨σm (t, ξ)󵄨󵄨󵄨 ≤ C|ξ|m ,

|ξ| ≥ 1,

t ∈ ℝd ,

or, equivalently, 󵄨 󵄨 c ≤ 󵄨󵄨󵄨σm (t, ξ)󵄨󵄨󵄨 ≤ C,

|ξ| = 1,

t ∈ ℝd ,

for some constants c > 0 and C > 0. A consequence of Lemma 1.6.36 and the symbol multiplication formula in Remark 1.6.7 is that the homogeneous components of the symbol of a product of two classical operators can be computed explicitly. d ′ m d ′ Example 1.6.38. If T ∈ Ψm cl (ℝ ) and T ∈ Ψcl (ℝ ) have symbols σ and σ with asymptotic expansions ′

∞

σ ∼ ∑ σm−j , j=0

∞

σ ′ ∼ ∑ σm′ −j , j=0

then TT ′ ∈ Ψm+m (ℝd ), and the symbol σ ′′ of TT ′ has the asymptotic expansion cl ′

∞

σ ′′ ∼ ∑( ∑

j=0 |α|+k+l=j

(−i)|α| α 𝜕 σ ⋅ 𝜕α σ ′ ′ ). α! ξ m−k t m −l

1.6 Pseudodifferential operators

� 115

Example 1.6.38 indicates that if classical operators T and T ′ have principal symbols ′ ′ ′ m d σm and σm ′ , respectively, then σm σm′ is a principal symbol for TT . If T ∈ Ψcl (ℝ ) has a principal symbol σm , then the principal symbol map d ∞ d d sym : Ψ∞ cl (ℝ ) → C (ℝ × {ξ ∈ ℝ : |ξ| ≥ 1}),

sym(T) = σm ,

is an algebraic homomorphism from operators to smooth functions. A noncommutative version of the symbol map will be considered in Chapter 5. An important feature in quantization is that under the principal symbol map, commutators of classical pseudodifferential operators map to the Poisson bracket. This follows from Example 1.6.38. ′ m ′ Example 1.6.39. If T ∈ Ψm cl and T ∈ Ψcl have principal symbols σm and σm′ , then ′

−1 [T, T ′ ] ∈ Ψm+m has the principal symbol cl ′

′ α α ′ α ′ α −i{σm , σm ′ } = −i( ∑ 𝜕ξ σm 𝜕t σm′ − 𝜕ξ σm′ 𝜕t σm ). |α|=1

Generally, for nonclassical pseudodifferential operators, a “first term” in an expansion of the symbol can only be defined using a broader equivalence class. To avoid confusion in terminology, the function that is the first term in an asymptotic expansion of a classical pseudodifferential operator will be called a principal symbol, whereas the equivalence class defined as follows will be called the principal symbol class. For a classical pseudodifferential operator, a principal symbol generates the principal symbol class of that operator. Definition 1.6.40. If σ1 and σ2 belong to S m (ℝd × ℝd ), then we write σ1 ∼ σ2 if σ1 − σ2 belongs to S m−1 (ℝd × ℝd ). This is an equivalence relation on the symbols of order m. If σ is the symbol of a pseudodifferential operator T ∈ Ψm (ℝd ), then the equivalence class [σ] ∈ S m (ℝd × ℝd )/S m−1 (ℝd × ℝd ) is called the principal symbol class of T. Proposition 1.6.4 implies that if T, T ′ ∈ Ψ∞ (ℝd ) have principal symbol classes [σ] and [σ ′ ], respectively, then TT ′ has principal the symbol class [σσ ′ ].

1.6.7 The noncommutative residue If T is a classical pseudodifferential operator of order m that is compactly supported on the left, then we can take, without loss, each term in the asymptotic expansion of its symbol σ ∼ ∑∞ j=0 σm−j to be a symbol with compact support in its first variable. If m is an integer, then the term σ−d makes sense (and is identically zero if m < −d). If m is a noninteger, then we set σ−d ≡ 0. Let ds denote the volume element of the (d − 1)-sphere

116 � 1 Bounded operators and pseudodifferential operators 𝕊d−1 = {s ∈ ℝd : |s| = 1}. When d = 1, ds is the counting measure on {−1, 1}. As remarked earlier, we scale the usual definition of the noncommutative residue by the reciprocal of d. Definition 1.6.41. The noncommutative residue of a classical pseudodifferential operad tor T ∈ Ψm cl (ℝ ) that is compactly supported on the left is the scalar value ResW (T) :=

1 ∫ ∫ σ−d (t, s)dtds, d(2π)d 𝕊d−1 ℝd

where σ−d ∈ S −d (ℝd × ℝd ) is the term in an asymptotic expansion of the symbol σ of T that is −d-homogeneous except near 0. Given that σ−d (t, ξ) is determined uniquely by T for all |ξ| ≥ 1, the definition of ResW (T) does not depend on the choice of an asymptotic expansion for σ. The assumption that T is compactly supported on the left is in place to ensure the convergence of the integral of σ−d in the t-variable. The noncommutative residue restricted to a classical pseudodifferential operator of order −d has properties similar to those observed in Section 1.1.2 for singular traces. Proposition 1.6.42. The noncommutative residue (a) is a linear functional on the linear space of all classical compactly pseudodifferential operators of order −d that are supported on the left; (b) vanishes on operators of order m < −d (those that are trace class), that is, if T ∈ d Ψm cl (ℝ ), m < −d, is compactly supported on the left, then ResW (T) = 0; m

m

(c) is a trace for T ∈ Ψcl 1 (ℝd ) compactly supported on the left and S ∈ Ψcl 2 (ℝd ) compactly supported on the left with m1 + m2 = −d, that is, ResW ([T, S]) = 0. The next example involves the noncommutative residue of an order −d classical pseudodifferential operator that is a product-convolution operator in the terminology of Chapter 1.5 and not a trace class operator. The noncommutative residue should be compared to the trace of trace class pseudodifferential operators as observed in Corollary 1.6.31 and to the trace of trace class product-convolution operators as observed in Corollary 1.5.23. Example 1.6.43. The Bessel potential m

(1 − Δ) 2 ∈ Ψm (ℝd ),

m ∈ ℝ,

1.6 Pseudodifferential operators �

117

m

introduced in Section 1.6.3 has the symbol (1 + |ξ|2 ) 2 ∈ S m (ℝd × ℝd ). Since m

m

(1 + |ξ|2 ) 2 = |ξ|m (1 + |ξ|−2 ) 2 ,

|ξ| ≥ 1,

m

m

by using the binomial series expansion for (1+|ξ|−2 ) 2 it follows that the symbol of (1−Δ) 2 has the asymptotic expansion ∞

m

(1 + |ξ|2 ) 2 ∼ ∑ ( j=0

m 2

j

) |ξ|m−2j .

m

Hence (1 − Δ) 2 is classical with principal symbol σm (t, ξ) = |ξ|m for t ∈ ℝd , |ξ| ≥ 1. If ϕ ∈ Cc∞ (ℝd ), then the multiplication operator Mϕ is a pseudodifferential operator of order 0 by Example 1.6.3. The operator Mϕ is classical, with principal symbol τ0 (t, ξ) = ϕ(t), t, ξ ∈ ℝd . By Example 1.6.38 m

d Mϕ (1 − Δ) 2 ∈ Ψm cl (ℝ )

is compactly supported on the left with principal symbol ρ given by ρ(t, ξ) = ϕ(t)|ξ|m ,

t ∈ ℝd ,

|ξ| ≥ 1.

d

Hence, when m = −d, the operator Mϕ (1 − Δ)− 2 is a classical pseudodifferential operator of order −d that is compactly supported on the left, and the principal symbol ρ reduces to ρ(t, s) = ϕ(s) on ℝd × 𝕊d−1 . Thus d

ResW (Mϕ (1 − Δ)− 2 ) =

Vol(𝕊d−1 ) ⋅ ∫ ϕ(s)ds. d(2π)d

(1.48)

ℝd m

When m < −d, the pseudodifferential operator Mϕ (1 − Δ) 2 restricted to L2 (ℝd ) is a trace class operator by Lemma 1.6.29. It is also an operator of product-convolution type, and by Corollary 1.5.23 m

Tr(Mϕ (1 − Δ) 2 ) = cm ⋅ Vol(𝕊d−1 ) ∫ ϕ(s)ds,

(1.49)

ℝd

where ∞

m

0 < cm = ∫ (1 + r 2 ) 2 r d−1 dr,

m < −d.

0

Chapter 2 generalizes the noncommutative residue on operators of order −d using traces on the weak trace class ideal ℒ1,∞ . In Chapter 3, we will demonstrate that (1.48)

118 � 1 Bounded operators and pseudodifferential operators continues to hold for the Laplace–Beltrami operator when the Euclidean space ℝd is curved by a Riemannian metric that is not flat, whereas (1.49) will no longer hold as terms from an asymptotic expansion involving curvature and derivatives of the metric are added. The underlying philosophy behind many of the subsequent chapters is that traces on the ideal ℒ1,∞ can be used to extract principal terms in asymptotic expansions of the trace of certain trace class operators, or act as substitutes for the principal terms for the trace of those trace class operators for which asymptotic expansions are not welldefined.

1.7 Notes Operators and inequalities For background on functional analysis and operators on separable Hilbert spaces, see [113, 273]. The references [113, 273] also introduce the theory of von Neumann algebras, and [221] introduces noncommutative Lp -spaces. Background to eigenvalues sequences, singular values, and the Calkin correspondence is provided in the end notes to Chapters 1 and 2 of Volume I. The modern theory of noncommutative measures and integration originated in the papers of Dixmier [110] and Segal [252]. Segal [252] developed a calculus for measurable operators, introduced the notions of integrable and square-integrable operators and proved extensions of the Riesz–Fischer, Radon– Nikodym, Lebesgue monotone convergence, and Fubini theorems. Segal was motivated by the work of von Neumann and Murray on rings of operators, quantum field theory, and harmonic analysis on groups. Important influences were the papers of Dye [122] on the extension of the Radon–Nikodym theorem to finite rings of operators and that of Ambrose [9] on standard rings. Segal’s work was not based on extending the notion of the singular value sequence of a compact operator. In [212, Chapter XV], von Neumann and Murray define a generalized eigenvalue function for self-adjoint operators in a finite von Neumann algebra analogous with the Courant–Fisher minimax characterization of the eigenvalues of a compact self-adjoint operator. They showed that the generalized eigenvalue function is a right inverse to the distribution function of the operator. This minimax characterization underlies Definition 1.2.12. Grothendieck [143] introduced a “decreasing rearrangement” for positive operators in a semifinite von Neumann algebra. In the case that the von Neumann algebra is the commutative von Neumann algebra L∞ (Ω, μ) for some σ-finite measure space (Ω, μ), the decreasing rearrangement introduced by Grothendieck coincides with the classical decreasing rearrangement. The first systematic presentation of generalized singular values was given by Ovčinnikov [216]. Some of his results had been obtained earlier for factors by Sonis [262]. Ovčinnikov applied his results to study noncommutative symmetric spaces containing as particular cases the noncommutative Lp spaces of Segal and Dixmier, as well as the symmetrically normed ideals of Schatten and Gohberg and Kreĭn [217]. Independently of the work of Ovčinnikov, Yeadon [289–291]

1.7 Notes

� 119

studied noncommutative Lp -spaces and symmetric norms, with a view to applications in noncommutative ergodic theory [292, 293]. A self-contained and comprehensive exposition of generalized s-numbers was given subsequently by Fack and Kosaki [129]. The Lorentz function spaces Lp,q were introduced by Lorentz [196, 197]. If 1 ≤ q ≤ p < ∞, then ‖ ⋅ ‖p,q is a norm, and if 1 < p < q < ∞, then it is a complete quasinorm, which is equivalent to a complete norm [23, 108]. We caution the reader on the many species of function spaces named Lorentz spaces. The terminology for a Lorentz ideal in Volume II, from the classical functional spaces Lp,q and sequences spaces lp,q , and that of Volume I are different [170]. Volume I follows Definition 5.7 in [23], where Lorentz functions spaces Mψ are defined for the quasiconcave function tψ(t)−1 , t > 0. Their difference is evident just for L2 , since there is no quasiconcave function tψ(t)−1 , t > 0, such that L2 = Mψ . Theorem 1.3.14 concerns operator Lipschitz estimates for the absolute value function. The study of operator Lipschitz estimates originates from the question of Kreĭn [179] whether 󵄩󵄩 󵄩 󵄩󵄩f (A) − f (B)󵄩󵄩󵄩ℒ1 ≤ ‖f ‖Lip ⋅ ‖A − B‖ℒ1 ,

A = A∗ ,

B = B∗ ∈ ℒ(H),

for a Lipschitz function f : ℝ → ℝ. Davies [100] showed that the absolute value function was not Lipschitz continuous on ℒ1 . The absolute value function is also not operator Lipschitz for the operator norm on ℒ(H) [172, 207]. Krein’s question was answered positively for ℒp , 1 < p < ∞, for all Lipschitz functions by Potapov and one of the authors [225] using double operator integrals. For a detailed study of commutator estimates for the absolute value function, we refer the reader to [117]. Theorem 1.3.14 is [117, Theorem 3.4]. A survey of operator Lipschitz functions with respect to the operator and trace class norms is given in [8]. The reference [59] bounds the quasinorm of the difference |A| − |B| ∈ ℒ1,∞ by the norm of A − B in ℒ1 . Theorem 1.3.3 has origins in mathematical physics. In 1970, Powers and Størmer [227, Lemma 4.1] used the estimate 1 1 󵄩󵄩 21 󵄩 󵄩󵄩A − B 2 󵄩󵄩󵄩ℒ2 ≤ ‖|A − B| 2 ‖ℒ2 ,

0 ≤ A, B ∈ ℒ(H),

to prove necessary and sufficient conditions for quasi-equivalence of gauge-invariant free states of the anticommutation algebra, [227]. For 0 ≤ A, B ∈ ℒ(H), 0 < θ < 1, and p > 1, Birman, Koplienko, and Solomyak [30] in 1975 proved that 󵄩󵄩 θ θ󵄩 θ 󵄩󵄩A − B 󵄩󵄩󵄩ℒp ≤ cp,θ ⋅ ‖|A − B| ‖ℒp for a constant cp,θ > 0. Birman and Solomyak [35, Theorem 4.1] proved the estimate for ℒp,∞ . Birman, Koplienko, and Solomyak proved the estimates using double operator integrals; Ando provided an alternative proof [10, Corollary 2]. The reference [114] noted the stronger estimate

120 � 1 Bounded operators and pseudodifferential operators Aθ − Bθ ≺≺ cp,θ ⋅ |A − B|θ , from which the estimates for the ideals ℒp and ℒp,∞ , 1 < p < ∞, follow. In 2018, Ricard [233] extended the Birman–Koplienko–Solomyak norm estimates to 0 < p < 1. Theorem 1.3.3 is proved in Theorem 6.1 in [163] by taking the functions f (s) = g(s) = |s|θ , s > 0, extending results of Aleksandrov and Peller on θ-Hölder functions [6, 7]. Theorem 1.3.4 can be deduced from Theorem 1.3.3 using the Cayley transform; see Corollary 7.1 in [163]. Theorem 1.3.5 was conjectured by Bikchentaev [27, p. 573] in the more general setting of τ-measurable operators and proved by one of the authors in [266]. The paper [266] provides applications of Theorem 1.3.5 to fully symmetric operator ideals, Golden– Thompson inequalities, and singular traces. Theorem 1.3.19 is a variant of [72, Chapter IV, Section 3.γ, Lemma 11], which states that for p > 1, 0 ≤ B ∈ ℒp,∞ (L2 (ℝd )), and a bounded function f ≥ 0 such that [B, Mf ] ∈ (ℒp,∞ )0 , we have p

p

1

1

p

Mf2 Bp Mf2 − (Mf2 BMf2 ) ∈ (ℒ1,∞ )0 . Replacing the assumption that [B, Mf ] ∈ (ℒp,∞ )0 with the formally stronger assump1

tion [B, Mf2 ] ∈ (ℒp,∞ )0 gives a restatement of Theorem 1.3.19. The result [72, Chapter IV, Section 3.γ, Lemma 11] was stated without proof. Theorem 1.3.19 was proved in [88] using the following weak operator integral representation for powers of a positive compact operator (see Lemma 5.2 in [88]). The following theorem also strengthens [49, Proposition 4.4]. 1

1

Theorem. Let 0 ≤ A, B ∈ ℒ(H), and let z ∈ ℂ with ℜ(z) > 1. Let Y := A 2 BA 2 . We define the mapping Tz : ℝ → ℒ(H) by 1

1

1

1

Tz (0) := Bz−1 [BA 2 , Az− 2 ] + [BA 2 , A 2 ]Y z−1 , 1

1

1

1

Tz (s) := Bz−1+is [BA 2 , Az− 2 +is ]Y −is + Bis [BA 2 , A 2 +is ]Y z−1−is ,

s ≠ 0.

We also define the function gz : ℝ → ℂ by z gz (0) := 1 − , 2 gz (t) := 1 −

z

t

z

e 2 t − e− 2 t

t

(e 2 − e− 2 )(e(

z−1 )t 2

+ e−(

z−1 )t 2

. )

Then (i) the mapping Tz : ℝ → ℒ(H) is continuous in the weak operator topology, and

1.7 Notes �

121

(ii) we have 1

1

z

Bz Az − (A 2 BA 2 ) = Tz (0) − ∫ Tz (s)(ℱ gz )(s) ds. ℝ

The integral formula in the generality stated above was proved in [271]. The integral formula has several applications, including an alternative approach to the results of Chapter 6; see [205]. Traces Background to traces on ideals and the notion of a singular trace are provided in the end notes to Chapters 1 and 2 of Volume I. Theorem 1.1.10 is proved in Theorems 6.1 and 6.2,and Corollary 7.1.4 of Volume I. Theorem 1.1.12 is a collation of results from Volume I. Theorem 1.1.12(a) follows from Theorem 5.1.2 in Volume I since the ideal ℒ1,∞ is quasi-Banach. Theorem 1.1.12(b) is Corollary 3.5.4 in Volume I, and Theorem 1.1.12(c) follows from the proof of that corollary. Theorem 1.1.12(d) follows from Lemmas 2.5.3 and 2.4.12 in Volume I. Theorem 1.1.12(e) is [191, Theorem 5.12]. Theorem 1.1.12(f) follows from Corollary 3.5.7, Theorems 6.1.2 and 6.1.3, and Example 6.3.6 in Volume I. Theorem 1.1.12(g) is Theorem 4.1.10 in Volume I. Theorem 1.1.13 is a restatement of Theorem 9.1.2 in Volume I. Submajorization and interpolation The partial order ≺ for real n-vectors, customarily termed Hardy–Littlewood–Pólya majorization, was introduced in the early 20th century by Muirhead [211], Lorenz [199], Dalton [97], and Schur [250]. It was subsequently applied by Hardy and Littlewood, and in the fundamental monograph by Hardy, Littlewood, and Pólya [149]. Hardy– Littlewood–Pólya majorization plays a central role in the study of function spaces, Banach lattices, and interpolation theory and has important applications in stochastic analysis, numerical analysis, geometric inequalities, matrix theory, statistical theory, optimization, and economic theory [203]. If x, y ∈ ℝn , then x ≺ y if and only if x belongs to the convex hull Ω(y) of {Py : P is a permutation matrix} [149] (see also [203, p. 10] and [228]). This result has an important noncommutative counterpart, which states that in the setting of n × n matrices, a Hermitian matrix A belongs to the doubly stochastic orbit of another Hermitian matrix B if and only if the vector λ(A) of eigenvalues of A is majorized by that of B in the sense of Hardy–Littlewood–Pólya (denoted by A ≺ B) [228]. This is closely related to the Birkhoff–von Neumann theorem identifying extreme points of doubly stochastic matrices with permutation matrices [28]. The Birkhoff–von Neumann theorem, combined with the result that x ≺ y if and only if there exists a double stochastic n × n matrix T such that x = Ty,

122 � 1 Bounded operators and pseudodifferential operators is the finite precursor of the relationship between submajorization, rearrangement invariance, and interpolation. For operators A, B ∈ ℒ(H) (respectively, functions f , g ∈ (L1 +L∞ )(Ω, ν) for a σ-finite measure space (Ω, ν)), the analogous statement is that A ≺≺ B (respectively, f ≺≺ g) if and only if there exists an absolute contraction T : ℒ(H) → ℒ(H) (respectively, T : (L1 + L∞ )(Ω, ν) → (L1 + L∞ )(Ω, ν)) such that A = T(B) (respectively, f = T(g)). The notion of majorization in the setting of Lebesgue-measurable functions on (0, 1) is due to Hardy, Littlewood, and Pólya [148] (see also [200, 243]). For Lebesguemeasurable functions on (0, ∞), majorization can be extended to functions belonging to L1 (0, ∞). For nonintegrable functions, where the equality ∞

∞

󵄨 󵄨 󵄨 󵄨 ∫ 󵄨󵄨󵄨f (s)󵄨󵄨󵄨ds = ∫ 󵄨󵄨󵄨g(s)󵄨󵄨󵄨ds 0

0

cannot be stipulated, submajorization has played an analogous role [203]. Hiai [155, 156, 158, 159] and many authors have contributed to extending the notion of majorization and subsequent analysis in semifinite von Neumann algebras. The reference [162] discusses results for the extreme points of the set of operators x ≺ y, y ∈ ℒ1 (ℳ, τ) and the Alberti– Ulhmann problem. Identifying the extremal points in the case of finite von Neumann algebras was solved in [98], extending commutative results of Ryff [244–246]. For interpolation and submajorization in classical function spaces, we refer to [23]. The terminology “rearrangement invariant Banach space” in [23] is equivalent to the terminology of a fully symmetric function space in this book. The result that exact interpolation spaces between L1 and L∞ are fully symmetric function spaces is due to Calderón [47, Theorem 3]; see also [23, Theorem 2.12]. The implications in Theorem 1.2.2 for the function space (L1 + L∞ )(Ω, ν), where (Ω, ν) is a σ-finite measure space, are observed in [47, Theorems 2 and 3]. Theorem 1.2.2 and Corollary 1.2.3, the analogies for bounded operators of Calderón’s result, were likely first observed by Russu [241, Theorem 1]. Theorem 2.2 of [116], used in Corollary 1.2.3, is the noncommutative version of [23, Proposition 2.11, p. 115] due originally to Calderón [47, Theorem 2]. Section 1.2 provides operator versions of the classical relationship between submajorization and interpolation. Prior results of Ovčinnikov and Yeadon (see citations above) and Russu [241] are collected and extended in [116]. Following Arazy’s result on unitary matrix spaces [11], submajorization is used in the proof of a functor transferring interpolation results on function spaces to noncommutative operator spaces [116, Theorem 3.2]. The p-convexification versions are introduced as background due to the importance of 2-convexification for the compactness of product-convolution operators in Section 1.5. For classical results on interpolation between Lp and L∞ , p > 1, see [198].

1.7 Notes

� 123

Double operator integrals Double operator integrals were first introduced by Daleckiĭ and Kreĭn [95, 96] to study differentiation of operator-valued functions. Motivated by notions of differentiability for perturbation theory in mathematical physics, the double operator integerals of Daleckiĭ and Kreĭn were treated rigorously in the 1960s by Birman and Solomyak [31, 32, 34] in the setting of compact operators on a separable Hilbert space. Birman and Solomyak’s survey [36] has further details. Double operator integrals in the Banach space setting were investigated in 2002 by de Pagter, Witvliet, and one of the authors [104]. There double operator integrals were defined in terms of continuous Schur multipliers of the form Tϕ := ∫ ϕ d(P ⊗ Q), ℝ2

where P, Q are spectral measures in a UMD-space X such that P ⊗ Q is finitely additive. Spectral measures in UMD-spaces were constructed in [103], which followed from work with Clément [63] on discrete Schauder decompositions over UMD-spaces. Theorem 1.4.11 was first derived for bounded self-adjoint operators in [101]. The fact that the divided difference f [1] in Theorem 1.4.11 belongs to the Birman–Solymyak class BS in Definition 1.4.5 has been investigated for various conditions on f [101, Theorem 5.16]. The reference [101] proved that if f is a function such that f [1] ∈ BS, with some additional smoothness requirements, then f (A) hasthe Gâteaux derivative given by the double operator integral TfA,A [1] , where A ∈ ℳ is a self-adjoint operator in a semifinite von Neumann algebra ℳ. Hence the analogy of the Daletskiĭ–Kreĭn formula for ℒ(H) (see [34]) holds for semifinite von Neumann algebras. The submajorization estimate in Theorem 1.4.11 was essentially proved in [102, Corollary 7.5]. Theorem 1.4.11 can be derived for any bounded function f such that f [1] (t, s), suitably defined when t = s, belongs to BS. At present, the best known sufficient conditions for f [1] ∈ BS were found by Arazy, Barton, and Friedman [12], and a practical sufficient condition more general than that in Lemma 1.4.12 is due to Peller [8]. The reference [223] derived extensions of Lipschitz and commutator estimates for unbounded self-adjoint operators using double operator integrals. Corollary 1.4.13 as stated remains true for unbounded operators D0 and D1 , provided that at least D0 X − XD1 ∈ ℒ(H). For details, see [223]. Estimates for product-convolution operators The Hausdorff–Young inequality can be proved from the fact that ‖ℱ f ‖∞ ≤ ‖f ‖1 , and

f ∈ L1 (ℝd ),

124 � 1 Bounded operators and pseudodifferential operators ‖ℱ f ‖2 = ‖f ‖2 ,

f ∈ L2 (ℝd ),

and interpolation using the Riesz–Thorin theorem. The inequality is an influential result in the historical development of Fourier analysis, functional analysis, and interpolation theory [45], formalizing the Fourier transform as a bounded operator between Riesz’s function spaces. William Young and his wife Grace Chisolm Young (a student of Klein) introduced the first version of the inequality in 1912–13 for Fourier series [294, 295] as an extension of the Riesz–Fischer theorem on square-integrable functions and Fourier series. They proved for p = 1+1/q, q an odd integer, that the Fourier series of an Lp -function is p∗ -summable, where p∗ is the Holder conjugate of p, and that any p-summable sequence is the Fourier series of some Lp∗ -function. For p > 2, Carleman [55] found a continuous function whose Fourier series was not p∗ -summable. Similarly, for p > 2, Hardy and Littlewood showed that a p-summable sequence need not be the Fourier series of a function [147]. Hausdorff [150] proved the result for any 1 < p ≤ 2 and introduced the inequality of norms. The proofs of Young and Hausdorff employ the Hölder inequality and Young’s convolution theorem. They do not use interpolation, which was introduced later by M. Riesz. The extension for the Fourier transform on ℝd and not Fourier series was first noted by Titchmarsh [275]. Calderón [47] used the Marcinkiewicz interpolation theorem to prove, for 1 < p < 2, the estimate for classical Lorentz function spaces ‖ℱ f ‖p∗ ,q ≤ ‖f ‖p,q ,

f ∈ Lp (ℝd ),

where p∗ is the Holder conjugate of p, and 1 ≤ q ≤ ∞. To associate product-convolution operators to pseudodifferential operators, the convolution operator is defined in Section 1.5 as a Fourier multiplier. Analogues of product-convolution operators in noncommutative geometry use products from the weak closure of noncommutative algebras and the C ∗ - or von Neumann algebra generated by the spectral projections of geometric operators such as the Laplacian. In the case of the noncommutative plane in Chapter 4, underneath the operator algebra the Moyal product plays the role of the convolution. Defined directly by convolution, without using the Fourier transform, the boundedness of product-convolution operators is a direct consequence of Young’s convolution theorem. The results of Section 1.5.5 on compactness estimates for product-convolution operators follow Section 4 in Simon’s book [260]. The statements of Theorem 1.5.20(a) and Theorem 1.5.22(a) are not sharp. Optimal constants in Theorem 1.5.20(a) are delivered by interpolation [260, Theorem 4.1]. For l1 (L2 )(ℝd ) and l2 (L2 )(ℝd ) = L2 (ℝd ), the conditions on the functions f and g in Theorem 1.5.22(a) are necessary and sufficient. If Mf g(∇) ∈ ℒ1 , then f and g must both belong to l1 (L2 )(ℝd ) [260, Proposition 4.7]. If Mf g(∇) ∈ ℒ2 , then f and g must both belong to L2 (ℝd ), which follows from Theorem 1.5.4. Theorem 1.5.20(b) was conjectured by Simon [257] and proved by Cwikel [93] (see also [33]). Theorem 1.5.22(b) was proved in [29]. For further results on compactness of product-convolution operators, see [29, 33, 133, 165, 283].

1.7 Notes

� 125

Our proof of compactness estimates uses submajorization and majorization. Submajorization is equivalent to exact interpolation for linear maps, as indicated in Section 1.2, but not necessarily equivalent for bilinear maps like the mapping from a pair of functions to a product-convolution operator. Assumption 1.5.9, as a sufficient condition to prove the submajorization estimate in Theorem 1.5.10, was introduced in [186]. The proof provided in Section 1.5.4 is shorter than that in [186], with an improved constant. For 0 < p < 2, the formulation of Assumption 1.5.16 and the majorization condition in the abstract setting (Theorem 1.5.17) follows the proof given for the Euclidean plane in [186]. Section 1.5 is based on [186]. The majorization technique allows proofs for quasi-Banach spaces within l1 , particularly the Lorentz spaces lp,q where 0 < p < 1, 0 < q ≤ ∞. Standard interpolation between Banach spaces is not available in this case, and the majorization method avoids the complexities of interpolation theory for quasi-Banach spaces [94, 281]. Weidl [283, 284] first suggested the tensor product form of the compactness estimates, making the symmetry between f and g overt. Weidl proved that t

1 μℒ(L2 (ℝd )),Tr (t, Mf g(∇)) ≤ cp ⋅ ∫ μ(s, f ⊗ g)2 ds, t 2

f ⊗ g ∈ (L2 + L∞ )(ℝd × ℝd )

0

for a constant cp > 0. Here μ(f ⊗g) denotes the decreasing rearrangement of the function f ⊗ g. Weidl’s inequality is not as strong as the statement Mf g(∇) ≺≺ cp ⋅ f ⊗ g, where the singular value function on operators and the decreasing rearrangement of functions are implicitly used on the respective sides of this relation. The trace formula for trace class operators The trace formula in Theorem 1.5.5 was established for pseudodifferential trace class operators by Shubin [256, Proposition 27.2]. Mehler’s formula in the proof of Theorem 1.5.5 was established in 1866 by Mehler [208] for real z ∈ (−1, 1). However, both the left- and right-hand sides of the formula are holomorphic functions of z in the unit disk. Since those holomorphic functions are equal on the interval (−1, 1), it follows from the uniqueness theorem that they are equal in the unit disk. Mercer [209] proved in 1909 that Tr(A) = ∫ kA (s, s)ds [a,b]

for a Hilbert–Schmidt operator A on L2 ([a, b]) with continuous positive-definite kernel kA ∈ C([a, b]2 ). A Hilbert–Schmidt operator A on L2 (Ω, ν) can be written as an integral operator for a kernel kA ∈ L2 (Ω × Ω, ν × ν) for any σ-finite locally compact space Ω with

126 � 1 Bounded operators and pseudodifferential operators σ-finite measure ν. Symbols are related to kernels through the Fourier transform. Since the Fourier transform is not available generally for σ-finite locally compact spaces, most extensions of Mercer’s theorem have concentrated on kernel formulas. Duflo [121, Theorem V.3.1.1] extended Mercer’s theorem to trace class operators on L2 (Ω, ν) with positive continuous kernels on Ω × Ω. Brislawn [42, 43] extended Duflo’s results by using averaging. Let A be a trace class operator on L2 (Ω, ν) with kernel kA ∈ L2 (Ω×Ω, ν ×ν). If {Xn }n∈ℕ is a sequence of partitions of Ω such that Xn+1 is a refinement of Xn and ω ∈ Ω, then denote by Cn (ω) the set in Xn such that ω ∈ Cn (ω). Let N denote the set of ω ∈ Ω such that ν(Cn (ω)) = 0 for some n ∈ ℕ. Define (Sn k)(x, y) =

1 ν(Cn (x))ν(Cn (y))

∫

k(v, u)dν(v)dν(u),

x, y ∈ N c ,

Cn (x)×Cn (y)

and (Sn k)(x, y) = 0 if x ∈ N or y ∈ N. Define k̃A (x, y) = lim (Sn kA )(x, y). n→∞

Brislawn [43, Theorem 3.1] proved that the function k̃A (x, x), x ∈ Ω, is almost everywhere defined and integrable and that Tr(A) = ∫ k̃A (x, x)dν(x). ℝd

If kA is continuous, then k̃A = kA almost everywhere. Calculus and regularity of pseudodifferential operators Classic references on pseudodifferential operators are [161] and [256]. The pseudodifferential operators we introduce correspond to the uniform class of Hörmander type (1,0) [242, Chapter 2], [256, Problem 3.1]. Chapter 2 of [242] contains an uncomplicated introduction to uniform symbols. Propositions 1.6.4 and 1.6.6 are the foundations of the pseudodifferential calculus. For the proof of both propositions for uniform symbols, see [242, Section 2.5]. A more general consideration than a uniform symbol is the notion of a proper pseudodifferential operator [256, §3]. The zero-order proper pseudodifferential operators do not define bounded operators from L2 (ℝd ) → L2 (ℝd ) generally; see [256, §6, §7]. The boundedness properties of uniform symbols make them a simpler object for the purposes of functional analysis. Boundedness properties of zero-order operators have a long history; see Calderón and Vaillancourt [48], Cordes [90], and Kato [173]. For the regularity theorem on the mapping between Sobolev spaces, Theorem 1.6.10, see [242, Theorem 2.6.11]. For an exposition of Sobolev regularity for proper pseudodifferential operators, see [256, §7].

1.7 Notes �

127

Complex powers of elliptic operators There are different definitions of ellipticity. Definition 1.6.23 states that a symbol σ is elliptic if a representative of the principal symbol class of σ is uniformly hypoelliptic of order (m, m) [256, Definition 5.1 and Proposition 5.1′ ]. The equivalent condition of ellipticity in Shubin requires, since uniform pseudodifferential operators are properly supported, that the full symbol be uniformly hypoelliptic of order (m, m) [256, Definition 5.3′′ ]. The uniform hypoellipticity of a representative of the principal symbol class is sufficient to prove elliptic regularity and hence sufficient for the purposes of Chapter 3 in this book. When σ is a classical symbol, the uniform hypoellipticity of order (m, m) of the symbol and uniform hypoellipticity of order (m, m) of the principal symbol are equivalent [256, Proposition 5.1]. In this case, for classical pseudodifferential operators, Definition 1.6.23 of an elliptic symbol and the equivalent definition of an elliptic symbol by Shubin coincide. The fact that a complex power of an elliptic pseudodifferential operator can be defined and is again a pseudodifferential operator is due originally to Seeley [251]. Seeley considered compact manifolds. Variations were later proved by Kumano-Go and Tsutsumi [181], Guillemin [146], and other authors. Expositions of the results for compact manifolds may be found in the books of Shubin [256] and Grubb [144]. The approach we take here essentially mirrors that of Beals [21], which is well suited to the calculus of operators on ℝd . Lemma 1.6.18 does not appear in standard references, as far as we could tell. The lemma confirms that when the elliptic pseudodifferential operator restricts to a positive operator on L2 (ℝd ), the restriction of its complex powers as a pseudodifferential operator coincides with the complex powers as defined by the functional calculus of self-adjoint operators. Unlike the earlier proof of Seeley, Beals’ proof was based on Theorem 1.6.12, which is now sometimes referred to as the Beals characterization of pseudodifferential operators or the Beals–Cordes characterization [21, 92, 274], [91, Theorem 10.1]. All of these constructions are compatible in the sense that the complex power of a pseudodifferential operator coincides with its complex power defined using spectral theory; see [256, Proposition 10.3] for the compact case. Compactly supported pseudodifferential operators A treatment of compactly supported pseudodifferential operators can be found in Shubin [256, §6]. In the study of pseudodifferential operators on ℝd that have Hilbert– Schmidt or trace class extensions, most texts favor the class Gm , m ∈ ℝ, of Shubin differential operators [256, IV]. The association of the class Gm to quantization, Weyl symbols, and anti-Wick symbols, makes the class Gm natural to study. Compactly supported on the left operators were previously called compactly based operators in [168] and [193]. Compactness estimates for pseudodifferential operators on ℝd that are not compactly supported on the left or right can be built up from Theorem 1.5.7 on product-

128 � 1 Bounded operators and pseudodifferential operators convolution operators [90]. Theorem 6.7 in [14] indicates that Op(σ) ∈ ℒp if σ ∈ Ws,p (ℝd × ℝd ) for s > n, 1 ≤ p < ∞. Here Ws,p denotes the Sobolev space s

Ws,p (ℝd × ℝd ) = {u ∈ Lp (ℝd × ℝd ) : (1 − Δℝd ×ℝd ) 2 u ∈ Lp (ℝd × ℝd )}, and Δℝd ×ℝd is the Laplacian operator for ℝd × ℝd . Other approaches involve the Weyl symbol [236] or the anti-Wick symbol [38, 46]. Pseudodifferential operators on manifolds The end notes to Chapters 2 and 3 discuss trace theorems for pseudodifferential operators on closed manifolds. An introduction to smooth Riemannian manifolds can be found in [185]. Shubin’s monograph considers pseudodifferential operators on arbitrary smooth manifolds [256, §4.3]. The noncommutative residue Asymptotic expansions and the notion of a classical operator and symbol are introduced in every text on pseudodifferential operators [256, §3.7], [242, §2.5.3]. Classical symbols σ(x, s) are a direct generalization of power series in s, making classical pseudodifferential operators the direct generalization of differential operators. The noncommutative residue originated in Wodzicki’s 1983 study of spectral asymmetry on closed manifolds [286, Section 7], although the form given of the integral over the trivial sphere bundle was coincidental to the original study of the residue of zeta functions associated with complex powers of elliptic pseudodifferential operators. More extensive results on manifolds, and that the residue arises from a density, are contained in [287]. The properties of the noncommutative residue in Proposition 1.6.42 were noticed by Wodzicki [286, Final Remarks 7.1.3] and Guillemin [146, Proposition 6.1]. The noncommutative residue defined by Wodzicki generalized the studies by Adler [3] and Manin [202]. Guillemin, separately, considered zeta functions associated with complex powers of elliptic operators and derived the noncommutative residue as the residue of the zeta function at z = 0 [146, Theorem 7.4]. The monograph [139] provides a different treatment of the noncommutative residue in terms of distributions.

2 Trace formulas 2.1 Introduction Let A ∈ ℒ2 (L2 (ℝd )) be a Hilbert–Schmidt operator on the Hilbert space L2 (ℝd ). As an integral operator, it has a symbol pA ∈ L2 (ℝd × ℝd ) such that d

(Au)(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ u)(ξ)dξ,

u ∈ L2 (ℝd ).

ℝd

Here ℱ denotes the Fourier transform defined in (1.16). From Theorem 1.5.5, essentially an extension of Mercer’s theorem, if A ∈ ℒ1 (L2 (ℝd )) and pA ∈ L1 (ℝd × ℝd ), then Tr(A) = (2π)−d

∫ pA (t, ξ)dtdξ, ℝd ×ℝd

and the trace of the trace class operator A can be calculated from its symbol. In this chapter, we introduce Laplacian modulated operators and prove that the trace of a weak trace class Laplacian modulated operator on ℝd can be calculated from its symbol. Laplacian modulated operators extend the notion of pseudodifferential operators of order −d and weak trace class product-convolution operators. Alain Connes’ result that the trace of a classical pseudodifferential operator A of order −d is a multiple of the noncommutative residue of A is derived as a corollary. Traces of Laplacian modulated operators Definition. A Laplacian modulated operator A : L2 (ℝd ) → L2 (ℝd ) is a Hilbert–Schmidt operator whose symbol pA ∈ L2 (ℝd × ℝd ) satisfies d 󵄨 󵄨2 ‖pA ‖mod := sup(1 + s) 2 ( ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ)

s>0

|ξ|>s

1/2

< ∞.

(2.1)

ℝd

In Section 2.3, we show that the set of Laplacian modulated operators with norm A 󳨃→ ‖pA ‖mod is a Banach space. The main result of this chapter is the following trace theorem, equating the calculation of a trace of a Laplacian modulated operator A to the asymptotic behavior of the sequence n 󳨃→

∫ 1 |ξ|s ℝd

by splitting the second integral into the regions |v| < |t|/2 and |v| ≥ |t|/2. Lemma 2.2.3. Lmod;1 (ℝd ) is a Banach space. Proof. Observe from Definition 2.2.1 that Lmod;1 (ℝd ) is a normed space. If fn , n ≥ 0, is a Cauchy sequence in Lmod;1 (ℝd ), then it is a Cauchy sequence in L1 (ℝd ). Thus it converges to f in L1 (ℝd ). For a given ε > 0, take N so large that 󵄨 󵄨 (1 + s)d ∫ 󵄨󵄨󵄨fn (t) − fm (t)󵄨󵄨󵄨dt ≤ ε |t|≥s

for all m, n > N and s > 0. Thus, for all s > 0 and n > N, we have

134 � 2 Trace formulas 󵄨 󵄨 󵄨 󵄨 (1 + s)d ∫ 󵄨󵄨󵄨fn (t) − f (t)󵄨󵄨󵄨dt = lim (1 + s)d ∫ 󵄨󵄨󵄨fn (t) − fm (t)󵄨󵄨󵄨dt ≤ ε. m→∞ |t|≥s

|t|≥s

Since ε > 0 is arbitrarily small, the statement follows. Theorem 2.2.4. Lmod;1 (ℝd ) is a Banach algebra. Proof. Lmod;1 (ℝd ) is a Banach space and an algebra. In fact, it is proved in Lemma 2.2.2 that ‖f ⋆ g‖mod;1 ≤ const ⋅(‖f ‖1 ⋅ ‖g‖mod;1 + ‖g‖1 ⋅ ‖f ‖mod;1 ) ≤ const ⋅‖g‖mod;1 ⋅ ‖f ‖mod;1 . Hence Lmod;1 (ℝd ) is a Banach algebra. Hence the modulated functions form a Banach subalgebra of the integrable functions L1 (ℝd ). The following result shows that, however, the algebras are generated quite differently. Lemma 2.2.5. The set of compactly supported integrable functions is not dense in Lmod;1 (ℝd ). Proof. Let f (t) = (1 + |t|)−2d , t ∈ ℝd , and let g ∈ L1 (ℝd ) be a compactly supported function. Fix s sufficiently large so that g(t) = 0 for |t| > s. Thus (1 + s)d ∫ |f − g|(t)dt = (1 + s)d ∫ (1 + |t|) |t|>s

|t|>s ∞ d

≥ const ⋅s ∫ (1 + |t|) s

−2d

−2d

dt

|t|d−1 d|t| ≥ const .

Thus the distance from f to the set of all compactly supported functions is positive.

2.2.2 Square modulated functions and behavior of the Fourier transform Definition 2.2.6. Define the Banach space Lmod;2 (ℝd ) consisting of the functions f ∈ L2 (ℝd ) such that |f |2 ∈ Lmod;1 (ℝd ) with norm ‖f ‖mod;2 := ‖|f |2 ‖1/2 . mod;1 Since each function f ∈ Lmod;2 (ℝd ) is square integrable, it is interesting to know whether the Fourier transform ℱ f belongs to Lmod;2 (ℝd ) in general. The answer is negative. Example 2.2.7. For every α ∈ (− 41 , 0), we have fα (t) := |t|α Kα (|t|),

−α− d2

(ℱ fα )(t) = const ⋅(1 + |t|2 )

,

t ∈ ℝd ,

2.2 The Banach algebra of modulated functions

�

135

where Kα is the Macdonald function (a modified Bessel function of the second kind, [1]) for α. In particular, fα ∈ Lmod;2 (ℝd ), but ℱ fα ∉ Lmod;2 (ℝd ). Proof. We confirm that the functions are related by the Fourier transform. Applying a unitary transform, we obtain, for t ∈ ℝd , ∫ ℝd

e−i⟨s,t⟩

d

(1 + |s|2 )α+ 2

ds = ∫ ℝd

e−is1 |t|

d

(1 + |s|2 )α+ 2

ds.

1

After substitution sk = vk (1 + s12 ) 2 , 2 ≤ k ≤ d, we obtain ∫ ℝd

e−i⟨s,t⟩ (1 + |s|2 )

α+ d2

ds = ∫ ℝd

e−is1 |t| (1 + s12 )

∞

= ∫ −∞

e−is1 |t| (1 +

−α− d2

⋅ (1 + |v|2 )

α+ 21

1

s12 )α+ 2

ds1 dv

ds1 ⋅ ∫ (1 + |v|2 )

−α− d2

dv = const ⋅|t|α Kα (|t|).

ℝd−1

Since ℱ fα ∈ L2 (ℝd ), it follows that fα ∈ L2 (ℝd ). Now observe that 󵄨 󵄨2 ∫ 󵄨󵄨󵄨(ℱ fα )(t)󵄨󵄨󵄨 dt > const ⋅ ∫ |t|−4α−2d dt = const ⋅(1 + s)−4α−d . |t|>s

|t|>s

Hence (ℱ fα )2 ∈ ̸ Lmod;1 (ℝd ). However, as the Macdonald function Kα decreases exponentially at +∞, [1], it follows that fα2 ∈ Lmod;1 (ℝd ). The situation is not improved even if we take a modulated function of compact support. Example 2.2.8. We have f (t) := χ|t|≤1 ,

t ∈ ℝd ,

(ℱ f )(t) = |t|−d/2 Jd/2 (|t|),

where Jd/2 is a Bessel function of the first kind [1]. In particular, f ∈ Lmod;2 (ℝd ) is a compactly supported function, and ℱ f ∉ Lmod;2 (ℝd ). Proof. We confirm that the functions are related by the Fourier transform. For s ∈ ℝd with |s| ≤ 1, we set v = (s2 , ⋅ ⋅ ⋅ , sd ) ∈ ℝd−1 . Observe that |v| ≤ (1 − s12 )1/2 and, for t ∈ ℝd , ∫ f (s)e−i⟨s,t⟩ ds = ∫ e−i⟨s,t⟩ ds = ∫ e−is1 |t| ds ℝd

|s|≤1

|s|≤1

1

= ∫e −1

1

−is1 |t|

(

∫

|v|≤(1−s12 )1/2

dv)ds1 = const ⋅ ∫ (1 − s12 )

e

(d−1)/2 −is1 |t|

−1

ds1 .

136 � 2 Trace formulas The latter integral is |t|−d/2 Jd/2 (|t|). Evidently, the compactly supported function f 2 is modulated. Since Jd/2 (2π|t|) ∼ |t|−1/2 ⋅ cos(|t| −

(d + 1)π ), 4

|t| → ∞,

the function ℱ f is strictly of order |t|−(1+d)/2 . It follows that (ℱ f )2 is not modulated.

2.3 Laplacian modulated operators In this section, we consider operators that are modulated with respect to the bounded positive operator d

(1 − Δ)− 2 : L2 (ℝd ) → L2 (ℝd ), where Δ is the Laplacian on ℝd . The notion of a modulated operator was introduced in Chapter 7 of Volume I. We show that such operators define the set of Laplacian modulated operators. We also show that Laplacian modulated operators extend the notion of pseudodifferential operators on ℝd of order −d acting on L2 (ℝd ). 2.3.1 Symbols and characterization of Laplacian modulated operators Recall from Theorem 1.5.4, a consequence of the Schmidt decomposition and the Plancherel theorem, that every Hilbert–Schmidt operator A on L2 (ℝd ) is an integral operator d

(Au)(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ u)(ξ)dξ,

t ∈ ℝd ,

u ∈ L2 (ℝd ),

(2.5)

ℝd d

with L2 -symbol pA ∈ L2 (ℝd × ℝd ) and ‖A‖2 = (2π)− 2 ‖pA ‖2 . The relationship between the kernel and symbol of a Hilbert–Schmidt operator was given in (1.24) and (1.23): pA (t, ξ) = ∫ ei⟨s,ξ⟩ kA (t, t + s)ds,

t, ξ ∈ ℝd ,

ℝd

and kA (t, s) = (2π)−d ∫ ei⟨t−s,ξ⟩ pA (t, ξ)dξ,

t, s ∈ ℝd .

ℝd

The next lemma states the essential properties of the symbol in terms of left multiplication by a product operator and right multiplication by a convolution operator.

2.3 Laplacian modulated operators

� 137

Lemma 2.3.1. Let A : L2 (ℝd ) → L2 (ℝd ) be a Hilbert–Schmidt operator. (a) For every f ∈ L∞ (ℝd ), we have pMf A = (f ⊗ 1)pA . (b) For every g ∈ L∞ (ℝd ), we have pAg(∇) = (1 ⊗ g)pA .

Let Δ be the Laplacian on ℝd . The Laplacian Δ is a classical second-order elliptic pseudodifferential operator such that ⟨Δu, v⟩ = ⟨u, Δv⟩,

u, v ∈ 𝒮 (ℝd ).

It follows from Theorem 1.6.25 that the restriction Δ : H 2 (ℝd ) → L2 (ℝd ) is self-adjoint and positive, with domain the Sobolev space H 2 (ℝd ). Subsequently, by d Theorem 1.6.19 there exists an elliptic pseudodifferential operator (1 − Δ)− 2 ∈ Ψ−d (ℝd ) of order −d such that the restriction d

(1 − Δ)− 2 : L2 (ℝd ) → L2 (ℝd ) is a bounded linear operator that coincides with the operator defined by the Borel functional calculus applied to the self-adjoint operator Δ|H 2 (ℝd ) . Normally, we will denote the restriction Δ|H 2 (ℝd ) by Δ when the context makes clear whether the Laplacian is treated as a pseudodifferential operator on tempered distributions 𝒮 (ℝd )′ → 𝒮 (ℝd )′ or an unbounded self-adjoint operator H 2 (ℝd ) → L2 (ℝd ). Following Chapter 7 of Volume I: Definition 2.3.2. A bounded operator A : L2 (ℝd ) → L2 (ℝd ) is said to be Laplacian modd

ulated if it is (1 − Δ)− 2 -modulated, that is,

1 d −1 󵄩 󵄩 ‖A‖mod := sup t 2 󵄩󵄩󵄩A(1 + t(1 − Δ)− 2 ) 󵄩󵄩󵄩ℒ < ∞,

t>0

2

(2.6)

where ‖ ⋅ ‖ℒ2 denotes the Hilbert–Schmidt norm on ℒ2 (L2 (ℝd )). Every Laplacian modulated operator is necessarily Hilbert–Schmidt (see Chapter 7 of Volume I or note that it follows from taking t = 1 in (2.6)) and has an L2 -symbol by Theorem 1.5.4. Laplacian modulated operators are equivalently defined by the following property of their symbols. Lemma 2.3.3. A Hilbert–Schmidt operator A : L2 (ℝd ) → L2 (ℝd ) with symbol pA is Laplacian modulated if and only if 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ⋅)󵄨󵄨󵄨 dt ∈ Lmod;1 (ℝd ),

ℝd

138 � 2 Trace formulas where Lmod;1 (ℝd ) is the Banach algebra of modulated functions. Moreover, if we define

‖pA ‖mod

1

2 󵄨 󵄨2 := sup(1 + s) ( ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ) , d 2

s>0

|ξ|>s ℝd

then the norms A 󳨃→ ‖A‖mod and A 󳨃→ ‖pA ‖mod are equivalent. d

Proof. Let V := (1 − Δ)− 2 . By Lemma 7.3.2 in Volume I the operator A is a Laplacian modulated operator if and only if the norm 1 󵄩 󵄩 |||A||| := sup s 2 󵄩󵄩󵄩AEV [0, s−1 ]󵄩󵄩󵄩2

s>0

is finite, where EV is the spectral measure of V . Since χ[0,s−1 ] (t) ≤ 2 ⋅ (1 + st)−1 ,

s, t > 0,

it follows from the functional calculus that 󵄩󵄩 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩AEV [0, s ]󵄩󵄩󵄩2 ≤ 2󵄩󵄩󵄩A(1 + sV ) 󵄩󵄩󵄩2 . Thus |||A||| ≤ 2 ⋅ ‖A‖mod .

(2.7)

From the proof of Lemma 7.3.2 in Volume I, for fixed k ∈ ℤ+ and s ∈ [2k , 2k+1 ], k−1

󵄩󵄩 󵄩 󵄩 −1 󵄩 −k 󵄩 k−j−1 −1 󵄩 ) 󵄩󵄩󵄩AEV (2−j−1 , 2−j ]󵄩󵄩󵄩2 . 󵄩󵄩A(1 + sV ) 󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩AEV [0, 2 ]󵄩󵄩󵄩2 + ∑ (1 + 2 j=0

Hence 2

k+1 2

k+1 󵄩󵄩 󵄩 −1 󵄩 −k 󵄩 󵄩󵄩A(1 + sV ) 󵄩󵄩󵄩2 ≤ 2 2 󵄩󵄩󵄩AEV [0, 2 ]󵄩󵄩󵄩2

k−1

+ ∑ (1 + 2k−j−1 ) 2 −1

(k−j+1) 2

j=0

j 󵄩 󵄩 2 2 ⋅ 󵄩󵄩󵄩AEV (2−j−1 , 2−j ]󵄩󵄩󵄩2

and k−j

k−1 1 1 2 2 󵄩 󵄩 󵄩 󵄩 sup s 󵄩󵄩󵄩A(1 + sV )−1 󵄩󵄩󵄩2 ≤ 2 2 (1 + ∑ ) ⋅ sup s 2 󵄩󵄩󵄩AEV [0, s−1 ]󵄩󵄩󵄩2 . k−j−1 s>0 s∈[2k ,2k+1 ] j=0 1 + 2 1 2

Substituting n = k − j − 1, we have

2.3 Laplacian modulated operators �

k−1

∑

j=0

k−j

139

n

k−1 k−1 1 1 1 1 2 2 22 −n 2 ∑ 2 ∑ 2 2 ≤ 2 2 (1 + 2 2 ). = 2 ≤ 2 n k−j−1 1+2 1+2 n=0 n=0

Thus ‖A‖mod ≤ 5 ⋅ |||A|||.

(2.8)

It follows from (2.7) and (2.8) that A 󳨃→ |||A||| and A 󳨃→ ‖A‖mod are equivalent norms. We now identify the norm |||A||| using the symbol of the Hilbert–Schmidt operator A. Let gs (ξ) := χ

d

{(1+|ξ|2 ) 2 >s}

ξ ∈ ℝd .

,

It follows from Theorem 1.5.4 and Lemma 2.3.1 that 󵄩󵄩 󵄩󵄩AE

d

(1−Δ)− 2

󵄩 󵄩 󵄩 [0, s−1 ]󵄩󵄩󵄩2 = 󵄩󵄩󵄩Ags (∇)󵄩󵄩󵄩2 d d 󵄩 󵄩 = (2π)− 2 ‖pAgs (∇) ‖2 = (2π)− 2 󵄩󵄩󵄩pA (1 ⊗ gs )󵄩󵄩󵄩2 .

Hence |||A|||2 = (2π)−d sup s s>0

∫ d

󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ.

(1+|ξ|2 ) 2 >s ℝ

d

Since d

{(1 + |ξ|2 ) 2 > s} ⊂ {|ξ|d > s},

s > 0,

we have 󵄨 󵄨2 |||A|||2 ≤ (2π)−d sup(1 + s)d ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ s>0

|ξ|>s ℝd

by replacing s with sd and using s < 1 + s. For the reverse inequality, note that 󵄨 󵄨2 sup (1 + s)d ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ ≤ 3d ⋅ ‖pA ‖22 = 3d (2π)d ⋅ ‖A‖22 .

0 0 and (1 + s) ≤ 2 ⋅ s. Then we have 󵄨 󵄨2 sup(1 + s)d ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ ≤ 2d C1−1 ⋅ sup sd s≥2

s≥2C1

|ξ|>s ℝd

∫

󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ.

d d (1+|ξ|2 ) 2 >sd ℝ

Thus there is a constant C > 0 such that 󵄨 󵄨2 sup(1 + s)d ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ ≤ C ⋅ sup s s≥2

s>0

|ξ|>s ℝd

∫ d

󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ.

(1+|ξ|2 ) 2 >s ℝ

(2.10)

d

From (2.9) and (2.10) we obtain 󵄨 󵄨2 sup(1 + s)d ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ ≤ (2π)d ⋅ (3d + C) ⋅ |||A|||2 . s>0

|ξ|>s ℝd

The first statement of the lemma now follows since A is Laplacian modulated if and only if 󵄨 󵄨2 ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ = O((1 + s)−d ),

s ≥ 0.

|ξ|>s ℝd

The second statement of the lemma follows since we have shown that the norms ‖A‖mod and |||A||| are equivalent and that the norms |||A||| and ‖pA ‖mod are equivalent. Remark 2.3.4. The set of Laplacian modulated operators is a Banach space with norm A 󳨃→ ‖A‖mod by results in Chapter 7 of Volume I. The symbols of Laplacian modulated operators form a Banach space S mod with norm ‖pA ‖mod

1

2 󵄨 󵄨2 := sup(1 + s) ( ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ) . d 2

s>0

|ξ|>s ℝd

With the equivalent norm in Lemma 2.3.3 on the Banach space of Laplacian modulated operators, the symbol map provides an isometric isomorphism between the Banach space of Laplacian modulated operators and the Banach space of their symbols S mod .

2.3 Laplacian modulated operators

� 141

2.3.2 Examples of Laplacian modulated operators d

By Lemma 1.6.18 and Theorem 1.6.19 the operator (1 − Δ)− 2 : L2 (ℝd ) → L2 (ℝd ) defined d

by the functional calculus is the restriction of the pseudodifferential operator (1−Δ)− 2 ∈ Ψ−d (ℝd ). Since the operations d

(1 − Δ)− 2 : L2 (ℝd ) → H d (ℝd ) and d

(1 − Δ)− 2 : H −d (ℝd ) → L2 (ℝd ) are bounded, and for f ∈ L2 (ℝd ), Mf : L∞ (ℝd ) ⊃ H d (ℝd ) → L2 (ℝd ) and Mf : L2 (ℝd ) → L1 (ℝd ) ⊂ H −d (ℝd ) are bounded, the operators considered in Section 1.5, d

Mf (1 − Δ)− 2 ,

d

f ∈ L2 (ℝd ),

(1 − Δ)− 2 Mf ,

are everywhere defined and bounded on L2 (ℝd ). d

Lemma 2.3.5. Let f ∈ L2 (ℝd ) and h(ξ) := (1 + |ξ|2 )− 2 , ξ ∈ ℝd . Then (a) A := Mf (1 − Δ)

− d2

∈ ℒ2 with symbol

pA (t, ξ) = f (t)h(ξ),

t, ξ ∈ ℝd ,

and 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dt = ‖f ‖22 ⋅ 󵄨󵄨󵄨h(ξ)󵄨󵄨󵄨 ,

ξ ∈ ℝd .

ℝd d

(b) B := (1 − Δ)− 2 Mf ∈ ℒ2 with symbol pB (t, ξ) = (2π)−d ∫ ei⟨t,ξ−s⟩ h(s)(ℱ f )(ξ − s)ds, ℝd

and

t, ξ ∈ ℝd ,

142 � 2 Trace formulas 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pB (t, ξ)󵄨󵄨󵄨 dt = (|h|2 ⋆ |ℱ f |2 )(ξ),

ξ ∈ ℝd ,

ℝd

where ℱ denotes the Fourier transform. If, in addition, f ∈ L1 (ℝd ), then ∫ pB (t, ξ)dt = ∫ pA (t, ξ)dt = h(ξ) ⋅ ∫ f (t)dt, ℝd

ℝd

ξ ∈ ℝd .

ℝd

Proof. The operators A and B = A∗ are Hilbert–Schmidt by Theorem 1.5.6. We also have pA (t, ξ) = f (t)h(ξ) by Theorem 1.5.6, and statement (a) follows. For (b), let eξ (t) := e−i⟨t,ξ⟩ , t, ξ ∈ ℝd . Then ((1 − Δ)−d/2 Mf eξ )(t) = (2π)−d ∫ e−i⟨t,s⟩ h(s)(ℱ (feξ ))(s)ds ℝd

= (2π)

−d

∫ e−i⟨t,s⟩ h(s)(ℱ f )(ξ − s)ds. ℝd

The statement for the symbol pB (t, ξ) = e−ξ (t)(Beξ )(t), t, ξ ∈ ℝd , now follows. Given the symbol, we can calculate that 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pB (t, ξ)󵄨󵄨󵄨 dt = (|h|2 ⋆ |ℱ f |2 )(ξ)

ℝd

and ∫ pB (t, ξ)dt = h(ξ)(ℱ f )(0) = h(ξ) ⋅ ∫ f (t)dt ℝd

ℝd

if f ∈ L1 (ℝd ). The Banach space Lmod;2 (ℝd ) of square-modulated functions was introduced in Definition 2.2.6. Lemma 2.3.6. d (a) If f ∈ L2 (ℝd ), then Mf (1 − Δ)− 2 is Laplacian modulated.

d

(b) If ℱ f ∈ Lmod;2 (ℝd ), where ℱ denotes the Fourier transform, then (1−Δ)− 2 Mf is Laplacian modulated. d (c) There exists f ∈ Lmod;2 (ℝd ) such that (1 − Δ)− 2 Mf is not Laplacian modulated. d

Proof. Let h(ξ) := (1 + |ξ|2 )− 2 . Then h ∈ Lmod;2 (ℝd ).

2.3 Laplacian modulated operators �

143

d

(a) For A := Mf (1 − Δ)− 2 , from Lemma 2.3.5(a) we have 󵄨 󵄨2 󵄨 󵄨2 ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ = ‖f ‖22 ⋅ ∫ 󵄨󵄨󵄨h(ξ)󵄨󵄨󵄨 dξ⋅ = O((1 + s)−d ).

|ξ|>s ℝd

|ξ|>s

From Lemma 2.3.3 it follows that A is Laplacian modulated. d (b) For B := (1 − Δ)− 2 Mf , from Lemma 2.3.5(b) we have 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pB (t, ⋅)󵄨󵄨󵄨 dt = |h|2 ⋆ |ℱ f |2 .

ℝd

We have h, ℱ f ∈ Lmod;2 (ℝd ). Thus the convolution belongs to Lmod;1 (ℝd ) by Lemma 2.2.2. From Lemma 2.3.3 it follows that B is Laplacian modulated. (c) Let fα be the function constructed in Example 2.2.7. We have ℱ fα = h1+2α/d and fα ∈ d

Lmod;2 (ℝd ). If B = (1 − Δ)− 2 Mf , then from Lemma 2.3.5(b) it follows that 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pB (t, ⋅)󵄨󵄨󵄨 dt = |h|2 ⋆ 󵄨󵄨󵄨ℱ −1 fα 󵄨󵄨󵄨 = h2 ⋆ h2+4α/d .

ℝd

Note that for a constant C > 0, h2 ⋆ h2+4α/d ≥ C ⋅ h2+4α/d ∉ Lmod;1 (ℝd ). By Lemma 2.3.3, B is not Laplacian modulated. Remark 2.3.7. If f ∈ 𝒮 (ℝd ), then f ∈ Lmod;2 (ℝd ). Hence ℱ f ∈ Lmod;2 (ℝd ), and the Hilbert–Schmidt operator d

(1 − Δ)− 2 Mf is Laplacian modulated for every f ∈ 𝒮 (ℝd ). 2.3.3 Pseudodifferential operators are Laplacian modulated Laplacian modulated operators extend pseudodifferential operators on ℝd of order −d acting on L2 (ℝd ). Definition 2.3.8. An operator A : L2 (ℝd ) → L2 (ℝd ) is said to be (a) compactly supported on the left if there exists ϕ ∈ Cc∞ (ℝd ) such that Mϕ A = A. (b) compactly supported on the right if there exists ψ ∈ Cc∞ (ℝd ) such that A = AMψ . (c) compactly supported if it is compactly supported on the left and from the right.

144 � 2 Trace formulas The proof of the following lemma, like its equivalent in Section 1.6.5, is omitted. Lemma 2.3.9. A Laplacian modulated operator A : L2 (ℝd ) → L2 (ℝd ) is (a) compactly supported on the left if and only if its symbol pA is compactly supported in the first variable. (b) compactly supported if and only if its integral kernel kA is compactly supported. The next lemma shows that the Banach space of Laplacian modulated operators is a bimodule for Ψ0 (ℝd ). Lemma 2.3.10. Let A be a Laplacian modulated operator, and let T, S ∈ Ψ0 (ℝd ). Then the operator TAS is Laplacian modulated. Proof. It follows from Definition 2.3.2 that the set of Laplacian modulated operators is a d left ideal of ℒ(L2 (ℝd )). Denote for brevity V = (1 − Δ)− 2 . It follows from Corollary 1.6.5 that [S, V ] ∈ Ψ−d−1 (ℝd ). Thus [S, V ]V −1 ∈ Ψ−d−1 (ℝd ) ⋅ Ψd (ℝd ) ⊂ Ψ−1 (ℝd ) ⊂ Ψ0 (ℝd ) by Proposition 1.6.4. In particular, by Theorem 1.6.10 the operators S and [S, V ]V −1 are bounded when restricted to L2 (ℝd ). We have AS(1 + tV )−1 = A(1 + tV )−1 ⋅ S + A ⋅ [S, (1 + tV )−1 ] = A(1 + tV )−1 ⋅ S − A(1 + tV )−1 ⋅ [S, V ]V −1 ⋅ tV (1 + tV )−1 . Thus 󵄩󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩 −1 󵄩 󵄩󵄩AS(1 + tV ) 󵄩󵄩󵄩2 ≤ 󵄩󵄩󵄩A(1 + tV ) 󵄩󵄩󵄩2 ⋅ (‖S‖∞ + 󵄩󵄩󵄩[S, V ]V 󵄩󵄩󵄩∞ ). The statement that AS is Laplacian modulated now follows from Definition 2.3.2. Theorem 2.3.11. If T ∈ Ψ−d (ℝd ) is compactly supported on the left or on the right, then the restrictions of T and T † to L2 (ℝd ) are Laplacian modulated operators. Proof. Assume that T is compactly supported on the left. We have T = Mϕ T for some ϕ ∈ Cc (ℝd ). Set d

A := Mϕ (1 − Δ)− 2 ,

d

S := (1 − Δ) 2 T,

so that T = AS.

Since T ∈ Ψ−d (ℝd ), it follows that S ∈ Ψ0 (ℝd ). By combining Lemma 2.3.10 and Lemma 2.3.6 we infer that the restriction of T is Laplacian modulated. Next, T = Mϕ T, and therefore T † = T † Mϕ for some ϕ ∈ Cc∞ (ℝd ). Since T † ∈ Ψ−d (ℝd ), d

it follows that R := T † (1 − Δ) 2 is in Ψ0 (ℝd ) and therefore is bounded. Obviously,

2.4 Eigenvalues of Laplacian modulated operators

�

145

d

T † = R ⋅ (1 − Δ)− 2 Mϕ . d

By Remark 2.3.7 the operator (1 − Δ)− 2 Mϕ is Laplacian modulated. Hence so is the restriction of T † . The statement in the case of T compactly supported on the right follows from T † being compactly supported on the left.

2.4 Eigenvalues of Laplacian modulated operators The following eigenvalue theorem is central to Theorem 2.1.1. It associates the symbol of a compactly supported Laplacian modulated operator with partial sums of eigenvalues of the operator. Consequently, by Theorem 2.3.11 it provides a new estimate for partial sums of eigenvalues of compactly supported pseudodifferential operators. Theorem 2.4.1. Let A : L2 (ℝd ) → L2 (ℝd ) be a compactly supported Laplacian modulated operator with symbol pA . Then A ∈ ℒ1,∞ , and we have n

∑ λ(j, A) − (2π)−d

j=0

∫ 1 |ξ|≤n d

∫ pA (t, ξ)dtdξ = O(1),

n ≥ 0,

(2.11)

ℝd

where λ(A) is an eigenvalue sequence of A (the eigenvalues of A in any order such that their absolute values are decreasing). We prove the theorem in the rest of this section. The proof relies on the compact support of the Laplacian modulated operator, so that we can use the torical Laplacian. The torical Laplacian has compact resolvent, allowing us to use the main theorem on modulated operators, Theorem 7.1.3 in Volume I. In the following, we identify the Hilbert spaces L2 (𝕋d ) and L2 ([−π, π]d ). Let zm , m ∈ ℤd , be the orthonormal basis of L2 ([−π, π]d ) given by zm (t) = ei⟨m,t⟩ , t ∈ [−π, π]d . The torical Laplacian Δ𝕋d : H2 ([−π, π]d ) → L2 ([−π, π]d ) is defined by Δ𝕋d zm = −|m|2 zm ,

m ∈ ℤd ,

with domain H2 ([−π, π]d ) := {h ∈ L2 ([−π, π]d ) : {|m|2 ⟨h, zm ⟩}m∈ℤd ∈ l2 (ℤd )}. If required, we extend zm to a multiperiodic function on ℝd . Lemma 2.4.2. Let A : L2 (ℝd ) → L2 (ℝd ) be a Laplacian modulated operator. For every Schwartz function ϕ ∈ 𝒮 (ℝd ), we have 󵄩 󵄩2 ∑ 󵄩󵄩󵄩A(zm ϕ)󵄩󵄩󵄩 = O(s−d ),

|m|>s

s > 0.

146 � 2 Trace formulas Proof. Since (ℱ (zm ϕ))(s) = (ℱ ϕ)(s − m), s ∈ ℝd , m ∈ ℤd , from (2.5) and the Cauchy– Schwarz inequality it follows that 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨 󵄨 󵄨 −d 󵄨󵄨(A(zm ϕ))(t)󵄨󵄨󵄨 ≤ (2π) ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 󵄨󵄨󵄨(ℱ ϕ)(ξ − m)󵄨󵄨󵄨dξ ⋅ ∫ 󵄨󵄨󵄨(ℱ ϕ)(ξ − m)󵄨󵄨󵄨dξ ℝd

󵄨 󵄨2 󵄨 󵄨 = const ⋅ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 󵄨󵄨󵄨(ℱ ϕ)(ξ − m)󵄨󵄨󵄨dξ.

ℝd

ℝd

Observe that ℱ ϕ is a Schwartz function. In particular, |(ℱ ϕ)(ξ)| ≤ const ⋅⟨ξ⟩−2d . If |v − m| ≤ 1/2, then ξ ∈ ℝd ,

󵄨󵄨 󵄨 −2d ≤ const ⋅⟨ξ − v⟩−2d , 󵄨󵄨(ℱ ϕ)(ξ − m)󵄨󵄨󵄨 ≤ const ⋅⟨ξ − m⟩ and therefore 󵄨󵄨 󵄨2 󵄨 󵄨2 −2d 󵄨󵄨(A(zm ϕ))(t)󵄨󵄨󵄨 ≤ const ⋅ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 ⟨ξ − v⟩ dξ.

(2.12)

ℝd

Define functions f , h ∈ Lmod;1 (ℝd ) by setting 󵄨 󵄨2 f (ξ) := ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dt,

h(ξ) := ⟨ξ⟩−2d ,

ξ ∈ ℝd .

ℝd

For every v with |v − m| ≤ 1/2, from (2.12) it follows that 󵄩󵄩 󵄩2 󵄩󵄩A(zm ϕ)󵄩󵄩󵄩 ≤ const ⋅(f ⋆ h)(v). Therefore 󵄩 󵄩2 ∑ 󵄩󵄩󵄩A(zm ϕ)󵄩󵄩󵄩 ≤ const ⋅ ∑

|m|>s

∫

(f ⋆ h)dv ≤ const ⋅

|m|≥s |v−m|≤1/2

∫

(f ⋆ h)dv.

|v|≥s−1/2

It follows from Lemma 2.2.2 that f ⋆ h ∈ Lmod;1 . Hence 󵄩 󵄩2 ∑ 󵄩󵄩󵄩A(zm ϕ)󵄩󵄩󵄩 ≤ const ⋅‖f ⋆ h‖mod;1 ⋅ O(s−d ).

|m|>s

The previous lemma allows us to prove that a compactly supported Laplacian modulated operator is torical Laplacian modulated. Theorem 1.5.27 shows that the torical d Laplacian has compact resolvent and (1 − Δ𝕋d )− 2 ∈ ℒ1,∞ . The following result allows us to use Theorem 7.1.3 in Volume I. Denote by Q : L2 (ℝd ) → L2 ([−π, π]d ) the idempotent given by restriction.

2.4 Eigenvalues of Laplacian modulated operators �

147

Theorem 2.4.3. Let A : L2 (ℝd ) → L2 (ℝd ) be a Laplacian modulated operator. If A is compactly supported in the cube [−π, π]d , then QAQ∗ : L2 ([−π, π]d ) → L2 ([−π, π]d ) is d

(1 − Δ𝕋d )− 2 -modulated.

Proof. By the assumption on A there exists a Schwartz function ϕ : ℝd → ℝ supported in [−π, π]d such that A = Mϕ AMϕ . Let zm , m ∈ ℤd , be the eigenbasis of Δ𝕋d . We have QAQ∗ zm = QA(zm ϕ), where zm is extended to a multiperiodic function on ℝd on the right-hand side. Therefore on the Hilbert space L2 ([−π, π]d ), we have 󵄩󵄩 ∗ 󵄩󵄩QAQ E (1−Δ

d

)− 2 𝕋d

󵄩2 [0, t −1 ]󵄩󵄩󵄩2 = (2π)−d

∑

2

1+|m|2 ≥t d

󵄩󵄩 󵄩2 ∗ 󵄩󵄩QAQ zm 󵄩󵄩󵄩

󵄩 󵄩2 ≤ (2π)−d ∑ 󵄩󵄩󵄩QAQ∗ zm 󵄩󵄩󵄩 1

|m|≥t d

󵄩 󵄩2 = (2π)−d ∑ 󵄩󵄩󵄩A(zm ϕ)󵄩󵄩󵄩 = O(t −1 ). 1

|m|≥t d d

In the last equality, we used that ‖zm ‖l2 = (2π) 2 for all m ∈ ℤd . The statement now follows from Lemma 7.3.2 in Volume I. Given Theorem 2.4.3, Theorem 7.1.3 in Volume I states that n

∑ λ(j, QAQ∗ ) − (2π)−d

j=0

∑ ⟨QAQ∗ zm , zm ⟩ = O(1),

|m|≤n1/d

n ≥ 0,

(2.13)

for some eigenvalue sequence λ(QA∗ Q) of QAQ∗ . The next steps in proving Theorem 2.4.1 involve showing that n

n

j=0

j=0

∑ λ(j, A) − ∑ λ(j, QAQ∗ ) = O(1),

n ≥ 0,

(2.14)

for any eigenvalue sequence λ(A) of A, and ∑ ⟨QAQ∗ zm , zm ⟩ −

|m|≤n1/d

∫ |ξ|≤n1/d

∫ pA (t, ξ)dtdξ = O(1),

n ≥ 0.

(2.15)

ℝd

Equation (2.14) is a consequence of the fact that A and QAQ∗ have the same list of nonzero eigenvalues with multiplicity. This can be seen from A = PAP, where P = χ[−π,π]d is a self-adjoint projection in ℒ(L2 (ℝd )), and UPAPU ∗ = QAQ∗ for a unitary operator U : L2 (ℝd ) → L2 ([−π, π]d ).

148 � 2 Trace formulas The next two lemmas are purely technical, and the difference in (2.15) is demonstrated to be bounded in Lemma 2.4.7. Lemma 2.4.4. Let A : L2 (ℝd ) → L2 (ℝd ) be a Laplacian modulated operator compactly supported in [−π, π]d . If A = Mϕ AMϕ for a Schwartz function ϕ supported in [−π, π]d , then d

⟨QAQ∗ zm , zm ⟩ = (2π)− 2

∫ pA (t, ξ)(ℱ ϕt,ξ )(−m)dtdξ,

m ≥ 0,

ℝd ×ℝd

where ϕt,ξ (s) := e−i⟨ξ,s⟩ ϕ(t + s),

t, s, ξ ∈ ℝd .

Proof. Let m ≥ 0. Recall that AQ∗ zm = A(ϕzm ). By definition d

(A(ϕzm ))(t) = (2π)− 2 ∫ ei⟨t,ξ⟩ pA (t, ξ)(ℱ (ϕzm ))(ξ)dξ,

t ∈ ℝd .

ℝd

Thus, because pA (t, ξ) has compact support when t ∈ [−π, π]d , d

∫ ei⟨t,ξ−m⟩ pA (t, ξ)(ℱ (ϕzm ))(ξ)dtdξ.

⟨AQ∗ zm , Q∗ zm ⟩ = (2π)− 2

ℝd ×ℝd

Since (ℱ (ϕzm ))(ξ) = (ℱ ϕ)(ξ − m), it follows that d

∫ ei⟨t,ξ−m⟩ pA (t, ξ)(ℱ ϕ)(ξ − m)dtdξ.

⟨AQ∗ zm , Q∗ zm ⟩ = (2π)− 2

ℝd ×ℝd

Since ei⟨t,ξ−m⟩ (ℱ ϕ)(ξ − m) = (ℱ ϕt,ξ )(−m),

t, ξ ∈ ℝd ,

the statement follows. Lemma 2.4.5. Let ϕ be a Schwartz function on ℝd . (a) If |ξ| ≥ s, then ∑ (ℱ ϕt,ξ )(m) = |m|≤s

uniformly over t, ξ, and s. (b) If |ξ| ≤ s, then

O(1) , ⟨s − |ξ|⟩d

t, ξ ∈ ℝd ,

s > 0,

2.4 Eigenvalues of Laplacian modulated operators

∑ (ℱ ϕt,ξ )(m) = |m|>s

O(1) , ⟨s − |ξ|⟩d

t, ξ ∈ ℝd ,

�

149

s > 0,

uniformly over t, ξ, and s. Proof. Let s > 0. Observe that ℱ ϕ is a Schwartz function, and therefore |ℱ ϕ(ξ)| ≤ const ⋅⟨ξ⟩−2d , ξ ∈ ℝd . For |ξ| ≥ s, we have 󵄨󵄨 󵄨󵄨 −d 󵄨󵄨 󵄨 i⟨t,ξ−m⟩ (ℱ ϕ)(ξ − m)󵄨󵄨󵄨 ≤ const ⋅ ∑ ⟨m − ξ⟩−2d ≤ const ⋅⟨s − |ξ|⟩ . 󵄨󵄨 ∑ e 󵄨󵄨 󵄨󵄨 |m|≤s |m|≤s For |ξ| ≤ s, we have 󵄨󵄨 󵄨󵄨 −d 󵄨󵄨 󵄨 i⟨t,ξ−m⟩ (ℱ ϕ)(ξ − m)󵄨󵄨󵄨 ≤ const ⋅ ∑ ⟨m − ξ⟩−2d ≤ const ⋅⟨s − |ξ|⟩ . 󵄨󵄨 ∑ e 󵄨󵄨 󵄨󵄨 |m|>s |m|>s Proposition 2.4.6. If A : L2 (ℝd ) → L2 (ℝd ) is a Laplacian modulated operator compactly supported on the left with symbol pA , then ∫

󵄨 󵄨 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ = O(1),

s ≥ 0.

s≤|ξ|≤2s ℝd

In particular, 󵄨 󵄨 ∫ ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ = O(log(s)),

s ≥ 2.

|ξ|≤s ℝd

Proof. It follows from Lemma 2.3.9 that the symbol pA of A is compactly supported in its first variable. Fix a compact set K ⊂ ℝd such that pA (t, ξ) = 0 for t ∉ K. Observe that m(K × {ξ ∈ ℝd : s ≤ |ξ| ≤ 2s}) = const ⋅m(K)sd for the Lebesgue measure m. Using the Cauchy–Schwarz inequality, we obtain ∫

󵄨 󵄨 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ =

s≤|ξ|≤2s ℝd

∫

󵄨 󵄨 ∫󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ

s≤|ξ|≤2s K

1

≤ const ⋅(m(K)sd ≤ const ⋅‖pA ‖mod .

∫

2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨 dtdξ)

s≤|ξ|≤2s ℝd

150 � 2 Trace formulas Lemma 2.4.7. Let A : L2 (ℝd ) → L2 (ℝd ) be a Laplacian modulated operator compactly supported in [−π, π]d . If pA is the symbol of A, then ∑ ⟨QAQ∗ zm , zm ⟩ − ∫ ∫ pA (t, ξ)dtdξ = O(1), |m|≤s

|ξ|≤s

s > 0.

(2.16)

ℝd

Proof. Let ϕ be a Schwartz function supported in [−π, π]d and such that AMϕ = Mϕ A = A. By Lemma 2.4.4 we have d

⟨QAQ∗ zm , zm ⟩ = (2π)− 2

pA (t, ξ)(ℱ ϕt,ξ )(−m)dtdξ,

∫

m ≥ 0.

ℝd ×[−π,π]d

Thus, for s > 0, d

∑ ⟨QAQ∗ zm , zm ⟩ = (2π)− 2 |m|≤s

that

∫ ℝd ×[−π,π]d

pA (t, ξ)( ∑ (ℱ ϕt,ξ )(m))dtdξ. |m|≤s

Since ϕ vanishes outside [−π, π]d , it follows from the Poisson summation formula d

d

∑ (ℱ ϕt,ξ )(m) = (2π) 2 ∑ ϕt,ξ (2πm) = (2π) 2 ϕ(t),

m∈ℤd

m∈ℤd

t ∈ [−π, π]d .

It follows from Lemma 2.4.5 that 󵄨󵄨 󵄨󵄨 d const 󵄨󵄨 󵄨 , 󵄨󵄨 ∑ (ℱ ϕt,ξ )(m) − (2π) 2 ϕ(t)χ[0,s] (|ξ|)󵄨󵄨󵄨 ≤ 󵄨󵄨 󵄨󵄨 ⟨s − |ξ|⟩d |m|≤s

t, ξ ∈ ℝd .

Therefore 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∑ ⟨Azm , zm ⟩ − ∫ ∫ pA (t, ξ)ϕ(t)dtdξ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 |m|≤s d |ξ|≤s ℝ

≤ const ⋅

∫ ℝd ×[−π,π]d

−d 󵄨󵄨 󵄨 󵄨󵄨pA (t, ξ)󵄨󵄨󵄨⟨s − |ξ|⟩ dtdξ.

Obviously,

⟨s − |ξ|⟩

Thus

−d

2d s−d , |ξ| ≤ 21 s, { { { 1 ≤ {1, s ≤ |ξ| ≤ 2s, 2 { { d −d {2 |ξ| , |ξ| ≥ 2s.

2.4 Eigenvalues of Laplacian modulated operators �

∫ ℝd ×[−π,π]d

151

−d 󵄨󵄨 󵄨 󵄨󵄨pA (t, ξ)󵄨󵄨󵄨⟨s − |ξ|⟩ dtdξ

≤ 2d s−d ∫

󵄨 󵄨 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ

|ξ|≤ 21 s [−π,π]d

+

∫ 1 s≤|ξ|≤2s 2

󵄨 󵄨 ∫ 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ + 2d ∫

[−π,π]d

∫

|ξ|≥2s [−π,π]d

󵄨 󵄨 |ξ|−d 󵄨󵄨󵄨pA (t, ξ)󵄨󵄨󵄨dtdξ.

The first and second summands are bounded by Lemma 2.4.6. The third summand is bounded by the Cauchy–Schwarz inequality. In other words, we have ∑ ⟨QAQ∗ zm , zm ⟩ − ∫ ∫ pA (t, ξ)ϕ(t)dtdξ = O(1). |m|≤s

|ξ|≤s ℝd

Taking into account that pA (t, ξ)ϕ(t) = pA (t, ξ), we complete the proof. We now have the estimates required to prove Theorem 2.4.1. Proof of Theorem 2.4.1. Without loss, we can assume that A is compactly supported in [−π, π]d . Observe that d

V := (1 − Δ𝕋d )− 2 ∈ ℒ1,∞ . By Theorem 2.4.3 the operator QAQ∗ is V -modulated. Theorem 7.1.3 in Volume I then indicates that QAQ∗ ∈ ℒ1,∞ (L2 ([−π, π]d )). Observe that A = PAP, where P := χ[−π,π]d . Then μ(A) = μ(PAP) = μ(QAQ∗ ) by unitary invariance of singular values. Hence A ∈ ℒ1,∞ (L2 (ℝd )). Let xj , j ≥ 0, be a rearrangement of the sequence zm , m ∈ ℤd , according to increasing 1

|m|. For a given n ≥ 0, let xn = zmn with |mn | ∼ n d . By Theorem 7.1.3 in Volume I n

n

∑ λ(j, QAQ∗ ) = (2π)−d ∑⟨QAQ∗ xj , xj ⟩ + O(1)

j=0

j=0

= (2π)

−d

∑ ⟨QAQ∗ zm , zm ⟩ + O(1). 1

|m|≤n d

The proof is complete with (2.14) and (2.15), of which the latter was proved in Lemma 2.4.7.

152 � 2 Trace formulas

2.5 Trace formulas for Laplacian modulated operators We prove Theorems 2.1.1 and 2.1.2, restated further as Theorem 2.5.1, which allow a trace of a Laplacian modulated operator to be computed using the symbol of that operator.

2.5.1 Computing traces from symbols Recall from Chapter 6 of Volume I that the set of normalized positive traces on the twosided ideal ℒ1,∞ of ℒ(H) are in bijective correspondence with the set of Banach limits on l∞ . From the same chapter, the set of Dixmier traces on ℒ1,∞ corresponds to the set of normalized positive traces that are monotone for Hardy–Littlewood submajorization and is in bijective correspondence with the set of factorizable Banach limits on l∞ . The notation Trω is used for the Dixmier trace associated with the extended limit ω on l∞ using the formula introduced in Section 1.1 of Chapter 1. Theorem 2.5.1 (Trace theorem). Suppose A : L2 (ℝd ) → L2 (ℝd ) is a compactly supported Laplacian modulated operator with symbol pA . Then A ∈ ℒ1,∞ and: (a) φ(A) =

1 ⋅ (φ ∘ diag)( (2π)d

∫ pA (t, ξ)dtdξ)

∫

1 nd

1 0, we infer that m

‖pMf T ‖1 ≤ c ⋅ ‖f ‖1 ∫ (1 + |ξ|2 ) 2 dξ < ∞. ℝd

The next lemma is an adjustment of Theorem 2.4.1 for the Laplacian modulated operator Mf T, which is only known to be compactly supported on the left. Lemma 3.2.5. Let T ∈ Ψ−d (ℝd ), and let f ∈ L2 (ℝd ) be a compactly supported function. We have n

∑ λ(j, Mf T) − (2π)−d

j=0

∫ f (t)pT (t, ξ)dtdξ = O(1),

∫ 1 |ξ|≤n d

n ≥ 0.

(3.2)

ℝd

Proof. Let ϕ ∈ Cc∞ (ℝd ) be such that fϕ = f . We have Mf T = Mfϕ T = Mf TMϕ + Mf [Mϕ , T]. By Corollary 1.6.5 we have [Mϕ , T] ∈ Ψ−d−1 (ℝd ). By Lemma 3.2.3 we have Mf [Mϕ , T] ∈ ℒ1 . By Theorem 5.1.5 in Volume I we have n

n

j=0

j=0

∑ λ(j, Mf T) = ∑ λ(j, Mf TMϕ ) + O(1).

By Theorem 2.4.1 we have n

∑ λ(j, Mf TMϕ ) − (2π)−d

j=0

∫ 1 |ξ|≤n d

∫ pMf TMϕ (t, ξ)dtdξ = O(1), ℝd

n ≥ 0.

174 � 3 Integration formulas Next, by Lemma 2.3.5 and Proposition 1.6.4, pMf [T,Mϕ ] = pMf TMϕ − pMf Mϕ T = pMf TMϕ − pMf T , and 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ ∫ pMf TMϕ (t, ξ)dtdξ − ∫ ∫ pMf T (t, ξ)dtdξ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 1 1 d d |ξ|≤n d ℝ

|ξ|≤n d ℝ

󵄨󵄨 󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨 ∫ ∫ pMf [T,Mϕ ] (t, ξ)dtdξ 󵄨󵄨󵄨 ≤ ‖pMf [T,Mϕ ] ‖1 = O(1), 󵄨󵄨 󵄨󵄨 1 d |ξ|≤n d ℝ

where the last statement follows from Lemma 3.2.4. Lemma 3.2.6. Let a(t), t ∈ ℝd , be a family of positive-definite d × d matrices such that ⟨a(t)ξ, ξ⟩ > c,

c > 0,

t ∈ ℝd .

|ξ| = 1,

Then 1 ∈ L∞ (ℝd ), √det(a) and for every compactly supported f ∈ L1 (ℝd ), ∫ 1 |ξ|≤n d

∫ ℝd

f (t)

d

(1 + ⟨a(t)ξ, ξ⟩) 2

dtdξ =

f (t) Vol(𝕊d−1 ) log(n + 1) ⋅ ∫ dt + O(1). d √det(a(t)) ℝd

Proof. Let w be a normalized eigenvector of a(t) with eigenvalue λ. Then λ = ⟨a(t)w, w⟩ > c. Hence det(a(t))

− 21

d−1

− 21

= ∏ λ(i, a(t)) i=0

d

< c− 2 , 1

where λ(a(t)) is an eigenvalue sequence of a(t). Making the substitution a 2 (t)ξ =: v, we have ∫

∫

1 |ξ|≤n d

ℝd

=∫ ℝd

f (t)

d

(1 + ⟨a(t)ξ, ξ⟩) 2 f (t) ( √det(a(t))

1 |a− 2

dtdξ

∫ 1 (t)v|≤n d

1

d

(1 + |v|2 ) 2

dv)dt

3.2 Integration formula for elliptic differential operators and the curved plane

=∫ ℝd

f (t) ( ∫ √det(a(t))

1

+∫ ℝd

−∫ ℝd

f (t)

√det(a(t))

(

dv)dt 1

∫ 1 (t)v|≤n d 1 |v|>n d

1 |a− 2

f (t) ( √det(a(t))

d

(1 + |v|2 ) 2

1 |v|≤n d

1

∫ 1 (t)v|>n d 1 |v|≤n d

1 |a− 2

d

dv)dt

d

dv)dt.

(1 + |v|2 ) 2

(1 + |v|2 ) 2

Obviously, 󵄨󵄨 f (t) 󵄨󵄨 ( 󵄨󵄨 ∫ 󵄨󵄨 √det(a(t)) d ℝ

1

∫ 1

1

|a− 2 (t)v|≤n d

󵄨󵄨 󵄨 dv)dt 󵄨󵄨󵄨 󵄨󵄨 2 (1 + |v| ) d 2

1 |v|>n d

󵄨󵄨 |f (t)| 󵄨 ≤ 󵄨󵄨󵄨 ∫ ( 󵄨󵄨 √det(a(t)) d ℝ

∫ 1

1

|a− 2 (t)v|≤n d

󵄨󵄨 󵄨 |v|−d dv)dt 󵄨󵄨󵄨 󵄨󵄨

1 |v|>n d

󵄨󵄨 |f (t)| 󵄨 = 󵄨󵄨󵄨 ∫ ( 󵄨󵄨 √det(a(t)) d ℝ

∫ 1

|a− 2 (t)v|≤1 |v|>1

󵄨󵄨 󵄨 |v|−d dv)dt 󵄨󵄨󵄨 󵄨󵄨

and 󵄨󵄨 f (t) 󵄨󵄨 ( 󵄨󵄨 ∫ 󵄨󵄨 √det(a(t)) d ℝ

1

∫ 1

1

|a− 2 (t)v|>n d 1 |v|≤n d

󵄨󵄨 |f (t)| 󵄨 ≤ 󵄨󵄨󵄨 ∫ ( 󵄨󵄨 √det(a(t)) d ℝ

∫ 1

1

|a− 2 (t)v|>n d

ℝ

∫ 1

|a− 2 (t)v|>1 |v|≤1

A combination of these equations yields

1 |ξ|≤n d

∫ ℝd

󵄨󵄨 󵄨 |v|−d dv)dt 󵄨󵄨󵄨 󵄨󵄨

1 |v|≤n d

󵄨󵄨 |f (t)| 󵄨 = 󵄨󵄨󵄨 ∫ ( 󵄨󵄨 √det(a(t)) d

∫

󵄨󵄨 󵄨 dv)dt 󵄨󵄨󵄨 󵄨󵄨 2 (1 + |v| ) d 2

f (t)

d

(1 + ⟨a(t)ξ, ξ⟩) 2

dtdξ

󵄨󵄨 󵄨 |v|−d dv)dt 󵄨󵄨󵄨. 󵄨󵄨

�

175

176 � 3 Integration formulas

=∫ ℝd

f (t) ( ∫ √det(a(t))

1

d

(1 + |v|2 ) 2

1 |v|≤n d

dv)dt + O(1).

Proof of Theorem 3.2.1. By hypothesis T + 1 : H 2 (ℝd ) → L2 (ℝd ) is self-adjoint and positive as an unbounded operator on L2 (ℝd ). Since T is positive, h ∈ H 2 (ℝd ).

⟨(T + 1)h, h⟩ ≥ ⟨h, h⟩,

d

By Theorem 1.6.19 there is a pseudodifferential operator (T + 1)− 2 ∈ Ψ−d (ℝd ) whose d restriction to L2 (ℝd ) coincides with the operator (T +1)− 2 defined by the Borel functional calculus of the self-adjoint operator T on L2 (ℝd ). Suppose f ∈ L2 (ℝd ) has compact support. By Lemma 3.2.3 we have d

Mf (T + 1)− 2 ∈ ℒ1,∞ , which proves statement (a). By Lemma 3.2.5 we have n

d

∑ λ(j, Mf (T + 1)− 2 ) − (2π)−d

j=0

∫ f (t)σ(t, ξ)dtdξ = O(1),

∫ 1 |ξ|≤n d

n ≥ 0,

(3.3)

ℝd

d

where σ is the symbol of (T + 1)− 2 ∈ Ψ−d (ℝd ). By Theorem 1.6.22 we have − d2

σ(t, ξ) − (⟨a(t)ξ, ξ⟩ + 1)

∈ S −d−1 (ℝd × ℝd ).

Hence f (t)(σ(t, ξ) − (⟨a(t)ξ, ξ⟩ + 1)

− d2

) ∈ L1 (ℝd × ℝd ),

and ∫ f (t)σ(t, ξ)dtdξ =

∫ 1 |ξ|≤n d

ℝd

∫ 1 |ξ|≤n d

− d2

∫ f (t)(⟨a(t)ξ, ξ⟩ + 1)

dtdξ + O(1).

ℝd

By Lemma 3.2.6 we have ∫ 1 |ξ|≤n d

∫ f (t)σ(t, ξ)dtdξ = ℝd

Combining (3.3) and (3.4), we have

f (t) Vol(𝕊d−1 ) log(n + 1) ⋅ ∫ dt + O(1). d √det(a(t)) ℝd

(3.4)

3.2 Integration formula for elliptic differential operators and the curved plane n

d

∑ λ(j, Mf (T + 1)− 2 ) = (

j=0

�

177

f (t) Vol(𝕊d−1 ) ⋅∫ dt) log(n + 1) + O(1). d(2π)d √det(a(t)) d ℝ

Statement (b) in Theorem 3.2.1 now follows from Theorem 1.1.13(a).

3.2.2 Integration theorem on a curved plane Let (s, ξ) 󳨃→ ⟨g(t)s, ξ⟩,

t, s, ξ ∈ ℝd ,

be a Riemannian metric on ℝd with matrix of coefficients g(t), t ∈ ℝd . We say the Riemannian metric g is bounded above and below if for all t ∈ ℝd , c1 < ⟨g(t)ξ, ξ⟩ < c2 ,

|ξ| = 1,

for some constants 0 < c1 ≤ c2 . Definition 3.2.7. Let g be a Riemannian metric that is smooth and bounded above and below. The differential operator Δg : 𝒮 (ℝd ) → 𝒮 (ℝd ), defined by (Δg h)(t) =

d 1 ∑ 𝜕i (√det(g)(g −1 )ij 𝜕j h)(t), √det(g(t)) i,j=1

h ∈ 𝒮 (ℝd ),

is called the Laplace–Beltrami operator associated with the metric g. We identify Δg with its unique continuous extension Δg : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝd ) as described before in Definition 1.6.2. Remark 3.2.8. The operator −Δg is an elliptic differential operator of order 2 with principal symbol σ2 (t, ξ) = ⟨g(t)−1 ξ, ξ⟩, by Example 1.6.35.

t, ξ ∈ ℝd ,

178 � 3 Integration formulas The volume of a Borel subset E of the plane ℝd with the metric g is given by volg (E) := ∫ √det(g(t))dt. E

The operator U : L2 (ℝd ) → L2 (ℝd , volg ), given by U := Mdet(g)−1/4 , is unitary. As a pseudodifferential operator of order 0 (see Example 1.6.3 and Theorem 1.6.10), Mdet(g)−1/4 : H r (ℝd ) → H r (ℝd ),

r ∈ ℝ,

is a continuous linear operator with continuous inverse Mdet(g)1/4 . Lemma 3.2.9. Let g be a Riemannian metric on ℝd bounded from above and below. Then −Δg |H 2 (ℝd ) = UTg |H 2 (ℝd ) U ∗ , where Tg is an elliptic differential operator with principal symbol t, ξ ∈ ℝd ,

σ2 (t, ξ) = ⟨g(t)−1 ξ, ξ⟩,

and Tg |H 2 (ℝd ) is a positive self-adjoint operator on L2 (ℝd ) with domain H 2 (ℝd ). Proof. By Example 1.6.35, Tg := Mdet(g)1/4 (−Δg )Mdet(g)−1/4 is an elliptic differential operator of order 2 with principal symbol t, ξ ∈ ℝd .

σ2 (t, ξ) = ⟨g(t)−1 ξ, ξ⟩, The operator

Tg : 𝒮 (ℝd ) → 𝒮 (ℝd ) is symmetric and positive on 𝒮 (ℝd ) in the inner product of L2 (ℝd ), since ⟨Tg h, h⟩ = − ∑ ∫ det(g(t)) i,j

ℝd

− 41

1

1

𝜕i (det(g(t)) 2 (g −1 )ij 𝜕j (det(g)− 4 h))(t)h(t)dt

3.2 Integration formula for elliptic differential operators and the curved plane

�

179

1 1 󵄨 1 󵄨2 = ∫ 󵄨󵄨󵄨g − 2 (t)(∇ det(g)− 4 h)(t)󵄨󵄨󵄨 det(g(t)) 2 dt > 0

ℝd

for all h ∈ 𝒮 (ℝd ). Thus, by Theorem 1.6.25, Tg |H 2 (ℝd ) is a self-adjoint positive operator on L2 (ℝd ) with domain H 2 (ℝd ). The statement follows since −Δg |UH 2 (ℝd ) = UTg |H 2 (ℝd ) U ∗ for the unitary operator U and UH 2 (ℝd ) = H 2 (ℝd ). By Lemma 3.2.9, −Δg |H 2 (ℝd ) is a self-adjoint and positive unbounded operator on the Hilbert space L2 (ℝd , volg ) with domain H 2 (ℝd ). Using the functional calculus of selfadjoint operators and Lemma 1.6.18, we have d

(1 − Δg |H 2 (ℝd ) )− 2 : L2 (ℝd , volg ) → UH d (ℝd ) = H d (ℝd ) ⊂ L∞ (ℝd ). We denote the product operator given by a function f ∈ L2 (ℝd , volg ) as an operator on the Hilbert space L2 (ℝd , volg ) by Mf = UMf U ∗ : L∞ (ℝd ) → L2 (ℝd , volg ). There is no conflict in notation since (UMf U ∗ u)(t) = (Mf u)(t),

u ∈ L∞ (ℝd ).

The composition d

Mf (1 − Δg |H 2 (ℝd ) )− 2

is an everywhere defined and bounded operator on L2 (ℝd , volg ). In the statement of Theorem 3.2.10, we denote −Δg |H 2 (ℝd ) by −Δg . Theorem 3.2.10. Let g be a Riemannian metric on ℝd bounded from above and below. (a) For every compactly supported f ∈ L2 (ℝd , volg ), we have d

Mf (1 − Δg )− 2 ∈ ℒ1,∞ (L2 (ℝd , volg )). (b) For every compactly supported f ∈ L2 (ℝd , volg ) and for every normalized trace φ on ℒ1,∞ , we have d

φ(Mf (1 − Δg )− 2 ) =

Vol(𝕊d−1 ) ⋅ ∫ f (t)d volg (t). d(2π)d ℝd

180 � 3 Integration formulas Proof of Theorem 3.2.10. Let U = M ator above. By Lemma 3.2.9,

1

det(g)− 4

: L2 (ℝd ) → L2 (ℝd , volg ) be the unitary oper-

d

d

Mf (1 − Δg )− 2 = U ⋅ Mf (1 + Tg )− 2 ⋅ U −1 , where Tg is a positive self-adjoint operator on L2 (ℝd ) with domain H 2 (ℝd ) that is the restriction of an elliptic differential operator with principal symbol t, ξ ∈ ℝd .

σ2 (t, ξ) = ⟨g(t)−1 ξ, ξ⟩, d

The operator (1 + Tg )− 2 belongs to ℒ1,∞ by Theorem 3.2.1(a). Thus 󵄩󵄩 −d 󵄩 󵄩󵄩Mf (1 − Δg ) 2 󵄩󵄩󵄩ℒ

1,∞ (L2 (ℝ

d ,vol

g ))

d 󵄩 󵄩 󵄩 󵄩 ≤ ‖U‖L2 (ℝd )→L2 (ℝd ,volg ) 󵄩󵄩󵄩Mf (1 + Tg )− 2 󵄩󵄩󵄩ℒ (L (ℝd )) 󵄩󵄩󵄩U −1 󵄩󵄩󵄩L (ℝd ,vol )→L (ℝd ) < ∞. 1,∞ 2 2 g 2

This proves part (a). By the trace property we have d

d

d

φ(Mf (1 − Δg )− 2 ) = φ(U ⋅ Mf (1 + Tg )− 2 ⋅ U −1 ) = φ(Mf (1 + Tg )− 2 ). Part (b) follows now from Theorem 3.2.1(b) with a(t) = g −1 (t), t ∈ ℝd . Theorem 3.2.10 is sufficient to prove a similar integration theorem for the Laplace– Beltrami operator on a compact Riemannian manifold without boundary. This is discussed in the end notes to Chapter 3 in Section 3.6.

3.3 Integration of square-integrable functions on 𝕋d Let 𝕋 denote the unit circle in the complex plane 𝕋 = {z ∈ ℂ : |z| = 1} with its canonical group multiplication. We write Lp (𝕋d ), p > 0, for the Lp -spaces of the d-torus 𝕋d = ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝕋 × ⋅⋅⋅ × 𝕋 d times

equipped with the normalized Haar measure. If zn , n ∈ ℤd , are the trigonometric basis elements of L2 (𝕋d ), then for r ≥ 0, denote r 󵄩 󵄩 H r (𝕋d ) := {h ∈ L2 (𝕋d ) : 󵄩󵄩󵄩(1 + |n|) hn 󵄩󵄩󵄩l < ∞, h = ∑ hn zn }. 2 n∈ℤd

3.3 Integration of square-integrable functions on 𝕋d

� 181

We can introduce the gradient d

∇𝕋d : H 1 (𝕋d ) → L2 (𝕋d ) and the Laplace operator

Δ𝕋d : H 2 (𝕋d ) → L2 (𝕋d ) on 𝕋d in terms of Fourier multipliers. The operator ∇𝕋d acts on a basis element zn by ∇𝕋d zn := (n1 zn , . . . , nd zn ),

n = (n1 , . . . , nd ) ∈ ℤd ,

and the operator Δ𝕋d acts on a basis element zn by Δ𝕋d zn := −|n|2 zn = −(|n1 |2 + ⋅ ⋅ ⋅ + |nd |2 )zn ,

n = (n1 , . . . , nd ) ∈ ℤd .

Given the domain H 2 (𝕋d ), the operator −Δ𝕋d : H 2 (𝕋d ) → L2 (ℝd ) is a self-adjoint positive unbounded operator on the Hilbert space L2 (ℝd ). For g ∈ l2 (ℤd ), Definition 1.5.25 in Section 1.5.6 defined the operator g(∇𝕋d ) : L2 (𝕋d ) → L∞ (𝕋d ) acting on a basis element zn by g(∇𝕋d )zn := g(n)zn ,

n ∈ ℤd .

If f ∈ L2 (ℝd ), then the composition of g(∇𝕋d ) with the multiplication operator Mf : L∞ (𝕋d ) → L2 (𝕋d ) forms the bounded operator Mf g(∇𝕋d ) : L2 (𝕋d ) → L2 (𝕋d ). Theorem 3.2.10 in Section 3.2 states that the noncommutative integral for the Laplace operator on the Euclidean plane ℝd extends the Lebesgue integral. Theorem 3.3.1 proves that a similar result holds for 𝕋d . Theorem 3.3.1. Let 𝕋d be the d-dimensional torus, and let Δ𝕋d be the Laplacian on 𝕋d . d

(a) If f ∈ L2 (𝕋d ), then Mf (1 − Δ𝕋d )− 2 ∈ ℒ1,∞ . Moreover,

d 󵄩 󵄩 cd ⋅ ‖f ‖L2 ≤ 󵄩󵄩󵄩Mf (1 − Δ𝕋d )− 2 󵄩󵄩󵄩ℒ ≤ Cd ⋅ ‖f ‖L2 1,∞

182 � 3 Integration formulas for constants 0 < cd ≤ Cd depending only on d. (b) For every f ∈ L2 (𝕋d ) and for every normalized trace φ on ℒ1,∞ , we have d

φ(Mf (1 − Δ𝕋d )− 2 ) =

Vol 𝕊d−1 ⋅ ∫ f (z)dz, d(2π)d 𝕋d

where dz is the Haar measure on 𝕋d . Theorem 3.3.1 is proved below using Lemma 3.3.4. Lemma 3.3.4 is a simplified form of Theorem 2.4.1. The simplified form uses the fact that for the Laplacian on the torus, the expectation values associated with the eigenvectors zn , n ∈ ℤd , of Δ𝕋d in the Hilbert space L2 (𝕋d ) are constant in n, ⟨Mf zn , zn ⟩ = ∫ f (z)zn z−n dz = ∫ f (z)dz, 𝕋d

f ∈ L2 (𝕋d ).

𝕋d

Section 2.3 introduced the Banach space of Laplacian modulated operators on L2 (ℝd ). A similar notion can be defined on the torus. d

Definition 3.3.2. A bounded operator T : L2 (𝕋d ) → L2 (𝕋d ) is said to be (1 − Δ𝕋d )− 2 modulated if 1 d −1 󵄩 󵄩 sup t 2 󵄩󵄩󵄩T(1 + t(1 − Δ𝕋d )− 2 ) 󵄩󵄩󵄩ℒ < ∞,

(3.5)

2

t>0

where ‖ ⋅ ‖ℒ2 denotes the Hilbert–Schmidt norm on ℒ2 (L2 (𝕋d )). For f ∈ L2 (𝕋d ) and g ∈ l2 (ℤd ), Theorem 1.5.26 in Section 1.5.6 states that Mf g(∇𝕋d ) ∈ ℒ2 and 󵄩󵄩 󵄩 −d 󵄩󵄩Mf g(∇𝕋d )󵄩󵄩󵄩ℒ2 = (2π) 2 ‖f ‖L2 ‖g‖l2 . d

The next lemma shows that the Hilbert–Schmidt operators of the form Mf (1 − Δ𝕋d )− 2 , d

f ∈ L2 (𝕋d ), are (1 − Δ𝕋d )− 2 -modulated.

d

Lemma 3.3.3. For every f ∈ L2 (𝕋d ), the Hilbert–Schmidt operator Mf (1 − Δ𝕋d )− 2 is (1 − d

Δ𝕋d )− 2 -modulated.

Proof. Set Δ = Δ𝕋d and ∇ = ∇𝕋d . Let t > 0. We have d

Mf (1 − Δ)− 2 ⋅

1

d

1 + t(1 − Δ)− 2

= Mf gt (∇),

3.3 Integration of square-integrable functions on 𝕋d

�

183

where − d2

gt (n) = (1 + |n|2 )

⋅

1

d

|n|2 )− 2

1 + t(1 +

=

1 t + (1 +

d

|n|2 ) 2

,

n ∈ ℤd .

By Theorem 1.5.26 we have 󵄩󵄩 1 󵄩󵄩 −d 󵄩󵄩Mf (1 − Δ) 2 ⋅ d 󵄩󵄩 1 + t(1 − Δ)− 2

󵄩󵄩 󵄩󵄩 −d 󵄩󵄩 = (2π) 2 ‖f ‖L2 ‖gt ‖l2 . 󵄩󵄩ℒ2

Now ‖gt ‖2l2 ≤ ∑ (t + |n|d )

−2

= O(t −1 ),

‖gt ‖2l2 ≤ ∑ (1 + |n|2 )

−d

= O(1) = O(t −1 ),

n∈ℤd n∈ℤd

t ≥ 1, t ≤ 1.

Therefore 󵄩󵄩 1 󵄩󵄩 −d 󵄩󵄩Mf (1 − Δ) 2 ⋅ d 󵄩󵄩 1 + t(1 − Δ)− 2

󵄩󵄩 󵄩󵄩 −1 󵄩󵄩 = O(t 2 ), 󵄩󵄩2

t > 0.

Modulated operators and partial sums of their eigenvalues were studied in Chapter 7 of Volume I. Using Theorem 7.1.3 from Volume I, a simpler version of Theorem 2.4.1 can be proved for the torus. For m ∈ ℤ+ , let b(m) := ∑ 1. l∈ℤd |l|2 ≤m

For n ∈ ℤ+ , choose m = m(n) ∈ ℤ+ such that b(m) ≤ n < b(m + 1). Such a choice is always possible because b(m) → ∞ as m → ∞. d

Lemma 3.3.4. If T is (1 − Δ𝕋d )− 2 -modulated, then T ∈ ℒ1,∞ , and n

∑ λ(k, T) = (2π)−d

k=0

∑ ⟨Tzl , zl ⟩ + O(1),

l∈ℤd |l|2 ≤m(n)

where zl , l ∈ ℤd , is the trigonometric basis of L2 (𝕋d ), and λ(T) is an eigenvalue sequence of T (the eigenvalues of T in any order such that their absolute values are decreasing).

184 � 3 Integration formulas Proof. Set Δ = Δ𝕋d . Let a : ℤ+ → ℤd be a bijection such that n → |a(n)| increases. Set d

fk (z) := (2π)− 2 za(k) ,

z ∈ 𝕋d .

k ∈ ℤ+ , d

Then {fk }k≥0 is an eigenbasis of the operator (1 − Δ)− 2 ordered so that Δ𝕋d fk = |a(k)|2 fk and |a(k)|2 is increasing. We have 󵄨󵄨 n 󵄨󵄨 b(m(n))−1 󵄨󵄨 󵄨 󵄨󵄨 ∑ λ(k, T) − ∑ λ(k, T)󵄨󵄨󵄨 ≤ (n − b(m(n)) + 1) ⋅ λ(b(m(n)), T) 󵄨󵄨 󵄨󵄨 󵄨󵄨k=0 󵄨󵄨 k=0

−1

≤ (b(m(n) + 1) − b(m(n))) ⋅ O(b(m(n)) ) = O(1).

By Theorem 7.1.3 in Volume I we have T ∈ ℒ1,∞ and b(m(n))−1

∑

k=0

λ(k, T) =

b(m(n))−1

∑

k=0

⟨Tfk , fk ⟩ + O(1).

Note that d

{fk }b(m(n))−1 = (2π)− 2 ⋅ {zm } k=0

m∈ℤd . |m|2 ≤m(n)

Combining these estimates completes the proof. The proof of Theorem 3.3.1 follows from Lemma 3.3.4. d

Proof of Theorem 3.3.1. Set Δ = Δ𝕋d . Let V := (1 − Δ)− 2 ∈ ℒ1,∞ . By Lemma 3.3.3, Mf V is V -modulated. (a) By Lemma 3.3.4 or Theorem 1.5.27, Mf V ∈ ℒ1,∞ . By Theorem 1.5.26, d

((2π)− 2 ‖g‖l2 ) ⋅ ‖f ‖L2 = ‖Mf V ‖ℒ2 ≤ ‖Mf V ‖ℒ1,∞ d

for the function g(n) := (1 + |n|2 )− 2 , n ≥ 0. This provides one of the inequalities required. Using the constant c2 in Theorem 1.5.27(c), we have ‖Mf V ‖ℒ1,∞ ≤ (c2 ‖g‖l1,∞ ) ⋅ ‖f ‖L2 . This proves the other inequality. (b) By Lemma 3.3.4 we have n

d

∑ λ(k, Mf (1 − Δ)− 2 ) = (2π)−d

k=0

∑

m∈ℤd |m|2 ≤m(n)

d

⟨Mf (1 − Δ)− 2 zm , zm ⟩ + O(1).

3.4 Integration of compactly supported square-integrable functions on ℝd

�

185

Clearly, d

⟨Mf (1 − Δ)− 2 zm , zm ⟩ = (1 + |m|2 )

− d2

⋅ ∫ f (z)dz. 𝕋d

Therefore n

d

∑ λ(k, Mf (1 − Δ)− 2 ) = (2π)−d

k=0

∑

d

m∈ℤ |m|2 ≤m(n)

(1 + |m|2 )

− d2

⋅ ∫ f (z)dz + O(1). 𝕋d

An elementary computation shows that ∑

m∈ℤd |m|2 ≤m(n)

(1 + |m|2 )

− d2

=

1 log(m(n)) ⋅ Vol(𝕊d−1 ) + O(1). 2

2

Since m(n) ≈ n d , it follows that n

Vol(𝕊d−1 ) log(n) ⋅ ∫ f (z)dz + O(1). d(2π)d

d

∑ λ(k, Mf (1 − Δ)− 2 ) =

k=0

𝕋d

The statement follows from Theorem 9.1.2(a) in Volume I.

3.4 Integration of compactly supported square-integrable functions on ℝd In this section, we prove the following integration formula involving the Laplacian Δ on the Euclidean plane ℝd by lifting the equivalent theorem on the torus (Theorem 3.3.1). The proof is an alternative to that of Section 3.2, without using powers of pseudodifferential operators or Laplacian modulated operators. Theorem 3.4.1. If f ∈ L2 (ℝd ) is a compactly supported function, then d

φ(Mf (1 − Δ)− 2 ) =

Vol(𝕊d−1 ) ⋅ ∫ f (t)dt d(2π)d ℝd

for every normalized trace φ on ℒ1,∞ . We identify the torus 𝕋d (as a measure space) with the interval [−π, π]d . Hence measurable functions on the torus are identified with measurable functions on [−π, π]d . The next lemma follows directly from the definition of the kernel of an integral operator.

186 � 3 Integration formulas Lemma 3.4.2. Let Q1 : L2 (ℝd ) → L2 ([0, 1]d ) and Q2 : L2 (𝕋d ) → L2 ([0, 1]d ) be the operators given by restriction. (a) If A ∈ ℒ(L2 (ℝd )) is an integral operator with integral kernel (t, s) → K(t, s),

t, s ∈ ℝd ,

then Q1 AQ1∗ ∈ ℒ(L2 ([0, 1]d )) is an integral operator with integral kernel (t, s) → K(t, s),

t, s ∈ [0, 1]d .

(b) If B ∈ ℒ(L2 (𝕋d )) is an integral operator with integral kernel (t, s) → K(t, s),

t, s ∈ 𝕋d ,

then Q2 BQ2∗ ∈ ℒ(L2 ([0, 1]d )) is an integral operator with integral kernel (t, s) → K(t, s),

t, s ∈ [0, 1]d .

In this section, ∇𝕋d is the gradient on the torus. For a ∈ l∞ (ℤd ), Definition 1.5.25 in Section 1.5.6 defined the operator a(∇𝕋d ) : L2 (𝕋d ) → L2 (𝕋d ). The inverse Fourier transform of a ∈ l2 (ℤd ) is denoted in this section by d

̌ := (2π)− 2 ∑ a(n)ei⟨t,n⟩ , a(t) n∈ℤd

t ∈ [−π, π]d .

Lemma 3.4.3. Let f1 ∈ L2 (ℝd ), f2 ∈ L∞ (ℝd ), and g ∈ L2 (ℝd ). Let h1 ∈ L2 (𝕋d ), h2 ∈ L∞ (𝕋d ), and a ∈ l2 (ℤd ). (a) Set A := Mf1 g(∇)Mf2 . The operator Q1 AQ1∗ has the integral kernel d

(t, s) → (2π)− 2 (ℱ −1 g)(t − s)f1 (t)f2 (s),

t, s ∈ [0, 1]d .

(b) Set B := Mh1 a(∇𝕋d )Mh2 . The operator Q2 BQ2∗ has the integral kernel d

̌ − s)h1 (t)h2 (s), (t, s) → (2π)− 2 a(t

t, s ∈ [0, 1]d .

Proof. (a) From Definition 1.5.1 we have d

(g(∇)ξ)(t) = (2π)− 2 ∫ (ℱ −1 g)(t − s)ξ(s)ds,

t ∈ ℝd .

ℝd

Thus g(∇) is an integral operator with integral kernel d

(t, s) → (2π)− 2 (ℱ −1 g)(t − s),

t, s ∈ ℝd .

3.4 Integration of compactly supported square-integrable functions on ℝd

� 187

It clearly follows that A is also an integral operator with integral kernel d

t, s ∈ ℝd .

(t, s) → (2π)− 2 (ℱ −1 g)(t − s)f1 (t)f2 (s),

Consequently, Q1 AQ1∗ is an integral operator with integral kernel d

t, s ∈ [0, 1]d .

(t, s) → (2π)− 2 (ℱ −1 g)(t − s)f1 (t)f2 (s), (b) From Definition 1.5.25 we have d

̌ − s)ξ(s)ds, (a(∇𝕋d )ξ)(t) = (2π)− 2 ∫ a(t

t ∈ 𝕋d .

𝕋d

Thus a(∇𝕋d ) is an integral operator with integral kernel d

̌ − s), (t, s) → (2π)− 2 a(t

t, s ∈ 𝕋d .

It clearly follows that B is also an integral operator with integral kernel d

̌ − s)h1 (t)h2 (s), (t, s) → (2π)− 2 a(t

t, s ∈ 𝕋d .

Consequently, Q2 BQ2∗ is an integral operator with integral kernel d

̌ − s)h1 (t)h2 (s), (t, s) → (2π)− 2 a(t

t, s ∈ [0, 1]d .

Bounded operators of the form Mf1 g(∇)Mf2 on L2 (ℝd ) restricted to L2 ([0, 1]d ) and bounded operators of the form Mh1 a(∇𝕋d )Mh2 on L2 (𝕋d ) restricted to L2 ([0, 1]d ) are identified as integral operators using Lemma 3.4.3. Lemma 3.4.4. Let f1 ∈ L2 (ℝd ), f2 ∈ L∞ (ℝd ), and g ∈ L2 (ℝd ). Let ϕ ∈ C ∞ (ℝd ) be supported in [−3, 3]d and such that ϕ|[−1,1]d = 1. Let d

a(n) := (2π)− 2

∫

e−i⟨n,t⟩ (ℱ −1 g)(t)ϕ(t)dt,

n ∈ ℤd .

[−π,π]d

Then Q1 ⋅ Mf1 g(∇)Mf2 ⋅ Q1∗ = Q2 ⋅ MQ2∗ Q1 f1 a(∇𝕋d )MQ2∗ Q1 f2 ⋅ Q2∗ . Proof. By Lemma 3.4.3(a) the left-hand side is an integral operator whose integral kernel is given by the formula d

(t, s) → (2π)− 2 (ℱ −1 g)(t − s)f1 (t)f2 (s),

t, s ∈ [0, 1]d .

188 � 3 Integration formulas By Lemma 3.4.3(b) the right-hand side is an integral operator whose integral kernel is given by the formula d

t, s ∈ [0, 1]d .

̌ − s)f1 (t)f2 (s), (t, s) → (2π)− 2 a(t

If t, s ∈ [0, 1]d , then t − s ∈ [−1, 1]d , so that ϕ(t − s) = 1. Thus t, s ∈ [0, 1]d .

̌ − s), (ℱ −1 g)(t − s) = a(t

Hence the integral kernel of the left-hand side coincides with that of the right-hand side. The statement follows from Lemma 3.4.2. Lemma 3.4.4 reduces the bounded operator of the form Mf1 g(∇)Mf2 to the operator Mh1 a(∇𝕋d )Mh2 on L2 (𝕋d ). The next lemma shows that a(∇𝕋d ) approximates the operator d

(1 − Δ𝕋d )− 2 up to a trace class operator. d

Lemma 3.4.5. Let g(t) := (1 + |t|2 )− 2 , t ∈ ℝd , and let a ∈ l2 (ℤd ) be as in Lemma 3.4.4. Then we have a(n) − g(n) = O((1 + |n|2 )

− d+1 2

),

n ∈ ℤd .

Proof. Since ϕ is supported in [−π, π]d , it follows that d

a(n) = (2π)− 2 ∫ (ℱ −1 g)(t)ϕ(t)e−i⟨n,t⟩ dt. ℝd

In other words, a(n) = (ℱ (ϕ ⋅ ℱ −1 g))(n),

n ∈ ℤd .

Set ψ = (1 − ϕ) ⋅ ℱ −1 g. This immediately implies that a(n) − g(n) = (ℱ ψ)(n),

n ∈ ℤd .

By Example 2.2.7 we have (ℱ −1 g)(t) = cd K0 (|t|),

t ∈ ℝd .

This function is smooth (except at 0) and rapidly decreasing at infinity. Since 1 − ϕ vanishes near 0, it follows that ψ is a smooth and rapidly decreasing function. In other words, ψ ∈ 𝒮 (ℝd ). This completes the proof, since then also ℱ ψ is of rapid decay. A unitary operator U : H1 → H2 between Hilbert spaces H1 and H2 provides a bijection between traces on ℒ1,∞ (H1 ) and ℒ1,∞ (H2 ) given by

3.4 Integration of compactly supported square-integrable functions on ℝd

φU (A) := φ(U ∗ AU),

�

189

A ∈ ℒ1,∞ (H2 ),

for a trace φ : ℒ1,∞ (H1 ) → ℂ. The bijection is independent of the unitary chosen, and therefore we write φU as φ when the context is clear. With the identification in Lemma 3.4.4, the next lemma shows that Theorem 3.3.1 in Section 3.3 can be used to transfer the integral theorem on 𝕋d to ℝd for compactly supported functions. Lemma 3.4.6. Let A ∈ ℒ1,∞ (L2 (ℝd )) be such that A = AMχ such that B = BMχ

[0,1]d

[0,1]d

. Let B ∈ ℒ1,∞ (L2 (𝕋d )) be

. If Q1 AQ1∗ = Q2 BQ2∗ , then φ(A) = φ(B) for every trace φ on ℒ1,∞ .

Proof. Let U1 : L2 (ℝd ) → L2 ([0, 1]d ) and U2 : L2 (ℝd ) → L2 (𝕋d ) be unitaries. Let φ be a trace on ℒ1,∞ (L2 (ℝd )). By the assumption on A and by the tracial property we have φU1 (Q1 AQ1∗ ) = φ(AQ1∗ Q1 ) = φ(AMχ

[0,1]d

) = φ(A).

By the assumption on B and by the tracial property we have φU1 (Q2 BQ2∗ ) = (φU2 )(U1 U2∗ ) (Q2 BQ2∗ ) = φU2 (BQ2∗ Q2 ) = φU2 (BMχ

[0,1]d

) = φU2 (B).

Since Q1 AQ1∗ = Q2 BQ2∗ , φU2 (B) = φ(A), and the statement follows. Proof of Theorem 3.4.1. Without loss of generality, f ∈ L2 (ℝd ) is supported on [0, 1]d . Let d

f1 := f and f2 := χ[0,1]d . Also, let g(t) := (1 + |t|2 )− 2 , t ∈ ℝd . Set A := Mf1 g(∇)Mf2 ,

B := MQ2∗ Q1 f1 a(∇𝕋d )MQ2∗ Q1 f2 .

By Theorem 1.5.20, A ∈ ℒ1,∞ (L2 (ℝd )). Clearly, A = AMχ

[0,1]d

. By Lemma 3.4.5,

a(∇𝕋d ) − g(∇𝕋d ) = (ℱ ψ)(∇𝕋d ), where ℱ ψ ∈ 𝒮 (ℝd ). Furthermore, {(ℱ ψ)(n)}n∈ℤd ∈ l1 (ℤd ), and hence by Theorem 1.5.26, MQ2∗ Q1 f1 (a(∇𝕋d ) − g(∇𝕋d )) = MQ2∗ Q1 f1 (ℱ ψ)(∇𝕋d ) ∈ ℒ1 . Since MQ2∗ Q1 f1 g(∇𝕋d ) ∈ ℒ1,∞ by Theorem 3.3.1, it follows that MQ2∗ Q1 f1 a(∇𝕋d ) ∈ ℒ1,∞ and B ∈ ℒ1,∞ (L2 (𝕋d )). Clearly, B = BMχ d . By Lemma 3.4.4 we have Q1 AQ1∗ = Q2 BQ2∗ . Thus, [0,1] by Lemma 3.4.6, φ(A) = φ(B) for every trace φ on ℒ1,∞ . By the tracial property we have d

d

φ(Mf (1 − Δ)− 2 ) = φ(Mf2 ⋅ Mf1 (1 − Δ)− 2 ) d

= φ(Mf1 (1 − Δ)− 2 ⋅ Mf2 ) = φ(Mf1 g(∇)Mf2 ) = φ(A).

190 � 3 Integration formulas By the preceding paragraph and by the tracial property we have d

φ(MQ2∗ Q1 f (1 − Δ𝕋d )− 2 ) = φ(MQ2∗ Q1 a(∇𝕋d )) = φ(MQ2∗ Q1 f2 ⋅ MQ2∗ Q1 f1 a(∇𝕋d )) = φ(MQ2∗ Q1 f1 a(∇𝕋d )MQ2∗ Q1 f2 ) = φ(B).

Therefore d

d

φ(Mf (1 − Δ)− 2 ) = φ(MQ2∗ Q1 f (1 − Δ𝕋d )− 2 ). The statement now follows from Theorem 3.3.1(b).

3.5 Integration formula on the noncommutative torus In this section, we prove the analogous statement of Theorem 3.3.1 for the noncommutative torus. We briefly recall the integral for the noncommutative torus and the Laplacian operator.

3.5.1 Definition of the noncommutative torus Let θ ∈ Md (ℝ), 1 < d ∈ ℕ, be an antisymmetric matrix. Let Aθ be the universal ∗-algebra generated by unitaries {Uk }dk=1 satisfying the conditions Uk2 Uk1 = eiθk1 ,k2 Uk1 Uk2 ,

1 ≤ k1 , k2 ≤ d.

Define a linear functional τθ : Aθ → ℂ by setting n

n

unless n = (n1 , . . . , nd ) = 0, n ∈ ℤd ,

n

n

when n = (n1 , . . . , nd ) = 0, n ∈ ℤd .

τθ (U1 1 ⋅ ⋅ ⋅ Ud d ) = 0 and τθ (U1 1 ⋅ ⋅ ⋅ Ud d ) = 1

It can be demonstrated that τθ is positive, that is, τθ (x ∗ x) ≥ 0 for x ∈ Aθ . We equip the linear space Aθ with an inner product defined by the formula ⟨x, y⟩ = τθ (xy∗ ),

x, y ∈ Aθ .

The action λ of Aθ on the pre-Hilbert space (Aθ , ⟨⋅, ⋅⟩) by left multiplication extends to a representation of Aθ as bounded operators on the completed Hilbert space L2 (𝕋dθ , τθ ). The closure in the weak operator topology of λ(Aθ ) in ℒ(L2 (𝕋dθ , τθ )) is denoted by L∞ (𝕋dθ ), and τθ extends to a faithful normal tracial state on L∞ (𝕋dθ ). The C ∗ -algebra

3.5 Integration formula on the noncommutative torus

� 191

C(𝕋dθ ) given by the closure of λ(Aθ ) in the uniform operator topology is isomorphic to the universal C ∗ -algebra constructed by Davidson (see pp. 166–170 in [99]). The Hilbert space L2 (𝕋dθ , τθ ) is isomorphic to the noncommutative Lorentz space ℒ2 (L∞ (𝕋dθ ), τθ ) defined in Example 1.2.14. For brevity, we denote Lp,q (𝕋dθ , τθ ) := ℒp,q (L∞ (𝕋dθ ), τθ ),

0 < p, q ≤ ∞.

From Example 1.2.14, τθ has an extension to L1 (𝕋dθ , τθ ), and the inner product on L2 (𝕋dθ , τθ ) can be written, using the extension, ⟨x, y⟩ = τθ (xy∗ ),

x, y ∈ L2 (𝕋dθ , τθ ).

Let zn , n ∈ ℤd , be the trigonometric basis elements of L2 (𝕋d ) from Section 3.3. When θ = 0, L∞ (𝕋d0 ) is spatially ∗-isomorphic to L∞ (𝕋d ) using the representation (Uk h)(z) := zk h(z), h ∈ L2 (𝕋d ), 1 ≤ k ≤ d, on the Hilbert space L2 (𝕋d ). Under this spatial ∗-isomorphism, τ0 (f ) =

1 1 (ℱ f )(0) = ∫ f (z)dz, (2π)d/2 (2π)d

f ∈ L∞ (𝕋d ),

𝕋d

and Lp (𝕋d0 , τ0 ) = Lp (𝕋d ), 0 < p < ∞. 3.5.2 Differential calculus of the noncommutative torus A natural differential calculus for the noncommutative torus is given as follows. Denote n

n

un := U1 1 ⋅ ⋅ ⋅ Ud d ,

n = (n1 , . . . , nd ) ∈ ℤd .

Define self-adjoint operators Dk , 1 ≤ k ≤ d, on the Hilbert space L2 (𝕋dθ , τθ ) by setting Dk : un → n k un ,

n = (n1 , . . . , nd ) ∈ ℤd ,

with domains dom(Dk ) = {h ∈ L2 (𝕋dθ , τθ ) : {nk ⟨h, un ⟩}n∈ℤd ∈ l2 (ℤd ), h = ∑ ⟨h, un ⟩un }. n∈ℤd

Define the Laplacian on the Hilbert space L2 (𝕋dθ , τθ ) by setting d

Δ := − ∑ D2k k=1

with domain

192 � 3 Integration formulas H 2 (𝕋dθ ) := {h ∈ L2 (𝕋dθ , τθ ) : {|n|2 ⟨h, un ⟩}n∈ℤd ∈ l2 (ℤd ), h = ∑ ⟨h, un ⟩un }. n∈ℤd

The self-adjoint operators Dk , k = 1, . . . , d, are essentially self-adjoint on H 2 (𝕋dθ ) and strongly commute: [Dk , Dj ]h = 0,

h ∈ H 2 (𝕋dθ ),

j, k ∈ {1, . . . , d},

with [eitDk , eisDk ] = 0,

t, s ∈ ℝ,

j, k ∈ {1, . . . , d}.

The operator −Δ is positive and self-adjoint on L2 (𝕋dθ , τθ ) with eigenvalues −Δun = |n|2 un ,

n ∈ ℤd .

For g ∈ l∞ (ℤd ), define the bounded operator g(∇) : L2 (𝕋dθ , τθ ) → L2 (𝕋dθ , τθ ) by g(∇)un = g(n)un ,

n ∈ ℤd .

3.5.3 Product-convolution operators for the noncommutative torus We prove the analogue of the Hausdorff–Young inequality for the noncommutative torus. Lemma 3.5.1. Let 𝕋dθ be the d-dimensional noncommutative torus. (a) If g ∈ lq (ℤd ) for q ∈ [1, 2], then the operator T(g) ∈ L2 (𝕋dθ , τθ ) defined by T(g) := ∑ g(n)un n∈ℤd

belongs to Lq∗ (𝕋dθ , τθ ), where q∗ is the Hölder conjugate of p, and 󵄩󵄩 󵄩 󵄩󵄩T(g)󵄩󵄩󵄩Lq∗ ≤ ‖g‖lq . (b) If g ∈ lq,2 (ℤd ) for q ∈ (1, 2), then the operator T(g) ∈ L2 (𝕋dθ , τθ ) defined by T(g) := ∑ g(n)un n∈ℤd

3.5 Integration formula on the noncommutative torus

�

193

belongs to Lq∗ ,2 (𝕋dθ , τθ ), where q∗ is the Hölder conjugate of q, and 󵄩󵄩 󵄩 󵄩󵄩T(g)󵄩󵄩󵄩Lq∗ ,2 ≤ ‖g‖lq,2 . Proof. Note that T : l2 (ℤd ) → L2 (𝕋dθ , τθ ) is a linear map. If g ∈ l1 (ℤd ) ⊂ l2 (ℤd ), then 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩T(g)󵄩󵄩󵄩L∞ ≤ ∑ 󵄨󵄨󵄨g(n)󵄨󵄨󵄨‖un ‖L∞ ≤ ‖g‖l1 , n∈ℤd

noting that L∞ (𝕋dθ ) ⊂ L2 (𝕋dθ , τ). If g ∈ l2 (ℤd ), then 󵄩󵄩 󵄩2 󵄩󵄩T(g)󵄩󵄩󵄩L = 2

󵄨 󵄨2 ∗ ) = ∑ 󵄨󵄨󵄨g(n)󵄨󵄨󵄨 = ‖g‖2l2 . ∑ g(n)g(m)τθ (un um

n,m∈ℤd

n∈ℤd

The spaces (lq , Lq∗ (𝕋dθ , τθ )) and (lq,2 , Lq∗ ,2 (𝕋dθ , τθ )) form interpolation pairs for the couples (l1 , L∞ (𝕋dθ )) and (l2 , L2 (𝕋dθ , τθ )) when 1 < p < 2. The statement for (lq , Lq∗ (𝕋dθ , τθ )) follows by complex interpolation; see, for example, the noncommutative version of the Riesz–Thorin theorem [115, p. 68]. The statement for (lq,2 , Lq∗ ,2 (𝕋dθ , τθ )) follows by real interpolation; see, for example, the noncommutative version of the Marcinkiewicz theorem [115, p. 70]. The operators in Lp (𝕋dθ , τθ ), 0 < p < ∞, are affiliated to L∞ (𝕋dθ ) but are not bounded operators on L2 (𝕋dθ , τθ ). To extend the abstract estimates of Section 1.5, we show that product-convolution type operators for the noncommutative torus are bounded. Lemma 3.5.2. Let 𝕋dθ be the d-dimensional noncommutative torus. (a) If p > 2, then λ(x)g(∇) ∈ ℒ(L2 (𝕋dθ , τθ )) for x ∈ Lp (𝕋dθ , τθ ) and g ∈ lp,∞ (ℤd ). (b) If p = 2, then λ(x)g(∇) ∈ ℒ(L2 (𝕋dθ , τθ )) for x ∈ L2 (𝕋dθ , τθ ) and g ∈ l2 (ℤd ). (c) If 0 < p < 2, then λ(x)g(∇) ∈ ℒ(L2 (𝕋dθ , τθ )) for x ∈ L2 (𝕋dθ , τθ ) and g ∈ lp,∞ (ℤd ).

194 � 3 Integration formulas Proof. With the Hausdorff–Young estimate in Lemma 3.5.1, the proof essentially repeats the commutative case. (a) Let v, h ∈ l2 (ℤd ). Note that g(∇)T(h) = ∑ g(n)h(n)un = T(gh). n∈ℤd

If g ∈ lp,∞ (ℤd ), then gh ∈ lq,2 (ℤd ),

q=

2p ∈ (1, 2), p+2

and 󵄨󵄨 󵄨 󵄩 ∗ 󵄩 󵄩 󵄩 󵄨󵄨⟨λ(x)g(∇)T(h), T(v)⟩󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩x T(v)󵄩󵄩󵄩Lq,2 󵄩󵄩󵄩T(gh)󵄩󵄩󵄩Lq∗ ,2 . From the Hölder inequality for Lorentz spaces and the continuous embedding of Lp in Lp,∞ , 󵄩󵄩 ∗ 󵄩 󵄩 󵄩 󵄩󵄩x T(v)󵄩󵄩󵄩Lq,2 ≤ ‖x‖Lp,∞ 󵄩󵄩󵄩T(v)󵄩󵄩󵄩L2 ≤ ‖x‖Lp ‖v‖l2 . From Lemma 3.5.1 and the Hölder inequality for Lorentz spaces, 󵄩󵄩 󵄩 󵄩󵄩T(gh)󵄩󵄩󵄩Lq∗ ,2 ≤ ‖gh‖lq,2 ≤ ‖g‖lp,∞ ‖h‖l2 . Hence 󵄨󵄨 󵄨 󵄨󵄨⟨λ(x)g(∇)T(h), T(v)⟩󵄨󵄨󵄨 ≤ ‖x‖Lp ‖g‖lp,∞ ‖h‖l2 ‖v‖l2 , and the operator λ(x)g(∇) is everywhere defined and bounded. (b). If x ∈ L2 (𝕋dθ , τθ ) and g ∈ l2 (ℤd ), then from Lemma 3.5.1 and the Hölder inequality, 󵄩󵄩 󵄩 󵄩󵄩T(gh)󵄩󵄩󵄩L∞ ≤ ‖gh‖l1 ≤ ‖g‖l2 ‖h‖l2 . Hence 󵄨󵄨 󵄨 󵄩 ∗ 󵄩 󵄩 󵄩 󵄨󵄨⟨λ(x)g(∇)T(h), T(v)⟩󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩x T(v)󵄩󵄩󵄩L1 󵄩󵄩󵄩T(gh)󵄩󵄩󵄩L∞ ≤ ‖x‖L2 ‖v‖l2 ‖g‖l2 ‖h‖l2 , and the operator λ(x)g(∇) is everywhere defined and bounded. (c) From (b), λ(x)g(∇) is everywhere defined and bounded for x ∈ L2 (𝕋dθ , τθ ) and g ∈ l2 (ℤd ). For 0 < p < 2, if g ∈ lp,∞ (ℤd ), then g ∈ l2 (ℤd ). The statement follows. The next theorem is the equivalent of Theorem 1.5.27 for product-convolution operators on the noncommutative torus.

3.5 Integration formula on the noncommutative torus

�

195

Theorem 3.5.3. Let 𝕋dθ be the d-dimensional noncommutative torus. (a) Let p ≥ 2. If x ∈ Lp (𝕋dθ , τθ ) and g ∈ lp (ℤd ), then λ(x)g(∇) ∈ ℒp , and 󵄩󵄩 󵄩 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒp ≤ 13 ⋅ ‖x‖Lp ‖g‖lp . (b) Let p > 2. If x ∈ Lp (𝕋dθ , τθ ) and g ∈ lp,∞ (ℤd ), then λ(x)g(∇) ∈ ℒp,∞ , and p 󵄩󵄩 󵄩 ⋅ ‖x‖Lp ‖g‖lp,∞ . 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒp,∞ ≤ 18√ p−2 (c) Let 0 < p ≤ 2. If x ∈ L2 (𝕋dθ , τθ ) and g ∈ lp (ℤd ), then λ(x)g(∇) ∈ ℒp , and 󵄩󵄩 󵄩 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒp ≤ ‖x‖L2 ‖g‖lp . (d) Let 0 < p < 2. If x ∈ L2 (𝕋dθ , τθ ) and g ∈ lp,∞ (ℤd ), then λ(x)g(∇) ∈ ℒp,∞ , and there exists a constant cp > 0 such that 󵄩󵄩 󵄩 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒp,∞ ≤ cp ⋅ ‖x‖L2 ‖g‖lp,∞ . Proof. Let x ∈ L2 (𝕋dθ , τθ ). Then 󵄩󵄩 󵄩2 ∗ ∗ ∗ 2 󵄩󵄩λ(x)un 󵄩󵄩󵄩L2 = τθ (xun un x ) = τθ (xx ) = ‖x‖L2 . Let 0 < p ≤ 2. Note that (see, e. g., [137, p. 95]) p p 󵄩󵄩 󵄩p 󵄩 󵄩p 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒp ≤ ∑ 󵄩󵄩󵄩λ(x)g(∇)un 󵄩󵄩󵄩L2 = ‖x‖L2 ‖g‖l (ℤd ) , p n∈ℤd

using the previous display. This proves (c). Let pj , j ≥ 1, be the sequence of projections in L∞ (𝕋dθ ) given by p1 = 1, pj = 0, j ≥ 2. Let qk , k ≥ 1, be the sequence of projections in

196 � 3 Integration formulas l∞ (ℤd ) given by the elementary vector with 1 in the n(k)th position, where n : ℕ → ℤd is an enumeration. Then, by the above display, 󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩λ(pj x)(gqk )(∇)󵄩󵄩󵄩ℒp ≤ ‖pj x‖L2 󵄨󵄨󵄨g(n(k))󵄨󵄨󵄨 = ‖pj x ⊗ gqk ‖L2 (𝕋d )⊗l2 (ℤd ) , θ

j, k ∈ ℕ.

Hence Assumption 1.5.16 is satisfied, which also implies Assumption 1.5.9. Statements (a) and (b) for x ∈ L∞ (𝕋dθ ) follow from Corollary 1.5.15. Statement (d) for x ∈ L∞ (𝕋dθ ) follows from Corollary 1.5.18. Lemma 3.5.2 states that λ(x)g(∇) is everywhere defined and bounded in the cases where x ∈ Lp (𝕋dθ , τθ ). An approximation similar to the argument in the proof of Theorem 1.5.20 proves the statements for x ∈ Lp (𝕋dθ , τθ ) and is omitted. 3.5.4 Integration of square-integrable functions on the noncommutative torus The following theorem extends Theorem 3.3.1 to noncommutative tori. Theorem 3.5.4. Let 𝕋dθ be the d-dimensional noncommutative torus. (a) If x ∈ L2 (𝕋dθ , τθ ), then d

λ(x)(1 − Δ)− 2 ∈ ℒ1,∞ with d 󵄩 󵄩 cd ⋅ ‖x‖L2 ≤ 󵄩󵄩󵄩λ(x)(1 − Δ)− 2 󵄩󵄩󵄩ℒ ≤ Cd ⋅ ‖x‖L2 1,∞

for constants 0 < cd ≤ Cd depending only on d. (b) For every x ∈ L2 (𝕋dθ , τθ ) and for every normalized trace φ on ℒ1,∞ we have d

φ(λ(x)(1 − Δ)− 2 ) =

Vol 𝕊d−1 ⋅ τθ (x), d

where τθ also denotes the extension of the finite normal semifinite trace τθ on L∞ (𝕋dθ ) to L1 (𝕋dθ , τθ ). Proof. Set − d2

g(k) := (1 + |k|2 )

,

k ∈ ℤd .

Then g ∈ l1,∞ (ℤd ). (a) We have 󵄩󵄩 󵄩2 󵄩 󵄩2 2 2 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒ2 = ∑ 󵄩󵄩󵄩λ(x)g(∇)un 󵄩󵄩󵄩L2 = ‖x‖L2 ‖g‖l2 . n∈ℤd

3.6 Notes

� 197

Then 󵄩 󵄩 󵄩 󵄩 ‖g‖l2 ‖x‖L2 = 󵄩󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒ ≤ 󵄩󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒ . 2 1,∞ If c2 is the constant from Theorem 3.5.3(d), we have 󵄩󵄩 󵄩 󵄩󵄩λ(x)g(∇)󵄩󵄩󵄩ℒ1,∞ ≤ (c2 ‖g‖l1,∞ ) ⋅ ‖x‖L2 . Setting Cd := max{c2 ‖g‖1,∞ , ‖g‖−1 2 }, the statement is shown. if

d

(b) By (7.8) on p. 246 in Volume I, T ∈ ℒ(L2 (𝕋dθ , τθ )) is (1 − Δ)− 2 -modulated if and only ∑ ‖Tuk ‖22 = O(n−1 ),

|k|1/d >n

n → ∞.

We have −d 󵄩 󵄩2 ∑ 󵄩󵄩󵄩λ(x)g(∇)uk 󵄩󵄩󵄩2 = ‖x‖22 ⋅ ∑ (1 + |k|2 ) = ‖x‖22 ⋅ O(n−1 ).

|k|1/d >n

|k|>n1/d

d

Hence λ(x)g(∇) is (1 − Δ)− 2 -modulated. Note that ⟨λ(x)g(∇)uk , uk ⟩ = g(k)τθ (x),

k ∈ ℤd .

d

Since λ(x)g(∇) is (1 − Δ)− 2 -modulated, by Corollary 7.1.4(a) in Volume I we have φ(λ(x)g(∇)) = τθ (x) ⋅ (φ ∘ diag)({g(k)}k∈ℤd ) =

Vol(𝕊d−1 ) ⋅ τθ (x) d

for every normalized trace φ on ℒ1,∞ . The last equality follows from the fact ∞

(φ ∘ diag)({g(k)}k∈ℤd ) = (φ ∘ diag)({μ(n, g)}n=0 ), since φ ∘ diag is a symmetric functional on l1,∞ , and from Weyl’s asymptotic law on the torus [61], μ(n, g) =

Vol(𝕊d−1 ) 1 1 ⋅ + o( ), d n n

n → ∞.

3.6 Notes Noncommutative integral Let D : dom(D) → H be a self-adjoint unbounded operator on a Hilbert space H. Alain Connes introduced the formula [72, p. 545]

198 � 3 Integration formulas Trω (A⟨D⟩−p ),

A ∈ ℒ(H),

as a noncommutative integral with density (also termed an infinitesimal of order one) p ⟨D⟩−p = (1 + D2 )− 2 ∈ ℒ1,∞ for some p > 0. The operator D acts as a Dirac operator in p

Connes’ noncommutative geometry, and the condition that (1 + D2 )− 2 ∈ ℒ1,∞ imposes spectral behavior analogous to Weyl’s law. The term “noncommutative integral” was established by Connes and expositions such as [22, 85, 139, 278], where [22, p. 34] states “This led Connes to introduce the Dixmier trace as the correct operator theoretical substitute for integration of infinitesimals of order one in non-commutative geometry.”

Connes proved Corollary 2.5.8 for Dixmier traces and closed manifolds. The fact that the noncommutative integral d

A 󳨃→ Trω (A(1 − ΔX )− 2 ),

A ∈ Ψ0cl (X),

is a quantization of integration of classical zero-order symbols over the sphere bundle can be found in [22, 68, 192]. Here ΔX is the Laplace–Beltrami operator on a compact Riemannian manifold X of dimension d without boundary. For details on Liouville measure and the sphere bundle, see [62, VII]. Heuristically, the pseudodifferential operad tor A(1 − ΔX )− 2 is a quantization of a(v)dv for the zero-order operator A with principal symbol a is a standard idea of pseudodifferential theory; see, for example, [73, 142]. For further observations that Connes’ trace theorem can recover the integral of a smooth function and the Liouville measure, see [139, Corollary 7.21], [22, §1.1], [182, p. 98], [232]. Translating between integral formulas for closed manifolds and the integral formulas in Section 3.2 for the curved plane is described below. Integral formula for functions that are not smooth and traces that are not Dixmier traces Deriving an integral formula from the noncommutative residue and Connes’ original trace theorem for a classical pseudodifferential operator requires smooth (compactly supported) functions. Extension to continuous (compactly supported) functions is proved through positivity of the Dixmier trace as a trace on ℒ1,∞ . The noncommutative integral on compactly supported continuous functions is sufficient to associate a unique measure to the noncommutative integral using the Riesz–Markov theorem [231, Theorem IV.14]. Since traces on ℒ1,∞ are not normal semifinite traces, however, the identification of the integral on continuous functions is not sufficient to show that d

f 󳨃→ Trω (Mf (1 − ΔX )− 2 ) is the integral according to this measure for an arbitrary f ∈ L∞ (X).

3.6 Notes

� 199

Showing that the noncommutative integral of an arbitrary function f ∈ L∞ (X) on a closed Riemannian manifold X equals the integral of f given by the volume form led to the development of Laplacian modulated operators. The fact that the noncommutative integral of an arbitrary function f ∈ L∞ (X) on a closed Riemannian manifold is equal to the integral of f was proved explicitly for Dixmier traces in [190] by the first and third authors with Potapov. Zeta-function residues (Chapter 8 in Volume I) were used in the proof. Proposition 12 of [70] uses a form of zeta function residues to give an equivalent proof. Furthermore, [190] proved the result for f ∈ L2 (X) and gave the bounds in Theorem 3.3.1(a) in terms of ‖f ‖2 , which holds in general for closed manifolds. Proving that the noncommutative integral of Mf , f ∈ L2 (X), given an arbitrary trace on ℒ1,∞ is equal to the integral of f given by the volume form of X required a different method. Nigel Kalton suggested to the authors an alternative proof using commutators and eigenvalues [168], which is the origin of Laplacian modulated operators and the approach described in Chapters 2 and 3. Dixmier traces are not singled out by applications especially. Chapters 2 and 3 show that trace theorems concerning the noncommutative residue and integration hold for all traces on ℒ1,∞ . This was first observed in [168]. Equally, Chapter 4 on integration for the noncommutative plane, Chapter 5 on extending the notion of the principal symbol and the Liouville integral, Chapter 6 on the quantum calculus and densities from forms, and Chapter 7 on Connes’ character formula in noncommutative geometry prove trace formulas for all traces or all continuous traces on ℒ1,∞ . The set of all continuous traces on ℒ1,∞ is larger than the set of Dixmier traces on ℒ1,∞ (see Chapter 6 of Volume I). Integration formulas for the plane and closed manifolds Theorem 2.5 of [190] proved the equivalent of Theorem 3.2.10 for the Laplace–Beltrami operator on a closed manifold. The paper [190] only considered Dixmier traces associated with zeta-functions (see Chapter 8 in Volume I). Theorem 3.2.10 and its equivalent for the Laplace–Beltrami operator on a closed manifold, an arbitrary trace, and f ∈ L2 (X) or f ∈ L2 (ℝd ) of compact support was proved in [168, Corollary 7.24]. Theorem 3.1.1 was proved in [168, Corollary 6.38]. Transferring from compactly supported Laplacian modulated operators on ℝd to Laplace–Beltrami modulated operators on a closed d-dimensional manifold is possible using a partition of unity {ϕi }∞ i=1 subordinate to a chart for which the metric g is bounded above and below on the chart. With the partition of unity, the operators d

Mϕi Mf (1 − ΔX )− 2 Mϕj on L2 (X) can be transferred using chart maps into a compactly supported operator of the form d

Mϕi Mf (1 + T)− 2 Mϕj + ℒ1

200 � 3 Integration formulas on L2 (ℝd ), where T is elliptic with principal symbol ⟨g −1 (u)s, s⟩ and satisfies the conditions of Theorem 3.2.10. Since a trace φ on ℒ1,∞ vanishes on ℒ1 and d

d

φ(Mϕi Mf (1 + T)− 2 Mϕj ) = φ(Mϕi ϕj f (1 + T)− 2 ) =

Vol(𝕊d−1 ) ∫ ϕi ϕj fd volg , d(2π)d X

the integral formula for the manifold X follows by summing over i and j. In this case the d condition f ∈ L2 (X) is necessary and sufficient for Mf (1 − ΔX )− 2 ∈ ℒ1,∞ [193, p. 359]. The reference [168] first used chart maps to transfer results on compactly supported Laplacian modulated operators on ℝd to Laplace–Beltrami modulated operators on the closed manifold X. In Chapter 11 of [193], the same method was used. Integration formulas for functions that are not compactly supported For the noncompact manifold ℝd , the product-convolution estimates of Birman and d Solomyak [29] stated in Section 1.5.5 indicate that Mf (1 − Δ)− 2 ∈ ℒ1,∞ when f belongs to the function space l1 (L2 )(ℝd ) = {f : ∑ ‖fχQm ‖2 < ∞, Qm = unit cube in ℝd translated by m}. m∈ℤd

Setting the norm ‖f ‖l1 (L2 ) = ∑ ‖fχQm ‖2 , m∈ℤd

d

f ∈ l1 (L2 )(ℝ𝕕 )

on l1 (L2 )(ℝd ), we have (see Theorem 1.5.22 in Section 1.5.5) 󵄩󵄩 −d 󵄩 󵄩󵄩Mf (1 − Δ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ c‖f ‖l1 (L2 ) ,

f ∈ l1 (L2 )(ℝd ),

c > 0.

Compactly supported functions in L2 (ℝd ) are dense in l1 (L2 )(ℝd ). Using continuity properties of a positive trace φ and the Birman–Solomyak estimate, the condition of compact support can be removed [267, Prop. 4.1]. Summarizing, we have the following: Theorem. Let f ∈ l1 (L2 )(ℝd ) be as above, and let Δ be the Laplacian on ℝd . Then Mf (1 − d

Δ)− 2 ∈ ℒ1,∞ , and

d

φ(Mf (1 − Δ)− 2 ) =

Vol 𝕊d−1 ⋅ ∫ f (t)dt, d(2π)d

f ∈ l1 (L2 )(ℝd ),

ℝd

for every positive normalized trace φ on ℒ1,∞ . The condition of compact support has not yet been removed in the statement for traces on ℒ1,∞ that are not continuous.

3.6 Notes

� 201

Failure of the formula for integrable functions Theorem 3.3.1 indicates that the noncommutative integral on 𝕋d is limited to the integration of square-integrable functions. Oversights in [128] and [139, Corollary 7.22] incorrectly extended the result to f ∈ L1 (𝕋d ) before [190] and [168]. To investigate how traces can recover integration of functions on a closed manid fold X that are not square integrable, the product Mf (1 − ΔX )− 2 needs to be replaced by a symmetrized version such as d

d

(1 − ΔX )− 4 Mf (1 − ΔX )− 4 . It was proved in [190, Theorem 5.9] that d

d

Trω ((1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) =

Vol(𝕊d−1 ) ⋅ ∫ fd volg , d(2π)d

f ∈ Lp (X), p > 1,

X

for Dixmier traces Trω . In [190, Lemma 5.7], it was proved that there exists f ∈ L2 (X) such that d

Mf (1 − ΔX )− 4 ∈ ̸ ℒ2,∞ , or, equivalently, there exists f ∈ L1 (X) such that d

d

(1 − ΔX )− 4 Mf (1 − ΔX )− 4 ∈ ̸ ℒ1,∞ . The symmetric form of the noncommutative integral does not extend to integrable functions. Symmetric noncommutative integration formulas and Cwikel estimates at the critical dimension The failure at L1 (X) relates to Cwikel estimates and Sobolev estimates at the critical dimension [195]. In [195], it was computed for ℝd that 󵄩󵄩 −d 󵄩 󵄩󵄩Mf (1 − Δ) 4 󵄩󵄩󵄩ℳ(2) ≤ cd ‖f ‖Λ1 (ℝd )(2) , 1,∞

f ∈ Λ1 (ℝd ) , (2)

and d

d

φ((1 − Δ)− 4 Mf (1 − Δ)− 4 ) =

Vol(𝕊d−1 ) ⋅ ∫ f (t)dt, d(2π)d

f ∈ Λ1 (ℝd ),

ℝd

for all continuous traces φ on ℳ1,∞ . Here ℳ1,∞ is the smallest ideal solid under Hardy– Littlewood submajorization that contains ℒ1,∞ (see Chapter 2 of Volume I), ℳ(2) 1,∞ is the 2-convexification (which is larger than the ideal ℒ2,∞ ), and

202 � 3 Integration formulas ∞

Λ1 (ℝd ) = {f ∈ L0 (ℝd ) : ∫ μ(s, f )(1 + log+ (s−1 ))ds < ∞} 0

is a Lorentz space containing (Lp ∩ L1 )(ℝd ), p > 1, but strictly contained by L1 (ℝd ). The space Λ1 (ℝd )(2) also denotes the 2-convexification. The function μ(s, f ), s > 0, is the decreasing rearrangement of the measurable function f . In [195], it is shown that Λ1 (ℝd )(2) plays the role of the function space Lp (ℝd ), p > 2, in a p = 2 extension of Cwikel’s estimate for p > 2; see Theorem 1.5.20 in Section 1.5.5 and [93]. The same result for the symmetric noncommutative integral on a closed manifold X is proved in Section 5.4 of [195]. In the case of a compact manifold, the Lorentz space Λ1 (X) is equivalent to the Zygmund space L log L(X) [195, Lemma 4.5], [269, Theorem 1.1], [269, Theorem 1.3]: 󵄩󵄩 −d 󵄩 󵄩󵄩Mf (1 − ΔX ) 4 󵄩󵄩󵄩ℳ(2) ≤ cd ‖f ‖L log L(X)(2) , 1,∞

f ∈ L log L(X)(2) ,

and d

d

φ((1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) =

Vol(𝕊d−1 ) ⋅ ∫ fd volg , d(2π)d

f ∈ L log L(X),

X

for any continuous normalized trace φ on ℳ1,∞ . The approach of [195] uses tensor multipliers and is limited to the 2-convexification of the Banach ideal ℳ1,∞ . The approach pioneered by Solomyak [261, Lemma 2.1 and Theorem 2.1] strengthens the Cwikel estimate and the integration formula for a compact manifold X to the 2-convexification of the quasi-Banach ideal ℒ1,∞ [269, Theorem 1.1], [270, Theorem 1.3]: 󵄩󵄩 −d 󵄩 󵄩󵄩Mf (1 − ΔX ) 4 󵄩󵄩󵄩ℒ

2,∞

≤ cd ‖f ‖L log L(X)(2) ,

f ∈ L log L(X)(2) .

Strengthening the Cwikel estimate at the critical dimension to the ideal ℒ2,∞ for the plane ℝd is fundamentally different. For ℝd , for any symmetric quasi-Banach space E(ℝd ) of functions, there exists 0 ≤ f ∈ E(ℝd )(2) such that [270, Theorem 1.2] d

Mf (1 − Δ)− 4 ∈ ̸ ℒ2,∞ . In particular, and unlike the compact case, when taking the function space E(ℝd ) = Λ1 (ℝd ), the Cwikel estimate at the critical value p = 2 for ℳ(2) 1,∞ above cannot be im-

proved to ℒ2,∞ . The Zygmund space L log L(ℝd ) on ℝd is strictly contained within Λ1 (ℝd ). A substitute result for the quasi-Banach ideal ℒ1,∞ in the Euclidean case is provided by [270, Theorem 1.3]: if f ∈ L log L(ℝd )(2) and 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 log(1 + |t|)dt < ∞,

ℝd

3.6 Notes

� 203

then 1

2 󵄩󵄩 󵄨 󵄨2 −d 󵄩 󵄩󵄩Mf (1 − Δ) 4 󵄩󵄩󵄩ℒ2,∞ ≤ cd ‖f ‖L log L(ℝd )(2) + ( ∫ 󵄨󵄨󵄨f (t)󵄨󵄨󵄨 log(1 + |t|)dt) .

ℝd

Noncommutative integration and asymptotic spectral behavior The Cwikel estimate at the critical dimension on a closed manifold X and the associated integral formula on ℒ1,∞ or ℳ1,∞ implies that, when 0 ≤ f ∈ L log L(X), d

d

M − lim tμ(t, (1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) = t→∞

Vol(𝕊d−1 ) ⋅ ∫ fd volg . d(2π)d X

Here M : L∞ (ℝd+ ) → L∞ (ℝd+ ) is the logarithmic mean operator t

(Mf )(t) =

1 ds , ∫ f (s) log(1 + t) 1+s

t > 0,

f ∈ L∞ (ℝd+ ),

0

and, when the limit of the right-hand side exists, M − lim f (t) = lim (Mf )(t), t→∞

t→∞

f ∈ L∞ (ℝ+ ),

indicates logarithmic mean convergence. Logarithmic mean convergence is weaker than Cesàro convergence. Asymptotic behavior of this kind is known broadly as local Weyl asymptotics, where in this case the convergence is in the logarithmic mean limit rather than in the ordinary limit. In the terminology of Chapter 9 of Volume I, the opd d erator (1 − ΔX )− 4 Mf (1 − ΔX )− 4 , 0 ≤ f ∈ L log L(X), is Dixmier measurable; see also [194, Theorem 4.2]. No extended limits are required to calculate the Dixmier trace or, indeed, recover the integral of the positive function f on the manifold X from the eigenvalues d d of the positive operator (1 − ΔX )− 4 Mf (1 − ΔX )− 4 . Theorem 1.3 in [270] improves the convergence estimate for asymptotic spectral behavior and proves that, when 0 ≤ f ∈ L log L(X), d

d

lim tμ(t, (1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) =

t→∞

Vol(𝕊d−1 ) ⋅ ∫ fd volg . d(2π)d

(3.6)

X

This implies that d

d

φ((1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) =

Vol(𝕊d−1 ) ⋅ ∫ fd volg , d(2π)d

0 ≤ f ∈ L log L(X),

X

for every continuous normalized trace φ on ℒ1,∞ . This follows from Theorem 6.11 in Volume I, since

204 � 3 Integration formulas

ωθ (a) = θ(

1 log(2)

2⋅(2n −1)

∑

k=2n −1

ak ), k

a = {ak }∞ k=0 ∈ l∞ ,

is an extended limit on l∞ for a Banach limit θ on l∞ , and for every positive normalized trace φ on ℒ1,∞ there exists a Banach limit θφ on l∞ such that φ(A) = ωθφ (nμ(n, A)),

0 ≤ A ∈ ℒ1,∞ .

Every continuous trace is a linear combination of four positive traces by Theorem 4.1.10 in Volume I. d d It is still unknown whether the operator (1−ΔX )− 4 Mf (1−ΔX )− 4 is uniquely traceable on ℒ1,∞ , that is, whether d

d

φ((1 − ΔX )− 4 Mf (1 − ΔX )− 4 ) =

Vol(𝕊d−1 ) ⋅ ∫ fd volg , d(2π)d

f ∈ L log L(X),

X

for every normalized trace φ on ℒ1,∞ that is not continuous. The remainder term in spectral asymptotics describing the rate of convergence determines unique traceability (see Chapter 9 of Volume I). For 0 ≤ A ∈ ℒ1,∞ , M(tμ(t, A))(s) = c + O(

1 ), log(1 + s)

s > 0,

for some constant c is equivalent to φ(A) = c for all normalized traces φ on ℒ1,∞ . Weyl asymptotics and unique traceability are inequivalent conditions. Integration formula for the torus Theorem 3.3.1 for Dixmier traces and smooth functions on the torus follows from Connes’ original trace formula [68, Theorem 1]. In Proposition 12 of [70], Connes used a form of zeta function residues to prove the statement of Theorem 3.3.1 for Dixmier traces on ℒ1,∞ and f ∈ L∞ (𝕋d ). The method presented in Section 3.3 is based on expectation values and modulated operators. A proof based on the logarithmic mean of expectation values originated in [192]. Lemma 3.3.4 on modulated operators originated in [168]. Integration formula for the noncommutative torus The noncommutative torus has equivalent presentations to the universal algebra generated by unitaries satisfying Weyl relations [222, 234]. In [84], it is presented as an isospectral deformation; see also [288]. In [234], it is presented as a cross product of the action

3.6 Notes

� 205

of ℤd on C(𝕋d ). Usually, the emphasis is on irrational values for the parameter θ [234]. The noncommutative torus appears in many applications, including string theory; see, for example, [83, 177]. The noncommutative torus has a concrete representation π1 on the Hilbert space l2 (ℤd ) using the Fourier dual of the discrete Moyal product; the realizations are π1 (um )h(n) = e−i⟨m,θn⟩ h(n − m),

n, m ∈ ℤd ,

h ∈ l2 (ℤd ).

The Moyal product itself goes back to Groenewold and Moyal in the 1940s [141, 210]. Representation using the Fourier dual of the Moyal product is the approach used in Chapter 4 for the noncommutative plane. The differential calculus of the noncommutative torus, including connections, curvature, a pseudodifferential calculus, and index pairings, was developed originally by Connes [64] in 1980; see also presentations in [76], [139, Sect. 12.3], [72, p. 758], [87]. In Proposition 13 of [70], Connes used a form of zeta function residue to prove the statement of Theorem 3.5.4(b) for x ∈ L∞ (𝕋dθ ) and Dixmier traces on ℒ1,∞ .

�

Part II: The principal symbol mapping in noncommutative geometry

4 Integration formula for the noncommutative plane 4.1 Introduction Connes’ integration formula for Euclidean space, proven in Theorem 3.2.1, or, equivalently, Theorem 3.4.1, states that for all f ∈ Cc∞ (ℝd ) and all normalized traces φ on ℒ1,∞ , we have d

φ(Mf (1 − Δ)− 2 ) =

Vol(𝕊d−1 ) ⋅ ∫ f (t) dt. d(2π)d ℝd

Here Δ : 𝒮 (ℝd )′ → 𝒮 (ℝd )′ is the Laplacian operator acting on tempered distributions. It follows from Theorem 1.6.25 that the restriction Δ : H 2 (ℝd ) → L2 (ℝd ) is self-adjoint and positive as an unbounded operator on L2 (ℝd ), with domain the Sobolev space H 2 (ℝd ). By Theorem 1.6.19 the restriction of the pseudodifferential operd ator (1 − Δ)− 2 ∈ Ψ−d (ℝd ), d

(1 − Δ)− 2 : L2 (ℝd ) → L2 (ℝd ), is a bounded linear operator that coincides with the operator defined by the Borel functional calculus applied to the self-adjoint restriction of Δ. The operator Mf : L2 (ℝd ) → L2 (ℝd ) given by (Mf u)(t) := f (t)u(t), u ∈ L2 (ℝd ), t ∈ ℝd , is a product operator. The operator d

Mf (1 − Δ)− 2 ∈ ℒ1,∞ , viewed as a product-convolution operator as discussed in Section 1.5, or as a pseudodifferential operator as discussed in Section 1.6, or, more generally, as a Laplacian modulated operator as introduced in Section 2.3, is a weak trace class operator on the Hilbert space L2 (ℝd ). Taking Guillemin’s view of the noncommutative residue, in Chapter 3, we showed that Connes’ formula is an extension of Weyl’s asymptotic law. The asymptotic behavior d of the eigenvalues of the compact operator Mf (1 − Δ)− 2 , when arranged in order of decreasing absolute value, is of order n−1 as n → ∞, and the coefficient of the order n−1 term, extracted by a trace on the ideal ℒ1,∞ , is the Lebesgue integral of the function f . https://doi.org/10.1515/9783110700176-004

210 � 4 Integration formula for the noncommutative plane Connes’ motivation for introducing the integral formula was to demonstrate that the notion of an integral or, more broadly, the notion of Weyl asymptotics and the noncommutative residue can be extended beyond the commutative ∗-algebra Cc∞ (ℝd ) represented as product operators on L2 (ℝd ). This chapter considers Connes’ integration formula when the commutative ∗-algebra of product operators associated with functions on Euclidean space is replaced by the noncommutative ∗-algebra of operators associated with the so-called noncommutative plane. The noncommutative plane Noncommutative geometry begins with the observation that to do geometry on a space X, we typically only need certain algebras of functions on X, such as the spaces of measurable functions, continuous functions, or smooth functions given the pointwise product. From this point of view, it is possible to study certain noncommutative algebras A using the language of geometry, as though A were a space of functions on a fictitious “noncommutative space”. This is illustrated with one of the best studied noncommutative spaces, the Moyal plane, sometimes called the quantum plane. The algebra Poly(ℝ2 ) of complex polynomials on the plane ℝ2 may be presented as the universal associative unital ∗-algebra over the complex numbers generated by two commuting self-adjoint generators X and Y , corresponding to the coordinate functions of ℝ2 , that is, n

Poly(ℝ2 ) := { ∑ cj,k X j Y k : cj,k ∈ ℂ}, j,k=0

where the generators X and Y are taken to be self-adjoint and commuting. This algebra admits a natural noncommutative deformation. For a positive parameter ℏ > 0, the algebra Poly(ℝ2ℏ ) is defined to be the universal associative unital ∗-algebra over the complex numbers generated by two self-adjoint generators X and Y obeying the relation XY − YX = iℏ. The algebra Poly(ℝ2ℏ ) is noncommutative for all ℏ > 0 and hence cannot be an algebra of scalar-valued functions on some space. Nevertheless, we can study the algebra Poly(ℝ2ℏ ) as though it is an algebra of functions on an exotic “space” ℝ2ℏ , which naturally deforms ℝ2 . Similarly, a natural extension of the algebra of polynomials on Poly(ℝd ) can be defined as the universal associative unital ∗-algebra over the complex numbers on d self-adjoint generators {X1 , . . . , Xd } obeying the relations [Xj , Xk ] = iθj,k ,

1 ≤ j, k ≤ d,

4.1 Introduction

� 211

where θ = {θj,k }dj,k=1 is a d × d real antisymmetric matrix. This defines the algebra Poly(ℝdθ ) of polynomial functions on the noncommutative Euclidean space ℝdθ . In mathematical physics the noncommutative plane ℝ2ℏ plays a prominent role in the phase space portrait of quantum mechanics. The one-dimensional momentum p = −iℏ𝜕x and position q = Mx operators obey [q, p] = iℏ and hence generate the algebra Poly(ℝ2ℏ ).

Calculus of the noncommutative plane From the point of view of analysis, it is desirable to realize Poly(ℝ2ℏ ) as an algebra of operators on a Hilbert space. From this realization we can also describe other spaces that are analogous to the algebra of essentially bounded functions and Sobolev spaces on ℝ2 . A formal computation using the Baker–Campbell–Hausdorff formula shows that if X and Y are self-adjoint linear operators on a Hilbert space H obeying [X, Y ] = iℏ, then the unitaries U(t) := exp(itX),

V (t) := exp(itY ),

t ∈ ℝ,

obey the relation U(t)V (s) = eitsℏ V (s)U(t),

t, s ∈ ℝ.

More generally, if {X1 , . . . , Xd } are d self-adjoint operators obeying the relation [Xj , Xk ] = iθj,k ,

1 ≤ j, k ≤ d,

where θ = {θj,k }dj,k=1 is a d × d real anti-symmetric matrix, then the family of operators U(t) := exp(it1 X1 + it2 X2 + ⋅ ⋅ ⋅ + itd Xd ),

t = (t1 , . . . , td ) ∈ ℝd ,

formally obeys i

U(t + s) = e 2 ⟨θt,s⟩ U(t)U(s),

t, s ∈ ℝd .

In Section 4.2, we show that if a family U(t), t ∈ ℝd , of noncommuting unitaries satisfies this formal relationship, then θ must be nondegenerate, real, and antisymmetric, and d must be an even integer. A concrete representation of the family of operator U(t), t ∈ ℝd , as unitaries on the Hilbert space L2 (ℝd ) is given by i

(U(t)u)(s) := e− 2 ⟨t,θs⟩ u(s − t),

u ∈ L2 (ℝd ),

s, t ∈ ℝd .

The von Neumann algebra generated by this family of unitaries {U(t)}t∈ℝd on the Hilbert space L2 (ℝd ) is taken as the definition of the noncommutative algebra of bounded operators L∞ (ℝdθ ).

212 � 4 Integration formula for the noncommutative plane We will also describe the differential calculus for ℝdθ , which is based on a family of d derivations {𝜕j }dj=1 of the algebra L∞ (ℝdθ ) satisfying 𝜕j (U(t)) = itj U(t),

t ∈ ℝd ,

1 ≤ j ≤ d.

Note that the same notation is used for the derivations of the algebra L∞ (ℝdθ ) and the usual partial derivatives on ℝd introduced in Section 1.6. If the context does not make clear which operation is being referred to by the partial derivative notation, then the operation will be described explicitly in statements. There also exists an integral calculus for ℝdθ in the form of a faithful semifinite normal trace τθ on the von Neumann algebra L∞ (ℝdθ ), which plays the role of the integral on L∞ (ℝd ). Following Section 1.2.3 and the subsequent calculations in sections below, using the trace τθ , we define the associated Lp -spaces, 1 ≤ p < ∞, when θ ≠ 0, 1

Lp (ℝdθ ) := {x ∈ L∞ (ℝdθ ) : ‖x‖Lp := τθ (|x|p ) p < ∞}. Note that our presentation of the noncommutative plane works entirely in the Fourier dual picture. Instead of deforming the commutative algebra 𝒮 (ℝd ) of Schwartz functions with pointwise product, we deform the commutative algebra 𝒮 (ℝd ) of Schwartz functions with convolution. The algebra L∞ (ℝdθ ) is defined densely by a deformed convolution on 𝒮 (ℝd ), i

(f ⋆θ u)(s) := ∫ e− 2 ⟨t,θs⟩ f (t)u(s − t) dt,

f , u ∈ 𝒮 (ℝd ).

ℝd

Integration formula for the noncommutative plane The main purpose of this chapter is in proving Connes’ integration formula for noncommutative Euclidean space. Later, in Section 5.5, this will be extended to a trace formula for bounded operators on L2 (ℝd ) that are the equivalent of principal terms of zero-order pseudodifferential operators for the noncommutative plane. The first step in an integral formula for the noncommutative plane is proving product-convolution-type compactness estimates for the noncommutative plane or, rather, since the picture is in the Fourier dual, convolution-product type compactness estimates. For the Euclidean plane, Theorem 1.5.22 in Section 1.5 states that d

x(1 − Δℱ )− 2 ∈ ℒ1,∞ , where x = ℱ −1 Mf ℱ : L1 (ℝd ) → L2 (ℝd ) for a function f ∈ l1 (L2 (ℝd )) acting by convolution, and Δℱ : ℱ −1 H 2 (ℝd ) → L2 (ℝd ),

(Δℱ u)(t) := |t|2 u(t),

u ∈ ℱ −1 H 2 (ℝd ),

4.1 Introduction

� 213

is the Fourier dual of the Laplacian, Δℱ = ℱ −1 Δℱ . Here H 2 (ℝd ) is the Sobolev space from Section 1.6.7. It is clear that d

(1 − Δℱ )− 2 : L2 (ℝd ) → L1 (ℝd ), d

so that the composition x(1 − Δℱ )− 2 is a bounded linear operator on L2 (ℝd ). In the notad

tion of Section 1.5 the operator x(1 − Δℱ )− 2 is the convolution-product operator f (∇)Mg d

where g(ξ) := (1 + |ξ|2 )− 2 , ξ ∈ ℝd . When x ∈ L∞ (ℝdθ ) acts on the Hilbert space L2 (ℝd ) by a deformed convolution, we have the following analogous convolution-product compactness estimate. Define the space W1d (ℝdθ ) := {x ∈ L∞ (ℝdθ ) : ‖x‖W d := ∑ ‖𝜕α (x)‖L1 < ∞}, 1

|α|≤d

where for a multiindex α = (α1 , . . . , αd ), the linear operation 𝜕α on the von Neumann algebra L∞ (ℝdθ ) is defined by α

α

𝜕α := 𝜕1 1 ∘ ⋅ ⋅ ⋅ ∘ 𝜕d d on the domain dom(𝜕α ) := {x ∈ L∞ (ℝdθ ) : 𝜕α (x) ∈ L∞ (ℝdθ )}. The space W1d (ℝdθ ) is dense in L∞ (ℝdθ ) in the norm topology of L∞ (ℝdθ ) and is analogous to a subspace of the integrable functions on the Euclidean plane that are d times weakly differentiable with bounded weak partial derivatives. This is stronger than the condition f ∈ l1 (L2 (ℝd )) in Theorem 1.5.22 for an integrable function f on the Euclidean plane. Theorem 4.1.1 (Compactness estimates). Let d ∈ 2ℕ, and let θ ∈ Md (ℝ) be nondegenerate and antisymmetric. If x ∈ W1d (ℝdθ ), then (a) x(1 − Δℱ )−

d+1 2

∈ ℒ1 , and

󵄩󵄩 − d+1 󵄩 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ ≤ cd,θ ⋅ ‖x‖W d 1

1

for a constant cd,θ > 0 depending on θ; d

(b) x(1 − Δℱ )− 2 ∈ ℒ1,∞ , and

󵄩󵄩 −d 󵄩 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ cd,θ ⋅ ‖x‖W d 1 for a constant cd,θ > 0 depending on θ.

214 � 4 Integration formula for the noncommutative plane Theorem 4.1.1 is proved in Section 4.3. Theorem 4.1.1 ensures that the “convolutiond product” operator x(1 − Δℱ )− 2 is a weak trace class operator on the Hilbert space L2 (ℝd ) when x ∈ W1d (ℝdθ ). The integration formula for the noncommutative plane shows that this weak trace class operator is uniquely traceable for positive traces on ℒ1,∞ and that the trace is a multiple of the faithful normal semifinite trace τθ on the von Neumann algebra L∞ (ℝdθ ). Theorem 4.1.2 (Integration formula). Let d ∈ 2ℕ, and let θ ∈ Md (ℝ) be nondegenerate and antisymmetric. If x ∈ W1d (ℝdθ ), then d

x(1 − Δℱ )− 2 ∈ ℒ1,∞ (L2 (ℝd )), and d

φ(x(1 − Δℱ )− 2 ) =

Vol(𝕊)d−1 ⋅ τθ (x) d(2π)d

for every normalized continuous trace on ℒ1,∞ . Theorem 4.1.2 is proved in Section 4.6. Section 4.2.2 discusses the normalization of the trace τθ such that if x = Lθ (f ) is the Weyl transform Lθ of a Schwartz function f ∈ 𝒮 (ℝd ), then τθ (Lθ (f )) = ∫ f (t)dt,

f ∈ 𝒮 (ℝd ).

ℝd

4.2 Definition of the noncommutative plane Let θ ∈ Md (ℝ) be a d × d real matrix with det(θ) ≠ 0. In this section, we define the von Neumann algebra L∞ (ℝdθ ) of bounded operators on the Hilbert space L2 (ℝd ) associated with a convolution product deformed by θ and a faithful normal semifinite trace τθ on L∞ (ℝdθ ). We also define a differential calculus on dense subalgebras of L∞ (ℝdθ ) using the Weyl transform of the partial derivatives on ℝd . 4.2.1 Canonical commutation relations The algebra L∞ (ℝdθ ) of essentially bounded functions on the noncommutative Euclidean space is the von Neumann algebra generated by a strongly continuous family of unitary operators {U(t)}t∈ℝd acting on the Hilbert space L2 (ℝd ), d ∈ ℕ, satisfying the commutation relation i

U(t + s) = e− 2 ⟨t,θs⟩ U(t)U(s),

t, s ∈ ℝd ,

(4.1)

4.2 Definition of the noncommutative plane

� 215

where θ is a fixed d × d matrix with det(θ) ≠ 0. Ultimately, we will define L∞ (ℝdθ ) as the von Neumann algebra generated by a particular fixed family of unitary operators acting on the Hilbert space L2 (ℝd ) and satisfying (4.1). First, we investigate properties that any such family must satisfy. Lemma 4.2.1. If a family {U(t)}t∈ℝd of unitary operators acting on the Hilbert space L2 (ℝd ) satisfies (4.1), then θ is real and antisymmetric. In particular, d is even. Proof. We have i

e− 2 ⟨t,θs⟩ = U(t + s)U(s)∗ U(t)∗ . The left-hand side is scalar operator, and the right-hand side is a unitary one. A scalar operator is unitary if and only if its absolute value is the identity operator. Thus 󵄨󵄨 − 2i ⟨t,θs⟩ 󵄨󵄨 󵄨󵄨e 󵄨󵄨 = 1,

t, s ∈ ℝd .

So θ must be real. Next, combining the fact that i

U(t + s) = e− 2 ⟨t,θs⟩ U(t)U(s), i

U(−s − t) = e− 2 ⟨s,θt⟩ U(−s)U(−t), with the equality U(t)−1 = U(−t), it follows from multiplying the left- and right-hand sides of these equations that ⟨t, θs⟩ + ⟨s, θt⟩ = 0. Therefore θ has to be antisymmetric. Since θ is antisymmetric, it follows that the transpose of θ is −θ. Hence det(θ) = det(−θ) = (−1)d det(θ). Since θ is nondegenerate (i. e., det(θ) ≠ 0), it follows that d is even. Lemma 4.2.2. Let d ∈ 2ℕ, and let θ be a fixed nondegenerate antisymmetric real d × d matrix. A strongly continuous family of unitaries defined by setting i

(Uθ (t)u)(s) := e− 2 ⟨t,θs⟩ u(s − t),

u ∈ L2 (ℝd ),

s, t ∈ ℝd ,

(4.2)

satisfies the commutation relation (4.1). Proof. Since the group of translations is strongly continuous, it follows that the family {U(s)}s∈ℝd of unitaries is strongly continuous. Moreover, for all s, t ∈ ℝd , we have

216 � 4 Integration formula for the noncommutative plane i

i

(Uθ (s)Uθ (t)u)(v) = e− 2 ⟨s,θv⟩ e− 2 ⟨t,θ(v−s)⟩ u(v − t − s) i

i

= e 2 ⟨t,θs⟩ e− 2 ⟨s+t,θv⟩ u(v − t − s) i

= e 2 ⟨t,θs⟩ (Uθ (s + t)u)(v) for every u ∈ L2 (ℝd ). Definition 4.2.3. Let d ∈ 2ℕ, and let θ be a fixed nondegenerate antisymmetric real d × d matrix. The von Neumann subalgebra in ℒ(L2 (ℝd )) generated by the family of unitary operators {Uθ (t)}t∈ℝd introduced in (4.2) is called the noncommutative plane and denoted by L∞ (ℝdθ ). We sometimes write U(t) instead of Uθ (t) when the use of the unitaries in (4.2) and the matrix θ are clear from the context. Remark 4.2.4. The classical case of Euclidean space ℝd is recovered from a family of unitaries satisfying (4.2) when θ = 0 (although this is not covered by Definition 4.2.3 due to the nondegeneracy assumption). In this case, L∞ (ℝd0 ) is the representation of L∞ (ℝd ) on L2 (ℝd ) by Fourier multipliers. Proof. When θ = 0, U(t) is the operator of translation by t, (U(t)u)(s) = u(s − t),

u ∈ L2 (ℝd ),

s ∈ ℝd .

In terms of the Fourier transform ℱ , we have U(t) = ℱ −1 Met ℱ , where et (s) := e−i⟨t,s⟩ , s, t ∈ ℝd , is an exponential function. Since ℱ is unitary, it follows that the von Neumann algebra generated by {U(t)}t∈ℝd is equal to ℱ ML∞ (ℝd ) ℱ . −1

Therefore the von Neumann algebra generated by {U(t)}t∈ℝd is ∗-isomorphic to L∞ (ℝd ). In what follows, we will exclusively consider the nondegenerate case, thats is, det(θ) ≠ 0. While this excludes the commutative case, it simplifies computations. Characterization of canonical commutation relation algebras The noncommutative Euclidean space L∞ (ℝdθ ) can be characterized as the von Neud

d

mann algebra ℒ(L2 (ℝ 2 )) of bounded operators on the Hilbert space L2 (ℝ 2 ). To show this, we modify the original family {Uθ (t)}t∈ℝd and the matrix θ so that the commutation relation (4.1) becomes simpler. It will follow from this reduction that the von Neumann d algebras L∞ (ℝdθ ) are ∗-isomorphic to ℒ(L2 (ℝ 2 )).

4.2 Definition of the noncommutative plane

� 217

Recall that a spatial ∗-isomorphism of von Neumann algebras ℳ1 ⊆ ℒ(H1 ) and ℳ2 ⊆ ℒ(H2 ) is a unitary isomorphism W : H1 → H2 such that x 󳨃→ W ∗ xW , x ∈ ℳ1 , is a ∗-isomorphism from ℳ1 to ℳ2 . Proposition 4.2.5. Let d ∈ 2ℕ, and let θ be a fixed nondegenerate antisymmetric real d×d d matrix. Define the nondegenerate antisymmetric real d×d matrix Ω acting on ℝd = (ℝ2 )⊕ 2 by d

Ω := S ⊕ 2 ,

S := (

0 1

−1 ). 0

Then there exists a spatial ∗-isomorphism ρθ : L∞ (ℝdθ ) → L∞ (ℝdΩ ). Proof. Since the matrix θ is antisymmetric and real, it follows from basic linear algebra ̃ ∗ , where θ̃ consists of d blocks that there exists an orthogonal matrix Q such that θ = QθQ 2 as follows: 0 θ1

−θ1 0

( ( ( ̃θ = ( ( ( (

0 θ2

−θ2 0

..

.

0 θd

(

) ) ) ) ) ) ) −θ d 2

0 )

2

with θk > 0. Define also the matrix θ1−1/2 ( N := (

θ1−1/2

θ2−1/2

θ2−1/2

(

) ). ..

.)

We have that N = N ∗ , [N, Q] = 0, and ̃ = N ∗ Q∗ θQN. Ω = N θN Introducing the unitary operator W on L2 (ℝd ) by setting (Wu)(t) := det(N)1/2 ⋅ u(QNt),

t ∈ ℝd ,

u ∈ L2 (ℝd ),

(4.3)

218 � 4 Integration formula for the noncommutative plane we obtain that i

(WUθ (QNt)u)(s) = det(N)1/2 e− 2 ⟨QNt,θQNs⟩ u(QNs − QNt) i

= det(N)1/2 e− 2 ⟨t,Ωs⟩ u(QN(s − t)) = (UΩ (t)Wu)(s) for s, t ∈ ℝd and u ∈ L2 (ℝd ). The statement follows from Uθ (QNt) = W ∗ UΩ (t)W ,

t ∈ ℝd ,

and letting ρθ be the spatial ∗-isomorphism implemented by the unitary W . d

The algebra ℒ(L2 (ℝ 2 )) is associated with the canonical commutation relations by the following standard presentation. Using this presentation, we show that the algebra d L∞ (ℝdΩ ) is ∗-isomorphic to the algebra ℒ(L2 (ℝ 2 )) (see [138, Theorem 2] and [134, Proposition 2.13]). For 1 ≤ k ≤ n, we denote by Mk and Dk the unbounded self-adjoint linear operators on L2 (ℝn ) given by (Mk u)(t) := tk u(t),

t = (t1 , . . . , tn ) ∈ ℝn ,

u ∈ L2 (ℝn ),

(4.4)

with domains dom(Mk ) = {u ∈ L2 (ℝn ) : ‖Mk u‖2 < ∞}, and (Dk u)(t) := −i(𝜕k u)(t),

t = (t1 , . . . , tn ) ∈ ℝn ,

u ∈ L2 (ℝn ),

(4.5)

with domains dom(Dk ) = {u ∈ L2 (ℝn ) : ‖𝜕k u‖2 < ∞}. Clearly, [Dk , Ml ] = −iδk,l on dom(Dk ) ∩ dom(Ml ), 1 ≤ k, l ≤ n. The semigroups exp(isMk ), 1 ≤ k ≤ n, s ∈ ℝ, and exp(isDk ), 1 ≤ k ≤ n, s ∈ ℝ, are respectively the operators of modulation and translation on L2 (ℝn ) given by (exp(isMk )u)(t) = eistk u(t),

t = (t1 , . . . , tn ) ∈ ℝn ,

u ∈ L2 (ℝn ),

and (exp(isDk )u)(t) = u(t1 , . . . , tk + s, . . . , tn ),

t = (t1 , . . . , tn ) ∈ ℝn ,

u ∈ L2 (ℝn ).

The following relations are easily checked for t, s ∈ ℝ and 1 ≤ l, k ≤ n: exp(isMk ) exp(itMl ) = exp(itMl ) exp(isMk ), exp(isDk ) exp(itDl ) = exp(itDl ) exp(isDk ),

exp(isMk ) exp(itDl ) = e

−itsδk,l

⋅ exp(itDl ) exp(isMk ).

The Baker–Campbell–Hausdorff formula also supplies the identity

(4.6)

4.2 Definition of the noncommutative plane

� 219

i

exp(isMk + itDl ) = e 2 stδk,l ⋅ exp(isMk ) exp(itDl ) i

= e− 2 stδk,l ⋅ exp(itDl ) exp(isMk ),

s, t ∈ ℝ.

d

Let Ω = S ⊕ 2 . We formally define a representation rΩ of the generators {X1 , . . . , Xd } of Poly(ℝdΩ ) by rΩ (X2j−1 ) := Dj ,

rΩ (X2j ) := Mj ,

1≤j≤

d . 2

Then [rΩ (Xk ), rΩ (Xl )] = iΩk,l , and hence rΩ defines a realization of the algebra Poly(ℝdΩ ) d

as an algebra of unbounded linear operators on L2 (ℝ 2 ) acting on the common core d

𝒮 (ℝ 2 ).

To define the representation of L∞ (ℝdΩ ) rigorously, we show that the mapping satisfying d 2

s = (s1 , . . . , sd ) ∈ ℝd ,

rΩ : UΩ (s) 󳨃→ exp(i(∑ s2j Mj + s2j−1 Dj )), j=1

d

provides a unique ∗-isomorphism rΩ from L∞ (ℝdΩ ) to ℒ(L2 (ℝ 2 )). Theorem 4.2.6. Let Ω be the antisymmetric real nondegenerate d × d matrix from Proposition 4.2.5. There exists a spatial ∗-isomorphism r0 of the algebra L∞ (ℝdΩ ) acting on the d

d

Hilbert space L2 (ℝd ) and the algebra ℒ(L2 (ℝ 2 )) ⊗ 1 acting on the Hilbert space L2 (ℝ 2 ) ⊗ d 2

d 2

d 2

L2 (ℝ ), where 1 : L2 (ℝ ) → L2 (ℝ ) is the identity operator. For s = (s1 , . . . , sd ) ∈ ℝd , we have r0 (UΩ (s)) = e

d

2 s M +s i(∑j=1 2j j 2j−1 Dj )

⊗ 1, d

where the operators Mj and Dj are the unbounded operators on L2 (ℝ 2 ) described above. Proof. Introduce the quadratic form h on ℝd by setting d 2

h(s) := ∑ s2j s2j−1 , j=1

s ∈ ℝd .

(4.7)

Now define the operator V ∈ ℒ(L2 (ℝd )) by setting (Vu)(s) := Evidently,

1 (2π)

d 2

∫ ei(− ℝd

h(s) −⟨s,Ωt⟩+h(t)) 2

u(t)dt,

s ∈ ℝd ,

u ∈ L2 (ℝd ).

220 � 4 Integration formula for the noncommutative plane V =M

e−

ih 2

WMeih ,

where s ∈ ℝd ,

(Wu)(s) := (ℱ −1 u)(Ωs),

u ∈ L2 (ℝd ).

Since W is a unitary operator, we have that V is a unitary operator. Standard computations show that UΩ (s) = Ve

d

2 s q(M )+s i(∑j=1 2j j 2j−1 q(Dj ))

s ∈ ℝd .

V ∗,

d

2 Here the exponent ∑j=1 s2j q(Mj ) + s2j−1 q(Dj ) acts on 𝒮 (ℝd ), and q is given by

(q(Dk )u)(t) := −i𝜕k u(t1 , . . . , td ),

1≤k≤

d , 2

(q(Mk )u)(t) := −itk u(t1 , . . . , td ),

1≤k≤

d . 2

and

d

Note that acting on the dense subspace spanned by pairs u1 ⊗ u2 , u1 , u2 ∈ L2 (ℝ 2 ), within L2 (ℝd ), (V ∗ U(s)V )(u1 ⊗ u2 ) = e

d

2 s q(M )+s i(∑j=1 2j j 2j−1 q(Dj ))

= (e

d 2 s M +s i(∑j=1 2j j 2j−1 Dj )

(u1 ⊗ u2 )

u1 ) ⊗ u2 .

Hence, defining the spatial ∗ -isomorphism x ∈ L∞ (ℝdΩ ),

r0 (x) := V ∗ xV , we have r0 (UΩ (s)) = e

d

2 s M +s i(∑j=1 2j j 2j−1 Dj )

⊗ 1,

and r0 (L∞ (ℝdΩ )) is the von Neumann algebra generated by r0 (UΩ (s)), s ∈ ℝd . Since r0 (L∞ (ℝdΩ )) contains the operators eitMj ⊗ 1 and eitDj ⊗ 1 for t ∈ ℝ and 1 ≤ j ≤ d2 , we have d

r0 (L∞ (ℝdΩ )) = ℒ(L2 (ℝ 2 )) ⊗ 1.

4.2 Definition of the noncommutative plane

� 221

Let ι be the ∗-isomorphism d

d

ι : ℒ(L2 (ℝ 2 )) ⊗ 1 → ℒ(L2 (ℝ 2 )) given by d

ι(A ⊗ 1) = A,

A ∈ ℒ(L2 (ℝ 2 )),

d

d

where A ⊗ 1 acts on L2 (ℝd ) = L2 (ℝ 2 ) ⊗ L2 (ℝ 2 ). Let ρθ be the spatial ∗-isomorphism from Proposition 4.2.5. Set rθ := ι ∘ r0 ∘ ρθ . Then d

rθ : L∞ (ℝdθ ) → ℒ(L2 (ℝ 2 )) is a ∗-isomorphism. Lp -spaces of the noncommutative plane

d

Having established the ∗-isomorphism rθ : L∞ (ℝdθ ) → ℒ(L2 (ℝ 2 )), we now equip L∞ (ℝdθ ) d

with a faithful normal semifinite trace τθ . Let Tr be the standard trace on ℒ(L2 (ℝ 2 )). We set 1

τθ (x) := ((2π)d ⋅ det(θ)) 2 ⋅ Tr(rθ (x)),

x ∈ L∞ (ℝdθ ).

(4.8)

In particular, we can define symmetric function spaces on L∞ (ℝdθ ), which, under d

the isomorphism rθ , correspond to symmetric ideals of compact operators on L2 (ℝ 2 ).

Definition 4.2.7. Let E be a symmetric sequence space. The symmetric ideal E(L∞ (ℝdθ ), τθ ) in L∞ (ℝdθ ) defined in Example 1.2.14 is denoted by E(ℝdθ ). In particular, the Lp -ideals associated with L∞ (ℝdθ ) are denoted by Lp (ℝdθ ), 0 < p < ∞, and rθ (Lp (ℝdθ )) = ℒp . d

Remark 4.2.8. Since the algebra L∞ (ℝdθ ) is isomorphic to ℒ(L2 (ℝ 2 )) when det(θ) ≠ 0, we have Lp (ℝdθ ) ⊂ Lq (ℝdθ ),

p < q.

The inclusion does not hold when θ = 0, and this demonstrates the difference between the noncommutative and commutative theories.

222 � 4 Integration formula for the noncommutative plane We can also define an algebra in L∞ (ℝdθ ) equivalent to the continuous functions vanishing at infinity. Definition 4.2.9. Define the C ∗ -algebra C0 (ℝdθ ) as the C ∗ -algebra whose image under the d

isomorphism rθ is the algebra of compact operators on L2 (ℝ 2 ).

The constant for the trace in (4.8) is chosen such that the Weyl transform Lθ introduced in the next section is an isometry of Hilbert spaces, Lθ : L2 (ℝd ) → L2 (ℝdθ ). This will be shown in Lemma 4.2.17.

4.2.2 Distributions and the Schwartz space of the noncommutative plane In this section, we define algebras of smooth bounded functions on noncommutative Euclidean space, which manifest as subalgebras of L∞ (ℝdθ ). The key tool is the Weyl transform. Definition 4.2.10. For f ∈ 𝒮 (ℝd ), denote by Uθ (f ) ∈ L∞ (ℝdθ ) the operator Uθ (f ) := ∫ f (t)Uθ (t) dt. ℝd

Denote by Lθ (f ) the composition of Uθ with the Fourier transform, that is, d

Lθ (f ) := (2π)− 2 ∫ (ℱ f )(ξ)Uθ (ξ) dξ. ℝd

The mapping Lθ : 𝒮 (ℝd ) → ℒ(L2 (ℝd )) is called the Weyl transform. When it is not necessary to specify the matrix θ, we will write U and L for Uθ and Lθ , respectively. The integrals defining U(f ) and L(f ) converge in the weak operator topology in the sense of Definition 1.4.1. This follows from Lemma 1.4.2 since the unitary family {U(t)}t∈ℝd is strongly continuous and f and ℱ f are integrable. The conditions of Lemma 1.4.2 are satisfied for the functions t 󳨃→ f (t)U(t) and ξ 󳨃→ (ℱ f )(ξ)U(ξ). Lemma 1.4.2 also supplies the estimates ‖U(f )‖∞ ≤ ‖f ‖1 ,

‖L(f )‖∞ ≤ ‖ℱ f ‖1 .

It follows that for every f ∈ 𝒮 (ℝd ), the operators U(f ) and L(f ) belong to the weak closure of {U(t)}t∈ℝd , that is, U(f ) and L(f ) belong to L∞ (ℝdθ ).

4.2 Definition of the noncommutative plane

� 223

Remark 4.2.11. Recalling the formal relationship ξ ∈ ℝd ,

U(ξ) = exp(iξ1 X1 + iξ2 X2 + ⋅ ⋅ ⋅ + iξd Xd ),

where {X1 , . . . , Xd } are the noncommuting generators of Poly(ℝdθ ), we have d

L(f ) = (2π)− 2 ∫ exp(iξ1 X1 + ⋅ ⋅ ⋅ + iξd Xd )(ℱ f )(ξ) dξ. ℝd

Thus we have the heuristic interpretation of the mapping f 󳨃→ L(f ) as a functional calculus for the noncommuting d-tuple (X1 , . . . , Xd ), and L(f ) = “f (X1 , . . . , Xd )”. Recall that by Theorem 4.2.6 there is a ∗-isomorphism d

rθ : L∞ (ℝdθ ) → ℒ(L2 (ℝ 2 )). Most of the important properties of Uθ (f ) and Lθ (f ) follow from their representation as d

integral operators on L2 (ℝ 2 ). The next lemma shows that when f is a Schwartz class d

function, rθ (Uθ (f )) is an integral operator on L2 (ℝ 2 ) with Schwartz class kernel, and moreover every integral operator with Schwartz class kernel is of the form rθ (Uθ (f )). The computations are simplified when d = 2 and θ = S. 2 Lemma 4.2.12. Let S = ( 01 −1 0 ). For all f ∈ 𝒮 (ℝ ), the operator rS (US (f )) ∈ ℒ(L2 (ℝ)) has

the integral kernel k ∈ 𝒮 (ℝ2 ) given by

i

k(s1 , s2 ) = ∫ f (s2 − s1 , s)e 2 (s1 +s2 )s ds,

s1 , s2 ∈ ℝ.

ℝ

Equivalently, 1

k(s1 , s2 ) = (2π) 2 ((1 ⊗ ℱ −1 )(f ))(s2 − s1 ,

s1 + s2 ), 2

s1 , s2 ∈ ℝ,

where ℱ is the one-dimensional Fourier transform. Every Schwartz class function k ∈ 𝒮 (ℝ2 ) is an integral kernel of an operator rS (US (f )) for some unique f ∈ 𝒮 (ℝ2 ). Proof. From relation (4.1), writing (t1 , t2 ) as (t1 , 0) + (0, t2 ), t = (t1 , t2 ) ∈ ℝ2 , we have i

US (t1 , t2 ) = e− 2 t1 t2 US (t1 , 0)US (0, t2 ). Since rS (US (t1 , 0)) = eit1 Dx1 and r(US (0, t2 )) = eit2 M1 , we have i

rS (US (t)) = e− 2 t1 t2 eit1 Dx1 eit2 M1 ,

t ∈ ℝ2 .

224 � 4 Integration formula for the noncommutative plane Thus the action of rS (US (t)) on u ∈ L2 (ℝ) is given by i

(rS (US (t))u)(s) = e− 2 t1 t2 eit2 (s+t1 ) u(s + t1 ) i

= e 2 t2 (t1 +2s) u(s + t1 ),

s ∈ ℝ.

Therefore, for u ∈ L2 (ℝ) and s ∈ ℝ, we have i

(rS (US (f ))u)(s) = ∫ f (t1 , t2 )e 2 t2 (t1 +2s) u(s + t1 ) dt1 dt2 ℝ2

i

= ∫ f (t1 − s, t2 )e 2 t2 (t1 +s) u(t1 ) dt1 dt2 ℝ2 i

= ∫(∫ f (t1 − s, t2 )e 2 t2 (t1 +s) dt2 )u(t1 ) dt1 . ℝ

ℝ

Relabeling the variables supplies the formula i

k(s1 , s2 ) = ∫ f (s2 − s1 , s)e 2 (s1 +s2 )s ds,

s1 , s2 ∈ ℝ.

ℝ

Rewriting the integral as the inverse Fourier transform in the second variable, we also have the formula 1

k(s1 , s2 ) = (2π) 2 (1 ⊗ ℱ −1 )(f )(s2 − s1 ,

s1 + s2 ). 2

Since f is Schwartz class, K is also Schwartz class. Since the operation 1 ⊗ ℱ −1 is an isomorphism of 𝒮 (ℝ2 ), it follows that for every k ∈ 𝒮 (ℝ2 ), there exists a unique f ∈ 𝒮 (ℝ2 ) such that k is the kernel of r(U(f )). Remark 4.2.13. Since LS (f ) = (2π)−1 US (ℱ f ), Lemma 4.2.12 implies that rS (LS (f )) ∈ ℒ(L2 (ℝ)) has the integral kernel 1

k(u1 , u2 ) = (2π)− 2 (ℱ ⊗ 1)(f )(u2 − u1 ,

u1 + u2 ). 2

In the next lemma, we will use the fact that if A is an integral operator on L2 (ℝd ) with Schwartz class kernel, that is, (Au)(t) = ∫ kA (t, s)u(s) ds,

u ∈ L2 (ℝ),

ℝ

where kA ∈ 𝒮 (ℝ2 ), then A is trace class with trace class norm of A being bounded by a sum of Schwartz seminorms of kA . To see this, we may take a representation of kA of the form

4.2 Definition of the noncommutative plane

∞

� 225

s1 , s2 ∈ ℝ 2 ,

kA (s1 , s2 ) = ∑ λj aj (s1 )bj (s2 ), j=1

where aj , bj are Schwartz class functions on ℝ satisfying ‖aj ‖2 = ‖bj ‖2 = 1, and {λj }∞ j=1 is a scalar sequence in l1 . This follows from the characterization of 𝒮 (ℝ2 ) as the projective tensor product of 𝒮 (ℝ) with itself; see, for example, [276, Theorem 51.6]. From the decomposition of kA , the operator A is trace class, and ∞

Tr(A) = ∑ λj ∫ aj (u)bj (u) du = ∫ kA (u, u) du. j=1

ℝ

ℝ

Moreover, the mapping kA → A is continuous from 𝒮 (ℝ2 ) to ℒ1 (L2 (ℝ)). Equivalently, there exist N > 0 and CN > 0 such that ‖A‖ℒ1 ≤ CN ⋅

pα,β (kA ),

∑ |α|+|β|≤N

where pα,β are the Schwartz seminorms defined in Section 1.6.1. Lemma 4.2.14. For all f ∈ 𝒮 (ℝd ), we have Uθ (f ) ∈ L1 (ℝdθ ), and there exist n ≥ 1 and a constant cn,θ that depends on θ such that ‖Uθ (f )‖L1 ≤ cn,θ ⋅

pα,β (f ).

∑ |α|+|β|≤n

The trace τθ of Uθ (f ) is given by τθ (Uθ (f )) = (2π)d ⋅ f (0),

f ∈ 𝒮 (ℝd ).

Equivalently, Lθ (f ) ∈ L1 (ℝdθ ), and τθ (Lθ (f )) = ∫ f (t) dt,

f ∈ 𝒮 (ℝd ).

ℝd

Proof. First, assume that d = 2 and consider θ=S=(

0 1

−1 ). 0

By definition, US (f ) ∈ L1 (ℝ2S ) if and only if rS (US (f )) ∈ ℒ1 (L2 (ℝ)). By Lemma 4.2.12, rS (US (f )) has the integral kernel 1

k(u1 , u2 ) := (2π) 2 (1 ⊗ ℱ −1 )(f )(u2 − u1 ,

u1 + u2 ). 2

226 � 4 Integration formula for the noncommutative plane Since the Fourier transform is a continuous isomorphism of the Schwartz space, we have that k ∈ 𝒮 (ℝ2 ), and the mapping f 󳨃→ k is continuous from 𝒮 (ℝ2 ) to 𝒮 (ℝ2 ). Thus by the discussion preceding the theorem we have rS (US (f )) ∈ ℒ1 (L2 (ℝ)), and there exist N ≥ 0 and a constant cn > 0 such that 󵄩 󵄩 ‖US (f )‖1 = 󵄩󵄩󵄩rS (US (f ))󵄩󵄩󵄩1 ≤ cn

pα,β (f ).

∑ |α|+|β|≤n

This proves the first part of the theorem when θ = S. To see the formula for the trace, by definition we have τS (US (f )) = Tr(rS (US (f ))) and 1

Tr(rS (US (f ))) = ∫ k(u, u) du = (2π) 2 ∫(1 ⊗ ℱ −1 )f (0, u) du. ℝ

ℝ

By the Fourier inversion theorem this delivers the formula f ∈ 𝒮 (ℝ2 ).

Tr(rS (US (f ))) = 2π ⋅ f (0, 0), d

Now d ∈ 2ℕ and Ω := S ⊕ 2 . Then d 2

UΩ (t) = ⨂ US (t2k−1 , tk ), k=1

t = (t1 , . . . , td ) ∈ ℝd , d

under the canonical isomorphism between L2 (ℝd ) and L2 (ℝ2 )⊗ 2 . Suppose the Schwartz function f is of the form d

2 f = ⊗k=1 fk ,

fk ∈ 𝒮 (ℝ2 ),

1≤k≤

d . 2

Then d 2

UΩ (f ) = ⨂ US (fk ). k=1

It follows that d 2

󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩rΩ (UΩ (f ))󵄩󵄩󵄩1 ≤ ∏ 󵄩󵄩󵄩rS (US (fk ))󵄩󵄩󵄩1 ≤ cn,Ω ⋅ k=1

∑ |α|+|β|≤n

pα,β (f )

4.2 Definition of the noncommutative plane

� 227

for a constant cn,Ω . Similarly, d 2

d

Tr(rΩ (UΩ (f ))) = ∏ 2π ⋅ fk (0, 0) = (2π) 2 ⋅ f (0). k=1

The statements follow for θ = Ω since the linear span of all Schwartz functions of the form above are dense in the Fréchet topology of 𝒮 (ℝd ). From the proof of Proposition 4.2.5 it follows that there exist an orthogonal d × d matrix Q and a Hermitian d × d matrix N such that the unitary W : L2 (ℝd ) → L2 (ℝd ),

1

u ∈ L2 (ℝd ),

W (u) = det(N) 2 u ∘ QN,

satisfies Uθ (QNt) = W ∗ UΩ (t)W ,

t ∈ ℝd .

Since W is based on a linear transformation of coordinates, we have that W : 𝒮 (ℝd ) → 𝒮 (ℝd ) is a continuous linear operator. From the properties of the weak operator integral and the substitution t 󳨃→ N −1 Q−1 t, Uθ (f ) = W ∗ ∫ f (t)UΩ (N −1 Q−1 t)dt W ℝd

1

= det(N) ⋅ W ∗ ∫ f (NQt)UΩ (t)dt W = det(N) 2 ⋅ W ∗ UΩ (Wf )W . ℝd

From (4.9) we have 1

1

‖Uθ (f )‖L1 ≤ det(N) 2 ⋅ ‖UΩ (Wf )‖L1 ≤ det(N) 2 cn,Ω ⋅

∑ |α|+|β|≤n

≤ cn,θ ⋅

∑

pα,β (f ),

|α|+|β|≤n

where 1

cn,θ := det(N) 2 ⋅ cn,Ω ⋅ Wn with Wn > 0 the smallest constant such that for all f ∈ 𝒮 (ℝd ), ∑ |α|+|β|≤n

pα,β (W (f )) ≤ Wn ⋅

∑ |α|+|β|≤n

pα,β (f ).

pα,β (W (f ))

(4.9)

228 � 4 Integration formula for the noncommutative plane It also follows from (4.9) that 1

ρθ (Uθ (f )) = det(N) 2 ⋅ UΩ (Wf ), where ρθ is the spatial ∗-isomorphism associated to the unitary W . Since rθ = rΩ ∘ ρθ , 1

rθ (Uθ (f )) = det(N) 2 ⋅ rΩ (UΩ (Wf )), and therefore 1

Tr(rθ (Uθ (f ))) = det(N) 2 ⋅ Tr(rΩ (UΩ (Wf ))) 1

d

d

= det(N) 2 (2π) 2 ⋅ (Wf )(0) = det(N)(2π) 2 ⋅ f (0). 1

The statement follows since det(N) = det(θ)− 2 . The second property of the Weyl transform is that it is an isometry of L2 -spaces, which is a consequence of the following: Corollary 4.2.15. For all f , g ∈ 𝒮 (ℝd ), we have τθ (U(f )∗ U(g)) = (2π)d ⋅ ∫ f (t)g(t) dt ℝd

and τθ (L(f )∗ L(g)) = ∫ f (t)g(t) dt. ℝd

Proof. Let f , g ∈ 𝒮 (ℝd ). First, we observe that U(f )∗ U(g) = U(h), where h ∈ 𝒮 (ℝd ) is given by the integral i

h(t) = ∫ e 2 ⟨θt,s⟩ f (s − t)g(s) ds,

t ∈ ℝd .

ℝd

Hence U(f )∗ U(g) ∈ L1 (ℝdθ ) by Lemma 4.2.14. The statement for U follows from the formula in Lemma 4.2.14, τθ (U(h)) = (2π)d ⋅ h(0) = (2π)d ⋅ ∫ f (t)g(t) dt. ℝd d

The statement for L follows from L(f ) = (2π)− 2 U(ℱ f ) and the Plancherel theorem.

4.2 Definition of the noncommutative plane

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Remark 4.2.16. The bilinear map i

(f , g) 󳨃→ f ⋆θ g := ∫ e− 2 ⟨θt,s⟩ f (t − s)g(s) ds ℝd

is called the twisted convolution of Schwartz functions f and g. A consequence of Corollary 4.2.15 is that the Weyl transform L admits a unique extension to L2 (ℝd ) given by L : L2 (ℝd ) → L2 (ℝdθ ) such that τθ (L(f )L(g)∗ ) = ⟨f , g⟩, f , g ∈ L2 (ℝd ). Equivalently, U admits an extension U : L2 (ℝd ) → L2 (ℝdθ ) such that τθ (U(f )U(g)∗ ) = (2π)d ⋅ ⟨f , g⟩, f , g ∈ L2 (ℝd ). The maps L and U are both surjective, providing us with an explicit description of the space L2 (ℝdθ ). Lemma 4.2.17. The Weyl transform Lθ admits a unique extension to an isometry of Hilbert spaces Lθ : L2 (ℝd ) → L2 (ℝdθ ). Proof. It follows from Corollary 4.2.15 that on 𝒮 (ℝd ), L is injective, and similarly that L is an isometry. We only need to show that L is surjective or, equivalently, that the image of L is dense. It suffices to consider d = 2 and θ = S. By Theorem 1.5.4 every A ∈ ℒ2 (L2 (ℝ)) admits a unique integral kernel kA ∈ L2 (ℝ2 ) with ‖A‖2 = ‖kA ‖L2 (ℝ2 ) . It follows that the set of all operators with Schwartz class kernel kA ∈ 𝒮 (ℝ2 ) is dense in ℒ2 (L2 (ℝ)). Lemma 4.2.12 implies that every kA ∈ 𝒮 (ℝ2 ) is equal to US (f ) for some f ∈ 𝒮 (ℝ2 ), and thus L(S(ℝ2 )) is dense in L2 (ℝ2S ). Definition 4.2.18. The Schwartz space 𝒮 (ℝdθ ) of the noncommutative plane is defined as the image of the classical Schwartz space under Uθ , that is, d

d

𝒮 (ℝθ ) := Uθ (𝒮 (ℝ )).

Equivalently, 𝒮 (ℝdθ ) = Lθ (𝒮 (ℝd )). Remark 4.2.19. The map Uθ : 𝒮 (ℝd ) → 𝒮 (ℝdθ ) is a representation of the algebra 𝒮 (ℝd ) of Schwartz functions given the product of the twisted convolution,

230 � 4 Integration formula for the noncommutative plane f , g ∈ 𝒮 (ℝd ).

Uθ (f )Uθ (g) = Uθ (f ⋆θ g),

Similarly, the map Lθ : 𝒮 (ℝd ) → 𝒮 (ℝdθ ) is a representation of the algebra 𝒮 (ℝd ) of Schwartz functions given the Moyal product deforming pointwise products, f , g ∈ 𝒮 (ℝd ).

Lθ (f )Lθ (g) = Lθ (f ⋅θ g),

Here the Moyal product is related to the twisted convolution by d

f , g ∈ 𝒮 (ℝd ).

f ⋅θ g = (2π)− 2 ℱ −1 (ℱ f ⋆θ ℱ g),

Since Uθ is injective, 𝒮 (ℝdθ ) is linearly isomorphic to 𝒮 (ℝd ). This allows us to define a topology on 𝒮 (ℝdθ ) as the image of the canonical Fréchet topology on 𝒮 (ℝd ) under U = Uθ . The topological dual of 𝒮 (ℝdθ ) is denoted 𝒮 ′ (ℝdθ ). We can extend the Weyl transform to distributions. If T ∈ 𝒮 ′ (ℝd ), then denote by U(T) ∈ 𝒮 ′ (ℝdθ ) the functional defined by (U(T), U(f )) = (2π)d ⋅ (T, f ),

f ∈ 𝒮 (ℝd ).

Similarly, L(T) is the functional defined by (L(T), L(f )) = (T, f ), where f ∈ 𝒮 (ℝd ). We mention without proof that if eξ (t) := ei⟨ξ,t⟩ , t, ξ ∈ ℝd , is an exponential basis function considered as a distribution, then L(eξ ) = U(ξ),

ξ ∈ ℝd .

Lemma 4.2.20. The map (ι(x))(z) := τθ (xz),

x ∈ L∞ (ℝdθ ),

z ∈ 𝒮 (ℝdθ ),

defines an embedding ι : L∞ (ℝdθ ) → 𝒮 ′ (ℝdθ ). Proof. If z = U(f ) where f ∈ 𝒮 (ℝd ), then 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(ι(x))(z)󵄨󵄨󵄨 = 󵄨󵄨󵄨τθ (xU(f ))󵄨󵄨󵄨 ≤ ‖x‖∞ ‖U(f )‖1 . By Lemma 4.2.14 we have ‖U(f )‖1 ≤ cN ⋅ ∑|α|+|β|≤N pα,β (f ), and hence ι(x) is a continuous linear functional on 𝒮 (ℝdθ ), that is, ι(x) ∈ 𝒮 ′ (ℝdθ ).

From hereon, when required, we identify L∞ (ℝdθ ) and its subspaces such as C0 (ℝdθ ) and Lp (ℝdθ ), 0 < p < ∞, with their image inside 𝒮 ′ (ℝdθ ). Unlike the commutative case, Lp (ℝdθ ) for p < 1 is a space of distributions.

4.2 Definition of the noncommutative plane

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4.2.3 Differentiation for the noncommutative plane In addition to the noncommutative integral τθ , there is a natural notion of differentiation on ℝdθ obtained from the Weyl transform of the partial derivates on ℝd . Derivatives The derivative operators 𝜕1 , . . . , 𝜕d on ℝd are the generators of translations in the basis directions. We can make a similar assertion about derivatives on ℝdθ , although to do so, we must define what is meant by translation on the noncommutative plane. We will now define the translation action t 󳨃→ Tt ∈ Aut(L∞ (ℝdθ )), as being the unique automorphism defined on the generators {U(s)}s∈ℝd by Tt (U(s)) = ei⟨t,s⟩ U(s),

t, s ∈ ℝd .

(4.10)

In terms of the heuristic definition U(s) = exp(is1 X1 +⋅ ⋅ ⋅+isd Xd ), we have Tt (Xj ) = Xj +tj , where s = (s1 , . . . , sd ), t = (t1 , . . . , td ) ∈ ℝd . To see that such an automorphism exists, let W (t), t ∈ ℝd , denote the unitary operator on L2 (ℝd ) given by (W (t)u)(r) := ei⟨t,r⟩ u(r),

t, r ∈ ℝd ,

u ∈ L2 (ℝd ),

and define Tt (A) := W (t)AW (−t),

t ∈ ℝd ,

A ∈ ℒ(L2 (ℝd )).

Note that if u ∈ L2 (ℝd ), r, s, t ∈ ℝd , then (W (t)U(s)W (−t)u)(r) = ei⟨t,r⟩ (U(s)e−i⟨t,⋅⟩ u)(r) i

= ei⟨t,r⟩ e− 2 ⟨s,θ(r−s)⟩ e−i⟨t,r−s⟩ u(r − s) i

= ei⟨t,s⟩ e− 2 ⟨s,θr⟩ u(r − s) = ei⟨t,s⟩ (U(s)u)(r). Thus W (t)U(s)W (−t) = ei⟨t,s⟩ U(s),

s, t ∈ ℝd ,

and Tt has the required property. Since Tt maps the span of the generating set {U(t)}t∈ℝd into itself, it follows that Tt uniquely extends to a ∗-automorphism of L∞ (ℝdθ ). For nondegenerate θ, translation has the following description, which has no classical analogy since the operators x ∈ L∞ (ℝdθ ) and U(t), t ∈ ℝd , commute in the classical case.

232 � 4 Integration formula for the noncommutative plane Lemma 4.2.21. For every x ∈ L∞ (ℝdθ ) and every t ∈ ℝ, Tt (x) = U(−θ−1 t)xU(θ−1 t). Proof. Since the family {U(s)}s∈ℝd generates L∞ (ℝdθ ) in the weak operator topology and the two sides of the desired equality are weakly continuous and linear in x, it suffices to consider x = U(s) with s ∈ ℝd . We have already verified that Tt (U(s)) = ei⟨t,s⟩ U(s). On the other hand, from (4.1), U(−t)U(s)U(t) = ei⟨θt,s⟩ U(s),

t, s ∈ ℝd .

Thus U(−θ−1 t)U(s)U(θ−1 t) = ei⟨t,s⟩ U(s). This completes the proof. It follows from (4.10) that Tt : 𝒮 (ℝdθ ) → 𝒮 (ℝdθ ), since Tt (U(f )) = ∫ f (s)ei⟨t,s⟩ U(s)ds = U(fet ),

t ∈ ℝd ,

f ∈ 𝒮 (ℝd ),

ℝd

where et is the exponential function et (s) = ei⟨t,s⟩ , t, s ∈ ℝd . Translation can be extended to an automorphism of distributions x ∈ 𝒮 ′ (ℝdθ ) by (Tt (x), U(f )) := (x, U(fet )),

t ∈ ℝd ,

f ∈ 𝒮 (ℝd ).

Translation on the noncommutative plane can alternatively be defined by the Weyl transform of translation on ℝd . Remark 4.2.22. Denote by t 󳨃→ Tt also the action of ℝd on 𝒮 ′ (ℝd ) by translation. Equation (4.10) is equivalent to Tt (Lθ (f )) = Lθ (Tt (f )),

t ∈ ℝd ,

f ∈ 𝒮 (ℝd ),

where translation of the operator Lθ (f ) appears on the right-hand side, and translation of the Schwartz function on ℝd appears on the left-hand side of the equality. We now define derivations of the von Neumann algebra L∞ (ℝdθ ) that will correspond to the Weyl transform of the partial derivatives on ℝd . We use the same notation for the derivation of the algebra L∞ (ℝdθ ) and the partial derivative of a function, since

4.2 Definition of the noncommutative plane

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the context should make clear whether the operation is applied to an operator or a function. Definition 4.2.23. For 1 ≤ k ≤ d, the derivation 𝜕k on L∞ (ℝdθ ) is defined as the generator of the semigroup t 󳨃→ Ttek , t > 0, of ∗-automorphism of L∞ (ℝdθ ), where ek is the basis vector (0, 0, . . . , 1, . . . , 0) with 1 in the kth position and zero elsewhere. That is, for x ∈ L∞ (ℝdθ ), we define 𝜕k (x) := lim+

Ttek (x) − x

t→0

t

,

where the limit is taken in the strong operator topology, and the domain of 𝜕k is the set dom(𝜕k ) := {x ∈ L∞ (ℝdθ ) : 𝜕k (x) ∈ L∞ (ℝdθ )}. The limit in the strong operator topology belongs to L∞ (ℝdθ ) since Ttek , t > 0, is a semigroup of automorphisms. For a multiindex α ∈ ℤd+ , we abbreviate α

α

𝜕α := 𝜕1 1 ⋅ ⋅ ⋅ 𝜕d d with domain dom(𝜕α ) := {x ∈ L∞ (ℝdθ ) : 𝜕α (x) ∈ L∞ (ℝdθ )}. Let 1 ≤ k ≤ d. The unbounded derivation 𝜕k on the von Neumann algebra L∞ (ℝdθ ) is spatially generated, meaning that 𝜕k (x) = [Mk , x], where the operator (Mk u)(t) = tk u(t),

t = (t1 , . . . , td ) ∈ ℝd ,

u ∈ dom(Mk ),

introduced in (4.4) with domain dom(Mk ) = {u ∈ L2 (ℝd ) : tk u(t) ∈ L2 (ℝd )}, is the self-adjoint generator of the group of unitaries t → W (tek ) acting on the Hilbert space L2 (ℝd ). In terms of the generators t → U(t), it follows from (4.10) that the partial derivation in Definition 4.2.23 satisfies 𝜕j (U(t)) = itj U(t),

t ∈ ℝd .

(4.11)

234 � 4 Integration formula for the noncommutative plane In terms of the representation f → U(f ) and the Weyl transform f → L(f ) of a Schwartz function f ∈ 𝒮 (ℝd ), the description of the derivations {𝜕j }dj=1 is simple. Theorem 4.2.24. For 1 ≤ j ≤ d, the derivations 𝜕j of L∞ (ℝdθ ) restrict to derivations 𝜕j : 𝒮 (ℝdθ ) → 𝒮 (ℝdθ ) with 𝜕j (U(f )) = U(iMj f ),

f ∈ 𝒮 (ℝd ),

1 ≤ j ≤ d,

where (Mj f )(t) = tj f (t) is the product operator like before, and 𝜕j (L(f )) = L(𝜕j f ),

1 ≤ j ≤ d,

f ∈ 𝒮 (ℝd ),

where 𝜕j f on the right-hand side of the formula is the usual partial derivative of f . It is immediate that {𝜕j }dj=1 mutually commute on 𝒮 (ℝdθ ). This description permits us

to extend the derivations {𝜕1 , . . . , 𝜕d } of L∞ (ℝdθ ) to linear endomorphisms of 𝒮 ′ (ℝdθ ). Definition 4.2.25. For all T ∈ 𝒮 ′ (ℝdθ ) and 1 ≤ j ≤ d, denote by 𝜕j (T) the distribution (𝜕j (T), x) = (T, 𝜕j (x)),

x ∈ 𝒮 (ℝdθ ).

The notation for partial derivatives on ℝdθ is chosen to be identical to partial derivatives on ℝd . The Weyl transform is an explicit topological isomorphism L : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝdθ ) under which the partial derivatives on ℝd are mapped to partial derivations on ℝdθ . Sobolev spaces As in the commutative case, with the notation of a partial derivative of a tempered distribution and the embedding of Lp -spaces in the space of tempered distributions, we can define Sobolev spaces for the noncommutative plane. Definition 4.2.26. Let k ∈ ℤ+ and 1 ≤ p ≤ ∞. The Sobolev space Wpk (ℝdθ ) is the subset of x ∈ 𝒮 ′ (ℝdθ ) such that x ∈ Lp (ℝdθ ) and 𝜕α (x) ∈ Lp (ℝdθ ) for all |α| ≤ k. The Sobolev norm ‖x‖Wpk is defined by ‖x‖Wpk := ∑ ‖𝜕α (x)‖p . |α|≤k

Lemma 4.2.27. Let k ∈ ℤ+ and 1 ≤ p ≤ ∞. The Sobolev space Wpk (ℝdθ ) is a Banach space.

4.2 Definition of the noncommutative plane

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Proof. Let {xn }∞ n=0 be a Cauchy sequence in the Sobolev norm ‖ ⋅ ‖Wpk . In particular, for all |α| ≤ k, the sequence {𝜕α (xn )}n=0 ∞

is Cauchy in Lp (ℝdθ ). Denote by yα the limit in Lp (ℝnθ ), yα := lim 𝜕α (xn ). n→∞

We will show that y0 ∈ Wpk (ℝdθ ) and 𝜕α (y0 ) = yα for all |α| ≤ k. Since the sequence

d ′ d {𝜕α (xn )}∞ n=0 converges in Lp (ℝθ ), a forteriori it converges in 𝒮 (ℝθ ), that is, for every d z ∈ 𝒮 (ℝθ ), we have

(𝜕α (xn ), z) → (yα , z). However, by definition (𝜕α (xn ), z) = (xn , 𝜕α (z)), and therefore (yα , z) = lim (xn , 𝜕α (z)) = (y0 , 𝜕α (z)), n→∞

that is, yα = 𝜕α (y0 ). Since yα ∈ Lp (ℝdθ ), it follows that y0 ∈ Wpk (ℝdθ ) and xn → y0 in Wpk (ℝdθ ).

Since the Weyl transform is an isometric isomorphism between L2 (ℝd ) and L2 (ℝdθ ), and, moreover, it exchanges the derivatives on ℝd with those on ℝdθ , we immediately obtain the following description of L2 -Sobolev spaces. Lemma 4.2.28. The Weyl transform L : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝdθ ) restricts to an isomorphism of Sobolev spaces L : H k (ℝd ) → W2k (ℝdθ ) for every k ∈ ℤ+ . In the following lemma, we use the fact that 𝒮 (ℝdθ ) contains an orthonormal basis of L2 (ℝdθ ) consisting of idempotent elements of 𝒮 (ℝdθ ). The existence of such a basis demonstrates the difference between the commutative and noncommutative cases: the Schwartz space 𝒮 (ℝd ) given the pointwise product or the convolution product contains no idempotents besides 0.

236 � 4 Integration formula for the noncommutative plane Lemma 4.2.29. Let k ∈ ℤ+ and 1 ≤ p ≤ ∞. (a) If f ∈ 𝒮 (ℝd ), then U(f ) ∈ Wpk (ℝdθ ).

(b) The Schwartz space 𝒮 (ℝdθ ) is dense in Lp (ℝdθ ) when p ≠ ∞. In particular, Wpk (ℝdθ ) is dense in Lp (ℝdθ ) when p ≠ ∞.

Proof. (a) Let f ∈ 𝒮 (ℝd ). It is sufficient to verify that 𝜕α (U(f )) ∈ L1 (ℝdθ ) for every multiindex α ∈ ℤd+ . We use the results of [134]. By [134, Lemma 2.4] there exists a sequence {ekl }k,l∈ℤd/2 ⊂ +

𝒮 (ℝdθ ) such that

∗ (i) ek1 l1 ek2 l2 = δl1 ,k2 ek1 l2 and ekl = elk ; (ii) τθ (ekk ) = 1; (iii) ∑k∈ℤd/2 ekk = 1 in the strong operator topology. +

By Proposition 2.5 in [134] we have U(f ) =

∑ ckl ekl ,

k,l∈ℤd/2 +

∑ |ckl | < ∞,

k,l∈ℤd/2 +

where the series converges in the L1 -norm. Thus U(f ) ∈ L1 (ℝdθ ). Let fα (t) = (it)α f (t), t ∈ ℝd . By Theorem 4.2.24, 𝜕α (U(f )) = U(fα ). Since fα ∈ 𝒮 (ℝd ), it follows that 𝜕α (U(f )) ∈ L1 (ℝdθ ). This proves (a). To prove (b), note that for every x ∈ L1 (ℝdθ ), ∑ ekk xell = ( ∑ ekk )x( ∑ ell ) → x

k,l≤N

k≤N

l≤N

d/2 in the ℒ1 -norm as N → ∞ (here N ∈ ℤd/2 + , and, respectively, N → ∞ in ℤ+ ). Note that d d ekk xell is a scalar multiple of ekl ∈ 𝒮 (ℝθ ). Since 𝒮 (ℝθ ) is a linear space, it follows that

∑ ekk xell ∈ 𝒮 (ℝdθ ) ⊂ W1d (ℝdθ ).

k,l≤N

This proves (b) for p = 1. The statement is similarly proved for p > 1. In the next proposition, we consider the properties of the Sobolev spaces Wpk (ℝdθ ). The properties of the noncommutative Sobolev spaces are simpler than those of commutative Sobolev spaces due to the nesting property observed in Remark 4.2.8. Proposition 4.2.30. Let k ∈ ℤ+ and 1 ≤ p ≤ ∞. (i) The Sobolev space Wpk (ℝdθ ) is a ∗-algebra.

(ii) Let q, r ≥ 1 be such that p1 + q1 = r1 . If y ∈ Wpk (ℝdθ ) and z ∈ Wqk (ℝdθ ), then yz ∈ Wrk (ℝdθ ). (iii) If p < q, then Wpk (ℝdθ ) ⊂ Wqk (ℝdθ ).

4.2 Definition of the noncommutative plane

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Proof. Given the nesting of Lp (ℝdθ ) observed in Remark 4.2.8, (iii) is immediate from the definition of Wpk (ℝdθ ). Statement (ii) is a straightforward consequence of the Hölder inequality for Schatten norms and the Leibniz rule 𝜕α (yz) = ∑ cβ,γ 𝜕β (y) ⋅ 𝜕γ (z). β,γ≥0 β+γ=α

Here y, z ∈ Wpk (ℝdθ ). Note that this identity for y, z ∈ 𝒮 (ℝdθ ) is a consequence of the Leibniz rule for the derivatives on ℝdθ and may be extended to Wpk (ℝdθ ) by continuity and the density of 𝒮 (ℝdθ ) in Wpk (ℝdθ ), as proved in Lemma 4.2.29.

The first statement follows from (iii) and (ii). Let y, z ∈ Wpk (ℝdθ ). If 1 ≤ p ≤ 2, then

z ∈ Wpk (ℝdθ ) ⊂ Wpk∗ (ℝdθ ), where p∗ is the Hölder conjugate of p. Hence yz ∈ W1k (ℝdθ ) ⊂ Wpk (ℝdθ ). Similarly, if 2 ≤ p ≤ ∞, then yz ∈ W kp (ℝdθ ) ⊂ Wpk (ℝdθ ) since r = 2

p 2

≥ 1. Thus

Wpk (ℝdθ ) is an algebra and thus a ∗-algebra, which follows from 𝜕α (x ∗ ) = (−1)α 𝜕α (x), x ∈ Wpk (ℝdθ ).

Remark 4.2.31. Proposition 4.2.30 implies that Sobolev embedding theorems become largely trivial for the noncommutative plane ℝdθ . For example, a classical Sobolev embedding theorem states that for s > d2 , we have W2s (ℝd ) ⊂ L∞ (ℝd ). This should be compared with Proposition 4.2.30(iii), which gives the embedding W2k (ℝdθ ) ⊂ L∞ (ℝdθ ) for k ≥ 0. In fact, there is no loss of smoothness since k W2k (ℝdθ ) ⊂ W∞ (ℝdθ ),

k ∈ ℤ+ .

This further demonstrates the counterintuitive nature of ℝdθ for nondegenerate θ. We note a final equivalence between norms on Sobolev spaces. Let d

Ω := S ⊕ 2 ,

S := (

0 1

−1 ), 0

be the block diagonal d × d matrix from Proposition 4.2.5. Then there is a spatial ∗-isomorphism ρθ : L∞ (ℝdθ ) → L∞ (ℝdΩ )

238 � 4 Integration formula for the noncommutative plane such that f ∈ 𝒮 (ℝd ),

ρθ (Uθ (f )) = UΩ (Wf ), where 1

(Wu)(t) := det(N) 2 u(QNt),

u ∈ L2 (ℝd ),

t ∈ ℝd ,

for the orthogonal matrix Q and diagonal matrix N such that Ω = (QN)∗ θQN. Lemma 4.2.32. Let k ∈ ℕ and 1 ≤ p ≤ ∞. The spatial ∗-isomorphism ρθ restricts to an invertible bounded mapping ρθ : Wpk (ℝdθ ) → Wpk (ℝdΩ ) such that ‖ρθ (x)‖W k (ℝd ) ≤ cd,p,k,θ ⋅ ‖x‖W k (ℝd ) p

p

Ω

θ

for a constant depending on d, on p, k ∈ ℕ, and on θ. Proof. By construction 1

‖x‖Lp (ℝd ) = det(θ) 2p ⋅ ‖ρθ (x)‖Lp (ℝd ) . Ω

θ

From (4.9), for a multiindex α such that 1 ≤ |α| ≤ k, 𝜕Ωα (ρθ (Uθ (f ))) = 𝜕Ωα (UΩ (Wf )) = ∫ (Wf )(s)(is)α UΩ (s)ds. ℝd 1

Since (Wf )(t) = det(N) 2 f (QNt), using a change of variables, we have α

∫ (Wf )(s)(is)α UΩ (s)ds = ∫ f (s)(iQ−1 N −1 s) UΩ (Q−1 N −1 s)ds. ℝd

ℝd

For each multiindex β such that |β| = |α|, there are coefficients cβ depending on the entries of the matrix Q−1 N −1 such that α

(iQ−1 N −1 s) = ∑ cβ ⋅ (is)β . |β|=|α|

Hence ∫ (Wf )(s)(is)α UΩ (s)ds = ∑ cβ ∫ f (s)(is)β UΩ (Q−1 N −1 s)ds ℝd

|β|=|α|

ℝd

4.2 Definition of the noncommutative plane

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∑ cβ ∫ f (s)(is)β ρθ (Uθ (s))ds = ρθ ( ∑ cβ ∫ f (s)(is)β Uθ (s)ds) |β|=|α|

|β|=|α|

ℝd

ℝd β

= ρθ ( ∑ cβ 𝜕θ (Uθ (f ))). |β|=|α|

In summary, β

𝜕Ωα (ρθ (Uθ (f ))) = ρθ ( ∑ cβ 𝜕θ (Uθ (f ))). |β|=|α|

Therefore, for each multiindex α with 1 ≤ |α| ≤ k, −1 󵄩󵄩 α 󵄩 󵄩 β 󵄩 󵄩󵄩𝜕Ω (ρθ (Uθ (f )))󵄩󵄩󵄩Lp (ℝd ) ≤ det(θ) 2p ck,θ ⋅ ∑ 󵄩󵄩󵄩𝜕θ (Uθ (f ))󵄩󵄩󵄩Lp (ℝd ) , Ω θ |β|=|α|

where ck,θ := max |cβ | 1≤|β|≤k

is a constant. It is evident that ρθ is a linear and injective map 𝒮 (ℝdθ ) → 𝒮 (ℝdΩ ). By construction ρθ is surjective, since for every f ∈ 𝒮 (ℝd ), UΩ (f ) = ρθ (Uθ (W −1 f )), where W −1 f ∈ 𝒮 (ℝd ). The statement now follows from Lemma 4.2.29 since 𝒮 (ℝdθ ) and 𝒮 (ℝdΩ ) are dense in Wpk (ℝdθ ) and Wpk (ℝdΩ ), respectively.

4.2.4 Additional symmetries of the noncommutative plane We have already introduced the group {Tt }t∈ℝd of translation automorphisms of L∞ (ℝdθ ). In terms of the heuristic description of ℝdθ as a space with coordinates {X1 , . . . , Xd }, the translation Tt shifts the generators {X1 , . . . , Xd } by scalars, that is, Tt (Xj ) = Xj + tj ,

1 ≤ j ≤ d, t = (t1 , . . . , td ) ∈ ℝd .

This induces a symmetry of the noncommutative plane ℝdθ because the relation [Xj , Xk ] = iθj,k is preserved under Tt , that is, [Tt (Xj ), Tt (Xk )] = [Xj + tj , Xk + tk ] = [Xj , Xk ],

1 ≤ j, k ≤ d.

Unlike Euclidean space, the symmetries of ℝdθ are quite restricted. For example, there is no action of dilation, since for r ≠ ±1, the variables {rX1 , . . . , rXd } do not satisfy the commutation relation [rXj , rXk ] = iθj,k , unless θ = 0.

240 � 4 Integration formula for the noncommutative plane If we consider a general linear change of variables, d

Xj′ := ∑ Aj,k Xj + tj , k=1

where A = {Aj,k }dj,k=1 is a real matrix, and t = {tj }dj=1 is a vector, we see that the variables {X1′ , X2′ , . . . , Xd′ } obey [Xj′ , Xk′ ] = iθj,k if and only if AθA∗ = θ. The set of all matrices obeying this relation is a Lie group, which forms a group of linear symmetries of ℝdθ . We define the symplectic group Sp(θ, d) := {A ∈ Md (ℝ) : A∗ θA = θ}. If, in particular, Ω is as in Proposition 4.2.5, then Sp(Ω, d) is the usual symplectic group Sp(d, ℝ). The rigorously defined action of Sp(θ, d) on L∞ (ℝdθ ) is given in the following lemma. Lemma 4.2.33. Define a unitary action A 󳨃→ WA of the special orthogonal group SO(d) on L2 (ℝd ) by u ∈ L2 (ℝd ).

WA u := u ∘ A−1 ,

If A ∈ Sp(θ, d) ∩ SO(d), then conjugation by WA defines a trace-preserving group of ∗-automorphisms of L∞ (ℝdθ ). Proof. Recall from Lemma 4.2.2 that i

(U(t)u)(s) = e− 2 ⟨t,θs⟩ u(s − t). Similarly to the proof of Proposition 4.2.5, i

(WA−1 U(t)u)(s) = (U(t)u)(As) = e− 2 ⟨t,θAs⟩ u(As − t) i

= e− 2 ⟨A

−1

t,A∗ θAs⟩

u(A(s − A−1 t))

= (U(A−1 t)WA−1 u)(s) since A∗ θA = θ. In other words, WA−1 U(t)WA = U(A−1 t). Since the family {U(t)}t∈ℝd generates L∞ (ℝdθ ), it follows that WA−1 xWA ∈ L∞ (ℝdθ ),

x ∈ L∞ (ℝdθ ).

4.3 Product-convolution estimates for the noncommutative plane

� 241

Hence the conjugation by WA defines a group of ∗-automorphisms of L∞ (ℝdθ ). To see that this group of ∗-isomorphisms is trace preserving, let x := ∫ f (s)U(s)ds ∈ 𝒮 (ℝdθ ), ℝd

and note that WA−1 xWA = ∫ f (s)U(A−1 s)ds = ∫ f (As)U(s)ds. ℝd

ℝd

Thus τθ (WA−1 xWA ) = (2π)d ⋅ (WA−1 f )(0) = (2π)d ⋅ f (0) = τθ (x). Since 𝒮 (ℝdθ ) is dense in L1 (ℝdθ ), the action of conjugation by WA is trace preserving. Remark 4.2.34. The conjugation by the unitary operator WA on L2 (ℝd ) admits a simple expression in terms of the Weyl transform. We have WA∗ L(f )WA = L(WA (f )),

f ∈ 𝒮 (ℝd ),

A ∈ Sp(θ, d).

In this sense the action of Sp(θ, d) on 𝒮 (ℝdθ ) is the image under the Weyl transform of its action on 𝒮 (ℝd ).

4.3 Product-convolution estimates for the noncommutative plane In this section, we prove Theorem 4.1.1, which is a noncommutative analogue of productconvolution estimates in Section 1.5. The proof uses an algebraic argument based on properties of the Dirac operator on ℝd . Let Mk , k = 1, . . . , d, be the monomial product operators from equation (4.4). Definition 4.3.1. Define d

Δℱ := − ∑ Mk2 = ℱ −1 Δℱ k=1

with domain dom(Δℱ ) := {u ∈ L2 (ℝd ) : |t|2 u(t) ∈ L2 (ℝd )} = ℱ −1 H 2 (ℝd ), where Δ is the Laplacian on ℝd . The gradient ∇ℱ is defined as the vector of partial derivatives ∇ℱ := (M1 , M2 , . . . , Md ) = ℱ −1 ∇ℱ

(4.12)

242 � 4 Integration formula for the noncommutative plane with domain d

⋂ dom(Mk ) = ℱ −1 H 1 (ℝd ).

k=1

The next result corresponds to the statement in Theorem 1.5.6. Lemma 4.3.2. If x ∈ L2 (ℝdθ ) and g ∈ L2 (ℝd ), then xg(∇ℱ ) ∈ ℒ2 (L2 (ℝd )), and d

‖xg(∇ℱ )‖ℒ2 = (2π)− 2 ‖x‖L2 (ℝd ) ‖g‖L2 (ℝd ) . θ

Proof. By Lemma 4.2.17 there exists f ∈ L2 (ℝd ) such that d

x = (2π)− 2 U(f ), and by Corollary 4.2.15 d

(4.13)

‖x‖L2 (ℝd ) = (2π)− 2 ‖U(f )‖L2 (ℝd ) = ‖f ‖L2 (ℝd ) . θ

θ

Let u ∈ L2 (ℝd ). By definition we have g(∇ℱ )u = gu, and therefore xg(∇ℱ )u = x ⋅ gu is an integral operator with square-integrable kernel on ℝd × ℝd given by d

i

(t, s) 󳨃→ (2π)− 2 f (t − s)g(s)e 2 ⟨s,θt⟩ . Hence d

d

‖xg(∇ℱ )‖ℒ2 = (2π)− 2 ‖f ‖L2 (ℝd ) ‖g‖L2 (ℝd ) = (2π)− 2 ‖x‖L2 (ℝd ) ‖g‖L2 (ℝd ) , θ

where the last equality follows from (4.13). Product-convolution estimates for Lp (ℝdθ ) and Lp,∞ (ℝdθ ) now follow from the abstract product-convolution estimates in Theorem 1.5.10. We obtain from Lemma 4.3.2 the following, denoting by τ0 the Lebesgue integral on integrable functions on ℝd . ̄ ∞ (ℝd ), τθ ⊗ τ0 ), then Lemma 4.3.3. Let 2 < p < ∞. If x ⊗ g ∈ (ℒp,∞ ∩ ℒ∞ )(L∞ (ℝdθ )⊗L xg(∇ℱ ) ∈ ℒp,∞ (L2 (ℝd )), and there exists a constant cp dependent on p such that ‖xg(∇ℱ )‖ℒp,∞ ≤ cp ⋅ ‖x ⊗ g‖ℒp,∞ . Similarly, if x ⊗ g ∈ ℒp (L∞ (ℝdθ ) ⊗ L∞ (ℝd )), then xg(∇ℱ ) ∈ ℒp (L2 (ℝd )), and there exists a constant c > 0 such that ‖xg(∇ℱ )‖ℒp ≤ c ⋅ ‖x ⊗ g‖ℒp .

4.3 Product-convolution estimates for the noncommutative plane

� 243

Obtaining product-convolution estimates for ℒp and ℒp,∞ for 0 < p < 2 is more challenging. We make a direct estimate using the Fourier dual Dℱ of the canonical Dirac operator for ℝd . Dirac operators are of fundamental importance in both physics and geometry. However, for us, Dℱ will be a useful tool for proving compactness estimates for productconvolution operators on L2 (ℝdθ ) and in this sense has a technical use quite different from its origins. d

Definition 4.3.4. Let Nd := 2⌊ 2 ⌋ , and let {γj }dj=1 be Nd -dimensional self-adjoint matrices satisfying γi γk + γk γj = 2δj,k ,

1 ≤ j, k ≤ d.

These are called Clifford matrices, and the precise choice of matrices satisfying this relation is immaterial in this section. The Dirac operator Dℱ is the unbounded operator on the Hilbert space ℂN ⊗ L2 (ℝd ) defined by d

Dℱ := ∑ γj ⊗ Mj , j=1

where Mj is the product operator by the jth coordinate defined in (4.4). The operator Dℱ is a self-adjoint operator given the domain dom(Dℱ ) := ℂN ⊗ ℱ −1 H 1 (ℝd ) and has an invariant core Dℱ : ℂN ⊗ 𝒮 (ℝd ) → ℂN ⊗ 𝒮 (ℝd ). On the invariant core 𝒮 (ℝd ) the operators M1 , . . . , Md mutually commute, and the defining relation for the Clifford generators implies that d

D2ℱ = 1 ⊗ (∑ Mj2 ) = −1 ⊗ Δℱ , j=1

where 1 is an abbreviation for the identity operator on ℂNd . Hence the restriction D2ℱ : ℂNd ⊗ ℱ −1 H 2 (ℝd ) → ℂNd ⊗ L2 (ℝd ) is self-adjoint and positive as an unbounded operator on L2 (ℝd ). The first estimate we write in terms of the Dirac operator is the following: Lemma 4.3.5. If x ∈ L1 (ℝdθ ), then 󵄩󵄩 −1− d −1− d 󵄩 󵄩󵄩(Dℱ − i) 2 (1 ⊗ x)(Dℱ − i) 2 󵄩󵄩󵄩ℒ1 (ℂNd ⊗L2 (ℝd )) ≤ ‖x‖L1 .

244 � 4 Integration formula for the noncommutative plane Proof. Let x = u|x| be the polar decomposition of x. For brevity, denote D := Dℱ . We have 1

1

x = |x ∗ | 2 ⋅ u ⋅ |x| 2 . Therefore d

d

(D − i)−1− 2 (1 ⊗ x)(D − i)−1− 2 d

1

1

d

= (D − i)−1− 2 (1 ⊗ |x ∗ | 2 ) ⋅ (1 ⊗ u) ⋅ (1 ⊗ |x| 2 )(D − i)−1− 2 . By Hölder’s inequality we have d

d

‖(D − i)−1− 2 (1 ⊗ x)(D − i)−1− 2 ‖ℒ1

d 1 1 d 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(D − i)−1− 2 (1 ⊗ |x ∗ | 2 )󵄩󵄩󵄩ℒ ⋅ 󵄩󵄩󵄩(1 ⊗ |x| 2 )(D − i)−1− 2 󵄩󵄩󵄩ℒ . 2 2

By Lemma 4.3.2 we have 1 󵄩󵄩 󵄩 󵄩 ∗ 1󵄩 −1− d ∗ 1 󵄩 −1−d ∗ 1 󵄩 󵄩󵄩(D − i) 2 (1 ⊗ |x | 2 )󵄩󵄩󵄩ℒ2 = 󵄩󵄩󵄩(1 − Δℱ ) 2 4 |x | 2 󵄩󵄩󵄩ℒ2 = cd ⋅ 󵄩󵄩󵄩|x | 2 󵄩󵄩󵄩2 = cd ⋅ ‖x‖12 ,

where d

cd := (2π)− 2 ‖g‖L2 ,

− 21 − d4

g(ξ) := (1 + |ξ|2 )

∈ L2 (ℝd ),

ξ ∈ ℝd .

Similarly, 1 1 󵄩󵄩 󵄩 1 −1− d 󵄩 −1−d 󵄩 󵄩󵄩(1 ⊗ |x| 2 )(D − i) 2 󵄩󵄩󵄩2 = 󵄩󵄩󵄩|x| 2 (1 − Δ) 2 4 󵄩󵄩󵄩2 = cd ⋅ ‖x‖12 .

Combining the displays above, we have 󵄩󵄩 −1− d −1− d 󵄩 2 󵄩󵄩(D − i) 2 (1 ⊗ x)(D − i) 2 󵄩󵄩󵄩ℒ ≤ cd ⋅ ‖x‖1 , 1

and we complete the proof since a basic calculation shows that ‖g‖L2 ≤ 1. The next lemma is the core of the proof of Theorem 4.1.1(a). Lemma 4.3.6. If x ∈ W1k (ℝdθ ), k ∈ ℤ+ , then d 󵄩󵄩 󵄩 k−1− d2 (1 ⊗ x)(Dℱ − i)−k−1− 2 󵄩󵄩󵄩ℒ (ℂNd ⊗L (ℝd )) ≤ 2k ⋅ ‖x‖W k . 󵄩󵄩(Dℱ − i) 1 2 1

Proof. For brevity, denote D := Dℱ . Step 1. Let k, m ∈ ℤ+ . Note that (D − i)−k−m : ℂNd ⊗ L2 (ℝd ) → ℂNd ⊗ ℱ −1 H k+m (ℝd ).

� 245

4.3 Product-convolution estimates for the noncommutative plane

If A is a continuous linear operator A : ℂNd ⊗ ℱ −1 H k+m (ℝd ) → ℂNd ⊗ ℱ −1 H k (ℝd ), then (D − i)k A(D − i)−k−m : ℂNd ⊗ L2 (ℝd ) → ℂNd ⊗ L2 (ℝd ) is everywhere defined and bounded. We first show that if x ∈ W1k+m (ℝdθ ), then 𝜕α (x) : ℱ −1 H k+m (ℝd ) → ℱ −1 H k (ℝd ),

|α| = m,

so that γ ⊗ 𝜕α (x) : ℂNd ⊗ ℱ −1 H k+m (ℝd ) → ℂNd ⊗ ℱ −1 H k (ℝd ) for any matrix γ ∈ MN d (ℂ). First, we use the isomorphism U : L2 (ℝd ) → L2 (ℝdθ ), which by Lemma 4.2.28 restricts to U : ℱ −1 H k+m (ℝd ) → W2k+m (ℝdθ ). From the properties in Proposition 4.2.30, 𝜕α (x)U(h) ∈ W2k (ℝdθ ),

x ∈ W1k+m (ℝdθ ),

h ∈ ℱ −1 H k+m (ℝd ),

since 𝜕α (x) ∈ W1k (ℝdθ ) ⊂ W2k (ℝdθ ), U(h) ∈ W2k+m (ℝdθ ) ⊂ W2k (ℝdθ ), and W2k (ℝdθ ) is an algebra. Then U −1 (𝜕α (x)U(h)) ∈ ℱ −1 H k (ℝd ). Hence 𝜕α (x) = U −1 ∘ 𝜕α (x) ∘ U : ℱ −1 H k+m (ℝd ) → ℱ −1 H k (ℝd ) is continuous. Step 2. We prove the statement by induction on k. For k = 0, it is established in Lemma 4.3.5. Now suppose that the statement holds for k ≥ 0 and let us prove it for k + 1. Let x ∈ W1k+1 (ℝdθ ). We have d

d

(D − i)k− 2 (1 ⊗ x)(D − i)−k− 2 −2 d

d

d

d

= (D − i)k− 2 −1 (1 ⊗ x)(D − i)−k− 2 −1 + (D − i)k− 2 −1 [D, 1 ⊗ x](D − i)−k− 2 −2 . We set for brevity

246 � 4 Integration formula for the noncommutative plane d

d

T1 := (D − i)k− 2 −1 (1 ⊗ x)(D − i)−k− 2 −1 , d

d

T2 := (D − i)k− 2 −1 [D, 1 ⊗ x](D − i)−k− 2 −2 . By the induction hypothesis we have T1 ∈ ℒ1 (L2 (ℝd )) and ‖T1 ‖ℒ1 ≤ 2k ‖x‖W k ≤ 2k ‖x‖W k+1 , 1

1

where the last inequality follows directly from Definition 4.2.26. For T2 , using Definition 4.3.4, we obtain d

d

d

T2 = ∑(D − i)k− 2 −1 (γl ⊗ [Ml , x])(D − i)−k− 2 −2 . l=1

Since x ∈ W1k+1 (ℝdθ ), it follows that [Ml , x] = 𝜕l (x) ∈ W1k (ℝdθ ) and T2 is a bounded linear operator by Step 1. By the triangle inequality we have d

d

d

󵄩 󵄩 ‖T2 ‖ℒ1 ≤ ∑ 󵄩󵄩󵄩(D − i)k− 2 −1 (γl ⊗ 𝜕l (x))(D − i)−k− 2 −2 󵄩󵄩󵄩ℒ , 1

l=1

from which it follows by a short calculation that d

d d 󵄩 󵄩 ‖T2 ‖ℒ1 ≤ ∑ 󵄩󵄩󵄩(D − i)k− 2 −1 ⋅ (1 ⊗ 𝜕l (x))(D − i)−k− 2 −2 󵄩󵄩󵄩ℒ . 1

l=1

Applying the induction hypothesis to the element 𝜕l (x) ∈ W1k (ℝdθ ), we obtain d

‖T2 ‖ℒ1 ≤ 2k ∑ ‖𝜕l (x)‖W k ≤ 2k ‖x‖W k+1 . l=1

1

1

It follows that 󵄩󵄩 k− d −k− d −2 󵄩 k+1 󵄩󵄩(D − i) 2 (1 ⊗ x)(D − i) 2 󵄩󵄩󵄩ℒ1 ≤ 2 ‖x‖W k+1 . 1 This verifies the induction step and completes the proof. We require a final technical lemma before proving Theorem 4.1.1. Lemma 4.3.7. Let Ω be the block-diagonal d×d matrix from Proposition 4.2.5. There exists a family {Vt }t∈ℝd of unitary operators on L2 (ℝd ) such that (a) [Vt , x] = 0 for every x ∈ L∞ (ℝdΩ ) and every t ∈ ℝd , (b) Vt ∇ℱ Vt−1 = ∇ℱ + t for every t ∈ ℝd . Proof. Let Dk , Mk , 1 ≤ k ≤ d, be the partial differential and product operators in the kth coordinate on ℝd introduced in (4.4) and (4.5).

4.3 Product-convolution estimates for the noncommutative plane

� 247

Set d

Vt := ∏ eitk Ak , k=1

t ∈ ℝd ,

where 1 A2j−1 := D2j−1 + M2j , 2

1 A2j := D2j − M2j−1 , 2

1≤j≤

d . 2

To see the first statement, it suffices to show that [Vt , UΩ (s)] = 0 for all t, s ∈ ℝd . Furthermore, it suffices to show that [eitk Ak , UΩ (s)] = 0 for all 1 ≤ k ≤ d, tk ∈ ℝ, and all s ∈ ℝd . In other words, we need to show that [Ak , UΩ (s)] = 0 for all 1 ≤ k ≤ d and s ∈ ℝd . It suffices to show that [Ak , U(sl el )] = 0 for all 1 ≤ k, l ≤ d and sl ∈ ℝ, where el is the basis element of ℝd that is zero except the value 1 in the lth coordinate. Recall that U(s2j−1 e2j−1 ), U(s2j e2j ) s2j , s2j−1 ∈ ℝ, are unitary groups with generators 1 B2j−1 = −D2j−1 + M2j , 2

1 B2j = −D2j − M2j−1 . 2

It therefore suffices to show that [Ak , Bl ] = 0 for 1 ≤ k, l ≤ d. This is clearly seen from the fact that [Dk , Ml ] = −iδk,l , 1 ≤ k, l ≤ d, on the common core 𝒮 (ℝd ). We note that by the Weyl canonical commutation relations (4.6), eit2j A2j eit2j−1 A2j−1 = eit2j t2j−1 eit2j−1 A2j−1 eit2j A2j ,

1≤j≤

d , 2

and [eitk Ak , eitn An ] = 0,

1 ≤ k, n ≤ d,

for 1 ≤ k ≤ n ≤ d with k ≠ 2j − 1, n = 2j, for some 1 ≤ j ≤ d2 . It follows for s > 0 that eitk Ak eisMl = eisMl eitk Ak ,

l ≠ k,

(4.14)

and eitk Ak eisMk = eitk s eisMk eitAk ,

1 ≤ k ≤ d.

Therefore Vt eisMk = eitk s eisMk Vt ,

t ∈ ℝd ,

u ∈ ℝ.

In other words, Vt ∇ℱ Vt−1 = ∇ℱ + t for every t ∈ ℝd . This completes the proof of the second statement.

248 � 4 Integration formula for the noncommutative plane We are now ready to prove Theorem 4.1.1 involving trace class and weak trace class estimates for product-convolution type operators on L2 (ℝdθ ). Proof of Theorem 4.1.1. (a) For brevity, denote D := Dℱ and Δ := Δℱ . Let x ∈ L∞ (ℝdΩ ). Setting k = d2 + 1 in Lemma 4.3.6, we obtain d 󵄩󵄩 −d−2 󵄩 󵄩󵄩 ≤ 2 2 +1 ‖x‖ d , 󵄩󵄩(1 ⊗ x)(D − i) +1 󵄩 ℒ1 W2 1

d

x ∈ W12 (ℝdΩ ). +1

Clearly, d ∗ 󵄨2 󵄨󵄨 󵄨 −d−2 ∗ 󵄨󵄨2 ) 󵄨󵄨 = 󵄨󵄨󵄨(x(1 − Δ)− 2 −1 ) 󵄨󵄨󵄨 ⊗ 1. 󵄨󵄨((1 ⊗ x)(D − i)

Since 󵄩󵄩 󵄩󵄩 −d−2 󵄩 − d2 −1 󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩(1 ⊗ x)(D − i) 󵄩ℒ1 (ℂNd ⊗L2 (ℝd )) = 󵄩󵄩x(1 − Δ) 󵄩ℒ1 (L2 (ℝd )) , it follows that d d 󵄩󵄩 − d −1 󵄩 +1 +1 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1 ≤ 2 2 ‖x‖ d +1 ≤ 2 2 ‖x‖W d . 2 1 W 1

d

This has proven the statement for the operator (1 − Δ)− 2 −1 . We now prove the stated 1 ment for (1 − Δ)− 2 − 2 . By the spectral theorem we have x(1 − Δ)−

d+1 2

= ∑ x(1 − Δ)− n∈ℤd

d+1 2

χn+[0,1]d (∇ℱ ),

where the sum converges in strong operator topology. By the triangle inequality we have 󵄩󵄩 󵄩 󵄩 − d+1 󵄩 − d+1 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1 ≤ ∑ 󵄩󵄩󵄩x(1 − Δ) 2 χn+[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ1 . n∈ℤd

Clearly, x(1 − Δ)−

d+1 2

χn+[0,1]d (∇ℱ ) = xχn+[0,1]d (∇ℱ ) ⋅ (1 − Δ)−

d+1 2

χn+[0,1]d (∇ℱ ).

By Hölder’s inequality we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 − d+1 󵄩 − d+1 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1 ≤ ∑ 󵄩󵄩󵄩xχn+[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ1 󵄩󵄩󵄩(1 − Δ) 2 χn+[0,1]d (∇ℱ )󵄩󵄩󵄩∞ . n∈ℤd

Let Vn , n ∈ ℤd , be the unitary operator given by Lemma 4.3.7. We have Vn χn+[0,1]d (∇ℱ )Vn−1 = Vn χ[0,1]d (∇ℱ − n)Vn−1 = χ[0,1]d (∇ℱ ).

4.3 Product-convolution estimates for the noncommutative plane

� 249

Since Vn commutes with x, it follows that Vn xχn+[0,1]d (∇ℱ )Vn−1 = xVn χn+[0,1]d (∇ℱ )Vn−1 = xχ[0,1]d (∇ℱ ). Thus 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 − d+1 󵄩 − d+1 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1 ≤ 󵄩󵄩󵄩xχ[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ1 ∑ 󵄩󵄩󵄩(1 − Δ) 2 χn+[0,1]d (∇ℱ )󵄩󵄩󵄩∞ n∈ℤd

d 󵄩 󵄩 󵄩 󵄩 ≤ cd ⋅ 󵄩󵄩󵄩xχ[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ ≤ cd ⋅ 󵄩󵄩󵄩x(1 − Δ)− 2 −1 󵄩󵄩󵄩ℒ ≤ cd ⋅ ‖x‖W d 1

1

1

for a constant cd > 0 depending on d. The statement is proved for x ∈ W1d (ℝdΩ ). For general θ, we employ Proposition 4.2.5. Denote by W the unitary operator 1

(Wu)(t) := det(N) 2 u(QNt),

u ∈ L2 (ℝd ),

t ∈ ℝd ,

where Q is orthogonal, and N is diagonal. The unitary implements a spatial ∗-isomorphism ρθ between L∞ (ℝdθ ) and L∞ (ℝdΩ ). Hence, if x ∈ W1d (ℝdθ ), then 󵄩󵄩 󵄩 󵄩 −d 󵄩 −1 −d 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1 = 󵄩󵄩󵄩ρθ (x)W (1 − Δℱ ) 2 W 󵄩󵄩󵄩ℒ1 . Note that there is a bounded operator B such that d

d

B = (1 − Δℱ ) 2 W −1 (1 − Δℱ )− 2 W and d

‖B‖∞ ≤ 2 2 ⋅ ‖N‖d . Thus 󵄩󵄩 󵄩 󵄩 −1 −d −d 󵄩 󵄩󵄩ρθ (x)W (1 − Δℱ ) 2 W 󵄩󵄩󵄩ℒ1 ≤ 󵄩󵄩󵄩ρθ (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1 ⋅ ‖B‖∞ . We have 󵄩󵄩 −d 󵄩 󵄩󵄩ρθ (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1 ≤ cd ⋅ ‖ρθ (x)‖W d (ℝd ) 1 Ω since ρθ (x) ∈ W1d (ℝdΩ ) by Lemma 4.2.32. By Lemma 4.2.32, ‖ρθ (x)‖W d (ℝd ) ≤ cd,1,d,θ ⋅ ‖x‖W d (ℝd ) 1

Ω

1

θ

for a constant cd,1,d,θ > 0. Statement (a) follows from combining the above displays: d 󵄩󵄩 −d 󵄩 d 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1 ≤ 2 2 ‖N‖ cd cd,1,d,θ ⋅ ‖x‖W d (ℝd ) . 1 θ

250 � 4 Integration formula for the noncommutative plane (b) For brevity, denote Δ := Δℱ . For every n ∈ ℤd , we set d

Bn := x(1 − Δ)− 2 χn+[0,1]d (∇ℱ ),

An := xχn+[0,1]d (∇ℱ ).

Since the operators χn+[0,1]d (∇ℱ ) are pairwise orthogonal projections, Lemma 4.3.2 entails d 2 󵄩 󵄩 ∑ ‖Bn ‖2ℒ2 = 󵄩󵄩󵄩x(1 − Δ)− 2 󵄩󵄩󵄩ℒ = Cd ⋅ ‖x‖2ℒ2 < ∞ 2

n∈ℤd

for a constant Cd > 0. Since {Bn }n∈ℤd are disjoint from the right, Lemma 1.3.11 implies that d

μ2 ( ⨁ Bn ) ≺ μ2 ( ∑ Bn ) = μ2 (x(1 − Δ)− 2 ), n∈ℤd

(4.15)

n∈ℤd

where the direct sum notation is introduced in Definition 1.3.9. Using Lemma 1.3.8, we infer from (4.15) that 󵄩󵄩 󵄩󵄩 󵄩󵄩 −d 󵄩 . 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ 2󵄩󵄩󵄩 ⨁ Bn 󵄩󵄩󵄩 󵄩 󵄩ℒ1,∞ d n∈ℤ

Obviously, d

Bn = An ⋅ (1 − Δ)− 2 χn+[0,1]d (∇ℱ ),

n ∈ ℤd .

Thus d 󵄩 󵄩 μ(Bn ) ≤ μ(An ) ⋅ 󵄩󵄩󵄩(1 − Δ)− 2 χn+[0,1]d (∇ℱ )󵄩󵄩󵄩∞ ≤

cd

d

(1 + |n|2 ) 2

⋅ μ(An )

for a constant cd > 0. Consequently, d 󵄩󵄩 󵄩󵄩 󵄩󵄩 −d 󵄩 2 − . 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ cd ⋅ 󵄩󵄩󵄩 ⨁ (1 + |n| ) 2 An 󵄩󵄩󵄩 󵄩 󵄩ℒ1,∞

n∈ℤd

Let x ∈ W1d (ℝdΩ ). Let Vn , n ∈ ℤd , be the unitary operator given by Lemma 4.3.7. We have Vn An Vn−1 = xVn χn+[0,1]d (∇ℱ )Vn−1 = xχ[0,1]d (∇ℱ ) = A0 . Hence d 󵄩󵄩 󵄩󵄩 󵄩󵄩 −d 󵄩 2 − −1 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ cd ⋅ 󵄩󵄩󵄩 ⨁ (1 + |n| ) 2 Vn An Vn 󵄩󵄩󵄩 󵄩 󵄩ℒ1,∞

n∈ℤd

4.4 Smooth product-convolution estimates for the noncommutative plane

�

251

󵄩󵄩 󵄩󵄩 −d = cd ⋅ 󵄩󵄩󵄩 ⨁ (1 + |n|2 ) 2 A0 󵄩󵄩󵄩 󵄩 󵄩ℒ1,∞ d n∈ℤ

−d 󵄩 󵄩 = cd ⋅ 󵄩󵄩󵄩{(1 + |n|2 ) 2 }n∈ℤd ⊗ A0 󵄩󵄩󵄩ℒ . 1,∞

By Lemma 1.3.13 we have d 󵄩󵄩 󵄩 󵄩 −d 󵄩 2 − 󵄩󵄩x(1 − Δ) 2 󵄩󵄩󵄩ℒ1,∞ ≤ cd ⋅ 󵄩󵄩󵄩{(1 + |n| ) 2 }n∈ℤd 󵄩󵄩󵄩l1,∞ ‖A0 ‖ℒ1 .

Note that d+1 󵄩 󵄩 󵄩 󵄩 ‖A0 ‖ℒ1 = 󵄩󵄩󵄩xχ[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ ≤ bd ⋅ 󵄩󵄩󵄩x(1 − Δ)− 2 󵄩󵄩󵄩ℒ ≤ cd ⋅ ‖x‖W d , 1 1 1

where the last inequality follows for a constant cd > 0 from Theorem 4.1.1(a). The statement has been shown for x ∈ W1d (ℝdΩ ). The proof of the estimate for x ∈ W1d (ℝdθ ) using the mapping ρθ : W1d (ℝdθ ) → W1d (ℝdΩ ) follows the same steps as Theorem 4.1.1(a).

4.4 Smooth product-convolution estimates for the noncommutative plane In Chapter 5, we need product-convolution estimates for partial derivations of operators x ∈ 𝒮 (ℝdθ ). The estimates of singular values will involve the ideals ℒd,∞ and ℒ1 . Denote 1

δ(T) := [(1 − Δℱ ) 2 , T], where T ∈ ℒ(L2 (ℝd )) restricts to an endomorphism T : 𝒮 (ℝd ) → 𝒮 (ℝd ). Then δ is a bounded derivation on the algebra End(𝒮 (ℝd )). We set δl := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ δ ∘ δ ∘ ⋅ ⋅ ⋅ ∘ δ, l times

l ∈ ℕ,

where it is convenient to define δ0 := id. Theorem 4.4.1. If x ∈ 𝒮 (ℝdθ ), then 1

δl (x)(1 − Δℱ )− 2 ∈ ℒd,∞ ,

l ≥ 0.

Here δl (x) ∈ ℒ(L2 (ℝd )) denotes a bounded operator that restricts to the endomorphism δl (x) : 𝒮 (ℝd ) → 𝒮 (ℝd ). Furthermore, we have

252 � 4 Integration formula for the noncommutative plane 󵄩󵄩 l −1 󵄩 󵄩󵄩δ (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd,l ⋅ ‖x‖W 4l+2 , d

l ≥ 0,

for a constant cd,l > 0. A similar statement is available for ℒ1 . The proof is similar to that of Theorem 4.4.1 and is omitted. Theorem 4.4.2. If x ∈ 𝒮 (ℝdθ ), then δl (x)(1 − Δℱ )−

d+1 2

∈ ℒ1 ,

l ≥ 0,

where δl (x) ∈ ℒ(L2 (ℝd )) denotes a bounded operator that restricts to the endomorphism δl (x) : 𝒮 (ℝd ) → 𝒮 (ℝd ). Furthermore, we have 󵄩󵄩 l − d+1 󵄩 󵄩󵄩δ (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ1 ≤ cd,l ⋅ ‖x‖W 4l+2 , 1

l ≥ 0,

for a constant cd,l > 0. To prove Theorems 4.4.1 and 4.4.2, we use a weak operator integral formula for the bounded extension of δ(T), T ∈ End(𝒮 (ℝd )). 4.4.1 Integral formula for derivations of operators acting on invariant subspaces We prove a weak operator integral formula in the abstract setting in this section. Let H be a separable Hilbert space with a dense subspace 𝒮 . Let J :𝒮→𝒮 be an isomorphism such that ⟨ξ, Jη⟩ = ⟨Jξ, η⟩,

ξ, η ∈ 𝒮 .

Denote by J also the Friedrichs extension of J and suppose that J 2 ≥ 1, so that the Friedrichs extension of J −1 is a bounded operator on H. Let T :𝒮→𝒮 be an endomorphism and denote δ(T) := [J, T], and

T ∈ End(𝒮 ),

4.4 Smooth product-convolution estimates for the noncommutative plane

L(T) := J −1 [J 2 , T],

�

253

T ∈ End(𝒮 ).

The map L in this section does not denote the Weyl transform. We set δl := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ δ ∘ ⋅ ⋅ ⋅ ∘ δ,

l ∈ ℕ,

Ll := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ L ∘ ⋅ ⋅ ⋅ ∘ L,

l ∈ ℕ,

l times

where δ0 := id, and l times

with L0 := id. The next lemma determines whether the endomorphism δ(T) has a bounded extension to H in terms of bounded extensions of L(T) and L2 (T). Weak operator integrals are discussed in Section 1.4. Lemma 4.4.3. Let T ∈ ℒ(H) be such that T|𝒮 ∈ End(𝒮 ), and suppose that L(T), L2 (T) ∈ ℒ(H). Then δ(T) ∈ ℒ(H), and 1

λ 2 J2 1 1 dλ δ(T) = L(T) + ∫ L2 (T) . 2 π (λ + J 2 )2 λ + J2 ∞ 0

The integral here is understood in the sense of the weak operator topology. Proof. Since J is a positive self-adjoint operator on H with bounded inverse, we have the resolvent identity J −1 =

∞

1 1 −1 ∫ λ− 2 (λ + J 2 ) dλ. π

(4.16)

0

The integral converges in the norm topology of ℒ(H). Indeed, since J 2 ≥ 1, we have ‖(λ+J 2 )−1 ‖∞ ≤ (1+λ)−1 , and the resolvent identity proves that the function λ 󳨃→ (λ+J 2 )−1 is continuous in the operator norm. Hence (4.16) defines a Bochner integral in ℒ(H). From (4.16) we have ⟨J −1 u, v⟩ =

∞

1 1 −1 ∫ λ− 2 ⟨(λ + J 2 ) u, v⟩ dλ π

0

for all u, v ∈ H, and the integral converges due to the bound 󵄨󵄨 󵄨 2 −1 −1 󵄨󵄨⟨(λ + J ) u, v⟩󵄨󵄨󵄨 ≤ (1 + λ) ‖u‖H ‖v‖H . Replacing u with J 2 ξ and v with η, where ξ, η ∈ 𝒮 , yields

254 � 4 Integration formula for the noncommutative plane

⟨Jξ, η⟩ =

1 J2 1 ξ, η⟩ dλ, ∫ λ− 2 ⟨ π λ + J2

∞

ξ, η ∈ 𝒮 .

(4.17)

0

From (4.17) and the assumption that T : 𝒮 → 𝒮 we deduce that 1 J2 1 , T] dλ ∫ λ− 2 [ π λ + J2

∞

[J, T] =

0 ∞

1 1 −1 −1 ∫ λ 2 (λ + J 2 ) [J 2 , T](λ + J 2 ) dλ π

=

0 ∞

1 J 1 −1 L(T)(λ + J 2 ) dλ, ∫ λ2 2 π λ+J

=

0

where the integrals are correctly interpreted using the inner product on the invariant subspace 𝒮 as in (4.17). Applying the identity L(T)(λ + J 2 )

−1

= (λ + J 2 ) L(T) + [L(T), (λ + J 2 ) ], −1

−1

where the operators are treated as endomorphisms of 𝒮 , we have L(T)(λ + J 2 )

−1

= (λ + J 2 ) L(T) + (λ + J 2 ) [J 2 , L(T)](λ + J 2 ) −1

= (λ + J 2 ) L(T) + −1

−1

J −1 L2 (T)(λ + J 2 ) . λ + J2

Inserting this into the integral formula derived above for [J, T] yields [J, T] =

∞

1 J 1 L(T) dλ ∫ λ2 π (λ + J 2 )2

0

1 J2 1 −1 L2 (T)(λ + J 2 ) dλ. ∫ λ2 π (λ + J 2 )2

∞

+

0

Recall that ∞

1

∫ λ2 0

1 π dλ = . 2 2 (1 + λ)

By a change of variable, for every t > 0, we have ∞

1

∫ λ2 0

t π dλ = . 2 (t 2 + λ)2

−1

4.4 Smooth product-convolution estimates for the noncommutative plane

�

255

Hence by the spectral theorem, for all ξ, η ∈ 𝒮 , we have ∞

1

∫ λ2 ⟨ 0

J π L(T)ξ, η⟩ dλ = ⟨L(T)ξ, η⟩. 2 (J 2 + λ)2

Thus, for all ξ, η ∈ 𝒮 , we have 1 J2 1 1 −1 ⟨[J, T]ξ, η⟩ = ⟨L(T)ξ, η⟩ + ∫ λ 2 ⟨ L2 (T)(λ + J 2 ) ξ, η⟩ dλ. 2 2 2 π (λ + J )

∞ 0

By assumption, L(T) and L2 (T) are bounded on H, and therefore 󵄨󵄨 1 󵄨󵄨 J2 󵄨󵄨 2 󵄨󵄨 2 2 −1 L (T)(λ + J ) ξ, η⟩ 󵄨󵄨λ ⟨ 󵄨󵄨 󵄨󵄨 󵄨󵄨 (λ + J 2 )2 2 󵄩 󵄩󵄩 1 󵄩 󵄩 󵄩󵄩 J 󵄩󵄩 󵄩󵄩 2 −1 󵄩 ≤ λ 2 ‖ξ‖H ‖η‖H 󵄩󵄩󵄩L2 (T)󵄩󵄩󵄩∞ 󵄩󵄩󵄩 󵄩 󵄩(λ + J ) 󵄩󵄩󵄩∞ . 󵄩󵄩 (λ + J 2 )2 󵄩󵄩󵄩∞ 󵄩 By the functional calculus we have 󵄩󵄩 J 2 󵄩󵄩 t λt 1 󵄩󵄩 󵄩󵄩 ≤ sup ≤ λ−1 . 󵄩󵄩 󵄩󵄩 = sup 󵄩󵄩 (λ + J 2 )2 󵄩󵄩∞ t≥1 (λ + t)2 t≥ 1 λ2 (1 + t 2 )2 4 λ

Therefore 󵄨󵄨 1 󵄨󵄨 J2 1 󵄨󵄨 2 󵄨󵄨 󵄩󵄩 2 󵄩󵄩 2 2 −1 L (T)(λ + J ) ξ, η⟩ . 󵄨󵄨λ ⟨ 󵄨󵄨 ≤ 󵄩󵄩L (T)󵄩󵄩∞ ‖ξ‖H ‖η‖H 1 2 2 󵄨󵄨 󵄨 (λ + J ) 󵄨 λ 2 (1 + λ) The function of λ on the right side is integrable as a function of λ ∈ [0, ∞). It follows that the bilinear form ∞

1

(ξ, η) 󳨃→ ∫ λ 2 ⟨ 0

J2 −1 L2 (T)(λ + J 2 ) ξ, η⟩ dλ (λ + J 2 )2

defines a bounded linear operator on H and hence that δ(T) = [J, T] is bounded. The integral formula 1 J2 1 1 −1 ⟨[J, T]ξ, η⟩ = ⟨L(T)ξ, η⟩ + ∫ λ 2 ⟨ L2 (T)(λ + J 2 ) ξ, η⟩ dλ 2 2 2 π (λ + J )

∞ 0

is already proved for ξ, η ∈ 𝒮 , and since both sides are continuous, the statement extends to all u, v ∈ H for the extensions of the operators δ(T), L(T), and L2 (T).

256 � 4 Integration formula for the noncommutative plane Corollary 4.4.4. Let T ∈ ℒ(H) be such that T|𝒮 ∈ End(𝒮 ), and suppose that L(T), L2 (T) ∈ ℒ(H). Then δ(T) ∈ ℒ(H), and ‖δ(T)‖∞ ≤ c ⋅ (‖L(T)‖∞ + ‖L2 (T)‖∞ ) for an absolute constant c > 0. If, in addition, L(T), L2 (T) ∈ ℒd,∞ (H), then δ(T) ∈ ℒd,∞ (H), and ‖δ(T)‖d,∞ ≤ cd ⋅ (‖L(T)‖d,∞ + ‖L2 (T)‖d,∞ ) for a constant cd > 0 depending on d. Proof. By the Hölder inequality and the fact J ≥ 1 we have 󵄩󵄩 λ 21 J 2 1 󵄩󵄩󵄩󵄩 󵄩󵄩 2 L (T) 󵄩󵄩 󵄩 󵄩󵄩 (λ + J 2 )2 λ + J 󵄩󵄩󵄩d,∞ 1 󵄩󵄩 λ 2 J 2 󵄩󵄩 󵄩󵄩 1 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩 ⋅ L (T) ⋅ 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 d,∞ 󵄩󵄩 (λ + J)2 󵄩󵄩∞ 󵄩󵄩 λ + J 󵄩󵄩󵄩∞ 1 3 󵄩 󵄩 󵄩 󵄩 ≤ (1 + λ)− 2 ⋅ 󵄩󵄩󵄩L2 (T)󵄩󵄩󵄩d,∞ ⋅ (1 + λ)−1 = (1 + λ)− 2 ⋅ 󵄩󵄩󵄩L2 (T)󵄩󵄩󵄩d,∞ . Since ℒd,∞ is a Banach space in an equivalent norm, it follows that 󵄩󵄩 ∞ λ 21 J 2 dλ 󵄩󵄩󵄩󵄩 󵄩󵄩 L2 (T) 󵄩󵄩 ∫ 󵄩 2 2 λ + J 󵄩󵄩󵄩d,∞ 󵄩󵄩 (λ + J ) 0

󵄩󵄩 λ 21 J 2 1 󵄩󵄩󵄩󵄩 󵄩 2 ≤ bd ⋅ ∫ 󵄩󵄩󵄩 L (T) 󵄩 dλ 󵄩󵄩 (λ + J 2 )2 λ + J 󵄩󵄩󵄩d,∞ ∞ 0

∞

3 󵄩 󵄩 󵄩 󵄩 ≤ bd ⋅ 󵄩󵄩󵄩L2 (T)󵄩󵄩󵄩d,∞ ∫ (1 + λ)− 2 dλ = 2bd ⋅ 󵄩󵄩󵄩L2 (T)󵄩󵄩󵄩d,∞

0

for a constant bd > 0. Hence 2b 󵄩 2 󵄩 1󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩δ(T)󵄩󵄩󵄩d,∞ ≤ 󵄩󵄩󵄩L(T)󵄩󵄩󵄩d,∞ + d 󵄩󵄩󵄩L (T)󵄩󵄩󵄩d,∞ , 2 π and the second statement follows. The first statement is proved similarly using the ideal ℒ(H) with symmetric norm ‖ ⋅ ‖∞ where bd = 1, in the place of ℒd,∞ (H) and ‖ ⋅ ‖d,∞ . Lemma 4.4.5. Let T ∈ ℒ(H) be such that T|𝒮 ∈ End(𝒮 ), and suppose that Lk (T) ∈ ℒd,∞ (L2 (ℝd )) for 1 ≤ k ≤ m + 2l. We have 󵄩󵄩 l m 󵄩 󵄩󵄩δ (L (T))󵄩󵄩󵄩d,∞ ≤ cd,l,m ⋅ for a constant cd,l,m > 0.

∑

m+l≤k≤m+2l

󵄩󵄩 k 󵄩󵄩 󵄩󵄩L (T)󵄩󵄩d,∞

4.4 Smooth product-convolution estimates for the noncommutative plane

�

257

Proof. We prove the statement by induction on l. The base of induction (i. e., l = 0) is trivial. Suppose the statement holds for l. Let us prove it for l + 1. We write 󵄩󵄩 l+1 m 󵄩 󵄩 l m 󵄩 󵄩󵄩δ (L (T))󵄩󵄩󵄩d,∞ = 󵄩󵄩󵄩δ(δ (L (T)))󵄩󵄩󵄩d,∞ 󵄩 󵄩 󵄩 󵄩 ≤ cd (󵄩󵄩󵄩L(δl (Lm (T)))󵄩󵄩󵄩d,∞ + 󵄩󵄩󵄩L2 (δl (Lm (T)))󵄩󵄩󵄩d,∞ ), where the last inequality follows from Corollary 4.4.4. Since the operations L and δ commute for endomorphisms of 𝒮 , it follows that L(δl (Lm (T))) = δl (Lm+1 (T)),

L2 (δl (Lm (T))) = δl (Lm+2 (T)).

Thus 󵄩󵄩 l+1 m 󵄩 󵄩 󵄩 l m+1 l m+2 󵄩󵄩δ (L (T))󵄩󵄩󵄩d,∞ ≤ cd (‖δ (L (T))󵄩󵄩󵄩d,∞ + ‖δ (L (T))󵄩󵄩󵄩d,∞ ). By the inductive hypothesis we have 󵄩󵄩 l m+1 󵄩 󵄩󵄩δ (L (T))󵄩󵄩󵄩d,∞ ≤ cd,l,m+1 󵄩󵄩 l m+2 󵄩 󵄩󵄩δ (L (T))󵄩󵄩󵄩d,∞ ≤ cd,l,m+2

∑

󵄩󵄩 k 󵄩󵄩 󵄩󵄩L (T)󵄩󵄩d,∞ ,

∑

󵄩󵄩 k 󵄩󵄩 󵄩󵄩L (T)󵄩󵄩d,∞ .

m+1+l≤k≤m+1+2l m+2+l≤k≤m+2+2l

Setting cd,l+1,m := cd ⋅ max{cd,l,m+1 , cd,l,m+2 }, we complete the proof.

4.4.2 Product-convolution estimates for partial derivations on the noncommutative plane In this section, we prove Theorem 4.4.1. We start with an l = 0 version. 1

Lemma 4.4.6. If x ∈ Wd2 (ℝdθ ), then x(1 − Δℱ )− 2 ∈ ℒd,∞ , and 󵄩󵄩 −1 󵄩 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd ⋅ ‖x‖W 2 d for a constant cd > 0. Proof. Suppose first d > 2. We prove the stronger estimate 󵄩󵄩 −1 󵄩 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd ⋅ ‖x‖Ld ,

x ∈ Ld (ℝdθ ).

258 � 4 Integration formula for the noncommutative plane To employ Theorem 1.5.10, we consider the algebras 𝒜1 := L∞ (ℝdθ ) and 𝒜2 := L∞ (ℝd ). Let us represent the algebras 𝒜1 and 𝒜2 on L2 (ℝd ) by the representations π1 (x) := x,

x ∈ L∞ (ℝdθ ),

π2 (g) := g(∇ℱ ),

g ∈ L∞ (ℝd ).

By Lemma 4.3.2 Hypothesis 1.5.9 is satisfied in this case. It follows from Theorem 1.5.10 that μ2 (xg(∇)) ≺≺ cd ⋅ μ2 (x ⊗ g) where the singular value function of a compact operator is denoted in the left side, and the singular value function for the spatial tensor product L∞ (ℝdθ )⊗L∞ (ℝd ) with trace τθ ⊗ τ0 is denoted on the right side. Since d > 2, it follows that μ2 (A2 ) ≺≺ μ2 (A1 )

yields ‖A2 ‖d,∞ ≤ cd ⋅ ‖A1 ‖d,∞ .

Hence 󵄩󵄩 󵄩 󵄩󵄩xg(∇)󵄩󵄩󵄩ℒd,∞ (L2 (ℝd )) ≤ cd ‖x ⊗ g‖ℒd,∞ (L∞ (ℝd )⊗L∞ (ℝd )) ≤ cd ‖x‖Ld (ℝd ) ‖g‖Ld,∞ (ℝd ) . θ θ The last inequality follows from Remark 1.5.14. Setting −1

g(t) := (1 + |t|2 ) 2 ,

t ∈ ℝd ,

and noting that g ∈ Ld,∞ (ℝd ) ∩ L∞ (ℝd ), we derive the claimed estimate. Suppose now d = 2. If x ∈ W22 (ℝ2θ ), then by Proposition 4.2.30, |x|2 ∈ W12 (ℝ2θ ). By Theorem 4.1.1(b) we have 󵄩󵄩 2 󵄩 2󵄩 −1 󵄩 2 󵄩󵄩|x| (1 − Δℱ ) 󵄩󵄩󵄩ℒ1,∞ ≤ cd ⋅ 󵄩󵄩󵄩|x| 󵄩󵄩󵄩W 2 ≤ cd ⋅ ‖x‖W 2 . 1 2 Applying the Araki–Lieb–Thirring inequality (see Corollary 1.3.2) with 𝒥 = ℒ1,∞ (L2 (ℝd )), 1

r = 2, A = |x|, and B = (1 − Δℱ )− 2 , we infer

󵄩󵄩 󵄩󵄨 󵄩 2 − 1 󵄩2 − 1 󵄨2 󵄩 −1 󵄩 󵄩󵄩x(1 − Δℱ ) 2 󵄩󵄩󵄩ℒ2,∞ = 󵄩󵄩󵄩󵄨󵄨󵄨x(1 − Δℱ ) 2 󵄨󵄨󵄨 󵄩󵄩󵄩ℒ1,∞ ≤ cabs 󵄩󵄩󵄩|x| (1 − Δℱ ) 󵄩󵄩󵄩ℒ1,∞ . Combining the last two displays, we complete the proof. 1

If

We now use the results of Section 4.4.1 with H = L2 (ℝd ), 𝒮 = 𝒮 (ℝd ), and J = (1−Δℱ ) 2 . T : 𝒮 (ℝd ) → 𝒮 (ℝd ),

then in this case,

4.4 Smooth product-convolution estimates for the noncommutative plane

�

259

1

δ(T) = [(1 − Δℱ ) 2 , T], and 1

L(T) = (1 − Δℱ )− 2 [T, Δℱ ]. It follows from Remark 4.2.19 that f , u ∈ 𝒮 (ℝd ).

Uθ (f )u = f ⋆θ u, So, if x = Uθ (f ) ∈ 𝒮 (ℝdθ ), then

x : 𝒮 (ℝd ) → 𝒮 (ℝd ) is an endomorphism since the space of Schwartz functions is closed under twisted convolution. Proof of Theorem 4.4.1. Let l, m ∈ ℤ+ , and let x ∈ 𝒮 (ℝdθ ). Writing Δℱ = − ∑dk=1 Mk2 , we have d

[x, Δℱ ] = ∑ 2Mk ⋅ 𝜕k (x) − 𝜕k (x)2 k=1

as an endomorphism of 𝒮 (ℝd ). Hence d

1

L(x) = (1 − Δℱ )− 2 ⋅ ∑ 2Mk 𝜕k (x) − 𝜕k (x)2 . k=1

By iteration we have m

m

d

Lm (x) = (1 − Δℱ )− 2 ∑

∑ (Mk1 ⋅ ⋅ ⋅ Mkj ) ⋅ xk1 ,...,kj ,

j=0 k1 ,...,kj =1

where xk1 ,...,kj is a sum of derivations of x of order 2m − j. By applying the triangle inequality to (4.18), m

󵄩󵄩 m −1 󵄩 󵄩󵄩L (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd ⋅ ∑

d

∑

j=0 k1 ,...,kj =1

󵄩󵄩 −1 󵄩 󵄩󵄩xk1 ,...,kj (1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ .

By Lemma 4.4.6 we have 󵄩󵄩 −1 󵄩 ′ 󵄩󵄩xk1 ,...,kj (1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd ⋅ ‖xk1 ,...,kj ‖W 2 ≤ cd ⋅ ‖x‖W 2m−j+2 . d d Thus

(4.18)

260 � 4 Integration formula for the noncommutative plane 󵄩󵄩 m −1 󵄩 󵄩󵄩L (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd,m ⋅ ‖x‖W 2m+2 d for a constant cd,m > 0. Set 1

T = x(1 − Δℱ )− 2 : 𝒮 (ℝd ) → 𝒮 (ℝd ) and note that 1

Lm (T) = Lm (x)(1 − Δℱ )− 2 . We have shown that 󵄩󵄩 m 󵄩󵄩 󵄩󵄩L (T)󵄩󵄩ℒd,∞ ≤ cd,m ⋅ ‖x‖W 2m+2 . d Combining the latter estimate with Lemma 4.4.5, we obtain 󵄩󵄩 l −1 󵄩 󵄩󵄩δ (x)(1 − Δℱ ) 2 󵄩󵄩󵄩ℒd,∞ ≤ cd,l,1

∑

l+1≤k≤2l+1

󵄩󵄩 k 󵄩󵄩 󵄩󵄩L (T)󵄩󵄩ℒd,∞

≤ cd,l,1 ⋅ sup cd,k ⋅ k≤2l+1

∑

l+1≤k≤2l+1

‖x‖W 2k+2 ≤ bd,l ⋅ ‖x‖W 4l+2 d

d

for a constant bd,l > 0.

4.5 Integration formula up to a constant In this section, we prove that d

φ(x(1 − Δℱ )− 2 ) = cφ ⋅ τθ (x),

x ∈ W1d (ℝdθ ),

for a normalized continuous trace φ on ℒ1,∞ for a constant cφ . In Section 4.6, we identify the constant. For every ϕ ∈ L∞ (ℝd ), we define the bounded operator Tϕ : L2 (ℝdθ ) → L2 (ℝdθ ) by the formula Tϕ : ∫ f (s)U(s)ds 󳨃→ ∫ f (s)ϕ(s)U(s)ds, ℝd

f ∈ L2 (ℝd ).

ℝd

This operation is the noncommutative analogue of a Fourier multiplier. Lemma 4.5.1. If ϕ ∈ 𝒮 (ℝd ), then Tϕ : L1 (ℝdθ ) → L1 (ℝdθ ) is a bounded mapping. Proof. We claim that d

Tϕ (x) = (2π)− 2 ∫ (ℱ ϕ)(ξ)U(−θ−1 ξ)xU(θ−1 ξ)dξ, ℝd

x ∈ L2 (ℝdθ ).

4.5 Integration formula up to a constant

� 261

Since both sides of the equality define bounded operators on L2 (ℝdθ ), and since the set {U(f ) : f ∈ 𝒮 (ℝd )} is dense in L1 (ℝdθ ) (see Lemma 4.2.29), it suffices to establish the claim for x = ∫ f (s)U(s)ds,

f ∈ 𝒮 (ℝd ).

ℝd

Using the inverse Fourier transform, we write d

ϕ(t) = (2π)− 2 ∫ (ℱ ϕ)(ξ)ei⟨ξ,t⟩ dξ,

t ∈ ℝd ,

ℝd

that is, d

Tϕ (x) = (2π)− 2 ∫ ( ∫ f (t)(ℱ ϕ)(ξ)ei⟨ξ,t⟩ U(t)dξ)dt. ℝd ℝd

Since f , ℱ ϕ ∈ 𝒮 (ℝd ), it follows that 󵄩 󵄩 ∬ 󵄩󵄩󵄩f (t)(ℱ ϕ)(ξ)ei⟨ξ,t⟩ U(t)󵄩󵄩󵄩∞ dtdξ < ∞.

ℝd ×ℝd

By the Fubini theorem we have d

Tϕ (x) = (2π)− 2 ∬ f (t)(ℱ ϕ)(ξ)ei⟨ξ,t⟩ U(s)dtdξ. ℝd ×ℝd

It follows from (4.1) that ei⟨ξ,t⟩ U(t) = U(−θ−1 ξ)U(t)U(θ−1 ξ),

t, ξ ∈ ℝd .

Using the Fubini theorem again, we write the double integral as a repeated one: d

Tϕ (x) = (2π)− 2 ∫ (ℱ ϕ)(ξ)( ∫ f (t)U(−θ−1 ξ)U(t)U(θ−1 ξ)dt)dξ. ℝd

ℝd

Using the definition of x, we obtain ∫ f (t)U(−θ−1 ξ)U(t)U(θ−1 ξ)dt = U(−θ−1 ξ)xU(θ−1 ξ), ℝd

This proves the claim.

ξ ∈ ℝd .

262 � 4 Integration formula for the noncommutative plane Now we prove the lemma as follows: 󵄩󵄩 󵄩 󵄨 󵄨 󵄩 −1 −1 󵄩 󵄩󵄩Tϕ (x)󵄩󵄩󵄩L1 (ℝd ) ≤ ∫ 󵄨󵄨󵄨(ℱ ϕ)(ξ)󵄨󵄨󵄨 ⋅ 󵄩󵄩󵄩U(−θ ξ)xU(θ ξ)󵄩󵄩󵄩L1 (ℝd ) dξ θ θ ℝd

= ‖ℱ ϕ‖L1 (ℝd ) ‖x‖L1 (ℝd ) . θ

Lemma 4.5.2. Let t 󳨃→ V (t), t ∈ ℝd , be a bounded strongly continuous family of bounded mappings on L2 (ℝd ). For every A ∈ ℒ1 , the mapping t 󳨃→ V (−t)AV (t),

t ∈ ℝd ,

is continuous in the ℒ1 -norm. Proof. By Proposition 2.4.1 in [40] we have |V (−t) − V (−s)|2 → 0 in the strong (and, therefore, in weak) operator topology as t → s. By assumption the family {V (−t)}t∈ℝd is bounded. Suppose first A = p is a rank one projection and s, t ∈ ℝd . We have V (−t)pV (t) − V (−s)pV (s)

= V (−t)p ⋅ (pV (t) − pV (s)) + (V (−t)p − V (−s)p) ⋅ pV (s).

By the triangle and Hölder inequalities we have 󵄩󵄩 󵄩 󵄩󵄩V (−t)pV (t) − V (−s)pV (s)󵄩󵄩󵄩1 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩󵄩V (−t)p󵄩󵄩󵄩2 󵄩󵄩󵄩pV (t) − pV (s)󵄩󵄩󵄩2 + 󵄩󵄩󵄩V (−t)p − V (−s)p󵄩󵄩󵄩2 󵄩󵄩󵄩pV (s)󵄩󵄩󵄩2 . Clearly, 󵄩󵄩 󵄩 󵄨 󵄨2 󵄩󵄩V (−t)p − V (−s)p󵄩󵄩󵄩2 = Tr(p󵄨󵄨󵄨V (−t) − V (−s)󵄨󵄨󵄨 p) → 0,

t → s,

and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩V (−t)p󵄩󵄩󵄩2 = 󵄩󵄩󵄩V (−t)p󵄩󵄩󵄩∞ ≤ 󵄩󵄩󵄩V (−t)󵄩󵄩󵄩∞ ,

t ∈ ℝd .

Thus 󵄩󵄩 󵄩 󵄩󵄩V (−t)pV (t) − V (−s)pV (s)󵄩󵄩󵄩1 → 0,

t → s.

By the preceding paragraph the statement holds for a finite-rank operator A. Let now A ∈ ℒ1 be arbitrary. Fix ε > 0 and choose a finite-rank operator B such that ‖B − A‖1 < ε. Choose δ such that 󵄩󵄩 󵄩 󵄩󵄩V (−t)BV (t) − V (−s)BV (s)󵄩󵄩󵄩1 < ε,

|t − s| < δ.

4.5 Integration formula up to a constant

� 263

Then for |t − s| < δ, we have 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩V (−t)AV (t) − V (−s)AV (s)󵄩󵄩󵄩1 ≤ 󵄩󵄩󵄩V (−t)BV (t) − V (−s)BV (s)󵄩󵄩󵄩1 󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩V (−t)(A − B)V (t)󵄩󵄩󵄩1 + 󵄩󵄩󵄩V (−s)(A − B)V (s)󵄩󵄩󵄩1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≤ ε ⋅ (1 + 󵄩󵄩󵄩V (−t)󵄩󵄩󵄩∞ 󵄩󵄩󵄩V (t)󵄩󵄩󵄩∞ + 󵄩󵄩󵄩V (−s)󵄩󵄩󵄩∞ 󵄩󵄩󵄩V (s)󵄩󵄩󵄩∞ ). Thus 󵄩󵄩 󵄩 󵄩󵄩V (−t)AV (t) − V (−s)AV (s)󵄩󵄩󵄩1 → 0,

t → s.

Lemma 4.5.3. Let t 󳨃→ U(t), t ∈ ℝd , be as in Definition 4.2.3. For every x ∈ W1d (ℝdθ ), the mapping t 󳨃→ U(−t)xU(t),

t ∈ ℝd ,

is a continuous W1d (ℝdθ )-valued function. Moreover, 󵄩󵄩 󵄩 󵄩󵄩U(−t)xU(t)󵄩󵄩󵄩W d = ‖x‖W d , 1 1

t ∈ ℝd .

Proof. Since conjugation by U(t), t ∈ ℝd , preserves the domain of the derivations 𝜕j , 1 ≤ j ≤ d on L∞ (ℝdθ ), it follows that 𝜕j (U(−t)xU(t)) = 𝜕j (U(−t)) ⋅ xU(t) + U(−t) ⋅ 𝜕j (x) ⋅ U(t) + U(−t)x ⋅ 𝜕j (U(t))

= −itj U(−t)xU(t) + U(−t)𝜕j (x)U(t) + itj U(−t)xU(t) = U(−t)𝜕j (x)U(t).

Iterating the latter inequality, we obtain 𝜕α (U(−t)xU(t)) = U(−t)𝜕α (x)U(t),

t ∈ ℝd ,

|α| ≤ d.

By the latter equality and Definition 4.2.26 we have 󵄩󵄩 󵄩 󵄩 α 󵄩 󵄩󵄩U(−t)xU(t)󵄩󵄩󵄩W d = ∑ 󵄩󵄩󵄩𝜕 (U(−t)xU(t))󵄩󵄩󵄩1 1 |α|≤d

󵄩 󵄩 󵄩 󵄩 = ∑ 󵄩󵄩󵄩U(−t)𝜕α (x)U(t)󵄩󵄩󵄩1 = ∑ 󵄩󵄩󵄩𝜕α (x)󵄩󵄩󵄩1 = ‖x‖W d . 1 |α|≤d

|α|≤d

We now establish the continuity. By Theorem 4.2.6, (L∞ (ℝdθ ), τθ ) is ∗-isomorphic to d

(ℒ(L2 (ℝ 2 )), Tr) under the map rθ , and

󵄩󵄩 α 󵄩 󵄩 󵄩 α 󵄩󵄩𝜕 (U(−t)xU(t))󵄩󵄩󵄩L1 = cθ ⋅ 󵄩󵄩󵄩rθ (𝜕 (U(−t)xU(t)))󵄩󵄩󵄩ℒ1 ,

t ∈ ℝd ,

for a constant cθ dependent on θ. By Lemma 4.5.2, using the unitary family Vt := rθ (U(−t)), t ∈ ℝd , the mapping

264 � 4 Integration formula for the noncommutative plane t 󳨃→ rθ (𝜕α (U(−t)xU(t))) = rθ (U(−t))rθ (𝜕α (x))rθ (U(t)) is continuous in the ℒ1 -norm. This completes the proof. Lemma 4.5.4. Let t 󳨃→ U(t), t ∈ ℝd , be as in Definition 4.2.3. If F is a continuous linear functional on W1d (ℝdθ ) such that F(x) = F(U(−t)xU(t)),

x ∈ W1d (ℝdθ ),

t ∈ ℝd ,

then F = cF ⋅ τθ for some constant cF . Proof. Let T : W1d (ℝdθ ) → W1d (ℝdθ ) be defined by d

1

2

T(x) := (2π)− 2 ∫ U(−θ−1 t)xU(θ−1 t)e− 2 |t| dt, ℝd

where the integral is understood as a Bochner integral of a continuous W1d (ℝdθ )-valued function. The continuity and convergence of the integral follow from Lemma 4.5.3. For every x ∈ W1d (ℝdθ ), we have d

1

2

F(T(x)) = (2π)− 2 ∫ F(U(−θ−1 t)xU(θ−1 t))e− 2 |t| dt − d2

= (2π)

ℝd

1

2

∫ F(x)e− 2 |t| dt = F(x). ℝd

Thus F(x) = (F ∘ T)(x),

x ∈ W1d (ℝdθ ).

We claim that ‖T(x)‖W d ≤ cd ⋅ ‖x‖L1 for a constant cd > 0 and every x ∈ W1d (ℝdθ ). To 1 see this, by Lemma 4.2.17, x = ∫ f (s)U(s)ds ℝd 1

2

for some f ∈ L2 (ℝd ). If in the proof of Lemma 4.5.1, we take ϕ(t) := e− 2 |t| , t ∈ ℝd , then the argument given there yields 1

2

T(x) = ∫ f (s)U(s)e− 2 |s| ds. ℝd

By Theorem 4.2.24 we have

4.5 Integration formula up to a constant

1

2

𝜕α (T(x)) = ∫ f (s)U(s)(is)α e− 2 |s| ds,

� 265

α ∈ ℤd+ .

ℝd 1

2

Let ϕα (s) := (is)α e− 2 |s| , s ∈ ℝd . We have that 𝜕α ∘ T = Tϕα . By Lemma 4.5.1 we have 󵄩󵄩 α 󵄩 󵄩 󵄩 󵄩󵄩(𝜕 ∘ T)(x)󵄩󵄩󵄩L1 = 󵄩󵄩󵄩Tϕα (x)󵄩󵄩󵄩L1 ≤ ‖Tϕα ‖L1 →L1 ⋅ ‖x‖L1 . Thus 󵄩󵄩 󵄩 󵄩 α 󵄩 󵄩󵄩T(x)󵄩󵄩󵄩W d = ∑ 󵄩󵄩󵄩𝜕 (T(x))󵄩󵄩󵄩L1 ≤ ∑ ‖Tϕα ‖L1 →L1 ⋅ ‖x‖L1 1 |α|≤d

|α|≤d

In other words, T : W1d (ℝdθ ) → L1 (ℝdθ ) is a bounded operator. This proves the claim. For every x ∈ W1d (ℝdθ ), we have 󵄨󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨󵄨F(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨(F ∘ T)(x)󵄨󵄨󵄨 ≤ ‖F‖(W d )∗ ⋅ 󵄩󵄩󵄩T(x)󵄩󵄩󵄩W d ≤ cd ‖F‖(W d )∗ ⋅ ‖x‖L1 1 1 1 for a constant cd > 0. Thus the functional F on W1d (ℝdθ ) is bounded in the ‖ ⋅ ‖L1 -norm. By the Hahn–Banach theorem, F extends to a bounded functional on L1 (ℝdθ ). Hence there exists y ∈ L∞ (ℝdθ ) such that x ∈ W1d (ℝdθ ).

F(x) = τθ (xy), Clearly,

F(U(−t)xU(t)) = τθ (U(−t)xU(t)y) = τθ (xU(t)yU(−t)). Comparing the last two equalities, we obtain τθ (xU(t)yU(−t)) = τθ (xy),

x ∈ W1d (ℝdθ ).

By Lemma 4.2.29, W1d (ℝdθ ) is dense in L1 (ℝdθ ). It follows that y = U(t)yU(−t) for every t ∈ ℝd . In other words, y commutes with every U(t) and, therefore, with every element in L∞ (ℝdθ ). Since L∞ (ℝdθ ) is a factor by Theorem 4.2.6, it follows that y is a scalar operator. This completes the proof. The following proposition is Theorem 4.1.2 with a constant depending on the continuous trace. Proposition 4.5.5. If x ∈ W1d (ℝdθ ) and φ is a continuous trace on ℒ1,∞ , then d

φ(x(1 − Δℱ )− 2 ) = cφ ⋅ τθ (x) for some constant cφ .

266 � 4 Integration formula for the noncommutative plane Proof. By Theorem 4.1.1(b) the functional d

F : x → φ(x(1 − Δℱ )− 2 ),

x ∈ W1d (ℝdθ ),

is a well-defined bounded linear functional on W1d (ℝdθ ). Since φ is unitarily invariant, it follows that d

d

φ(x(1 − Δℱ )− 2 ) = φ(ei⟨t,∇ℱ ⟩ x(1 − Δℱ )− 2 e−i⟨t,∇ℱ ⟩ ),

t ∈ ℝd .

By the spectral theorem we have d

d

(1 − Δℱ )− 2 e−i⟨t,∇ℱ ⟩ = e−i⟨t,∇ℱ ⟩ (1 − Δℱ )− 2 , and so d

d

φ(x(1 − Δℱ )− 2 ) = φ(ei⟨t,∇ℱ ⟩ xe−i⟨t,∇ℱ ⟩ (1 − Δℱ )− 2 ),

t ∈ ℝd .

By Lemma 4.2.21 we have ei⟨t,∇ℱ ⟩ xe−i⟨t,∇ℱ ⟩ = U(−θ−1 t)xU(θ−1 t),

x ∈ L∞ (ℝdθ ).

Combining the preceding paragraphs, we obtain d

d

φ(x(1 − Δℱ )− 2 ) = φ(U(−θ−1 t)xU(θ−1 t)(1 − Δℱ )− 2 ). In other words, F(x) = F(U(θ−1 t)xU(θ−1 t)),

x ∈ W1d (ℝdθ ).

Applying Lemma 4.5.4 to the functional F, we conclude the argument.

4.6 Measurability in the noncommutative plane In this section, we identify the constant cφ in Proposition 4.5.5 by showing that there d

exists an operator x ∈ W1d (ℝdθ ) such that φ(x(1 − Δℱ )− 2 ) ≠ 0 does not depend on the choice of a normalized continuous trace φ on ℒ1,∞ .

Unique traceability of a symmetric operator d The first step is to consider h(∇ℱ )xh(∇ℱ ) instead of x(1 − Δℱ )− 2 , where x = Uθ (f ) for a function f ∈ 𝒮 (ℝd ) supported in [− 21 , 21 ]d . Here

4.6 Measurability in the noncommutative plane

d

2

h(t) := (1 + ∑ ⌊tk ⌋ ) k=1

− d4

,

t ∈ ℝd ,

�

267

(4.19)

and ⌊⋅⌋ is the floor function. Let A ∈ ℒ(L2 (ℝd )). If {pk }k∈ℤd ⊂ ℒ(L2 (ℝd )) is a family of pairwise orthogonal projections summing to the identity operator, then μ(A) = μ( ∑ pk Apl ⊗ Ek,l ), k,l∈ℤd

where {Ek,l }k,l∈ℤd ∈ ℒ(l2 (ℤd )) are matrix units. Here the singular value function on the left is computed for operators in ℒ(L2 (ℝd )) following Definition 1.2.12 using the trace TrL2 (ℝd ) , and the one on the right-hand side is computed for operators in ℒ(L2 (ℝd ) ⊗ l2 (ℤd )) using the trace TrL2 (ℝd ) ⊗ Trl2 (ℤd ) .

Lemma 4.6.1. Let f ∈ 𝒮 (ℝd ), and let h be as in (4.19). If x = Uθ (f ), then i

μ(h(∇ℱ )xh(∇ℱ )) = μ( ∑ e 2 ⟨l,θk⟩ Ak−l ⊗ h(k)h(l)Ek,l ), k,l∈ℤd

where the sum on the right-hand side is understood in the sense of strong convergence for operators in ℒ(L2 ([0, 1]d ) ⊗ l2 (ℤd )), and Am ∈ ℒ(L2 ([0, 1]d )) is defined by the formula i

i

(Am u)(t) := ∫ f (t + m − s)e 2 ⟨s,θt⟩ e− 2 ⟨m,θ(s+t)⟩ u(s)ds,

t ∈ [0, 1]d ,

u ∈ L2 ([0, 1]d ).

[0,1]d

Proof. For an arbitrary sequence {Um }m∈ℤd ⊂ ℒ(H) of unitaries on a Hilbert space H, define a unitary operator U ∈ ℒ(H ⊗ l2 (ℤd )) by the formula U = ∑ Um ⊗ Em,m , m∈ℤd

where the series converges in the strong operator topology. If A ∈ ℒ(H), then U −1 ( ∑ pk Apl ⊗ Ek,l )U = ∑ Uk−1 pk Apl Ul ⊗ Ek,l . k,l∈ℤd

k,l∈ℤd

Thus μ(A) = μ( ∑ Uk−1 pk Apl Ul ⊗ Ek,l ). k,l∈ℤd

Let A = h(∇ℱ )xh(∇ℱ ) and pk = χk+[0,1]d (∇ℱ ), k ∈ ℤd . Recall (see, e. g., the proof of Lemma 4.3.2) that x is an integral operator with integral kernel

268 � 4 Integration formula for the noncommutative plane i

(t, s) 󳨃→ f (t − s)e 2 ⟨s,θt⟩ ,

t, s ∈ ℝd .

Let k, l ∈ ℤd . Then pk Apl is an integral operator with integral kernel i

t, s ∈ ℝd .

(t, s) → h(t)h(s)f (t − s)e 2 ⟨s,θt⟩ χk+[0,1]d (t)χl+[0,1]d (s),

Since h(t) = h(k) and h(s) = h(l) for t ∈ k + [0, 1]d and s ∈ l + [0, 1]d , it follows that pk Apl is an integral operator with integral kernel i

t, s ∈ ℝd .

(t, s) → h(k)h(l) ⋅ f (t − s)e 2 ⟨s,θt⟩ χk+[0,1]d (t)χl+[0,1]d (s),

We now consider the concrete sequence {Um }m∈ℤd ⊂ ℒ(L2 (ℝd )) of unitaries defined

by the formula

i

(Um u)(t) := e 2 ⟨m,θt⟩ u(t − m),

u ∈ L2 (ℝd ),

t ∈ ℝd .

u ∈ L2 (ℝd ),

t ∈ ℝd .

We have i

(Um−1 u)(t) = e− 2 ⟨m,θt⟩ u(t + m),

Thus, (omitting the scalar factor h(k)h(l) to lighten the notations) for every u ∈ L2 (ℝd ), i

i

(pk Apl Ul u)(t) = ∫ f (t − s)e 2 ⟨s,θt⟩ χk+[0,1]d (t)χl+[0,1]d (s)e 2 ⟨l,θs⟩ u(s − l)ds ℝd

i

i

= χk+[0,1]d (t) ∫ f (t − s)e 2 ⟨s,θt⟩ χ[0,1]d (s − l)e 2 ⟨l,θs⟩ u(s − l)ds s→s+l

ℝd

i

i

= χk+[0,1]d (t) ∫ f (t − s − l)e 2 ⟨s+l,θt⟩ e 2 ⟨l,θs⟩ u(s)ds, [0,1]d

and i

(Uk−1 pk Apl Ul u)(t) = e− 2 ⟨k,θt⟩ ⋅ (pk Apl Ul u)(t + k) i

i

i

= e− 2 ⟨k,θt⟩ χk+[0,1]d (t + k) ∫ f (t + k − s − l)e 2 ⟨s+l,θ(t+k)⟩ e 2 ⟨l,θs⟩ u(s)ds [0,1]d

4.6 Measurability in the noncommutative plane

i

i

�

269

i

= e 2 ⟨l,θk⟩ ⋅ χ[0,1]d (t) ∫ f (t + k − s − l)e 2 ⟨s,θt⟩ e 2 ⟨l−k,θ(s+t)⟩ u(s)ds [0,1]d

=e

i ⟨l,θk⟩ 2

⋅ (Ak−l u)(t).

In the last equality, we identified Ak−l with an operator on ℒ(L2 (ℝd )) by viewing a function u ∈ L2 ([0, 1]d ) as a function u ∈ L2 (ℝd ) which is 0 outside [0, 1]d . The series in Lemma 4.6.1 converges in the strong operator topology on ℒ(L2 (ℝd ) ⊗ l2 (ℤ )), but traces on ℒ1,∞ are not continuous in the strong operator topology. We write d

i

∑ e 2 ⟨l,θk⟩ Ak−l ⊗ h(k)h(l)Ek,l = ∑ Am ⊗ Bm ,

k,l∈ℤd

m∈ℤd

(4.20)

where we formally set i

Bm := ∑ e 2 ⟨l,θk⟩ h(k)h(l)Ek,l .

(4.21)

k,l∈ℤd k−l=m

We would like the series on the right-hand side in (4.20) to converge in ℒ1,∞ . The following lemma delivers a criterion for an integral operator to fall into ℒ1 . It is used in the subsequent Lemma 4.6.3, which states the convergence of the series on the right-hand side in (4.20) in ℒ1,∞ . Lemma 4.6.2. (a) If A : L2 (𝕋d ) → L2 (𝕋d ) is an integral operator with integral kernel kA ∈ W12d+2 (𝕋d × 𝕋d ), then A ∈ ℒ1 and ‖A‖ℒ1 ≤ cd ⋅ ‖kA ‖W 2d+2 for a constant cd > 0. 1

(b) If A : L2 ([0, 1]d ) → L2 ([0, 1]d ) is an integral operator with integral kernel kA ∈ W12d+2 ([0, 1]d × [0, 1]d ), then A ∈ ℒ1 and ‖A‖ℒ1 ≤ cd ⋅ ‖kA ‖W 2d+2 for a constant cd > 0. 1

Proof. To prove the first statement, we note that by the Sobolev embedding theorem (see, e. g., Theorem 5.4 in [2]), W12d+2 (𝕋d × 𝕋d ) ⊂ W1d (𝕋d × 𝕋d ) ⊂ L2 (𝕋d × 𝕋d ). We consider the Fourier series of kA ∈ W12d+2 (𝕋d × 𝕋d ), kA (w, z) =

∑

m1 ,m2

∈ℤd

cm1 ,m2 wm1 zm2 ,

w, z ∈ 𝕋d .

The series converges in L2 (𝕋d × 𝕋d ). Set (Am1 ,m2 u)(w) := ⟨u, z−m2 ⟩wm1 ,

u ∈ L2 (𝕋d ).

270 � 4 Integration formula for the noncommutative plane It is an integral operator on L2 (𝕋d ) with integral kernel (w, z) → wm1 zm2 . In particular, Am1 ,m2 ∈ ℒ1 (L2 (𝕋d )) for every m1 , m2 ∈ ℤd . Hence A=

∑

m1 ,m2 ∈ℤd

cm1 ,m2 Am1 ,m2 ,

where the series converges in ℒ2 (L2 (𝕋d )). By the triangle inequality, we have ‖A‖1 ≤ ≤

∑

m1 ,m2 ∈ℤd

|cm1 ,m2 | d+1

sup (1 + |m1 |2 + |m2 |2 )

m1 ,m2

∈ℤd

|cm1 ,m2 | ⋅

∑

m1 ,m2 ∈ℤd

(1 + |m1 |2 + |m2 |2 )

−d−1

.

Observe that (1 + |m1 |2 + |m2 |2 )d+1 cm1 ,m2 is the (m1 , m2 )th Fourier coefficient of the function (1 − Δ𝕋2d )d+1 kA (here Δ𝕋2d is the classical Laplacian on the torus 𝕋2d ). Taking into account that the Fourier coefficients do not exceed the L1 -norm, we infer that d+1

(1 + |m1 |2 + |m2 |2 )

󵄩 󵄩 |cm1 ,m2 | ≤ 󵄩󵄩󵄩(1 − Δ𝕋2d )d+1 kA 󵄩󵄩󵄩1 = ‖kA ‖W 2d+2 . 1

Combining the last two displays, we arrive at ‖A‖1 ≤ ‖kA ‖W 2d+2 ⋅ 1

∑

m1 ,m2 ∈ℤd

(1 + |m1 |2 + |m2 |2 )

−d−1

.

This proves the first statement. For an integral operator S acting on the Hilbert space L2 ([−π, π]d ) with kernel kS ∈ 2d+2 W1 ([−π, π]d ×[−π, π]d ) supported in the interior of [−π, π]d , the first statement proves that ‖S‖1 ≤ bd ⋅ ‖kS ‖W 2d+2 1

for some constant bd > 0. To prove the second statement, note that by Theorem 4.28 in [2] there exists L ∈ W12d+2 (ℝd × ℝd ) such that L|[0,1]d ×[0,1]d = kA and such that ‖L‖W 2d+2 (ℝd ×ℝd ) ≤ cd ⋅ ‖kA ‖W 2d+2 ([0,1]d ×[0,1]d ) 1

1

for a constant cd > 0. Fix a function ϕ ∈ 𝒮 (ℝd ) compactly supported in [−3, 3]d × [−3, 3]d and such that ϕ = 1 on [0, 1]d × [0, 1]d . We have ‖L ⋅ ϕ‖W 2d+2 ([−π,π]d ×[−π,π]d ) ≤ cd′ ⋅ ‖kA ‖W 2d+2 ([0,1]d ×[0,1]d ) 1

1

for a constant cd′ > 0. Replacing L with L ⋅ ϕ if needed, we may assume without loss of generality that L is compactly supported in [−3, 3]d × [−3, 3]d and such that L|[0,1]d ×[0,1]d =

4.6 Measurability in the noncommutative plane

�

271

kA . Let S be the integral operator on L2 ([−π, π]d ) with kernel kS := L. We can equivalently view the integral operator A acting on L2 ([0, 1]d ) as the operator Mχ d SMχ d acting on (0,1)

L2 ([−π, π]d ). We have

‖A‖1 = ‖Mχ

(0,1)d

SMχ

(0,1)d

(0,1)

‖1 ≤ ‖S‖1

≤ bd ⋅ ‖kS ‖W 2d+2 ([−π,π]d ×[−π,π]d ) ≤ (bd cd′ ) ⋅ ‖kA ‖W 2d+2 ([0,1]d ×[0,1]d ) . 1

1

Lemma 4.6.3. Let f ∈ 𝒮 (ℝd ), and let x = Uθ (f ). The series on the right-hand side in (4.20) converges in ℒ1,∞ . Proof. Let Am ∈ ℒ(L2 ([0, 1]d )), m ∈ ℤd , be the bounded integral operator defined in Lemma 4.6.1 with kernel i

i

kAm (t, s) = f (t + m − s)e 2 ⟨s,θt⟩ e− 2 ⟨m,θ(s+t)⟩ ds,

s, t ∈ [0, 1]d .

By Lemma 4.6.2 we have ‖Am ‖ℒ1 ≤ cd ⋅ ‖kA ‖W 2d+2 ≤ cd (1 + |m|2 )

d+1

1

⋅ ‖f ‖W∞2d+2 (m + [−1, 1]2 )

(4.22)

for a constant cd > 0. For m ∈ ℤd , |Bm |2 =

=

i

i

∑ e− 2 ⟨l1 ,θk1 ⟩ h(k1 )h(l1 )El1 ,k1 ⋅ e 2 ⟨l2 ,θk2 ⟩ h(k2 )h(l2 )Ek2 ,l2

∑

k1 ,l1 ∈ℤd k2 ,l2 ∈ℤd k1 −l1 =m k2 −l2 =m

∑ h(k1 )2 h(l1 )2 El1 ,l1 .

k1 ,l1 ∈ℤd k1 −l1 =m

Thus d 󵄩 󵄩 ‖Bm ‖1,∞ = 󵄩󵄩󵄩{h(k)h(l)}k,l∈ℤd 󵄩󵄩󵄩1,∞ ≤ cd (1 + |m|2 ) 2

(4.23)

k−l=m

for a constant cd > 0. Let p ∈ (0, 21 ) and n1 , n2 ∈ ℤ+ be such that n1 < n2 . By the Aoki–Rolewicz theorem (Theorem 1.3 in [169]) applied to the ideal ℒ1,∞ we have 󵄩󵄩 󵄩p 󵄩󵄩 ∑ Am ⊗ Bm − ∑ Am ⊗ Bm 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩1,∞ d d m∈ℤ |m|≤n2

≤ cp ⋅

m∈ℤ |m|≤n1

∑

d

m∈ℤ n1 0. Since y ∈ C 1 (𝕊d−1 ) is Lipschitz, it follows that 󵄨󵄨 t n 󵄨󵄨󵄨 c 󵄨󵄨 󵄨󵄨y( ) − y( )󵄨󵄨󵄨 ≤ d ⋅ ‖y‖C 1 . |n| 󵄨󵄨 |n| 󵄨󵄨 |t| This completes the proof of the statement. The next step is to associate an Abel sum with the integral of y on 𝕊d−1 . Lemma 5.4.10. Let y ∈ C 1 (𝕊d−1 ). We have ∑ y(

n∈ℤd

2 n 1 1 −d )(1 + |n|2 ) 2 e−t|n| = ∫ y(s)ds ⋅ log( ) + O(1), |n| 2 t

t ∈ (0, 1).

𝕊d−1

Proof. To lighten the notations, denote z(t, s) := y(

2 s −d )(1 + |s|2 ) 2 e−t|s| , |s|

t > 0,

s ∈ ℝd .

Let n ∈ ℤd be such that |n| > 2d (in particular, we have n ∉ {−1, 0}d ). A combination of Lemmas 5.4.9 and 3.4.5 yields 󵄨 󵄨 a ‖y‖ + c ‖y‖ 1 sup 󵄨󵄨󵄨z(t, s) − z(t, n)󵄨󵄨󵄨 ≤ d ∞ d+1d C |n| d

s∈n+[0,1]

for constants ad , cd > 0. Thus 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫ 󵄨󵄨

n+[0,1]d

󵄨󵄨 a ‖y‖ + c ‖y‖ 1 󵄨 z(t, s)ds − z(t, n)󵄨󵄨󵄨 ≤ d ∞ d+1d C . 󵄨󵄨 |n|

We now write ∫ z(t, s)ds − ∑ z(t, n) = ∑ ( ∫ n∈ℤd

ℝd

n∈ℝd n+[0,1]d

z(t, s)ds − z(t, n))

= ∑ ( ∫ n∈ℝd n+[0,1]d |n|>2d

z(t, s)ds − z(t, n))

+ ∑ ( ∫ n∈ℝd n+[0,1]d |n|≤2d

Hence 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 ∫ z(t, s)ds − ∑ z(t, n)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n∈ℤd d ℝ

z(t, s)ds − z(t, n)).

312 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas

≤ ∑ n∈ℝd |n|>2d

ad ‖y‖∞ + cd ‖y‖C 1 󵄩 󵄩 + ∑ 2󵄩󵄩󵄩z(t, ⋅)󵄩󵄩󵄩∞ |n|d+1 d n∈ℝ |n|≤2d

≤ Cd ⋅ ‖y‖C 1

(5.7)

for a constant Cd > 0. We now compute the integral by passing to spherical coordinates: ∞

∫ z(t, s)ds = ∫ y(s)ds ⋅ ∫ (1 + r 2 ) ℝd

− d2 − tr2 d−1

e

r

dr.

(5.8)

t ∈ (0, 1).

(5.9)

0

𝕊d−1

Note that ∞

∫ (1 + r 2 )

− d2 − tr2 d−1

e

r

dr =

0

1 1 log( ) + O(1), 2 t

The required statement is a combination of (5.7), (5.8), and (5.9). The final step of the proof uses a heat trace formula from Volume I to associate the Abel sum with a trace. Lemma 5.4.11. If x ∈ L∞ (𝕋dθ ) and if y ∈ C(𝕊d−1 ), then d

φ(π1 (x)π2 (y)(1 − Δ)− 2 ) =

1 τ (x) ⋅ ∫ y(s)ds d θ 𝕊d−1

for every normalized continuous trace φ on ℒ1,∞ . Proof. Suppose first that y ∈ C 1 (𝕊d−1 ). d Our aim is to apply Theorem 9.1.3 in Volume I with V = (1 − Δ)− 2 ∈ ℒ1,∞ , A = π1 (x)π2 (y) ∈ ℒ(L2 (𝕋dθ )), and α = d2 . Note that the unitary generators {un }n∈ℤd are eigend

vectors for V = (1 − Δ)− 2 and also for π1 (y), π2 (y)un = y(

n )u , |n| n

n ∈ ℤd .

Thus −α

⟨AVe−(mV ) un , un ⟩ = y(

−2 2 n −d d )(1 + |n|2 ) 2 e−m (1+|n| ) ⋅ ⟨π1 (x)un , un ⟩. |n|

By definition, ⟨π1 (x)un , un ⟩ = τθ (xun ⋅ un∗ ) = τθ (x). Thus

5.4 Principal symbol and trace formula for noncommutative tori

−α

⟨AVe−(mV ) un , un ⟩ = y(

�

313

−2 2 n −d d )(1 + |n|2 ) 2 e−m (1+|n| ) ⋅ τθ (x). |n|

Summing over n ∈ ℤd , we obtain −α

Tr(AVe−(mV ) ) = e−m

−2 d

⋅ τθ (x) ⋅ ∑ y( n∈ℤd

−2 2 n −d d )(1 + |n|2 ) 2 e−m |n| . |n|

Using Lemma 5.4.10, we write −α

Tr(AVe−(mV ) ) =

1 τ (x) ⋅ ∫ y(s)ds ⋅ log(m) + O(1), d θ

m ∈ ℕ.

𝕊d−1

By Theorem 9.1.3 in Volume I, AV is universally measurable, and d

φ(π1 (x)π2 (y)(1 − Δ)− 2 ) = φ(AV ) =

1 τ (x) ⋅ ∫ y(s)ds d θ 𝕊d−1

for every normalized trace φ on ℒ1,∞ . This proves the statement for y ∈ C 1 (𝕊d−1 ). To remove the assumption that y ∈ C 1 (𝕊d−1 ), we use the continuity of φ. Note that both mappings y 󳨃→ ∫ y(s)ds 𝕊d−1

and d

y 󳨃→ φ(π1 (x)π2 (y)(1 − Δ)− 2 ) are continuous in the uniform norm on C(𝕊d−1 ). Hence, as we may approximate an ar1 bitrary y ∈ C(𝕊d−1 ) in the uniform norm by a sequence {yk }∞ k=0 of C -functions, the statement follows. Theorem 5.4.8 follows from Lemma 5.4.11 and an application of Theorem 5.2.8. d

Proof of Theorem 5.4.8. Let V := (1 − Δ)− 2 . Define the linear functionals ψ1 ∈ C(𝕋dθ )∗ and ψ2 ∈ C(𝕊d−1 )∗ by setting ψ1 := τθ ,

ψ2 :=

1 ∫ . d 𝕊d−1

Lemma 5.4.11 states that φ(π1 (x)π2 (y) ⋅ V ) = ψ1 (x)ψ2 (y),

x ∈ C(𝕋dθ ),

The statement now follows from Theorem 5.2.8.

y ∈ C(𝕊d−1 ).

314 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas

5.5 Principal symbol and trace formula for the noncommutative plane Section 5.3 proved that the C ∗ -algebra Π(ℂ + C0 (ℝd ), C(𝕊d−1 )) of operators on the Hilbert space L2 (ℝd ) generalizes principal terms in the asymptotic expansions of classical pseudodifferential operators of order zero. The algebra Π(ℂ + C0 (ℝd ), C(𝕊d−1 )) admits a symbol map that extends the notion of the principal symbol of classical pseudodifferential operators of order zero. This section considers an equivalent C ∗ -algebra and symbol map for the noncommutative plane introduced in Chapter 4.

5.5.1 Principal symbol map Let d ∈ 2ℕ be even. Let θ ∈ Md (ℝ) be an antisymmetric matrix with det(θ) ≠ 0. Define the strongly continuous family of unitaries on the Hilbert space L2 (ℝd ) by i

(U(t)v)(s) := e− 2 ⟨t,θs⟩ v(s − t),

v ∈ L2 (ℝd ),

s, t ∈ ℝd .

Let L∞ (ℝdθ ) denote the von Neumann algebra in ℒ(L2 (ℝd )) generated by the family of unitary operators {U(t)}t∈ℝd . In Chapter 4, we introduced the Weyl transform L : 𝒮 (ℝd ) → ℒ(L2 (ℝd )) in Definition 4.2.10 using the weak operator integral d

f ∈ 𝒮 (ℝd ).

L(f ) := (2π)− 2 ∫ (ℱ f )(ξ)U(ξ) dξ, ℝd

The range of the Weyl transform is d

d

𝒮 (ℝθ ) := {L(f ) : f ∈ 𝒮 (ℝ )} d

and is a ∗-algebra of bounded operators in ℒ(L2 (ℝ 2 )). On 𝒮 (ℝdθ ) the faithful normal semifinite trace τθ on L∞ (ℝdθ ) has the concrete form τθ (L(f )) = ∫ f (t)dt,

f ∈ 𝒮 (ℝd ).

ℝd

The noncommutative operator spaces Lp (ℝdθ ), 0 < p < ∞, were introduced in Definition 4.2.7. Lemmas 4.2.14 and 4.2.29(b) state that 𝒮 (ℝdθ ) is dense in the Banach algebra L1 (ℝdθ ) in the norm

5.5 Principal symbol and trace formula for the noncommutative plane

‖x‖L1 := τθ (|x|),

� 315

x ∈ L1 (ℝdθ ).

The closure of 𝒮 (ℝdθ ) in the operator norm topology of ℒ(L2 (ℝd )) is the C ∗ -algebra C0 (ℝdθ ) by Definition 4.2.9. The Laplacian Δℱ : ℱ −1 H 2 (ℝd ) → L2 (ℝd ) introduced in Chapter 4 and used further is the usual Laplacian on ℝd conjugated with the Fourier transform ℱ , (Δℱ u)(t) := −|t|2 u(t),

t ∈ ℝd ,

u ∈ ℱ −1 H 2 (ℝd ).

Similarly, in Chapter 4, we introduced Mk : dom(Mk ) → L2 (ℝd ), k = 1, . . . , d, as the conjugate of the partial derivatives on ℝd under the Fourier transform. Most simply, (Mk u)(t) := tk u(t),

t = (t1 , . . . , td ) ∈ ℝd ,

u ∈ dom(Mk ),

where dom(Mk ) := {u ∈ L2 (ℝd ) : tk u(t) ∈ L2 (ℝd )}. Denote d

d

∇ℱ := (M1 , . . . , Md ) : ⋂ dom(Mk ) → L2 (ℝd ) . k=1

For any g ∈ C(𝕊d−1 ), denote the homogeneous product operator 1

(g(∇ℱ (−Δℱ )− 2 )u)(t) := g(

t t1 , . . . , d )u(t), |t| |t|

t = (t1 , . . . , td ) ≠ 0 ∈ ℝd ,

u ∈ L2 (ℝd ).

The next theorem shows that the Calkin quotient map on ℒ(L2 (ℝd )), which in Section 5.3 was shown to implement the symbol map on principal terms of pseudodifferential operators in the C ∗ -algebra generated by ℂ + C0 (ℝd ) and homogeneous Fourier multipliers, also implements a principal symbol map for the C ∗ -algebra generated by ℂ + C0 (ℝdθ ) and homogeneous product operators. Fourier multipliers are replaced by product operators, as the picture of this section is the Fourier dual of the picture in Section 5.3. Theorem 5.5.1 (Principal symbol map). Let the C ∗ -algebras 𝒜1 := ℂ + C0 (ℝdθ ) and 𝒜2 := C(𝕊d−1 ) be represented as operators on the Hilbert space H := L2 (ℝd ) by π1 (x) := x,

1

π2 (g) := g(∇ℱ (−Δℱ )− 2 ),

x ∈ ℂ + C0 (ℝdθ ),

g ∈ C(𝕊d−1 ).

Let Π(ℂ + C0 (ℝdθ ), C(𝕊d−1 )) be the C ∗ -algebra in ℒ(H) generated by π1 (ℂ + C0 (ℝdθ )) and π2 (C(𝕊d−1 )). Then the Calkin quotient map on ℒ(H) implements a unique ∗-homomorphism sym : Π(ℂ + C0 (ℝdθ ), C(𝕊d−1 )) → C(𝕊d−1 , ℂ + C0 (ℝdθ ))

316 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas such that sym(π1 (x)) = x,

sym(π2 (g)) = g,

x ∈ ℂ + C0 (ℝdθ ),

g ∈ C(𝕊d−1 ),

where ℂ + C0 (ℝdθ ) is identified in C(𝕊d−1 , ℂ + C(ℝdθ )) as the constant functions. To prove Theorem 5.5.1 we verify the commutator and injectivity conditions of Theorem 5.2.6. Commutator condition In this section, we verify the commutator condition in Theorem 5.2.6. As in Section 5.4, denote 1

bk := Mk (−Δℱ )− 2 ,

k = 1, . . . , d,

defined directly by (bk u)(t) :=

tk u(t), |t|

t = (t1 , . . . , td ) ≠ 0 ∈ ℝd ,

u ∈ L2 (ℝd ).

Then bk is everywhere defined and bounded as an operator on L2 (ℝd ). Lemma 5.5.2. If x ∈ 𝒮 (ℝdθ ), then the operator [π1 (x), bk ] is compact. Proof. The proof follows Lemma 5.4.3 in Section 5.4. Set 1

ck := Mk ⋅ (1 − Δℱ )− 2 ,

1 ≤ k ≤ d, 1

which is defined everywhere and bounded since (1 − Δℱ )− 2 : L2 (ℝd ) → ⋂dk=1 dom(Mk ) is continuous. Let f ∈ 𝒮 (ℝd ), and let x = L(f ). Since x : ⋂dk=1 dom(Mk ) → ⋂dk=1 dom(Mk ) by Lemma 4.2.28, we can write 1

1

[ck , π1 (x)] = [Mk , π1 (x)] ⋅ (1 − Δℱ )− 2 + Mk ⋅ [(1 − Δℱ )− 2 , π1 (x)] and 1

1

1

[ck , π1 (x)] = [Mk , π1 (x)] ⋅ (1 − Δℱ )− 2 − ck [(1 − Δℱ ) 2 , π1 (x)] ⋅ (1 − Δℱ )− 2 since 1

d

(1 − Δℱ )− 2 : L2 (ℝd ) → ⋂ dom(Mk ). k=1

5.5 Principal symbol and trace formula for the noncommutative plane

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1

We can substitute the bounded extensions a := [(1−Δℱ ) 2 , π1 (x)] and 𝜕k (x) := [Mk , π1 (x)] into the formula. The boundedness of a follows from the proof of Theorem 4.4.1 in Chapter 4. We obtain 1

[ck , π1 (x)] = (𝜕k (x) − ck a) ⋅ (1 − Δℱ )− 2 for the bounded operator 𝜕k (x)−ck a ∈ ℒ(L2 (ℝd )). Theorem 4.4.1 states that (𝜕k (x)−ck a)⋅ 1 (1 − Δℱ )− 2 ∈ ℒd,∞ . Thus [ck , π1 (x)] ∈ 𝒞0 (H). Note from direct calculation that 1

1

1

(bk − ck ) ⋅ (1 − Δℱ ) 2 = bk ⋅ ((1 − Δℱ ) 2 + (−Δℱ ) 2 )

−1

∈ ℒ(L2 (ℝd )).

We write 1

1

[bk − ck , π1 (x)] = ((bk − ck ) ⋅ (1 − Δℱ ) 2 ) ⋅ (1 − Δℱ )− 2 π1 (x) 1

1

− π1 (x)(1 − Δℱ )− 2 ⋅ ((bk − ck ) ⋅ (1 − Δℱ ) 2 ⋅). Hence [bk − ck , π1 (x)] ∈ ℒ(L2 (ℝd )) ⋅ ℒd,∞ + ℒd,∞ ⋅ ℒ(L2 (ℝd )) ⊂ 𝒞0 (L2 (ℝd )) by Theorem 4.4.1. Writing [bk , π1 (x)] = [ck , π1 (x)] + [bk − ck , π1 (x)] ∈ 𝒞0 (L2 (ℝd )), we complete the proof. The next lemma is the commutator condition in Theorem 5.2.6. Lemma 5.5.3. If x ∈ ℂ + C0 (ℝdθ ), then [π1 (x), π2 (g)] is compact for every g ∈ C(𝕊d−1 ). Proof. The proof follows Lemma 5.4.3 in Section 5.4. Suppose x ∈ 𝒮 (ℝdθ ). Let m

y := ∏ bkl . l=1

Then m

j−1

m

j=1

l=1

l=j+1

[y, π1 (x)] = ∑(∏ bkl ) ⋅ [bkj , π1 (x)] ⋅ ( ∏ bkl ).

318 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas From Lemma 5.5.2 it follows that [y, π1 (x)] is compact. If y is in the ∗-algebra generated by {bk }dk=1 , then by the preceding paragraph and by linearity, [y, π1 (x)] is compact. Suppose now x is in C0 (ℝdθ ) and y := π2 (g). Choose sequences {xn }n≥0 in 𝒮 (ℝdθ ) and {yn }n≥0 in the ∗-algebra generated by {bk }dk=1 such that xn → x and yn → y in the uniform norm. We have [yn , π1 (xn )] → [y, π1 (x)] in the uniform norm. Since the sequence in the left-hand side consists of compact operators, it follows that its limit point on the right-hand side is also compact for each x ∈ C0 (ℝdθ ). The statement follows since [y, c ⋅ 1] = 0 for any constant c ∈ ℂ. Injectivity condition In this section, we verify the injectivity condition in Theorem 5.2.6 for the noncommutative plane. We start with the following technical lemmas. Lemma 5.5.4. Let H be a separable infinite-dimensional Hilbert space. If T ∈ 𝒞0 (H) and if {pk }k≥0 ⊂ ℒ(H) is a sequence of pairwise orthogonal projections, then ‖Tpk ‖∞ → 0 as k → ∞. Proof. Let ε > 0, and let T = T1 + T2 , where T1 is of finite rank, and ‖T2 ‖∞ < ε. As each pk is pairwise orthogonal and T1 ∈ ℒ2 , ∞

∑

k=0

‖T1 pk ‖22

󵄩󵄩 ∞ 󵄩󵄩2 󵄩󵄩 󵄩󵄩 = 󵄩󵄩󵄩T1 ∑ pk 󵄩󵄩󵄩 ≤ ‖T1 ‖22 < ∞. 󵄩󵄩 󵄩 󵄩 k=0 󵄩󵄩2

In particular, ‖T1 pk ‖2 → 0 as k → ∞. Since ‖T1 pk ‖∞ ≤ ‖T1 pk ‖2 , it follows that ‖T1 pk ‖∞ → 0 as k → ∞. By the triangle inequality we have lim sup ‖Tpk ‖∞ ≤ ε. k→∞

Since ε > 0 is arbitrarily small, the statement follows. Lemma 5.5.5. Let x ∈ L∞ (ℝdθ ). If x ⋅ χ[0,1]d (∇ℱ ) = 0, then x = 0. Proof. Let p ∈ L∞ (ℝdθ ) be a projection such that τθ (p) < ∞. We have px ⋅ χ[0,1]d (∇ℱ ) = 0. By Lemma 4.3.2 we have 󵄩 󵄩 0 = 󵄩󵄩󵄩px ⋅ χ[0,1]d (∇ℱ )󵄩󵄩󵄩ℒ = cd ⋅ ‖px‖L2 (ℝd ) ‖χ[0,1]d ‖L2 (ℝd ) 2

θ

for some nonzero constant cd . Thus ‖px‖L2 = 0, and px = 0.

5.5 Principal symbol and trace formula for the noncommutative plane

� 319

From Subsection 4.2.1, the von Neumann algebra L∞ (ℝdθ ) is ∗-isomorphic to d

ℒ(L2 (ℝ 2 )). If p is be the image of a finite-rank projection under this ∗-isomorphism,

then px = 0 for every such projection p implies that x = 0.

The next lemma gives an injectivity condition for the noncommutative plane. Lemma 5.5.6. Let xk ∈ L∞ (ℝdθ ) and yk ∈ C(𝕊d−1 ), 1 ≤ k ≤ n. Let π1 and π2 be as in Theorem 5.5.1. If n

∑ π1 (xk )π2 (yk ) ∈ 𝒞0 (L2 (ℝd )),

k=1

then necessarily n

∑ xk ⊗ yk = 0.

k=1

Proof. Fix s ∈ 𝕊d−1 and choose a sequence {mj }j≥0 ⊂ ℤd such that as j → ∞. It follows that

󵄨󵄨 t 󵄨󵄨 󵄨󵄨 󵄨 󵄨󵄨 − s󵄨󵄨󵄨 → 0, 󵄨 󵄨󵄨 |t| d 󵄨 t∈mj +[0,1] sup

mj |mj |

→ s and |mj | → ∞

j → ∞.

Since yk ∈ C(𝕊d−1 ), 1 ≤ k ≤ n, it follows that 󵄨󵄨 󵄨󵄨 t 󵄨 󵄨 sup 󵄨󵄨󵄨 yk ( )χmj +[0,1]d (t) − yk (s)χmj +[0,1]d (t)󵄨󵄨󵄨 󵄨 󵄨󵄨 |t| t∈ℝd 󵄨 󵄨󵄨 󵄨󵄨 t 󵄨 󵄨 = sup 󵄨󵄨󵄨 yk ( ) − yk (s)󵄨󵄨󵄨 → 0, j → ∞. 󵄨 |t| d 󵄨 󵄨󵄨 t∈m +[0,1] j

By the spectral theorem we have 1

yk (∇ℱ (−Δℱ )− 2 )χmj +[0,1]d (∇ℱ ) − yk (s)χmj +[0,1]d (∇ℱ ) → 0,

j → ∞,

in the uniform norm. By the definition of π2 we have π2 (yk )χmj +[0,1]d (∇ℱ ) − yk (s)χmj +[0,1]d (∇ℱ ) → 0

(5.10)

in the uniform norm as j → ∞. Assume initially that θ = Ω, where Ω is the block diagonal matrix in Proposition 4.2.5. By Lemma 5.5.4 we have that n

∑ π1 (xk )π2 (yk )χmj +[0,1]d (∇ℱ ) → 0

k=1

320 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas in the uniform norm as j → ∞. By (5.10) we have n

∑ π1 (xk )yk (s)χmj +[0,1]d (∇ℱ ) → 0

k=1

in the uniform norm as j → ∞. By Lemma 4.3.7 there exists a unitary operator Vj ∈ ℒ(L2 (ℝd )) that commutes with L∞ (ℝdΩ ) and such that Vj χmj +[0,1]d (∇ℱ )Vj−1 = χ[0,1]d (∇ℱ ). Thus n

n

k=1

k=1

∑ π1 (xk )yk (s)χ[0,1]d (∇ℱ ) = Vj ⋅ ( ∑ π1 (xk )yk (s)χmj +[0,1]d (∇ℱ )) ⋅ Vj−1 → 0

in the uniform norm as j → ∞. The left-hand side does not depend on j, and therefore n

∑ π1 (xk )yk (s)χ[0,1]d (∇ℱ ) = 0.

k=1

Taking into account that π1 = id, we write n

( ∑ xk ⋅ yk (s)) ⋅ χ[0,1]d (∇ℱ ) = 0. k=1

By Lemma 5.5.5 we have n

∑ xk ⋅ yk (s) = 0.

k=1

Since s ∈ 𝕊d−1 is arbitrary, the statement follows when θ = Ω. If θ is now a general matrix and if xk belongs to L∞ (ℝdθ ) and is such that n

∑ π1 (xk )π2 (yk ) ∈ 𝒞0 (L2 (ℝd )),

k=1

then, setting xk′ := Wxk W −1 ∈ L∞ (ℝdΩ ) where W is the unitary implementing the spatial isomorphism in Proposition 4.2.5, n

∑ π1 (xk′ )π2 (Wyk ) ∈ 𝒞0 (L2 (ℝd )).

k=1

Thus, as the result has been established for L∞ (ℝdΩ ),

5.5 Principal symbol and trace formula for the noncommutative plane

� 321

n

∑ xk′ ⋅ yk (s′ ) = 0.

k=1

Here, if s = |t|−1 t for some t ∈ ℝd , t ≠ 0, then 󵄨 󵄨−1 s′ = 󵄨󵄨󵄨Q−1 N −1 t 󵄨󵄨󵄨 (Q−1 N −1 t). Since yk (s′ ) is a scalar, n

n

k=1

k=1

∑ xk ⋅ yk (s′ ) = W −1 ⋅ ∑ xk′ ⋅ yk (s′ ) ⋅ W = 0.

Since s′ ∈ 𝕊d−1 is arbitrary, the statement follows.

5.5.2 Trace formula In this section, we establish Connes’ trace theorem for the analogue of principal terms of pseudodifferential operators on noncommutative Euclidean space. An operator T ∈ Π(C(ℝdθ ), C(𝕊d−1 )) is compactly supported on the right if there exists z ∈ 𝒮 (ℝdθ ) such that T = Tz. Theorem 5.5.7. Let Π(ℂ + C0 (ℝdθ ), C(𝕊d−1 )) ⊂ ℒ(L2 (ℝd )) be the C ∗ -algebra generated by the C ∗ -algebras π1 (𝒜1 ) and π2 (𝒜2 ) as in the statement of Theorem 5.5.1. If T ∈ Π(ℂ + C0 (ℝdθ ), C(𝕊d−1 )) is compactly supported on the right, then d

T(1 − Δℱ )− 2 ∈ ℒ1,∞ . Furthermore, for every normalized continuous trace φ on ℒ1,∞ , we have d

φ(T(1 − Δℱ )− 2 ) =

1 ⋅ (τθ × ∫ )(sym(T)), d(2π)d 𝕊d−1

where (τθ × ∫ )(σ) := ∫ τθ (σ(s))ds, 𝕊d−1

σ ∈ C(𝕊d−1 , ℂ + C0 (ℝdθ )),

𝕊d−1

is the weight on the C ∗ -algebra C(𝕊d−1 , ℂ + C0 (ℝdθ )) given by composing the Lebesgue integral on C(𝕊d−1 ) with the faithful normal semifinite trace τθ on L∞ (ℝdθ ). The scheme of proof is similar to that of Section 5.3.2 but is complicated by the lack of rotation symmetry on ℝdθ . In Section 4.2.4 in Chapter 4, we discussed that the defining relation

322 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas [Xj , Xk ] = iθj,k ,

1 ≤ j, k ≤ d,

of the algebra of polynomials on ℝdθ is not invariant under the action of the rotation group SO(d) on generators X = {X1 , . . . , Xd } of the polynomial algebra. Under a linear change of variables, d

Xj′ := ∑ Aj,k Xj , k=1

where A = {Aj,k }dj,k=1 is a real matrix, the variables {X1′ , X2′ , . . . , Xd′ } obey [Xj′ , Xk′ ] = iθj,k if and only if A belongs to the symplectic group Sp(θ, d) := {A ∈ Md (ℝ) : A∗ θA = θ}. The proof of Theorem 5.5.7 involves identifying a continuous linear functional on the algebra C(𝕊d−1 , ℂ+C0 (ℝdθ )) that is invariant under the smaller symmetry group Sp(θ, d)∩ SO(d). Our first stage of the proof shows that rotation-invariant measures on spheres are characterized by the subgroup Sp(θ, d) ∩ SO(d) of rotations. d We identify the unit sphere in ℝd with the unit sphere in ℂ 2 d 2

d −1 2

{(z0 , . . . , z d −1 ) ∈ ℂ : ∑ |zk |2 = 1} 2

k=0

using the map d

γ : s = (s0 , s1 , . . . , sd−1 ) ∈ 𝕊d−1 󳨃→ γ(s) = (s0 + is1 , s2 + is3 , . . . , sd−2 + isd−1 ) ∈ ℂ 2 . Let U( d2 ) be the Lie group of all unitary matrices in dimension d2 , which we equip with the normalized Haar measure. Let a unitary matrix g ∈ U( d2 ) act on C(𝕊d−1 ) by the operator (Vg f )(s) := f ((γ−1 g −1 γ)(s)),

f ∈ C(𝕊d−1 ),

d g ∈ U( ). 2

For this section, we denote by m the linear functional on C(𝕊d−1 ) obtained via integration with respect to the rotation-invariant Lebesgue measure. The next lemma states that every U( d2 )-invariant measure on the sphere 𝕊d−1 is rotation invariant. Lemma 5.5.8. If l ∈ C(𝕊d−1 )∗ is such that l ∘ Vg = l for all g ∈ U( d2 ), then l = α ⋅ m for some α ∈ ℂ.

5.5 Principal symbol and trace formula for the noncommutative plane

� 323

Proof. Let f ∈ C(𝕊d−1 ). Define the function kf (s) := ∫ (Vg f )(s)dg,

s ∈ 𝕊d−1 ,

U( d2 )

where the integral is taken with respect to the normalized Haar measure on U( d2 ). The C(𝕊d−1 )-valued mapping g 󳨃→ Vg f ,

d g ∈ U( ), 2

is continuous and therefore Bochner integrable. Hence kf ∈ C(𝕊d−1 ). Since l ∈ C(𝕊d−1 )∗ and g 󳨃→ Vg f is Bochner integrable, it follows that l(kf ) = l( ∫ (Vg f )dg) = ∫ l(Vg f )dg = l(f ). U( d2 )

(5.11)

U( d2 )

Note that the action s 󳨃→ (γ−1 g −1 γ)(s) of U( d2 ) is transitive on 𝕊d−1 . For fixed s1 , s2 ∈ 𝕊d−1 , let h ∈ U( d2 ) be the matrix such that (γ−1 h−1 γ)(s1 ) = s2 . Then (γ−1 g −1 γ)(s2 ) = (γ−1 (hg)−1 γ)(s1 ),

d g ∈ U( ), 2

and ∫ (Vg f )(s2 )dg = ∫ (Vhg f )(s1 )dg U( d2 )

U( d2 )

= ∫ (Vg f )(s1 )d(h−1 g) = ∫ (Vg f )(s1 )dg. U( d2 )

U( d2 )

In the last equality, we used the property that dg is a Haar measure. Hence the function kf is a constant function on 𝕊d−1 . Therefore kf (s) =

1 ∫ kf (u)du, Vol(𝕊d−1 ) 𝕊d−1

It remains to note that ∫ kf (u)du = ∫ ( ∫ (Vg f )(s)dg)ds 𝕊d−1

𝕊d−1 U( d ) 2

s ∈ 𝕊d−1 .

324 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas

= ∫ ( ∫ (Vg f )(s)ds)dg = ∫ ( ∫ f (s)ds)dg = ∫ f (s)ds. U( d2 ) 𝕊d−1

U( d2 ) 𝕊d−1

𝕊d−1

Thus l(kf ) = (Vol(𝕊d−1 ))

−1

⋅ m(f ) ⋅ l(1).

(5.12)

The statement follows from (5.11) and (5.12). ′ We now associate the group U( d2 ) with the subgroup of symplectic rotations. Let Ek,l denote the matrix in M d (ℂ) that is zero except for 1 in the (k, l)-position, 1 ≤ k, l ≤ d2 . 2

Let Em,n denote the matrix in Md (ℝ) that is zero except for 1 in the (m, n)-position, 1 ≤ m, n ≤ d. Define the ℝ-linear map reald : M d (ℂ) → Md (ℝ) 2

by the formula ′ reald (Ek,l ) := E2k−1,2l−1 + E2k,2l ,

′ reald (iEk,l ) := E2k−1,2l − E2k,2l−1

for 1 ≤ k, l ≤ d2 , that is, if A ∈ M d (ℂ), then reald (A) is the underlying real d × d matrix 2

formed from the real and imaginary parts of the entries of A separately. Lemma 5.5.9. We have d reald (U( )) = Sp(Ω, d) ∩ SO(d), 2 where Ω is the matrix reald (iI d ). 2

Proof. The following properties are easy to check for every A, B ∈ M d (ℂ): 2

reald (AB) = reald (A) ⋅ reald (B),

reald (A∗ ) = (reald (A)) . ∗

By definition, reald (iI d ) = Ω. Thus, for every g ∈ U( d2 ), we have 2

(reald (g)) ⋅ Ω ⋅ reald (g) = reald (g ∗ ) ⋅ reald (iI d ) ⋅ reald (g) ∗

2

= reald (ig ⋅ I d ⋅ g) = reald (iI d ) = Ω. ∗

2

In other words, reald (g) ∈ Sp(Ω, d). Also,

2

5.5 Principal symbol and trace formula for the noncommutative plane

� 325

(reald (g)) ⋅ reald (g) = reald (g ∗ ) ⋅ reald (g) = reald (g ∗ ⋅ g) = reald (1) = 1. ∗

In other words, reald (g) ∈ SO(d). Thus d reald (U( )) ⊂ Sp(Ω, d) ∩ SO(d). 2 The reverse inclusion is proved similarly. by

Define the unitary action A 󳨃→ WA of the special orthogonal group SO(d) on L2 (ℝd ) WA u := u ∘ A−1 ,

u ∈ L2 (ℝd ).

Lemma 5.5.10. If l ∈ C(𝕊d−1 )∗ is such that l ∘ WA = l for every A ∈ Sp(θ, d) ∩ SO(d), then l = α ∘ m for some α ∈ ℂ. Proof. By comparing the definitions of Vg and WA we note that Vg = Wreald (g) ,

d g ∈ U( ). 2

Thus l = l ∘ Vg for every g ∈ U( d2 ). The statement when θ = Ω := reald (iI d ) follows from 2 Lemmas 5.5.8 and 5.5.9. Now let θ ∈ Md (ℝ) be an antisymmetric matrix with det(θ) ≠ 0. The following result of linear algebra is well known and easily follows from [255, Section 9.44]. There exists β ∈ SO(d) such that β−1 θβ = det(θ)Ω. If A ∈ Sp(θ, d), then β−1 Aβ ∈ Sp(Ω, d). Indeed, 1 ⋅ β−1 A∗ ⋅ θ ⋅ Aβ det(θ) 1 1 = ⋅ β−1 ⋅ A∗ θA ⋅ β = ⋅ β−1 ⋅ θ ⋅ β det(θ) det(θ)

(β−1 Aβ) ⋅ Ω ⋅ (β−1 Aβ) = β−1 A∗ ⋅ βΩβ−1 ⋅ Aβ = ∗

= Ω.

Conversely, if β−1 Aβ ∈ Sp(Ω, d), then A ∈ Sp(θ, d). Since A ∈ SO(d) if and only if β−1 Aβ ∈ SO(d), we have A ∈ Sp(θ, d) ∩ SO(d)

iff β−1 Aβ ∈ Sp(Ω, d) ∩ SO(d).

(5.13)

326 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas Now let l ∘ WA = l for every A ∈ Sp(θ, d) ∩ SO(d). It follows that l = l ∘ Wβ ∘ WB ∘ Wβ−1 for every B ∈ Sp(Ω, d) ∩ SO(d). Equivalently, (l ∘ Wβ ) ∘ WB = l ∘ Wβ for every B ∈ Sp(Ω, d) ∩ SO(d). Since the latter equality holds for every B ∈ Sp(Ω, d) ∩ SO(d), it follows from the case θ = Ω that l ∘ Wβ = α ⋅ m. Therefore l = (α ⋅ m) ∘ Wβ−1 = α ⋅ m since the Lebesgue measure on 𝕊d−1 is rotation invariant. Lemma 5.5.10 establishes that the invariance under the subgroup of rotations Sp(θ, d) ∩ SO(d) characterizes the Lebesgue measure on 𝕊d−1 . Let φ be a continuous trace on ℒ1,∞ . The second stage of the proof of Theorem 5.5.7 identifies the functional d

x ⊗ b 󳨃→ φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ),

x ∈ C0 (ℝdθ ),

b ∈ C(𝕊d−1 ),

(5.14)

on the algebraic tensor product of the C ∗ -algebras C0 (ℝdθ ) and C(𝕊d−1 ). The next lemma identifies the form of the functional (5.14) on the C0 (ℝdθ ) component of the algebraic tensor product. Observe that ⟨t, ∇ℱ ⟩ = ∑dk=1 tk Mk is self-adjoint on the domain ⋂{k:tk =0} ̸ dom(Mk )

for t = (t1 , . . . , td ) ≠ 0 ∈ ℝd and ⟨t, ∇ℱ ⟩ = 0 if t = 0. Since the operators Mk , 1 ≤ k ≤ d, strongly commute, d

ei⟨t,∇ℱ ⟩ = ∏ eitk Mk , k=1

t = (t1 , . . . , td ) ∈ ℝd .

Recall also from Definition 4.2.26 that the Sobolev space W1d (ℝdθ ) is defined as the subspace of x ∈ L1 (ℝdθ ) such that 󵄩 󵄩 ‖x‖W d := ∑ 󵄩󵄩󵄩𝜕α (x)󵄩󵄩󵄩L < ∞, 1 1 |α|≤d

where α

α

𝜕α (x) := (𝜕1 1 ∘ ⋅ ⋅ ⋅ ∘ 𝜕d d )(x)

5.5 Principal symbol and trace formula for the noncommutative plane

� 327

and 𝜕k (x) := [Mk , x], 1 ≤ αk ≤ d, 1 ≤ k ≤ d. The space 𝒮 (ℝdθ ) is dense in the Sobolev space W1d (ℝdθ ). In Lemma 4.5.4, we observed that the invariance of continuous linear functionals on the space W1d (ℝdθ ) can identify the trace τθ acting on W1d (ℝdθ ). Lemma 5.5.11. Let φ be a continuous trace on ℒ1,∞ . There is a continuous functional l ∈ C(𝕊d−1 )∗ such that for all x ∈ W1d (ℝdθ ) and for all b ∈ C(𝕊d−1 ), we have d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = τθ (x) ⋅ l(b). Proof. Let x ∈ W1d (ℝdθ ), and let b ∈ C(𝕊d−1 ). Since φ is unitarily invariant, it follows that d

d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = φ(ei⟨θt,∇ℱ ⟩ π1 (x)π2 (b)(1 − Δℱ )− 2 e−i⟨θt,∇ℱ ⟩ ) for every t ∈ ℝd . However, ∇ℱ commutes with Δℱ and with π2 (b). Thus d

d

ei⟨θt,∇ℱ ⟩ π1 (x)π2 (b)(1 − Δℱ )− 2 e−i⟨θt,∇ℱ ⟩ = ei⟨θt,∇ℱ ⟩ π1 (x)e−i⟨θt,∇ℱ ⟩ ⋅ π2 (b)(1 − Δℱ )− 2 . By Lemma 4.2.21 we have ei⟨θt,∇ℱ ⟩ π1 (x)e−i⟨θt,∇ℱ ⟩ = π1 (U(−t)xU(t)), where U(t) is the unitary in (4.2). Combining these three displays, we obtain d

d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = φ(π1 (U(−t)xU(t))π2 (b)(1 − Δℱ )− 2 )

(5.15)

for every t ∈ ℝd . For a fixed b ∈ C(𝕊d−1 ), define the linear functional Fb on W1d (ℝdθ ) by the formula d

Fb (x) := φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ),

x ∈ W1d (ℝdθ ). d

By the definition of π2 the operators π2 (b) and (1 − Δℱ )− 2 commute. Therefore 󵄨󵄨 󵄨 󵄩 −d 󵄩 󵄨󵄨Fb (x)󵄨󵄨󵄨 ≤ ‖φ‖ℒ1,∞ →ℂ 󵄩󵄩󵄩π1 (x)π2 (b)(1 − Δℱ ) 2 󵄩󵄩󵄩1,∞ d 󵄩 󵄩 ≤ ‖φ‖ℒ1,∞ →ℂ 󵄩󵄩󵄩π1 (x)(1 − Δℱ )− 2 󵄩󵄩󵄩1,∞ ‖b‖∞ . By Theorem 4.1.1 we have 󵄨󵄨 󵄨 󵄨󵄨Fb (x)󵄨󵄨󵄨 ≤ cd ‖φ‖ℒ1,∞ →ℂ ‖x‖W d (ℝd ) ‖b‖∞ , 1 θ so that Fb is continuous on W1d (ℝdθ ). By (5.15) we have Fb (U(−t)xU(t)) = Fb (x),

x ∈ W1d (ℝdθ ).

(5.16)

328 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas From Lemma 4.5.4 we can conclude that Fb is a scalar multiple of τθ . So d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = τθ (x) ⋅ l(b)

(5.17)

for some functional l on C(𝕊d−1 ). Clearly, the functional l is linear. Combining (5.16) and (5.17), we obtain 󵄨󵄨 󵄨 󵄨󵄨τθ (x) ⋅ l(b)󵄨󵄨󵄨 ≤ cd ‖φ‖ℒ1,∞ →ℂ ‖x‖W d (ℝd ) ‖b‖∞ , 1 θ

x ∈ W1d (ℝdθ ),

b ∈ C(𝕊d−1 ).

Taking a particular x ∈ W1d (ℝdθ ) such that τθ (x) ≠ 0, we infer that l is a continuous linear functional on C(𝕊d−1 ). The next lemma identifies the form of functional (5.14) on the C(𝕊d−1 ) component of the algebraic tensor product using the invariance under symplectic rotations. Lemma 5.5.12. Let l be the linear functional from Lemma 5.5.11. We have l ∘ WA−1 = l,

A ∈ Sp(θ, d) ∩ SO(d).

Proof. Fix x ∈ W1d (ℝdθ ) such that τθ (x) ≠ 0. Let A ∈ Sp(θ, d) ∩ SO(d). Since the operator WA from Lemma 4.2.33 is unitary, it follows from Lemma 5.5.11 that d

d

τθ (x)l(b) = φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = φ(WA∗ π1 (x)π2 (b)(1 − Δℱ )− 2 WA ). Since A ∈ SO(d), it follows that d

d

(1 − Δℱ )− 2 WA = WA (1 − Δℱ )− 2 ,

π2 (b)WA = WA π2 (WA−1 b).

Thus d

τθ (x)l(b) = φ(WA∗ π1 (x)WA π2 (WA−1 b)(1 − Δℱ )− 2 ). Since A ∈ Sp(θ, d), it follows from Lemma 4.2.33 that WA∗ π1 (x)WA = π1 (xA ) for some xA ∈ W1d (ℝdθ ). So d

τθ (x)l(b) = φ(π1 (xA )π2 (WA−1 b)(1 − Δℱ )− 2 ) = τθ (xA )l(WA−1 b), where the second equality uses Lemma 5.5.11. The statement follows since τθ (xA ) = τθ (x) by Lemma 4.2.33 and τθ (x) ≠ 0. We can now identify the functional in (5.14).

5.5 Principal symbol and trace formula for the noncommutative plane

� 329

Corollary 5.5.13. Let φ be a continuous normalized trace on ℒ1,∞ . For all x ∈ W1d (ℝdθ ) and all b ∈ C(𝕊d−1 ), we have d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) =

1 ⋅ τθ (x)m(b), d(2π)d

where m is the rotation-invariant Lebesgue measure on 𝕊d−1 , and τθ is the semifinite faithful normal trace on L∞ (ℝdθ ). Proof. Let l be the linear functional from Lemma 5.5.11. By Lemma 5.5.12 we have l = l ∘ WA for every A ∈ Sp(θ, d) ∩ SO(d). By Lemma 5.5.10 there exists α ∈ ℂ such that l = α ⋅ m. Thus d

φ(π1 (x)π2 (b)(1 − Δℱ )− 2 ) = α ⋅ τθ (x)m(b). Setting b = 1, we obtain d

φ(π1 (x)(1 − Δℱ )− 2 ) = α ⋅ τθ (x)m(1). Using Theorem 4.1.2, we conclude that α = (d(2π)d )−1 since m(1) = Vol(𝕊d−1 ). This completes the proof. An operator T ∈ Π(C(ℝdθ ), C(𝕊d−1 )) is compactly supported on the right if there is z ∈ W1d (ℝdθ ) such that T = Tz. Note that this is more general than the statement in Theorem 5.5.7. Proof of Theorem 5.5.7. We apply Theorem 5.2.8. Fix z ∈ W1d (ℝdθ ) and set d

V := π1 (z)(1 − Δℱ )− 2 . By Theorem 4.1.1 we have V ∈ ℒ1,∞ . Define continuous linear functionals on ℂ + C(ℝdθ ) and C(𝕊d−1 ) by setting, respectively, ψ1 (x) := τθ (xz), ψ2 (b) :=

x ∈ ℂ + C(ℝdθ ),

1 ⋅ m(b), d(2π)d

b ∈ C(𝕊d−1 ).

Suppose first that x ∈ ℂ + W1d (ℝdθ ), so that xz ∈ W1d (ℝdθ ). As established in d

Lemma 5.4.4, [π1 (x), π2 (b)] is compact, and we know that π2 (b) and (1 − Δℱ )− 2 commute. Since φ is continuous, it follows that φ(π1 (x)π2 (b) ⋅ V ) = φ(π2 (b)π1 (x) ⋅ V ) d

= φ(π1 (x) ⋅ V ⋅ π2 (b)) = φ(π1 (xz)π2 (b)(1 − Δℱ )− 2 ). Therefore

330 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas φ(π1 (x)π2 (b) ⋅ V ) = ψ1 (x)ψ2 (b),

x ∈ ℂ + W1d (ℝdθ ),

b ∈ C(𝕊d−1 ),

by Corollary 5.5.13. By Lemma 4.2.29 the subspace W1d (ℝdθ ) is dense in C0 (ℝdθ ) in the uniform norm. Since φ is a continuous trace and since V ∈ ℒ1,∞ , it follows that the mapping x 󳨃→ φ(π1 (x)π2 (b) ⋅ V ),

x ∈ ℂ + C0 (ℝdθ ),

is continuous in the uniform norm. By continuity we have φ(π1 (x)π2 (b) ⋅ V ) = ψ1 (x)ψ2 (b),

x ∈ ℂ + C0 (ℝdθ ), b ∈ C(𝕊d−1 ).

The conditions of Theorem 5.2.8 are satisfied, and therefore φ(TV ) = (ψ1 ⊗ ψ2 )(sym(T)),

T ∈ Π(ℂ + C(ℝdθ ), C(𝕊d−1 )).

Since z ∈ W1d (ℝdθ ) is arbitrary, choose z such that T = Tπ1 (z). Then d

TV = T(1 − Δℱ )− 2 ,

(ψ1 ⊗ ψ2 )(sym(T)) =

1 (τθ ⊗ m)(sym(T)), d(2π)d

and d

φ(T(1 − Δℱ )− 2 ) =

1 (τθ ⊗ m)(sym(T)). d(2π)d

5.6 Principal symbol and trace formula for SU(2) Connes’ trace formula, as originally stated by Connes, was stated for pseudodifferential operators on general compact manifolds. The C ∗ -algebraic approach to the principal symbol in Section 5.2 is suited to manifolds admitting a globally defined differential calculus, the most important examples of which are Lie groups.

5.6.1 Differential calculus of SU(2) To keep the discussion focused, we concentrate on SU(2), which is arguably the simplest non-abelian Lie group. The special unitary group SU(2) is the group of all complex 2 × 2 unitary matrices with unit determinant. Equivalently, a SU(2) := {( b

−b ) : a, b ∈ ℂ, |a|2 + |b|2 = 1} . a

5.6 Principal symbol and trace formula for SU(2)

�

331

Indeed, it is easy to see that any 2×2 unitary matrix must have the above form for unique a and b such that |a|2 + |b|2 = 1, the condition corresponding to the unit determinant condition. Topologically, SU(2) is a 3-sphere, because the mapping SU(2) ∋ (

a b

−b ) 󳨃→ (a, b) ∈ ℂ2 a

is a homeomorphism between SU(2) and the unit sphere in ℂ2 . Identifying SU(2) with 𝕊3 gives SU(2) the structure of a manifold. We equip SU(2) with its normalized Haar measure, which under the identification with 𝕊3 corresponds to a constant multiple of the Lebesgue measure on 𝕊3 . When we write L2 (SU(2)), we always implicitly mean the L2 -space defined by the Haar measure. Denote by λl and λr the left and right regular representations of SU(2) on L2 (SU(2)), respectively. These are defined by (λl (t)u)(s) := u(t −1 s),

(λr (t)u)(s) := u(st),

s, t ∈ SU(2),

u ∈ L2 (SU(2)).

Since L2 (SU(2)) is equipped with its Haar measure, the operation of right multiplication s 󳨃→ st is measure preserving for t ∈ SU(2). It follows that λl and λr are unitary representations of SU(2). Definition 5.6.1. Denote by σ1 , σ2 , and σ3 the matrices σ1 := (

1 0

0 ), −1

σ2 := (

0 i

−i ), 0

σ3 := (

0 1

1 ). 0

For j = 1, 2, 3, denote by Dj the derivation Dj : C ∞ (SU(2)) → C ∞ (SU(2)) defined by (Dj f )(s) :=

1 d 󵄨 f (e−itσj s)󵄨󵄨󵄨t=0 , i dt

s ∈ SU(2),

f ∈ C ∞ (SU(2)).

We call the triple {D1 , D2 , D3 } the canonical differential calculus of SU(2). Denote by Δ the operator Δ := −(D21 + D22 + D23 ). This is the Casimir operator of SU(2). For j = 1, 2, 3, the operators Dj and Δ admit unique self-adjoint extensions to L2 (SU(2)). As a self-adjoint operator on L2 (SU(2)), Dj is the generator of the unitary group

332 � 5 A C ∗ -algebraic approach to principal symbols and trace formulas t 󳨃→ λl (eitσj ),

t ∈ ℝ.

The domains of the self-adjoint extensions of Dj and Δ can be concretely described by the basis of L2 (SU(2)); see Theorem 5.6.3. The following lemma is a straightforward computation based on the identities eitσj = cos(t) + iσj sin(t) and [σ1 , σ2 ] = 2iσ3 . Lemma 5.6.2. We have the identities [D1 , D2 ] = 2iD3 , [Δ, Dj ] = 0,

[D2 , D3 ] = 2iD1 ,

j = 1, 2, 3.

[D3 , D2 ] = 2iD1 ,

These may be understood as equalities of linear endomorphisms of C ∞ (SU(2)). The Peter–Weyl theorem states that if G is a compact topological group, then L2 (G) (defined via the Haar measure) is unitarily equivalent to the direct sum of all Vπ ⊗ Vπ∗ , where π is an irreducible unitary representation of G on the finite-dimensional space Vπ . For every l ≥ 0, there exists a unique irreducible unitary representation of SU(2) of dimension l + 1 with several well-documented explicit constructions. We will need very little information on the nature of the Peter–Weyl decomposition. All that we need is the following result. l Theorem 5.6.3. There exists an orthonormal basis {ej,k }0≤j,k≤lε

u(s) ds, t−s

u ∈ Cc1 (ℝ).

Alternatively, H can be realized as the Fourier multiplier H = χ[0,∞) (∇) − χ(−∞,0) (∇), where ∇ = −i𝜕t is the one-dimensional gradient. The Plancherel theorem implies that H is unitary on L2 (ℝ), and H is easily verified to commute with all translations and dilations of ℝ. In the language of Connes’ quantized calculus, given f ∈ L∞ (ℝ), the commutator i[H, Mf ] -- . Explicitly, when f and u are suffiis termed the quantized differential of f , denoted df -ciently regular, df is expressed as -- )u)(t) = ((df

∞

f (t) − f (s) 1 u(s) ds, ∫ π t−s

t ∈ ℝ.

−∞

This is intended to be, in some sense, an operator-theoretic substitute for the differen-- makes sense for functions f that tial form df = 𝜕t fdt. Unlike the differential form df , df -- is that formal exneed not be differentiable or even continuous. Another feature of df z −1 --pressions such as |df | and exp(−|df | ) have natural interpretations as operators on L2 (ℝ), but there is no evident meaning to absolute values or fractional powers of a differential form df . We refer the reader to [72, Chapter 4] for further background and applications. -- , apart from its origins in An interesting feature of the quantized differential df quantized calculus, is the interplay between the differentiability of f and the compact-- . Heuristically, the singular value sequence μ(df -- ) is a measure of the size of the ness of df -infinitesimal df , and we should expect a high degree of regularity of f to imply a rapid -- ). Peller [220, Theorem 7.3, Ch. 6, p.2̃76] states that, for p > 0, the rate of decay of μ(df -operator df belongs to the Schatten ideal ℒp (L2 (ℝ)) if and only if f is in the Besov space 1

p Bp,p (ℝ). In this chapter, we consider analogous results for the Euclidean space ℝd and the weak Schatten ideal ℒd,∞ when d > 1.

https://doi.org/10.1515/9783110700176-006

350 � 6 Quantum differentiability for the Euclidean plane Quantized differential on Euclidean space -- when f is an essentially bounded function on the There is a standard definition of df d Euclidean space ℝ . The definition is given in terms of the Euclidean Dirac operator D, which we now define. For d ≥ 2, we define Dj := −i𝜕j : 𝒮 ′ (ℝd ) → 𝒮 ′ (ℝd ),

1 ≤ j ≤ d,

where 𝜕j is the distributional derivative on the jth coordinate as defined in Section 1.6.1. d

Let Nd := 2⌊ 2 ⌋ . The Dirac matrices {γ1 , γ2 , . . . , γd } are Nd ×Nd complex Hermitian matrices satisfying the anticommutation relation γj γk + γk γj = 2 ⋅ δj,k ,

1 ≤ j, k ≤ d,

where δ is the Kronecker delta. The Dirac operator on ℝd is the operator on vectors of tempered distributions given by d

D := ∑ γj ⊗ Dj : ℂNd ⊗ 𝒮 ′ (ℝd ) → ℂNd ⊗ 𝒮 ′ (ℝd ). j=1

(6.1)

Following Theorem 1.6.10, restricted to the Sobolev space ℂNd ⊗ H 1 (ℝd ), the operator D is an unbounded self-adjoint operator on the Hilbert space ℂNd ⊗ L2 (ℝd ), D : ℂNd ⊗ H 1 (ℝd ) → ℂNd ⊗ L2 (ℝd ). The subspace ℂNd ⊗ 𝒮 (ℝd ) is an invariant core for the Dirac operator. The anticommutation relations for the Dirac matrices and the mutual commutativity of the operators {D1 , . . . , Dd } on Schwartz functions imply D2 = −1 ⊗ Δ, where Δ is the Laplacian operator for ℝd . Now let sgn(t) :=

t , |t|

t ∈ ℝ \ {0}.

As the kernel of Δ is trivial, this allows us to define sgn(D) by the functional calculus for self-adjoint operators on the Hilbert space ℂNd ⊗ L2 (ℝd ). Since the function sgn is bounded, sgn(D) extends to a bounded linear operator on ℂNd ⊗ L2 (ℝd ), and d

Dj

j=1

√−Δ

sgn(D) = ∑ γj ⊗

.

6.1 Introduction

� 351

Definition 6.1.1. The quantized differential of f ∈ L∞ (ℝd ) is defined as -- := i[sgn(D), 1 ⊗ M ], df f

f ∈ L∞ (ℝd ),

(6.2)

where (Mf u)(t) = f (t)u(t),

u ∈ L2 (ℝd ),

t ∈ ℝd ,

is a product operator on L2 (ℝd ); see Section 1.5. Equivalently, the quantized differential is d

-- = ∑ γ ⊗ [R , M ], df j j f j=1

where Rj =

𝜕j √−Δ

is the Riesz transform. For sufficiently regular f ∈ L∞ (ℝd ) and u ∈

L2 (ℝd , ℂNd ), we have

-- )u)(t) = ((df

f (t) − f (s) 1 γ(t − s)u(s) ds, ∫ π Vol(𝕊d−1 ) |t − s|d+1

t ∈ ℝd ,

ℝd

where γ(ξ) := ∑dj=1 ξj γj for ξ = (ξ1 , . . . , ξd ) ∈ ℝd .

The analogy of Peller’s result for the Euclidean plane would be a characterization of the functions f ∈ L∞ (ℝd ) such that, for p ∈ (0, ∞), -- ∈ ℒ (ℂNd ⊗ L (ℝd )). df p 2 This problem turns out to be quite different for d ≥ 2 compared to d = 1. The most notable difference is the presence of a cut-off at p = d, which is absent in the onedimensional case. -- ∈ ℒ (ℂNd ⊗ L (ℝd )) The determination of necessary and sufficient conditions for df p 2 was resolved by Janson and Wolf [167] f in 1982. It required different methods to the -- ∈ ℒ (ℂNd ⊗ one-dimensional case. In [167], Theorem 1 on p. 303 states that for p ≤ d, df p -- ∈ ℒ (ℂNd ⊗ L (ℝd )) if and only if f L2 (ℝd )) if and only if f is constant. For p > d, df p 2 d

p coincides with a distribution belonging to the homogeneous Besov space Ḃ p,p (ℝd ). This result is analogous to that of Peller [220] when p > d but exhibits different behavior when p ≤ d. Of most interest are necessary and sufficient conditions on f for the commutator -df to belong to the operator ideal ℒd,∞ (ℂNd ⊗ L2 (ℝd )). This represents the boundary between the cut-off at p ≤ d and the regular behavior at p > d.

352 � 6 Quantum differentiability for the Euclidean plane Behavior of singular values at the critical dimension Let Ẇ d1 (ℝd ) denote the homogeneous Sobolev space consisting of all locally integrable functions f on ℝd such that Dk f ∈ Ld (ℝd ),

1 ≤ k ≤ d,

where Dk f denotes the kth distributional derivative of the locally integrable function f ; that is, if f ∈ Ẇ d1 (ℝd ) and (Dk f , ϕ) := ∫ f (t)(Dk ϕ)(t)dt,

ϕ ∈ Cc∞ (ℝd ),

1 ≤ k ≤ d,

ℝd

then there exists a function g ∈ Ld (ℝd ) such that (Dk f , ϕ) = (g, ϕ),

ϕ ∈ Cc∞ (ℝd ).

The distributional derivative of a locally integrable function generally is not a tempered distribution in the sense of Section 1.6, but in the cases considered, it can be identified with the tempered distribution defined by a function in Ld (ℝd ). Define d

‖f ‖Ẇ 1 (ℝd ) d

d 2

1 d

󵄨 󵄨2 := ( ∫ ( ∑ 󵄨󵄨󵄨(Dk f )(t)󵄨󵄨󵄨 ) dt) . ℝd

k=1

In this chapter, we prove the following theorem, which characterizes the essentially -- ∈ ℒ (ℂNd ⊗ L (ℝd )). bounded functions f on ℝd such that df d,∞ 2 -- ∈ ℒ (ℂNd ⊗ L (ℝd )) if and only Theorem 6.1.2. Let d ≥ 2, and let f ∈ L∞ (ℝd ). Then df d,∞ 2 1 d ̇ if f ∈ Wd (ℝ ), and there exists a constant cd > 0 such that -- ‖ cd−1 ⋅ ‖f ‖Ẇ 1 (ℝd ) ≤ ‖df d,∞ ≤ cd ⋅ ‖f ‖Ẇ 1 (ℝd ) . d

d

(6.3)

Theorem 6.1.2 illustrates the cut-off behavior between differentiability and decay of singular values in the case of the Euclidean space of dimension two or greater, which is not present in one dimension. Theorem 6.1.2 implies that even when f ≠ 0 is highly regular (for example, f belongs to the Schwartz class of functions), we cannot expect -- ) than any better decay on μ(df -- ) = O(k − d1 ), μ(k, df

k → ∞.

-- is supposed to act like a one-form in the quantized calculus, in which Recall that df -- |d ∈ ℒ case the compact operator |df 1,∞ is the equivalent of a density in Riemannian geometry. Given the demonstration in Chapters 3 and 4 of singular traces as integrals,

6.1 Introduction

� 353

-- |d is the equivalent in the quantized the following formula of the trace of the density |df calculus of integration of densities in Riemannian geometry. Theorem 6.1.3. Let d ≥ 2 and let D be the Dirac operator given in (6.1). Let f ∈ C0 (ℝd )+ℂ, and let g ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ) be real-valued. Then -- d = (1 ⊗ M )󵄨󵄨󵄨[sgn(D), M ]󵄨󵄨󵄨d ∈ ℒ , (1 ⊗ Mf )|dg| f 󵄨 g 󵄨 1,∞ and d 2

d

-- d ) = c ⋅ ∫ f (t)( ∑ 󵄨󵄨󵄨𝜕 g(t)󵄨󵄨󵄨2 ) dt φ((1 ⊗ Mf )|dg| d 󵄨 k 󵄨 ℝd

k=1

(6.4)

for every continuous normalized trace φ on ℒ1,∞ with constant Γ(d − 21 ) Nd 2⌊ 2 ⌋ 󵄨󵄨 󵄨󵄨 d2 cd := ⋅ e − s⟨s, e ⟩ ds = , ∫ 󵄨 󵄨 1 1 d 󵄨 󵄨 d−1 d(2π)d d(2π) 2 Γ( 2 )Γ(d + 1) d−1 d

𝕊

where e1 := (1, 0, . . . , 0) ∈ 𝕊d−1 . Using the constant function 1 ∈ C0 (ℝd ) + ℂ in Theorem 6.1.3, for real-valued f ∈ ∩ L∞ (ℝd ), we have

Ẇ d1 (ℝd )

-- |d ) = c ⋅ ‖f ‖d 1 φ(|df d Ẇ d

(6.5)

for the constant cd in Theorem 6.1.3. A consequence of (6.5) is the cut-off phenomenon proved by different methods by Janson and Wolff. -- ∈ ℒ for some p ≤ d, then f is constant. Corollary 6.1.4. If f ∈ L∞ (ℝd ) is such that df p -- ∈ ℒ , then |df -- |d ∈ ℒ . Proof. Assume without loss that f ∈ L∞ (ℝd ) is real-valued. If df d 1 1 d d By Theorem 6.1.2 we have f ∈ Ẇ d (ℝ ) ∩ L∞ (ℝ ). Let φ be a normalized continuous trace on ℒ1,∞ . Theorem 6.1.3 implies that -- |d ) = 0 ‖f ‖dẆ 1 = cd−1 ⋅ φ(|df d

since φ vanishes on trace class operators. Hence (D1 f , . . . , Dd f ) = ∇f = 0. The right-hand side inequality in (6.3) is proved for f ∈ Cc∞ (ℝd ) in Section 6.2 using the theory of double operator integrals. Combining the results of Section 6.3 with Theorem 5.3.5, we establish Theorem 6.1.3 for f ∈ Cc∞ (ℝd ) in Section 6.4. Section 6.5 contains approximation lemmas needed to extend the results to arbitrary f ∈ L∞ (ℝd ). With Theorem 6.1.3, the left-hand side inequality in (6.3) is proved in Section 6.6.

354 � 6 Quantum differentiability for the Euclidean plane

6.2 Compactness estimates for the quantized differential of a differentiable function The following theorem establishes the right-hand side inequality in (6.3) in the particular case of a smooth compactly supported function f . Approximation of a function f ∈ Wd1 (ℝd ) ∩ L∞ (ℝd ) by smooth functions is considered in Section 6.5. Theorem 6.2.1. Let d ≥ 2, and let D be the Dirac operator given in (6.1). If f ∈ Cc∞ (ℝd ), then [sgn(D), 1 ⊗ Mf ] ∈ ℒd,∞ , and there is a constant cd > 0 such that 󵄩󵄩 󵄩 󵄩󵄩[sgn(D), 1 ⊗ Mf ]󵄩󵄩󵄩d,∞ ≤ cd ⋅ ‖f ‖Ẇ 1 . d The proof of Theorem 6.2.1 uses double operator integrals, following from results in Section 1.4. A sequence of lemmas is used to identify a transformer that approximates the commutator [sgn(D), 1 ⊗ Mf ]. The estimate in Theorem 6.2.1 will then follow from submajorization inequalities involving the transformer since ℒd,∞ is a fully symmetric ideal of operators. Approximation by a double operator integral We start with a lemma based on Lemma 1.4.4. Lemma 6.2.2. Let h ∈ L∞ (ℝ) be such that ℱ h ∈ L1 (ℝ). Let ϕ(λ, μ) := h(λ − μ), λ, μ ∈ ℝ. We have TϕA,A (V ) ≺≺

1 ‖ℱ h‖1 ⋅ V , √2π

V ∈ ℒ(H),

where TϕA,A is the transformer associated with a self-adjoint operator A on the Hilbert

space L2 (ℝd ) (Definition 1.4.7), and ≺≺ denotes submajorization of bounded operators (Definition 1.2.1). Proof. We write h(λ − μ) =

1 ∫(ℱ h)(s)eis(λ−μ) ds. √2π ℝ

It follows from Definition 1.4.5 that ϕ ∈ BS. Thus TϕA,A (V ) =

1 ∫(ℱ h)(s)eisA Ve−isA ds, √2π ℝ

6.2 Compactness estimates for the quantized differential of a differentiable function

� 355

where the integral on the right side is a weak operator integral. The statement follows from Lemma 1.4.4. The next lemma is essentially a consequence of the fact that the function h(t) := has the Fourier transform

1 2 cosh(t)

(ℱ h)(ξ) =

√π

2√2 cosh( πλ ) 2

;

see [157, Appendix A.6]. In particular, observe that ℱ h is nonnegative, and therefore ∞

1

‖ℱ h‖L1 (ℝ) = ∫ (ℱ h)(ξ) dξ = (2π) 2 ⋅ h(0). −∞

Lemma 6.2.3. Let the function ψ : ℝ2+ → ℝ be given by the formula 1

ψ(λ, μ) :=

1

λ2 μ2 , λ+μ

λ, μ > 0.

Let A be a positive self-adjoint potentially unbounded operator on H with trivial kernel. Then TψA,A (V ) ≺≺ Proof. Set B :=

1 2

V ∈ ℒ(H).

log(A) and ϕ(λ, μ) := h(λ − μ),

Writing f (t) := ma 1.4.14 gives

1 ⋅ V, 2

1 2

h(λ) :=

1 , 2 cosh(λ)

log(t), we have ϕ(f (λ), f (μ)) = TψA,A (V ) = TϕB,B (V ),

1

1

λ2 μ2 λ+μ

λ, μ ∈ ℝ.

= ψ(λ, μ), and therefore Lem-

V ∈ ℒ(H).

By Lemma 6.2.2 we have TψA,A (V ) ≺≺

1 ‖ℱ h‖1 ⋅ V , √2π

V ∈ ℒ(H).

As observed in the discussion preceding the theorem, we have 1 1 1 ‖ℱ h‖1 = ∫(ℱ h)(s)ds = h(0) = . √2π √2π 2 ℝ

This completes the proof.

356 � 6 Quantum differentiability for the Euclidean plane In this section, we frequently use the notation 1

⟨A⟩ := (1 + |A|2 ) 2 for operators and complex vectors. When f ∈ Cc∞ (ℝd ) and D is the Dirac operator, the next lemma allows us to associate the commutator [D⟨D⟩−1 , 1 ⊗ Mf ] with the double operator integral Tψ⟨D⟩,⟨D⟩ (V ) for the bounded operator V in Lemma 6.2.5. Recall from Definition 1.4.10 the divided difference g [1] : ℝ2 → ℂ of a function g ∈ C 1 (ℝ), g [1] (λ, μ) := {

g(λ)−g(μ) , λ−μ ′

g (λ),

λ ≠ μ, λ = μ.

Lemma 6.2.4. Let g(λ) := λ⟨λ⟩−1 , t ∈ ℝ. We have g [1] = ψ1 ψ2 , where 1

ψ1 (λ, μ) :=

1

⟨λ⟩ 2 ⟨μ⟩ 2 , ⟨λ⟩ + ⟨μ⟩

ψ2 (λ, μ) :=

⟨λ⟩⟨μ⟩ − λμ + 1 3

3

⟨λ⟩ 2 ⟨μ⟩ 2

,

λ, μ ∈ ℝ.

Proof. We have g [1] (λ, μ) =

λ⟨μ⟩ − μ⟨λ⟩ λ2 ⟨μ⟩ − λμ⟨λ⟩ + λμ⟨μ⟩ − μ2 ⟨λ⟩ = . (λ − μ)⟨λ⟩⟨μ⟩ (λ2 − μ2 )⟨λ⟩⟨μ⟩

Also, by the definition of ⟨λ⟩ we have λ2 − μ2 = ⟨λ⟩2 − ⟨μ⟩2 = (⟨λ⟩ + ⟨μ⟩) ⋅ (⟨λ⟩ − ⟨μ⟩), Similarly, λ2 ⟨μ⟩ − λμ⟨λ⟩ + λμ⟨μ⟩ − μ2 ⟨λ⟩ = λ2 ⟨μ⟩ − μ2 ⟨λ⟩ − λμ(⟨λ⟩ − ⟨μ⟩)

= ⟨λ⟩2 ⟨μ⟩ − ⟨μ⟩2 ⟨λ⟩ − λμ(⟨λ⟩ − ⟨μ⟩) + (⟨λ⟩ − ⟨μ⟩) = (⟨λ⟩ − ⟨μ⟩) ⋅ (⟨λ⟩⟨μ⟩ − λμ + 1).

Thus g [1] (λ, μ) =

⟨λ⟩⟨μ⟩ − λμ + 1 1 ⋅ ⟨λ⟩⟨μ⟩ ⟨λ⟩ + ⟨μ⟩ 1

1

⟨λ⟩⟨μ⟩ − λμ + 1 ⟨λ⟩ 2 ⟨μ⟩ 2 = ⋅ 3 3 ⟨λ⟩ + ⟨μ⟩ ⟨λ⟩ 2 ⟨μ⟩ 2 = ψ2 (λ, μ) ⋅ ψ1 (λ, μ).

6.2 Compactness estimates for the quantized differential of a differentiable function

�

357

The following lemma describes the commutator [D⟨D⟩−1 , 1 ⊗ Mf ] as a double operator integral. Lemma 6.2.5. Let f ∈ Cc∞ (ℝd ), and let ψ be as in Lemma 6.2.3. We have [D⟨D⟩−1 , 1 ⊗ Mf ] = Tψ⟨D⟩,⟨D⟩ (V ), where 1

1

3

D

3

V := ⟨D⟩− 2 [D, 1 ⊗ Mf ]⟨D⟩− 2 + ⟨D⟩− 2 [D, 1 ⊗ Mf ]⟨D⟩− 2 −

⟨D⟩

3 2

[D, 1 ⊗ Mf ]

D

3

⟨D⟩ 2

.

Proof. Using the result and notation of Lemma 6.2.4, we have D,D [g(D), 1 ⊗ Mf ] = TgD,D [1] ([D, 1 ⊗ Mf ]) = Tψ ψ ([D, 1 ⊗ Mf ]) 1 2

= TψD,D (TψD,D ([D, 1 ⊗ Mf ])), 1

2

where the first equality is (1.15), and the last equality holds by Theorem 1.4.8. Since 1

λ

1

ψ2 (λ, μ) = ⟨λ⟩− 2 ⋅ ⟨μ⟩− 2 −

⟨λ⟩

3 2

⋅

λ ⟨λ⟩

3

3 2

3

+ ⟨λ⟩− 2 ⋅ ⟨μ⟩− 2 ,

λ, μ ∈ ℝ,

it follows that 1

1

TψD,D (X) = ⟨D⟩− 2 X⟨D⟩− 2 − 2

D ⟨D⟩

3 2

X

D ⟨D⟩

3

3 2

3

+ ⟨D⟩− 2 X⟨D⟩− 2

for every X ∈ ℒ(ℂNd ⊗ L2 (ℝd )). It follows from the definition of D that d

[D, 1 ⊗ Mf ] = ∑ γj ⊗ MDj f j=1

and [D, 1 ⊗ Mf ] is a bounded operator since Dj f ∈ L∞ (ℝd ), 1 ≤ j ≤ d. Hence 1

1

TψD,D ([D, 1 ⊗ Mf ]) = ⟨D⟩− 2 [D, 1 ⊗ Mf ]⟨D⟩− 2 2

3

3

+ ⟨D⟩− 2 [D, 1 ⊗ Mf ]⟨D⟩− 2 −

D ⟨D⟩

3 2

[D, 1 ⊗ Mf ]

D

3

⟨D⟩ 2

.

At the same time, Lemma 1.4.14 yields TψD,D (V ) = Tψ⟨D⟩,⟨D⟩ (V ) 1

for every V ∈ ℒ(ℂNd ⊗ L2 (ℝd )). A combination of these equalities yields the statement.

358 � 6 Quantum differentiability for the Euclidean plane Compactness estimates using double operator integral approximation We can now combine Lemmas 6.2.3 and 6.2.5 to obtain a compactness estimate. Lemma 6.2.6. If f ∈ Cc∞ (ℝd ), then [D⟨D⟩−1 , 1 ⊗ Mf ] ∈ ℒd,∞ , and there is a constant cd > 0 that depends only on d such that 󵄩󵄩 󵄩 −1 󵄩󵄩[D⟨D⟩ , 1 ⊗ Mf ]󵄩󵄩󵄩d,∞ ≤ cd ⋅ ‖f ‖Ẇ 1 . d Proof. Let g be as in Lemma 6.2.4, and let V be as in Lemma 6.2.5. By combining Lemmas 6.2.5 and 6.2.3 we obtain [g(D), 1 ⊗ Mf ] ≺≺ V . Since d > 1, the ideal ℒd,∞ is fully symmetric by Corollary 1.2.3 in Section 1.2.1. By Theorem 1.2.2, recalling that the quasinorm on ℒd,∞ is equivalent to a norm for which ℒd,∞ is a Banach space, we have that 󵄩󵄩 󵄩 ′ 󵄩󵄩[g(D), 1 ⊗ Mf ]󵄩󵄩󵄩d,∞ ≤ bd ⋅ ‖V ‖d,∞

(6.6)

for a constant b′d > 0. The triangle inequality for ℒd,∞ implies 1

1

󵄩󵄩 − 2 −2 󵄩 󵄩 ‖V ‖d,∞ ≤ b′′ d ⋅󵄩 󵄩⟨D⟩ [D, 1 ⊗ Mf ]⟨D⟩ 󵄩󵄩d,∞ . It follows from the definition of D that d

1

[D, 1 ⊗ Mf ] = ∑ γj ⊗ MDj f ,

⟨D⟩ = 1 ⊗ (1 − Δ) 2 .

j=1

Therefore 1

1

d

1

1

⟨D⟩− 2 [D, 1 ⊗ Mf ]⟨D⟩− 2 = ∑ γj ⊗ (1 − Δ)− 4 MDj f (1 − Δ)− 4 . j=1

By the triangle inequality in ℒd,∞ we have d

󵄩󵄩 − 21 󵄩 −1 󵄩 ′′′ −1 −1 󵄩 󵄩󵄩⟨D⟩ [D, 1 ⊗ Mf ]⟨D⟩ 2 󵄩󵄩󵄩d,∞ ≤ bd ⋅ ∑󵄩󵄩󵄩(1 − Δ) 4 MDj f (1 − Δ) 4 󵄩󵄩󵄩d,∞ . j=1

Substituting into (6.6), we obtain

6.2 Compactness estimates for the quantized differential of a differentiable function d

󵄩󵄩 󵄩 󵄩 −1 −1 󵄩 󵄩󵄩[g(D), 1 ⊗ Mf ]󵄩󵄩󵄩d,∞ ≤ bd ⋅ ∑󵄩󵄩󵄩(1 − Δ) 4 MDj f (1 − Δ) 4 󵄩󵄩󵄩d,∞ j=1

� 359

(6.7)

for a constant bd > 0. By Hölder’s inequality for weak ℒp -spaces (see Theorem 1.3.12) and Theorem 1.5.20 we have 󵄩󵄩 󵄩 −1 −1 󵄩 − 1 󵄩2 1 (1 − Δ) 4 󵄩 󵄩󵄩(1 − Δ) 4 MDj f (1 − Δ) 4 󵄩󵄩󵄩d,∞ ≤ 󵄩󵄩󵄩M 󵄩󵄩2d,∞ |Dj f | 2

1 2 󵄩 󵄩 ≤ ad ⋅ 󵄩󵄩󵄩|Dj f | 2 󵄩󵄩󵄩2d = ad ⋅ ‖Dj f ‖d

for a constant ad > 0 and each 1 ≤ j ≤ d. Substituting this into (6.7), we obtain d

󵄩󵄩 󵄩 󵄩󵄩[g(D), 1 ⊗ Mf ]󵄩󵄩󵄩d,∞ ≤ cd ⋅ ∑ ‖Dj f ‖d ≤ cd ⋅ ‖f ‖Ẇ 1 d

j=1

for a constant cd > 0. -- with the commuIn the next lemma, we approximate the quantized derivative df tator i[D⟨D⟩−1 , 1 ⊗ Mf ]. Lemma 6.2.7. If f ∈ Cc∞ (ℝd ), then [D⟨D⟩−1 − sgn(D), 1 ⊗ Mf ] ∈ ℒd , and 󵄩󵄩 󵄩 −1 󵄩󵄩[D⟨D⟩ − sgn(D), 1 ⊗ Mf ]󵄩󵄩󵄩ℒd ≤ cd ⋅ ‖f ‖Ld (ℝd ) for a constant cd > 0. Proof. Let g be as in Lemma 6.2.4. We have g(t) − sgn(t) = − sgn(t) ⋅

1 , √1 + t 2 ⋅ (|t| + √1 + t 2 )

t ∈ ℝ.

Set h(t) := (g(t) − sgn(t)) ⋅ (1 + t 2 ),

t ∈ ℝ.

Clearly, ‖h‖∞ ≤ 1. Since 󵄩󵄩 󵄩 󵄩 󵄩 2 −1 󵄩󵄩[g(D) − sgn(D), 1 ⊗ Mf ]󵄩󵄩󵄩d = 󵄩󵄩󵄩[h(D) ⋅ (1 + D ) , 1 ⊗ Mf ]󵄩󵄩󵄩d and, by the triangle inequality in ℒd , we have

360 � 6 Quantum differentiability for the Euclidean plane 󵄩󵄩 󵄩 2 −1 󵄩󵄩[h(D) ⋅ (1 + D ) , 1 ⊗ Mf ]󵄩󵄩󵄩d −1 −1 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩h(D) ⋅ (1 + D2 ) ⋅ (1 ⊗ Mf )󵄩󵄩󵄩d + 󵄩󵄩󵄩(1 ⊗ Mf ) ⋅ h(D) ⋅ (1 + D2 ) 󵄩󵄩󵄩d −1 −1 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(1 + D2 ) ⋅ (1 ⊗ Mf )󵄩󵄩󵄩d + 󵄩󵄩󵄩(1 ⊗ Mf ) ⋅ (1 + D2 ) 󵄩󵄩󵄩d 󵄩 󵄩 ≤ cd ⋅ 󵄩󵄩󵄩Mf (1 − Δ)−1 󵄩󵄩󵄩d for a constant cd > 0, it follows that 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩󵄩[g(D) − sgn(D), 1 ⊗ Mf ]󵄩󵄩󵄩d ≤ cd ⋅ 󵄩󵄩󵄩Mf (1 − Δ) 󵄩󵄩󵄩d for a constant cd > 0. The statement now follows from Theorem 1.5.20. Proof of Theorem 6.2.1 in the case f ∈ Cc∞ (ℝd ). It follows from Lemmas 6.2.6 and 6.2.7 that -- ‖ ‖df ℒd,∞ ≤ cd ⋅ (‖f ‖Ld + ‖f ‖Ẇ 1 )

(6.8)

d

for some constant cd > 0. Let (σt x)(s) := x( st ), s ∈ ℝd , t > 0, be the dilation action of ℝ+ on L∞ (ℝd ). We have σt Dk σt−1 = tDk ,

σt Δσt−1 = t 2 Δ,

1

1

σt (−Δ) 2 σt−1 = t(−Δ) 2 .

Hence sgn(D) is dilation invariant in the sense that (1 ⊗ σt ) ⋅ sgn(D) ⋅ (1 ⊗ σt−1 ) = sgn(D),

t > 0.

It follows immediately that -- ⋅ (1 ⊗ σ −1 ) = −i(sgn(D) ⋅ (1 ⊗ M ) − (1 ⊗ M ) ⋅ sgn(D)) = d(σ -- f ). (1 ⊗ σt ) ⋅ df σt f σt f t t d

Since t − 2 (1 ⊗ σt ) is a unitary operator on ℂNd ⊗ L2 (ℝd ), it follows that 󵄩󵄩 󵄩󵄩 󵄩󵄩 -󵄩󵄩 -- ‖ -‖df ℒd,∞ = 󵄩 󵄩(1 ⊗ σt ) ⋅ df ⋅ (1 ⊗ σt−1 )󵄩󵄩ℒd,∞ = 󵄩󵄩d(σt f )󵄩󵄩ℒd,∞ . Applying (6.8), we obtain -- ‖ ‖df ℒd,∞ ≤ cd (‖σt f ‖Ld + ‖σt f ‖Ẇ 1 ). d

Note that ‖σt f ‖Ẇ 1 = ‖f ‖Ẇ 1 , d

Thus

d

‖σt f ‖Ld = t‖f ‖Ld ,

t > 0.

6.3 Approximation of quantized differentials by principal terms

-- ‖ ‖df ℒd,∞ ≤ cd ⋅ (t‖f ‖Ld + ‖f ‖Ẇ 1 ),

�

361

t > 0.

d

Letting t → 0, we complete the proof.

6.3 Approximation of quantized differentials by principal terms The left-hand side of the inequality in (6.3) will be a consequence of Theorem 6.1.3, which proves the trace formula -- |d ) = c ⋅ ‖f ‖ ̇ 1 φ(|df d W d

for a constant cd > 0 and a normalized continuous trace φ on ℒ1,∞ . In Section 6.4, we prove the trace formula of Theorem 6.1.3 for f ∈ Cc∞ (ℝd ) by using the trace theorem for principal terms of pseudodifferential operators, Theorem 5.3.5. In this section, we show that quantized differentials can be approximated by products involving a principal term.

6.3.1 Approximation of differentials -- is approximately of the form A(1 − Δ)− 21 , where A To use Theorem 5.3.5, we show that df belongs to a C ∗ -algebra of principal terms of pseudodifferential operators in the sense of Chapter 5. Recall that (ℒd,∞ )0 denotes the closure of the finite-rank operators on L2 (ℝd ) in the quasinorm of ℒd,∞ . -- is the quantized differential of f , then Theorem 6.3.1. Let d ≥ 2, and let f ∈ Cc∞ (ℝd ). If df 1

-- − A(1 + D2 )− 2 ∈ (ℒ ) , df d,∞ 0 where d

Dj

j=1

√−Δ

A := i[D, 1 ⊗ Mf ] − i sgn(D) ⋅ (1 ⊗ ∑

MDj f ).

For the remainder of this chapter, we adopt the notation of Theorem 5.3.5. Denote 𝒜1 := C0 (ℝd ) + ℂ and 𝒜2 := C(𝕊d−1 ). Recall that π1 denotes the representation of 𝒜1 on L2 (ℝd ) as pointwise multipliers and π2 denotes the representation of 𝒜2 as homogeneous Fourier multipliers. The algebra Π(𝒜1 , 𝒜2 ) denotes the C ∗ -subalgebra of ℒ(L2 (ℝd )) generated by π1 (𝒜1 ) and π2 (𝒜2 ). We now prove Theorem 6.3.1. Generalizing Definition 1.6.26, we say that A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right (respectively, compactly supported) if there exists ϕ ∈ Cc∞ (ℝd ) such that

362 � 6 Quantum differentiability for the Euclidean plane A = A(1 ⊗ Mϕ ),

respectively, A = (1 ⊗ Mϕ )A(1 ⊗ Mϕ ).

Remark 6.3.2. If A is as in Theorem 6.3.1, then A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ), and A is compactly supported on the right. Proof. Indeed, we have d

[D, 1 ⊗ Mf ] = ∑ γj ⊗ MDj f ∈ MNd (ℂ) ⊗ π1 (𝒜1 ) ⊂ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ), j=1 d

Dj

j=1

√−Δ

sgn(D) = ∑ γj ⊗ d

Dj

j=1

√−Δ

1⊗∑

∈ MNd (ℂ) ⊗ π2 (𝒜2 ) ⊂ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ),

MDj f ∈ ℂ ⊗ Π(𝒜1 , 𝒜2 ).

Since MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is an algebra, the first statement follows. To see that A is compactly supported on the right, let ϕ ∈ Cc∞ (ℝd ) be such that fϕ = f . Then [D, 1 ⊗ Mf ](1 ⊗ Mϕ ) = 1 ⊗ MDj f Mϕ = 1 ⊗ MDj f = [D, 1 ⊗ Mf ]. Hence d

Dj

j=1

√−Δ

A(1 ⊗ Mϕ ) = i[D, 1 ⊗ Mf ](1 ⊗ Mϕ ) − i sgn(D) ⋅ (1 ⊗ ∑ d

Dj

j=1

√−Δ

= i[D, 1 ⊗ Mf ] − i sgn(D) ⋅ (1 ⊗ ∑

MDj f Mϕ )

MDj f ) = A,

and A is compactly supported on the right. For an operator T ∈ ℒ(L2 (ℝd )) such that T : 𝒮 (ℝd ) → 𝒮 (ℝd ), following Section 4.4.1, we denote 1

δ(T) := [(1 − Δ) 2 , T],

1

L(T) := (1 − Δ)− 2 [−Δ, T].

Both δ(T) and L(T) make sense as endomorphisms of 𝒮 (ℝd ). Lemma 6.3.3. Let d ≥ 2, and let T ∈ ℒ(L2 (ℝd )) be such that T : 𝒮 (ℝd ) → 𝒮 (ℝd ). Assume 1 that L2 (T)(1 − Δ)− 4 admits a bounded extension to L2 (ℝd ). Then δ(T) − 21 L(T) admits a bounded extension to L2 (ℝd ), and there exists a constant cd such that

6.3 Approximation of quantized differentials by principal terms

� 363

󵄩󵄩 󵄩󵄩 1 1 󵄩󵄩 󵄩 󵄩 󵄩 ≤ cd ⋅ 󵄩󵄩󵄩L2 (T)(1 − Δ)− 4 󵄩󵄩󵄩 2d ,∞ . 󵄩󵄩δ(T) − L(T)󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 2d ,∞ 3 2 3 Proof. From the proof of Lemma 4.4.3 we have 1

∞

1 1 λ 2 (1 − Δ) 2 dλ δ(T) − L(T) = ∫ L (T) , 2 π (1 + λ − Δ)2 1+λ−Δ 0

where the operator on the left side is the bounded extension of δ(T) − 21 L(T), and the integral on the right side is a weak operator integral. Since d > 1, the ideal ℒ 2d ,∞ admits 3 a Banach norm equivalent to the usual quasinorm. Switching to a norm, applying the triangle inequality and switching back to the usual quasinorm yields 󵄩󵄩 ∞ 1 󵄩 󵄩󵄩 λ 2 (1 − Δ) 2 dλ 󵄩󵄩󵄩󵄩 󵄩󵄩 ∫ L (T) 󵄩 󵄩󵄩 (1 + λ − Δ)2 1 + λ − Δ 󵄩󵄩󵄩󵄩 2d ,∞ 󵄩󵄩 0 3 ∞ 󵄩󵄩 λ 21 (1 − Δ) 󵄩󵄩󵄩 1 󵄩 󵄩󵄩 ≤ cd ⋅ ∫ 󵄩󵄩󵄩 L2 (T) dλ, 2 󵄩󵄩 (1 + λ − Δ) 1 + λ − Δ 󵄩󵄩󵄩 2d3 ,∞ 0

for a constant cd > 0. Clearly, 󵄩󵄩 λ 21 (1 − Δ) 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 2 L (T) 󵄩󵄩 󵄩 󵄩󵄩 (1 + λ − Δ)2 1 + λ − Δ 󵄩󵄩󵄩 2d3 ,∞ 󵄩󵄩 λ 21 (1 − Δ) 󵄩󵄩 󵄩 󵄩󵄩 󵄩󵄩 2 󵄩 −1 󵄩 −3 󵄩 ≤ 󵄩󵄩󵄩 󵄩 󵄩L (T)(1 + λ − Δ) 4 󵄩󵄩󵄩 2d ,∞ 󵄩󵄩󵄩(1 + λ − Δ) 4 󵄩󵄩󵄩∞ 󵄩󵄩 (1 + λ − Δ)2 󵄩󵄩󵄩∞ 󵄩 3 1 1 3 󵄩 󵄩 ≤ (1 + λ)− 2 󵄩󵄩󵄩L2 (T)(1 − Δ)− 4 󵄩󵄩󵄩 2d ,∞ (1 + λ)− 4 . 3 Thus 󵄩󵄩 ∞ 1 󵄩 󵄩󵄩 λ 2 (1 − Δ) 2 dλ 󵄩󵄩󵄩󵄩 󵄩󵄩 ∫ L (T) 󵄩 󵄩󵄩 (1 + λ − Δ)2 1 + λ − Δ 󵄩󵄩󵄩󵄩 2d ,∞ 󵄩󵄩 0 3 ∞

1 5 1 󵄩 󵄩 󵄩 󵄩 ≤ cd 󵄩󵄩󵄩L2 (T)(1 − Δ)− 4 󵄩󵄩󵄩 2d ,∞ ⋅ ∫ (1 + λ)− 4 dλ = 4cd ⋅ 󵄩󵄩󵄩L2 (T)(1 − Δ)− 4 󵄩󵄩󵄩 2d ,∞ . 3

3

0

Lemma 6.3.4. For every f ∈ Cc∞ (ℝd ), we have 1

d

δ(Mf )(1 − Δ)− 2 − ∑ j=1

Dj (−Δ)

1

1 2

MDj f (1 − Δ)− 2 ∈ ℒ 2d ,∞ . 3

364 � 6 Quantum differentiability for the Euclidean plane Proof. Set 1

T := Mf (1 − Δ)− 2 . Since f is smooth and compactly supported, T is an endomorphism of 𝒮 (ℝd ). Note that 1

δ(T) = δ(Mf )(1 − Δ)− 2 ,

1

L(T) = L(Mf )(1 − Δ)− 2 .

We will first show that 1

3

L2 (T)(1 − Δ)− 4 = L2 (Mf )(1 − Δ)− 4 ∈ ℒ 2d ,∞ , 3

so that Lemma 6.3.3 is applicable. Indeed, writing out −Δ as the sum of D2k and the Leibniz rule yield d

d

k=1

k=1

[−Δ, Mf ] = ∑ [D2k , Mf ] = ∑ Dk [Dk , Mf ] + [Dk , Mf ]Dk d

d

d

k=1

k=1

= ∑ Dk MDk f + MDk f Dk = 2 ∑ Dk MDk f − ∑ [Dk , MDk f ] k=1

d

= MΔf + 2 ∑ Dk MDk f . k=1

Similarly, d

d

d

k=1

k=1

j=1

[−Δ, [−Δ, Mf ]] = MΔ2 f + 2 ∑ Dk MDk Δf + 2 ∑ Dk MΔDk f + 2 ∑ Dk Dj MDj Dk f . Thus 3

3

L2 (Mf )(1 − Δ)− 4 = (1 − Δ)−1 [−Δ, [−Δ, Mf ]](1 − Δ)− 4 ∈ ℒ 2d ,∞ 3

by Theorem 1.5.20. By Lemma 6.3.3 we have 1 1 1 δ(Mf )(1 − Δ)− 2 − L(Mf )(1 − Δ)− 2 ∈ ℒ 2d ,∞ . 3 2

(6.9)

Note that 1

d

1

d

L(Mf ) = (1 − Δ)− 2 ∑ [D2k , Mf ] = (1 − Δ)− 2 MΔf + 2 ∑ k=1

and

k=1

Dk

1

(1 − Δ) 2

MDk f

6.3 Approximation of quantized differentials by principal terms

1

1

� 365

1

L(Mf )(1 − Δ)− 2 = (1 − Δ)− 2 MΔf (1 − Δ)− 2 d

Dk

+2∑

1

(1 − Δ)

k=1

1 2

MDk f (1 − Δ)− 2 .

By Hölder’s inequality and Theorem 1.5.20 we have 1

1

(1 − Δ)− 2 MΔf (1 − Δ)− 2 ∈ ℒ 2d ,∞ . 3

Thus d 1 1 Dk 1 L(Mf )(1 − Δ)− 2 − ∑ MDk f (1 − Δ)− 2 ∈ ℒ 2d ,∞ . 1 3 2 k=1 (1 − Δ) 2

(6.10)

Combining (6.9) and (6.10), we obtain 1

d

Dk

δ(Mf )(1 − Δ)− 2 − ∑

1

(1 − Δ)

k=1

1 2

MDk f (1 − Δ)− 2 ∈ ℒ 2d ,∞ . 3

The statement will be shown if we prove that d

∑

k=1

Dk

(1 − Δ)

d

1

1 2

Dk

MDk f (1 − Δ)− 2 − ∑

k=1

1

(−Δ)

1 2

MDk f (1 − Δ)− 2 ∈ ℒ 2d ,∞ . 3

To see this, note that 󵄩󵄩 󵄩 1 Dk Dk 󵄩󵄩 󵄩󵄩 󵄩 󵄩 − 21 󵄩 − )M (1 − Δ) ≤ cd ⋅ 󵄩󵄩󵄩(1 − Δ)−1 MDk f (1 − Δ)− 2 󵄩󵄩󵄩 d ,∞ 󵄩󵄩( 󵄩󵄩 Dk f 󵄩󵄩 (1 − Δ) 21 (−Δ) 21 󵄩󵄩 d ,∞ 2 2 for a constant cd > 0 since the operator (

Dk

(1 − Δ)

1 2

−

Dk

1

(−Δ) 2

)(1 − Δ)

on 𝒮 (ℝd ) has a bounded extension. We have 󵄩󵄩 󵄩 󵄩 −1 −1 󵄩 −1 󵄩󵄩(1 − Δ) MDk f (1 − Δ) 2 󵄩󵄩󵄩 d ,∞ ≤ 󵄩󵄩󵄩(1 − Δ) MDk f 󵄩󵄩󵄩 d ,∞ , 2 2 and it follows from Theorem 1.5.20 that (1 − Δ)−1 MDk f ∈ ℒ d ,∞ . This completes the proof.

2

With the estimate in Lemma 6.3.4, we can prove Theorem 6.3.1. Proof of Theorem 6.3.1. Let f ∈ Cc∞ (ℝd ). Since ℒd ⊂ (ℒd,∞ )0 , it follows from Lemma 6.2.7 that

366 � 6 Quantum differentiability for the Euclidean plane -- − i[D⟨D⟩−1 , M ] ∈ (ℒ ) . df f d,∞ 0 Note that i[D⟨D⟩−1 , Mf ] = i[D, Mf ]⟨D⟩−1 + D[⟨D⟩−1 , Mf ] = i[D, Mf ]⟨D⟩−1 − D⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 , where [D, Mf ] = ∑dj=1 1 ⊗ MDj f is the bounded extension to L2 (ℝd ) of the endomorphism [D, Mf ] of ℂNd ⊗ 𝒮 (ℝd ). Similarly, the endomorphism

1

[⟨D⟩, Mf ]⟨D⟩−1 = 1 ⊗ δ(Mf )(1 − Δ)− 2 of ℂNd ⊗ 𝒮 (ℝd ) has a bounded extension to L2 (ℝd ) by Lemma 6.3.4. Thus -- − i[D, M ]⟨D⟩−1 + D⟨D⟩−1 [⟨D⟩, M ]⟨D⟩−1 ∈ (ℒ ) . df f f d,∞ 0 The statement of the theorem follows by showing that d

Dj

j=1

√−Δ

D⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 − i sgn(D) ⋅ (1 ⊗ ∑

MDj f )⟨D⟩−1 ∈ (ℒd,∞ )0 .

(6.11)

We prove (6.11) in two parts. Let B := (1 − D⟨D⟩−1 )⟨D⟩ denote the bounded extension to L2 (ℝd ) of (1 − D⟨D⟩−1 )⟨D⟩ on ℂNd ⊗ 𝒮 (ℝd ). Then (1 − D⟨D⟩−1 )[⟨D⟩, Mf ]⟨D⟩−1 = B⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 .

(6.12)

We have 1

1

⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 = 1 ⊗ (1 − Δ)− 2 δ(Mf )(1 − Δ)− 2 . By Lemma 6.3.4, 1

1

d

(1 − Δ)− 2 δ(Mf )(1 − Δ)− 2 − ∑ j=1

Dj (−Δ)

1

1 2

1

(1 − Δ)− 2 MDj f (1 − Δ)− 2 ∈ ℒ 2d ,∞ .

However, for 1 ≤ j ≤ d, 󵄩󵄩 󵄩 −1 −1 󵄩 − 1 󵄩2 1 (1 − Δ) 2 󵄩 󵄩󵄩(1 − Δ) 2 MDj f (1 − Δ) 2 󵄩󵄩󵄩 d ≤ 󵄩󵄩󵄩M 󵄩󵄩d < ∞ 2 |Dj f | 2 by Theorem 1.5.20. Hence, for 1 ≤ j ≤ d,

3

6.3 Approximation of quantized differentials by principal terms

1

�

367

1

(1 − Δ)− 2 MDj f (1 − Δ)− 2 ∈ ℒ d ,∞ , 2

and, consequently, ⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 ∈ ℒ 2d ,∞ . 3

It now follows from (6.12) that D⟨D⟩−1 [⟨D⟩, Mf ]⟨D⟩−1 − [⟨D⟩, Mf ]⟨D⟩−1 ∈ (ℒd,∞ )0 .

(6.13)

Since 1

[⟨D⟩, Mf ]⟨D⟩−1 = 1 ⊗ δ(Mf )(1 − Δ)− 2 , it follows from Lemma 6.3.4 that d

Dj

j=1

√−Δ

[⟨D⟩, Mf ]⟨D⟩−1 − i sgn(D) ⋅ (1 ⊗ ∑

MDj f ) ∈ (ℒd,∞ )0 .

(6.14)

The combination of (6.13) and (6.14) proves (6.11).

6.3.2 Approximation of densities As in Theorem 5.3.1, let 𝒜1 := C0 (ℝd ) + ℂ and 𝒜2 := C0 (𝕊d−1 ), where 𝒜1 is represented on L2 (ℝd ) as pointwise multipliers, and 𝒜2 is represented on L2 (ℝd ) as dilation-invariant Fourier multipliers. By Theorem 6.3.1 we have 1

-- − A(1 + D2 )− 2 ∈ (ℒ ) , df d,∞ 0 where A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ). By Corollary 1.3.15 and Lemma 1.3.18 this implies that 1

-- |d − 󵄨󵄨󵄨A(1 + D2 )− 2 󵄨󵄨󵄨d ∈ (ℒ ) . |df 1,∞ 0 󵄨 󵄨 Hence, for any continuous trace φ on ℒ1,∞ , 1

-- |d ) = φ(󵄨󵄨󵄨A(1 + D2 )− 2 󵄨󵄨󵄨d ), φ(|df 󵄨 󵄨

(6.15)

since the trace vanishes on (ℒ1,∞ )0 . The next theorem complements Theorem 6.3.1. Recall that A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right if there exists ϕ ∈ Cc∞ (ℝd ) such that A = A(1 ⊗ Mϕ ).

368 � 6 Quantum differentiability for the Euclidean plane Theorem 6.3.5. If A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right, then 1 d 󵄨󵄨 2 − 󵄨d d 2 − 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − |A| (1 + D ) 2 ∈ (ℒ1,∞ )0 .

(6.16)

Theorem 6.3.5 says that for any continuous trace φ on ℒ1,∞ , − 1 󵄨d −d 󵄨 φ(󵄨󵄨󵄨A(1 + D2 ) 2 󵄨󵄨󵄨 ) = φ(|A|d (1 + D2 ) 2 ),

where A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right. Since MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is a C ∗ -algebra, we have |A|d ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ), and the formula -- |d ) = φ(|A|d (1 + D2 ) φ(|df

− d2

)

follows from Theorem 6.3.5 and (6.15). The format of the right side of the above equality -- |d by allows us to employ Theorem 5.3.5. In the next section, we compute the trace of |df calculating the principal symbol of |A|d . In this section, we prove Theorem 6.3.5 by a sequence of lemmas. Lemma 6.3.6. If A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported, then −r

[A, (1 + D2 ) 2 ] ∈ (ℒ d ,∞ )0 , r

r > 0.

Proof. Suppose first that A ∈ MNd (ℂ) ⊗ Alg(π1 (Cc∞ (ℝd )), π2 (C(𝕊d−1 ))), where the right-hand side of the display denotes the ∗-algebra generated, in the algebraic sense, by the ∗-algebras π1 (Cc∞ (ℝd )) and π2 (C(𝕊d−1 )). By the Leibniz rule it suffices r to prove the statement for A ∈ MNd (ℂ) ⊗ ℂ (in which case the commutator [A, (1 + D2 )− 2 ] vanishes), for A ∈ ℂ ⊗ π1 (Cc∞ (ℝd )) (which follows from Corollary 1.6.32), and for A ∈ ℂ ⊗ π2 (C(𝕊d−1 )) (in which case the commutator again vanishes). Consider now the general case. There exists a sequence {An }n≥0 ⊂ MNd (ℂ) ⊗ Alg(π1 (Cc∞ (ℝd )), π2 (C(𝕊d−1 ))) such that An → A in the uniform norm. Choose ϕ ∈ Cc∞ (ℝd ) such that A = (1 ⊗ Mϕ )A(1 ⊗ Mϕ ). Clearly, (1 ⊗ Mϕ )An (1 ⊗ Mϕ ) → (1 ⊗ Mϕ )A(1 ⊗ Mϕ ) = A

� 369

6.3 Approximation of quantized differentials by principal terms

in the uniform norm. We may assume without loss of generality that An = (1 ⊗ Mϕ )An (1 ⊗ Mϕ ). By the (quasi-)triangle inequality in ℒ d ,∞ and Hölder’s inequality we have r

− r2

−r 󵄩 󵄩 cr,d ⋅ 󵄩󵄩󵄩[An , (1 + D2 ) ] − [A, (1 + D2 ) 2 ]󵄩󵄩󵄩 d ,∞ −r 󵄩 󵄩 ≤ 󵄩󵄩󵄩(An − A) ⋅ (1 ⊗ Mϕ )(1 + D2 ) 2 󵄩󵄩󵄩 d ,∞

r

r

−r 󵄩 󵄩 + 󵄩󵄩󵄩(1 + D2 ) 2 (1 ⊗ Mϕ ) ⋅ (An − A)󵄩󵄩󵄩 d ,∞ r r 󵄩 󵄩 2 −2 󵄩 󵄩 ≤ 2‖An − A‖∞ 󵄩󵄩(1 ⊗ Mϕ )(1 + D ) 󵄩󵄩 d ,∞ r

for a constant cd,r > 0. Since for every n ≥ 0, we have −r

[An , (1 + D2 ) 2 ] ∈ (ℒ d ,∞ )0 , r

it follows that −r −r 󵄩 󵄩 −1 dist([A, (1 + D2 ) 2 ], (ℒ d ,∞ )0 ) ≤ 2cr,d ⋅ ‖An − A‖∞ 󵄩󵄩󵄩(1 ⊗ Mϕ )(1 + D2 ) 2 󵄩󵄩󵄩 d ,∞ . r

r

Passing to the limit as n → ∞, we obtain −r

dist([A, (1 + D2 ) 2 ], (ℒ d ,∞ )0 ) = 0. r

This completes the proof. The following lemma should be compared with Corollary 1.6.5, and may be viewed as an analogue of that result in the C ∗ -algebraic approach of Chapter 5. Lemma 6.3.7. If A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported, then − 21

A(1 + D2 )

− (1 + D2 )

− 41

− 41

A(1 + D2 )

∈ (ℒd,∞ )0 .

Proof. Choose ϕ ∈ Cc∞ (ℝd ) such that A = A(1 ⊗ Mϕ ). We have (1 + D2 )

− 41

− 41

A(1 + D2 ) − 41

= (1 + D2 )

− 41

A ⋅ (1 ⊗ Mϕ )(1 + D2 )

− 41

= [(1 + D2 )

, A] ⋅ (1 ⊗ Mϕ )(1 + D2 )

= [(1 + D2 )

, A] ⋅ (1 ⊗ Mϕ )(1 + D2 )

− 41

1 2 −2

− 41 − 41

− 41

− 41

+ A(1 + D2 )

(1 ⊗ Mϕ )(1 + D2 )

+ A(1 + D2 )

⋅ [1 ⊗ Mϕ , (1 + D2 )

− 41

1 2 −2

+ A ⋅ [(1 + D ) , 1 ⊗ Mϕ ] + A ⋅ (1 ⊗ Mϕ ) ⋅ (1 + D ) .

− 41

]

370 � 6 Quantum differentiability for the Euclidean plane By Lemma 6.3.6 and Hölder’s inequality the first three summands fall into (ℒd,∞ )0 . The 1

last summand equals A(1 + D2 )− 2 , exactly as needed.

Lemma 6.3.8. If A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported, then −1 m

− m2

(A(1 + D2 ) 2 ) − Am (1 + D2 )

∈ (ℒ d ,∞ )0 m

for every m ∈ ℤ+ . Proof. We prove the statement by induction on m. For m = 1, our operator is 0, and hence there is nothing to prove. This establishes the base of induction. Suppose the statement holds for m. Let us prove the statement for m + 1. We write − m+1 2

Am+1 (1 + D2 )

− Am (1 + D2 )

= Am ⋅ [A, (1 + D2 )

− m+1 2

− m2

⋅ A(1 + D2 ) − m2

] + Am (1 + D2 )

− 21 −1

⋅ [(1 + D2 ) 2 , A].

By Lemma 6.3.6 we have − m+1 2

[A, (1 + D2 )

] ∈ (ℒ

d ,∞ m+1

−1

[(1 + D2 ) 2 , A] ∈ (ℒd,∞ )0 .

)0 ,

By the Hölder inequality we have − m+1 2

] + Am (1 + D2 )

− m+1 2

− Am (1 + D2 )

Am ⋅ [A, (1 + D2 )

− m2

−1

⋅ [(1 + D2 ) 2 , A] ∈ (ℒ

d ,∞ m+1

)0 .

Thus Am+1 (1 + D2 )

− m2

⋅ A(1 + D2 )

− 21

∈ (ℒ

d ,∞ m+1

)0 .

∈ (ℒ

d ,∞ m+1

)0 .

d ,∞ m+1

)0 .

By the inductive assumption and the Hölder inequality we have Am (1 + D2 )

− m2

⋅ A(1 + D2 )

− 21

− 1 m+1

− (A(1 + D2 ) 2 )

Therefore − 1 m+1

(A(1 + D2 ) 2 )

− m+1 2

− Am+1 (1 + D2 )

∈ (ℒ

This establishes the step of induction, and the statement follows. Proof of Theorem 6.3.5. If A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right, then |A| is positive and compactly supported. Using the obvious equality 󵄨 󵄨 |AB| = 󵄨󵄨󵄨|A|B󵄨󵄨󵄨,

A, B ∈ ℒ(ℂNd ⊗ L2 (ℝd )),

6.4 Integration of quantum densities

� 371

we observe that equation (6.16) does not change if we replace A with |A|. Hence we may assume without loss of generality that A is positive and compactly supported. By Lemma 6.3.7 we have − 21

A(1 + D2 )

− (1 + D2 )

− 41

− 41

A(1 + D2 )

∈ (ℒd,∞ )0 .

By Lemma 1.3.16 we have 1 1 1 󵄨󵄨 2 − 󵄨 󵄨 2 − 2 − 󵄨 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − 󵄨󵄨󵄨(1 + D ) 4 A(1 + D ) 4 󵄨󵄨󵄨 ∈ (ℒd,∞ )0 .

Since A ≥ 0, it follows that 1 1 1 󵄨󵄨 2 − 󵄨 2 − 2 − 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − (1 + D ) 4 A(1 + D ) 4 ∈ (ℒd,∞ )0 .

By Lemma 6.3.7 we have 1 1 󵄨󵄨 2 − 󵄨 2 − 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − A(1 + D ) 2 ∈ (ℒd,∞ )0 .

Therefore 1 1 󵄨󵄨 2 − 󵄨d 2 − d 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − (A(1 + D ) 2 ) ∈ (ℒ1,∞ )0 .

The statement now follows from Lemma 6.3.8 (applied with m = d).

6.4 Integration of quantum densities In this section, we prove Theorem 6.1.3 in the particular case of a smooth compactly supported function. The central result is the following theorem. Theorem 6.4.1. Let d ≥ 2, and let D be the Dirac operator given in (6.1). Let f ∈ Cc∞ (ℝd ) be real-valued. Then -- |d = 󵄨󵄨󵄨[sgn(D), M ]󵄨󵄨󵄨d ∈ ℒ , |df f 󵄨 1,∞ 󵄨 and -- |d ) = c ⋅ ‖f ‖d 1 φ(|df d Ẇ d

for every continuous normalized trace φ on ℒ1,∞ . Here cd is the constant cd :=

Nd 󵄨 󵄨d ⋅ ∫ 󵄨󵄨󵄨e1 − s⟨s, e1 ⟩󵄨󵄨󵄨 2 ds, d d(2π) 𝕊d−1

where e1 := (1, 0, . . . , 0) ∈ 𝕊d−1 .

(6.17)

372 � 6 Quantum differentiability for the Euclidean plane The central part of the proof is the symbol mapping for principal terms introduced in Theorem 5.3.1 and the trace formula for principal terms in Theorem 5.3.5. In this section, we will lighten the notation by writing ∫ℝd ×𝕊d−1 for the integral with respect to the Lebesgue measure without explicitly writing the measure. The next lemma states the matrix-valued version of Theorem 5.3.5. Lemma 6.4.2. Let d ≥ 2, and let D be the Dirac operator given in (6.1). Let T ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) be compactly supported on the right. Then φ(T(1 + D2 )

− d2

)=

1 ⋅ (Tr ⊗ d(2π)d

∫ )((id ⊗ sym)(T)) ℝd ×𝕊d−1

for every continuous normalized trace φ on ℒ1,∞ . N

d Proof. Let {Ek,l }k,l=1 be matrix units in MNd (ℂ). We write

Nd

T = ∑ Ek,l ⊗ Tk,l , k,l=1

Tk,l ∈ Π(𝒜1 , 𝒜2 ),

1 ≤ k, l ≤ Nd .

Since T is compactly supported on the right, it follows that each Tk,l is compactly supported on the right. We now write − d2

T(1 + D2 )

Nd

d

= ∑ Ek,l ⊗ Tk,l (1 − Δ)− 2 . k,l=1

By Lemma 1.3.13 we have φ(T(1 + D2 )

− d2

Nd

d

) = ∑ Tr(Ek,l )φ(Tk,l (1 − Δ)− 2 ). k,l=1

By Theorem 5.3.5 we have d

φ(Tk,l (1 − Δ)− 2 ) =

1 ⋅ d(2π)d

sym(Tk,l ).

∫ ℝd ×𝕊d−1

Finally, φ(T(1 + D2 )

− d2

N

)=

= It remains to note that

d 1 ⋅ ∑ Tr(Ek,l ) ⋅ d d(2π) k,l=1

1 ⋅ (Tr ⊗ d(2π)d

∫

sym(Tk,l )

ℝd ×𝕊d−1 Nd

∫ )( ∑ Ek,l ⊗ sym(Tk,l )). ℝd ×𝕊d−1

k,l=1

6.4 Integration of quantum densities

� 373

Nd

∑ Ek,l ⊗ sym(Tk,l ) = (id ⊗ sym)(T).

k,l=1

Recall from Theorem 5.3.1 that sym : Π(ℂ + C0 (ℝd ), C(𝕊d−1 )) → C(𝕊d−1 , C0 (ℝd ) + ℂ). In the following lemma, we explicitly compute the symbol of the operator |A|d , where A is the operator featuring in Theorem 6.3.1. Lemma 6.4.3. Let A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) be as in Theorem 6.3.1. We have 󵄩 󵄩d ((id ⊗ sym)(|A|d ))(t, s) = 1 ⊗ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld , 2

(t, s) ∈ ℝd × 𝕊d−1 .

Proof. The symbol mapping sym from Theorem 5.3.5 is a homomorphism of C ∗ -algebras. Hence, so is the mapping id ⊗ sym. Thus 󵄨 󵄨d (id ⊗ sym)(|A|d ) = 󵄨󵄨󵄨(id ⊗ sym)(A)󵄨󵄨󵄨 . By the definition of A we have d

A = ∑ γk ⊗ A k , k=1

where d

Dk Dj

j=1

−Δ

Ak := M𝜕k f − ∑

M𝜕j f .

Thus d

(id ⊗ sym)(A) = ∑ γk ⊗ sym(Ak ), k=1

and d

󵄨󵄨 d 󵄨󵄨d 2 d 󵄨󵄨 󵄨󵄨 󵄨 󵄨2 d 󵄨 (id ⊗ sym)(|A| ) = 󵄨󵄨 ∑ γk ⊗ sym(Ak )󵄨󵄨󵄨 = 1 ⊗ ( ∑ 󵄨󵄨󵄨sym(Ak )󵄨󵄨󵄨 ) . 󵄨󵄨 󵄨󵄨 k=1 󵄨k=1 󵄨 Since sym is a ∗-homomorphism, it follows from the definition of Ak that d

Dk Dj

j=1

−Δ

sym(Ak ) = sym(MDk f ) − ∑ sym(

) ⋅ sym(MDj f ).

Appealing to the properties of sym stated in Theorem 5.3.1, we write

(6.18)

374 � 6 Quantum differentiability for the Euclidean plane

sym(MDk f ) = Dk f ⊗ 1,

sym(

Dk Dj

) = 1 ⊗ sk sj ,

−Δ

where sk : 𝕊d−1 → ℝ is the kth coordinate function. Therefore d

sym(Ak ) : (t, s) 󳨃→ (Dk f )(t) − ∑ sk sj (Dj f )(t). j=1

Finally, d 2

d

󵄨 󵄨2 󵄩 󵄩d ( ∑ 󵄨󵄨󵄨sym(Ak )󵄨󵄨󵄨 ) : (t, s) 󳨃→ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld . 2

(6.19)

k=1

The statement of the lemma follows by combining (6.18) and (6.19). Lemma 6.4.4. Let f ∈ Cc∞ (ℝd ) be real-valued, and let e1 := (1, 0, . . . , 0) ∈ 𝕊d−1 . Then 󵄩 󵄩d 󵄩 󵄩d ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld dtds = ∫ 󵄩󵄩󵄩e1 − ⟨e1 , s⟩s󵄩󵄩󵄩ld ds ⋅ ‖f ‖dẆ 1 . 2 2 d

ℝd ×𝕊d−1

𝕊d−1

Proof. By Fubini’s theorem the integral on the left is 󵄩 󵄩d ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld dtds 2

ℝd ×𝕊d−1

󵄩 󵄩d = ∫ ( ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld ds)dt. 2 ℝd 𝕊d−1

Now we use the fact that f is real-valued. For every t ∈ ℝd , and there exists an isometry R ∈ SO(d) such that Re1 = s → R−1 s yields

∇f (t) ‖∇f (t)‖ld

∇f (t) ‖∇f (t)‖ld 2

2

is a unit vector in ℝd ,

. The change of variable

󵄩 󵄩d 󵄩 󵄩d 󵄩 󵄩d ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld ds = 󵄩󵄩󵄩(∇f )(t)󵄩󵄩󵄩ld ∫ 󵄩󵄩󵄩e1 − ⟨e1 , s⟩s󵄩󵄩󵄩ld ds. 2 2 2

𝕊d−1

𝕊d−1

Thus 󵄩 󵄩d ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld dtds 2

ℝd ×𝕊d−1

󵄩 󵄩d 󵄩 󵄩d = ∫ 󵄩󵄩󵄩(∇f )(t)󵄩󵄩󵄩ld dt ⋅ ∫ 󵄩󵄩󵄩e1 − ⟨e1 , s⟩s󵄩󵄩󵄩ld ds. 2 2 ℝd

𝕊d−1

The first integral is on the right-hand side is exactly the definition of ‖f ‖dẆ 1 . d

6.4 Integration of quantum densities

� 375

Proof of Theorem 6.4.1 in the case where f ∈ Cc∞ (ℝd ). Let A ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) be as in Theorem 6.3.1. By Theorem 6.3.1 we have 1

-- − A(1 + D2 )− 2 ∈ (ℒ ) . df d,∞ 0 By Corollary 1.3.15 we have 1

-- | − 󵄨󵄨󵄨A(1 + D2 )− 2 󵄨󵄨󵄨 ∈ (ℒ ) . |df d,∞ 0 󵄨 󵄨 Lemma 1.3.18 yields 1 -- |d − 󵄨󵄨󵄨A(1 + D2 )− 2 󵄨󵄨󵄨d ∈ (ℒ ) . |df 1,∞ 0 󵄨 󵄨

By Remark 6.3.2 and Theorem 6.3.5 we have 1 d 󵄨󵄨 2 − 󵄨d d 2 − 󵄨󵄨A(1 + D ) 2 󵄨󵄨󵄨 − |A| (1 + D ) 2 ∈ (ℒ1,∞ )0 .

Combining the last two displays, we arrive at -- |d − |A|d (1 + D2 ) |df

− d2

∈ (ℒ1,∞ )0 .

Recall that A is compactly supported on the right. Since φ is continuous, it follows from Lemma 6.4.2 that − d2

φ(|A|d (1 + D2 )

)=

1 ⋅ (Tr ⊗ d(2π)d

∫ )((id ⊗ sym)(|A|d )). ℝd ×𝕊d−1

Therefore -- |d ) = φ(|df

1 ⋅ (Tr ⊗ d(2π)d

∫ )((id ⊗ sym)(|A|d )). ℝd ×𝕊d−1

By Lemma 6.4.3 we have -- |d ) = φ(|df =

1 ⋅ (Tr ⊗ d(2π)d Nd ⋅ d(2π)d

󵄩 󵄩d ∫ )(1 ⊗ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld )

ℝd ×𝕊d−1

2

󵄩 󵄩d ∫ 󵄩󵄩󵄩(∇f )(t) − ⟨(∇f )(t), s⟩s󵄩󵄩󵄩ld dtds. 2

ℝd ×𝕊d−1

Equality (6.17) in Theorem 6.4.1 now follows from Lemma 6.4.4. The proof of Theorem 6.4.1 can be adjusted to integrate product operators of continuous functions against quantum densities. The next result proves formula (6.4) in the case where g is real-valued and smooth.

376 � 6 Quantum differentiability for the Euclidean plane Corollary 6.4.5. Let d ≥ 2, and let D be the Dirac operator given in (6.1). Let f ∈ C0 (ℝd )+ℂ and g ∈ Cc∞ (ℝd ) be real-valued. Then -- d = (1 ⊗ M )󵄨󵄨󵄨[sgn(D), M ]󵄨󵄨󵄨d ∈ ℒ , (1 ⊗ Mf )|dg| f 󵄨 g 󵄨 1,∞ and d 2

d

-- ) = c ⋅ ∫ f (t)( ∑ 󵄨󵄨󵄨𝜕 g(t)󵄨󵄨󵄨2 ) dt φ((1 ⊗ Mf )|dg| d 󵄨 k 󵄨 d

k=1

ℝd

for every continuous normalized trace φ on ℒ1,∞ . Here cd is the constant cd :=

Nd 󵄨󵄨 󵄨󵄨 d2 ⋅ e − s⟨s, e ⟩ ∫ 󵄨 󵄨󵄨 ds, 1 1 󵄨 d(2π)d 𝕊d−1

where e1 := (1, 0, . . . , 0) ∈ 𝕊d−1 . Proof. Let f ∈ C0 (ℝd ) + ℂ and g ∈ Cc∞ (ℝd ) be real-valued. The product operator Mf : L2 (ℝd ) → L2 (ℝd ) belongs to the C ∗ -algebra Π(𝒜1 , 𝒜2 ) by construction, and hence (1 ⊗ Mf )|A|d ∈ MNd (ℂ) ⊗ Π(𝒜1 , 𝒜2 ) is compactly supported on the right with (id ⊗ sym)((1 ⊗ Mf )|A|d ) = (1 ⊗ f )(id ⊗ sym)(|A|d ). From the proof of Theorem 6.4.1 and the fact that Mf is bounded we have d

-- d − (1 ⊗ M )|A|d (1 + D2 )− 2 ∈ (ℒ ) . (1 ⊗ Mf )|dg| f 1,∞ 0 The statement now follows from Lemmas 6.4.2 and 6.4.3 since -- d ) = φ((1 ⊗ M )|A|d (1 − Δ)− d2 ) φ((1 ⊗ Mf )|dg| f and − d2

φ((1 ⊗ Mf )|A|d (1 + D2 )

)=

1 ⋅ (Tr ⊗ d(2π)d

∫ )((1 ⊗ f )(id ⊗ sym)(|A|d )) ℝd ×𝕊d−1

for every continuous normalized trace φ on ℒ1,∞ .

6.5 Approximation of homogeneous Sobolev functions by smooth functions

377

�

6.5 Approximation of homogeneous Sobolev functions by smooth functions Theorem 6.2.1 in Section 6.2 and Theorem 6.4.1 in Section 6.4 are established under the condition f ∈ Cc∞ (ℝd ). To prove Theorem 6.1.2, we need a number of lemmas for the approximation of homogeneous Sobolev functions by smooth functions. Lemma 6.5.1. Let ψ ∈ Cc∞ (ℝd ) be equal to 1 in a neighborhood of the origin. Define t ψn (t) := (σn ψ)(t) = ψ( ), n

t ∈ ℝd ,

n ≥ 1,

where σn is the dilation action of ℕ on Cc∞ (ℝd ), and d

ϕn (ξ) := (2π) 2 ⋅ (ℱ ψn )(ξ),

ξ ∈ ℝd ,

n ≥ 1,

where ℱ is the Fourier transform. Let f ∈ L∞ (ℝd ) and set fn := ψn ⋅ (ϕn ⋆ f ),

n ≥ 1.

Then fn ∈ Cc∞ (ℝd ) for every n ≥ 1. Proof. Since ϕn ∈ 𝒮 (ℝd ) and since f ∈ L∞ (ℝd ), it follows that f ∗ ϕn ∈ C ∞ (ℝd ) for every n ≥ 1. Since ψn is compactly supported, it follows that fn ∈ Cc∞ (ℝd ) for every n ≥ 1. Observe that, for all 1 ≤ p ≤ ∞ and n ≥ 1, we have ‖ψn ‖Lp = n

− dp

‖ϕn ‖Lp = n

d( p1 −1)

‖Dk ψn ‖Lp = n

d(1− p1 )

‖ψ‖Lp ,

(6.20)

‖ϕ1 ‖Lp ,

(6.21)

‖Dk ψ‖Lp .

(6.22)

The point of the sequence of smooth approximations of f ∈ L∞ (ℝd ) in Lemma 6.5.1 is the convergence in the Wd1 (ℝd ) norm when f has distributional derivatives. Lemma 6.5.2. Let f ∈ Wd1 (ℝd ) ∩ L∞ (ℝd ), and let {fn }n≥1 ⊂ Cc∞ (ℝd ) be as in Lemma 6.5.1. Then fn → f in Wd1 (ℝd ) as n → ∞. Proof. A well-known fact of harmonic analysis is that if f ∈ Ld (ℝd ), then f ∗ ϕn → f in the norm of Ld (ℝd ). See, for example, [140, Theorem 1.2.19]. Since Dk f ∈ Ld (ℝd ), it follows that Dk f ∗ ϕn → Dk f in Ld (ℝd ). By the triangle inequality we have 󵄩 󵄩 󵄩 󵄩 ‖fn − f ‖d ≤ 󵄩󵄩󵄩ψn ⋅ (f ∗ ϕn − f )󵄩󵄩󵄩d + 󵄩󵄩󵄩(1 − ψn ) ⋅ f 󵄩󵄩󵄩d 󵄩 󵄩 ≤ ‖ψn ‖∞ ‖f ∗ ϕn − f ‖d + 󵄩󵄩󵄩(1 − ψn ) ⋅ f 󵄩󵄩󵄩d

378 � 6 Quantum differentiability for the Euclidean plane 󵄩 󵄩 ≤ ‖ψ‖∞ ‖f ∗ ϕn − f ‖d + 󵄩󵄩󵄩(1 − ψn ) ⋅ f 󵄩󵄩󵄩d . Thus fn → f in Ld (ℝd ) as n → ∞. By the Leibniz rule we have Dk fn = (Dk ψn ) ⋅ (f ∗ ϕn ) + ψn ⋅ (Dk f ∗ ϕn ). Thus 󵄩 󵄩 ‖Dk fn − Dk f ‖d ≤ ‖Dk ψn ‖∞ ‖f ∗ ϕn ‖d + 󵄩󵄩󵄩ψn ⋅ (Dk f ∗ ϕn ) − Dk f 󵄩󵄩󵄩d . By Young’s inequality, (6.21), and (6.22) we have ‖Dk ψn ‖∞ ‖f ∗ ϕn ‖d ≤ ‖Dk ψn ‖∞ ‖f ‖d ‖ϕn ‖1 =

1 ‖D ψ‖ ‖f ‖ ‖ϕ ‖ . n k ∞ d 1 1

Also, decomposing ψn ⋅ (Dk f ∗ ϕn ) − Dk f as ψn ⋅ (Dk f ∗ ϕn ) − Dk f = (ψn − 1)Dk f + ψn (Dk f ∗ ϕn − Dk f ) and applying (6.20) with p = ∞ yield 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩ψn ⋅ (Dk f ∗ ϕn ) − Dk f 󵄩󵄩󵄩d ≤ 󵄩󵄩󵄩(1 − ψn ) ⋅ Dk f 󵄩󵄩󵄩d + ‖ψ‖∞ 󵄩󵄩󵄩(Dk f ∗ ϕn ) − Dk f 󵄩󵄩󵄩d . Combining the last three inequalities, we obtain that Dk fn → Dk f in Ld (ℝd ) as n → ∞. We conclude that fn → f in Wd1 (ℝd ). We can extend the approximation of Sobolev functions to approximation of homogeneous Sobolev functions. Lemma 6.5.3. For every f ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ), it is possible to find a sequence {fn }n≥1 ⊂ Wd1 (ℝd ) and a constant c such that (a) fn → f in Ẇ d1 (ℝd ), (b) fn → f − c uniformly on compact sets, and (c) supn≥1 ‖fn ‖∞ < ∞. To prove Lemma 6.5.3, we use Poincaré’s inequality for open subsets of ℝd , which we briefly recall here. If Ω ⊂ ℝd is an open set and 1 ≤ p ≤ ∞, then the homogeneous Sobolev space Ẇ p1 (Ω) on Ω is defined by Ẇ p1 (Ω) := {f ∈ L1,loc (Ω) : D1 f , D2 f , . . . , Dd f ∈ Lp (Ω)}, where D1 , . . . , Dd are interpreted in the distributional sense. We denote

6.5 Approximation of homogeneous Sobolev functions by smooth functions

d

‖f ‖Ẇ 1 (Ω) := ( ∑ p

k=1

p ‖Dk f ‖L (Ω) ) p

�

379

1 p

.

Similarly, the inhomogeneous Sobolev space Wp1 (Ω) is defined as Ẇ p1 (Ω) ∩ Lp (Ω). Poincaré’s inequality states that if Ω is bounded and connected with smooth boundary, then Ẇ p1 (Ω) = Wp1 (Ω), and there exists a constant Cp,Ω > 0 such that for all f ∈ Ẇ p1 (Ω), we have ‖f − c‖Lp (Ω) ≤ Cp,Ω ‖f ‖Ẇ 1 (Ω) , p

1 where c = |Ω| ∫Ω f (t) dt. Poincaré’s inequality is proved in classical reference works such as [126, Subsection 5.8.1, Theorem 1]. In the following proof, we will apply Poincaré’s inequality to the domain 2𝔹d \ 𝔹d , where 𝔹d is the unit ball in ℝd . Here we use essentially the assumption that d > 1, since in one dimension the annulus 2𝔹d \ 𝔹d is not connected.

Proof of Lemma 6.5.3. Let f ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ). Let η ∈ Cc∞ (ℝd ) be supported on 2𝔹d and such that η = 1 on 𝔹d . Let cn :=

1 |2n𝔹d \n𝔹d |

f (t) dt.

∫ 2n𝔹d \n𝔹d

Note that the sequence {cn }n≥1 is bounded because f ∈ L∞ (ℝd ). Define the sequence {fn }n≥0 ⊂ L∞ (ℝd ) by setting fn := σn η ⋅ (f − cn ). It is immediate that supn≥1 ‖fn ‖∞ is finite. We have ‖fn − f ‖dẆ 1 (ℝd ) = ‖fn − f ‖dẆ 1 (n𝔹d ) + ‖fn − f ‖dẆ 1 (2n𝔹d \n𝔹d ) d

d

+ ‖fn − f ‖dẆ 1 (ℝd \2n𝔹d ) .

d

d

On n𝔹d , we have fn = f − cn (because σn η = 1 on n𝔹d ). Hence ‖fn − f ‖Ẇ 1 (n𝔹d ) = ‖cn ‖Ẇ 1 = 0. d

d

(6.23)

On ℝd \2n𝔹d , we have fn = 0. Hence ‖fn − f ‖Ẇ 1 (ℝd \2n𝔹d ) = ‖f ‖Ẇ 1 (ℝd \2n𝔹d ) . d

By the Leibniz rule we have

d

(6.24)

380 � 6 Quantum differentiability for the Euclidean plane 1 σ (∇η) ⋅ (f − cn ) + σn η ⋅ ∇f . n n

∇(fn ) =

By Hölder’s inequality and the triangle inequality we have ‖fn − f ‖Ẇ 1 (2n𝔹d \n𝔹d ) ≤ ‖fn ‖Ẇ 1 (2n𝔹d \n𝔹d ) + ‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) d d d 󵄩󵄩 1 󵄩󵄩 󵄩󵄩 󵄩󵄩 ≤ 󵄩󵄩 σn (∇η)󵄩󵄩 ‖f − cn ‖Ld (2n𝔹d \n𝔹d ) 󵄩󵄩 n 󵄩󵄩∞ + ‖σn η‖∞ ‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) + ‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) d

d

1 = ‖η‖Ẇ 1,∞ ‖f − cn ‖Ld (2n𝔹d \n𝔹d ) + (1 + ‖η‖∞ )‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) . d n

Applying a similar identity to (6.20), we have 1 ‖f − cn ‖Ld (2n𝔹d \n𝔹d ) = ‖σ 1 f − cn ‖Ld (2𝔹d \𝔹d ) . n n Since, by definition, cn =

1

|2𝔹d \𝔹d |

∫ (σ 1 f )(t) dt, n

2𝔹d \𝔹d

it follows from the Poincaré inequality that ‖σ 1 f − cn ‖Ld (2𝔹d \𝔹d ) ≤ cd ‖σ 1 f ‖Ẇ 1 (2𝔹d \𝔹d ) . n

d

n

Thus 1 ‖f − cn ‖Ld (2n𝔹d \n𝔹d ) ≤ cd ‖σ 1 f ‖Ẇ 1 (2𝔹d \𝔹d ) = ‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) , d d n n that is, ‖fn − f ‖Ẇ 1 (2n𝔹d \n𝔹d ) ≤ cd′ ‖f ‖Ẇ 1 (2n𝔹d \n𝔹d ) . d

(6.25)

d

Combining (6.23), (6.24), and (6.25), we obtain ‖fn − f ‖dẆ 1 (ℝd ) ≤ cd′ d ‖f ‖dẆ 1 (2n𝔹d \n𝔹d ) + ‖f ‖dẆ 1 (ℝd \2n𝔹d ) ≤ cd′′ d ‖f ‖dẆ 1 (ℝd \n𝔹d ) . d

d

d

d

Thus fn → f in Ẇ d1 (ℝd ). Also, fn + cn = f on n𝔹d . Hence fn + cn → f uniformly on compact sets. Passing to a subsequence if needed, we can assume that cn → c. Thus fn → f − c uniformly on compact sets. The final lemma in this section identifies when f ∈ L∞ (ℝd ) belongs to the homogeneous Sobolev space by the uniform boundedness of its smooth approximations.

6.6 Trace formula for quantum densities on the Euclidean plane

�

381

Lemma 6.5.4. Let f ∈ L∞ (ℝd ), and let {fn }n≥1 ⊂ Cc∞ (ℝd ) be as in Lemma 6.5.1. If 1 < p < ∞ is such that sup ‖fn ‖Ẇ 1 < ∞, n≥1

p

(6.26)

then f ∈ Ẇ p1 (ℝd ). Proof. By assumption the sequence {∇fn }n≥1 is uniformly bounded in Lp (ℝd , ℂd ). Since 1 < p < ∞, it follows that Lp (ℝd , ℂd ) is reflexive and there is a subsequence of {∇fn }∞ n=1 that converges weakly to some g ∈ Lp (ℝd , ℂd ). Let ϕ ∈ 𝒮 (ℝd ). Recalling that (⋅, ⋅) denotes the distributional pairing, we have (fn , ϕ) = (ϕn ∗ f , ψn ϕ) = (f , ϕn ∗ (ψn ϕ)). The sequence {ℱ ψn ∗ (ψn ϕ)}n≥1 converges to ϕ in the norm of L1 (ℝd ); see, for example, [140, Theorem 1.2.19]. Thus 󵄨󵄨 󵄨 󵄨 󵄨 󵄩 󵄩 󵄨󵄨(fn , ϕ) − (f , ϕ)󵄨󵄨󵄨 = 󵄨󵄨󵄨(f , ϕn ∗ (ψn ϕ) − ϕ)󵄨󵄨󵄨 ≤ ‖f ‖∞ 󵄩󵄩󵄩ϕn ∗ (ψn ϕ) − ϕ󵄩󵄩󵄩1 . Thus (fn , ϕ) → (f , ϕ),

n → ∞.

In other words, the sequence {fn }n≥1 converges to f in the sense of tempered distributions. Consequently, the sequence {𝜕j fn }n≥1 converges to the distributional derivative 𝜕j f for every 1 ≤ j ≤ d in the sense of tempered distributions. Since weak convergence in Lp (ℝd ) implies convergence in the sense of tempered distributions, it follows that g = ∇f , and so ∇f ∈ Lp (ℝd , ℂd ).

6.6 Trace formula for quantum densities on the Euclidean plane In this section, we complete the proof of Theorems 6.1.2 and 6.1.3, showing that if f ∈ L∞ (ℝd ), d ≥ 2, then -- ∈ ℒ (ℂNd ⊗ L (ℝd )) df d,∞ 2 if and only if f ∈ Ẇ d1 (ℝd ). Further, if g ∈ L∞ (ℝd ) ∩ Ẇ d1 (ℝd ) is real-valued and f ∈ C0 (ℝd ) + ℂ, then we have the trace formula

382 � 6 Quantum differentiability for the Euclidean plane d 2

d

-- d ) = c ⋅ ∫ f (t)( ∑ 󵄨󵄨󵄨𝜕 g(t)󵄨󵄨󵄨2 ) dt φ((1 ⊗ Mf )|dg| d 󵄨 k 󵄨 ℝd

k=1

for every continuous trace φ on ℒ1,∞ . Here cd is the constant in the statement of Theorem 6.1.3. Theorem 6.2.1 proved the following lemma when f ∈ Cc∞ (ℝd ). The lemma provides the right side of (6.3) for f ∈ Wd1 (ℝd ) ∩ L∞ (ℝd ). Lemma 6.6.1. Let d ≥ 2, and let f ∈ Wd1 (ℝd ) ∩ L∞ (ℝd ). Then -- ∈ ℒ , df d,∞ and there exists a constant cd > 0 such that -- ‖ ‖df ℒd,∞ ≤ cd ⋅ ‖f ‖Ẇ 1 (ℝd ) . d

Proof. Let f ∈ Wd1 (ℝd ) ∩ L∞ (ℝd ), and let {fn }n≥1 be as in Lemma 6.5.1. By Lemma 6.5.1 we have fn ∈ Cc∞ (ℝd ) for n ≥ 1. By Lemma 6.5.2 we have that fn → f in Wd1 (ℝd ) as n → ∞. In particular, the sequence {fn }n≥1 is Cauchy in Wd1 (ℝd ). Since fn → f as n → ∞ in Ld (ℝd ), it follows that fn → f as n → ∞ in measure. Passing to a subsequence, we may assume that fn → f almost everywhere as n → ∞. By Young’s inequality, d

d

‖fn ‖∞ ≤ (2π) 2 ‖ψn ‖∞ ‖f ‖∞ ‖ℱ ψn ‖1 ≤ (2π) 2 ‖ψ‖∞ ‖f ‖∞ ‖ℱ ψ‖1 ,

n ≥ 1,

where the final inequality uses (6.20) and (6.21). By the dominated convergence theorem -- → df -we have that Mfn → Mf as n → ∞ in the strong operator topology. Therefore df n as n → ∞ in the strong operator topology. Since fn ∈ Cc∞ (ℝd ), the case of Theorem 6.1.2 already proved gives 󵄩󵄩 -󵄩󵄩 -󵄩󵄩 -- )󵄩󵄩󵄩 󵄩󵄩d(fn ) − d(f m 󵄩d,∞ = 󵄩 󵄩d(fn − fm )󵄩󵄩d,∞ ≤ cd ‖fn − fm ‖Ẇ 1 , d

m, n ≥ 1.

-- )} Since the sequence {fn }n≥1 is Cauchy in Ẇ d1 (ℝd ), it follows that the sequence {d(f n n≥1 is a Cauchy sequence in ℒd,∞ . Hence the latter sequence converges in ℒd,∞ . Denote its -- ) → A in the strong operator topology. By the limit by A. In particular, we have that d(f n --- ) → df -- in ℒ . We now have uniqueness of the limit, A = df , and therefore d(f n d,∞ 󵄩 -- 󵄩󵄩 -- ‖ ‖df lim 󵄩󵄩󵄩d(f sup ‖fn ‖Ẇ 1 = cd ‖f ‖Ẇ 1 . d,∞ = n→∞ n )󵄩 󵄩d,∞ ≤ cd lim d d n→∞ The following theorem provides the right-hand side inequality in (6.3) in Theorem 6.1.2 without the restriction of f being smooth and compactly supported.

6.6 Trace formula for quantum densities on the Euclidean plane

� 383

Theorem 6.6.2. If f ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ), d ≥ 2, then -- ∈ ℒ , df d,∞ and there exists a constant cd > 0 such that -- ‖ ‖df ℒd,∞ ≤ cd ⋅ ‖f ‖Ẇ 1 . d

Proof. Let f ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ), and let {fn }n≥0 be a sequence defined in Lemma 6.5.3. By that lemma, fn ∈ Wd1 (ℝd ) for n ≥ 1, and fn → f in Ẇ d1 (ℝd ) as n → ∞. In particular, the sequence {fn }n≥1 is Cauchy in Ẇ d1 (ℝd ). By Lemma 6.6.1 we have 󵄩󵄩 -󵄩󵄩 -- − df -- ‖ ‖df n m d,∞ = 󵄩 󵄩d(fn − fm )󵄩󵄩d,∞ ≤ cd ‖fn − fm ‖Ẇ 1 , d

m, n ≥ 1.

-- } is a Since the sequence {fn }n≥1 is Cauchy in Ẇ d1 (ℝd ), it follows that the sequence {df n n≥0 Cauchy sequence in ℒd,∞ . Hence the latter sequence converges in ℒd,∞ . Denote its limit by A. By Lemma 6.5.3, fn → f − c uniformly on bounded sets as n → ∞. Since supn≥1 ‖fn ‖∞ < ∞, it follows from the dominated convergence theorem that Mfn → Mf −c -- → df -- in the strong operator topology in the strong operator topology as n → ∞. Thus df n -as n → ∞. By the preceding paragraph, dfn → A as n → ∞ also in the strong operator -- and, therefore, df -- → df -- as n → ∞ in ℒ . topology. By uniqueness of the limit, A = df n d,∞ We now have -- ‖ -‖df d,∞ = lim ‖dfn ‖d,∞ ≤ cd lim sup ‖fn ‖Ẇ 1 = cd ⋅ ‖f ‖Ẇ 1 , n→∞

n→∞

d

d

where the last inequality follows by Lemma 6.5.3. Now we complete the proof of Theorem 6.1.3. Proof of Theorem 6.1.3. Let f ∈ C0 (ℝd ) + ℂ, and denote the linear functional on L1 (ℝd ) associated with f by (f , h) := ∫ f (t)h(t)dt,

h ∈ L1 (ℝd ).

ℝd

Now let h ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ). Denote the function d

󵄨2

1 2

󵄨 ‖∇h‖l2 : t 󳨃→ ( ∑ 󵄨󵄨󵄨(𝜕k h)(t)󵄨󵄨󵄨 ) , k=1

By assumption, ‖∇h‖dl2 ∈ L1 (ℝd ).

t ∈ ℝd .

384 � 6 Quantum differentiability for the Euclidean plane Let g ∈ Ẇ d1 (ℝd ) ∩ L∞ (ℝd ) be real-valued. Let {gn }n≥1 be the sequence of smooth approximations of g in Lemma 6.5.1. By Lemma 6.5.1 we have gn ∈ Cc∞ (ℝd ) for n ≥ 1, and by Corollary 6.4.5 we have -- |d ) = c ⋅ (f , ‖∇g ‖d ), φ((1 ⊗ Mf )|dg n d n l2

n ≥ 1,

(6.27)

for the constant cd from Theorem 6.4.1. By Lemma 6.5.2 we have gn → g in Ẇ d1 (ℝd ) as n → ∞. By Theorem 6.6.2 we have --- as n → ∞ in ℒ . By Theorem 1.3.14 we have |dg -- | → |dg| -- as n → ∞ in ℒ . dgn → dg d,∞ n d,∞ d d -- | → |dg| -- as n → ∞ in ℒ . Since M is a bounded From Corollary 1.3.17 we obtain |dg n 1,∞ f operator on L2 (ℝd ), -- |d → (1 ⊗ M )|dg| -- d , (1 ⊗ Mf )|dg n f

n → ∞,

in ℒ1,∞ . Therefore -- d ) = lim φ((1 ⊗ M )|dg -- |d ) φ((1 ⊗ Mf )|dg| f n n→∞

(6.28)

for the normalized continuous trace φ on ℒ1,∞ . By Lemma 6.5.2 we have gn → g in Ẇ d1 (ℝd ) as n → ∞, and hence (h, ‖∇gn ‖dl2 ) → (h, ‖∇g‖dl2 ),

n → ∞,

for all h ∈ L1 (ℝd )∗ = L∞ (ℝd ). Since f is bounded, (f , ‖∇gn ‖dl2 ) → (f , ‖∇g‖dl2 ),

n → ∞.

(6.29)

Combining (6.27), (6.28), and (6.29), the statement of Theorem 6.1.3 is shown. The next lemma is used to prove the left-hand side inequality in (6.3). -- ∈ Lemma 6.6.3. Let f ∈ L∞ (ℝd ), and let {fn }n≥1 be the sequence in Lemma 6.5.1. If df -- ∈ ℒ ℒd,∞ , then df n d,∞ for n ≥ 1. Moreover, we have -- ‖ -‖df n ℒd,∞ ≤ cψ ⋅ (‖f ‖L∞ + ‖df ‖ℒd,∞ ),

n ≥ 1,

for a constant cψ that depends on d and the function ψ in Lemma 6.5.1. Proof. By the definition of fn and the Leibniz rule, for every n ≥ 1 we have -- = (2π) d2 ⋅ ((dψ -- ) ⋅ M -df n n f ∗ℱ ψn + Mψn ⋅ d(f ∗ ℱ ψn )). By the triangle inequality and (6.21) we have 󵄩󵄩 -󵄩󵄩 -- ‖ -‖df n d,∞ ≤ cd ⋅ (‖dψn ‖d,∞ ⋅ ‖f ∗ ℱ ψn ‖∞ + ‖ψn ‖∞ 󵄩 󵄩d(f ∗ ℱ ψn )󵄩󵄩d,∞ )

6.6 Trace formula for quantum densities on the Euclidean plane

� 385

󵄩 -󵄩 ≤ cψ ⋅ (‖f ‖∞ + 󵄩󵄩󵄩d(f ∗ ℱ ψn )󵄩󵄩󵄩d,∞ ) for constants cd , cψ > 0. Therefore 󵄩󵄩 -󵄩󵄩 -- ‖ ‖df n d,∞ ≤ cψ ⋅ (‖f ‖∞ + sup󵄩 󵄩d(f ∗ ℱ ψn )󵄩󵄩d,∞ ). n≥1

(6.30)

Let t 󳨃→ Ut , t ∈ ℝd , be the group action of translation on locally integrable functions on ℝd ; that is, for t, s ∈ ℝd , (Ut f )(s) := f (s − t) for a locally integrable function f . By the definition of convolution we have (f ∗ ℱ ψn )(s) = ∫ (Ut f )(s)(ℱ ψn )(t)dt,

s ∈ ℝd .

ℝd

Note that D commutes with translations and so does sgn(D). Since MUt f = Ut Mf Ut−1 , it -- f ) = (1 ⊗ U )(df -- )(1 ⊗ U )−1 . Since follows that d(U t t t 󵄩 -󵄩󵄩 󵄨󵄨 󵄨󵄨 --∫ 󵄩󵄩󵄩d(U t f )ℱ ψn (t)󵄩 󵄩∞ dt = ‖df ‖∞ ⋅ ∫ 󵄨󵄨ℱ ψn (t)󵄨󵄨dt = ‖df ‖∞ ‖ℱ ψ‖1 ,

ℝd

ℝd

where (6.21) is used in the last equality, then by Lemma 1.4.2 -- ∗ ℱ ψ ) = ∫ d(U -- f )ℱ ψ (t) dt d(f n t n ℝd

-- ) ⋅ (1 ⊗ U −1 ) ⋅ ℱ ψ (t)dt = ∫ (1 ⊗ Ut ) ⋅ (df n t ℝd

where the right side denotes the weak operator integral. Applying Lemma 1.4.4 to the finite signed Borel measure ℱ ψn (t)dt, we obtain 󵄩󵄩 -󵄩 -- ‖ ‖ℱ ψ ‖ = b ⋅ ‖df -- ‖ ‖ℱ ψ‖ 󵄩󵄩d(f ∗ ℱ ψn )󵄩󵄩󵄩d,∞ ≤ bd ⋅ ‖df d,∞ n 1 d d,∞ 1

(6.31)

for a constant bd > 0. The statement follows by combining (6.30) and (6.31). The following theorem provides the left-hand side inequality in (6.3) in Theorem 6.1.2. -- ∈ ℒ , then f ∈ Ẇ 1 (ℝd ). Furthermore, Theorem 6.6.4. Let f ∈ L∞ (ℝd ) and d ≥ 2. If df d,∞ d there is a constant cd > 0 depending only on d such that -- ‖ . ‖f ‖Ẇ 1 ≤ cd ⋅ ‖df d,∞ d

Proof. Suppose f ∈ L∞ (ℝd ) is real-valued. Let {fn }n≥1 be the sequence in Lemma 6.5.1. By Lemma 6.5.1 we have fn ∈ Cc∞ (ℝd ) for n ≥ 1.

386 � 6 Quantum differentiability for the Euclidean plane By Theorem 6.4.1 we have -- |d ) ≤ ‖df -- ‖d . cd ⋅ ‖fn ‖dẆ 1 = φ(|df n n d,∞ d

By Lemma 6.6.3 we have -- ‖ )d , cd ⋅ ‖fn ‖dẆ 1 ≤ cψd ⋅ (‖f ‖∞ + ‖df d,∞ d

n ≥ 1.

Hence sup ‖fn ‖Ẇ 1 < ∞. n≥1

d

By Lemma 6.5.4 we obtain f ∈ Ẇ d1 (ℝd ). By Theorem 6.1.3 we have -- |d ) ≤ ‖df -- ‖d . cd ‖f ‖dẆ 1 = φ(|df d,∞ d

This proves the statement for real-valued f ∈ L∞ (ℝd ). -- ∈ ℒ . Since d(ℜf -- ) = Now let f ∈ L∞ (ℝd ) be complex valued and such that df d,∞ ---ℜ(df ) and d(ℑf ) = ℑ(df ), the statement follows from the real-valued case. Proof of Theorem 6.1.2. Theorem 6.6.2 provides the implication that if f ∈ Ẇ d1 (ℝd ) ∩ -- ∈ ℒ , and we have the right side of (6.3). L∞ (ℝd ), then df d,∞ -- ∈ L , then Theorem 6.6.4 provides the implication that if f ∈ L∞ (ℝd ) and df d,∞ 1 d ̇ f ∈ Wd (ℝ ). Theorem 6.6.4 also proves the left side of (6.3).

6.7 Notes The contents of this chapter are primarily based on the paper [189]. An alternative approach was later developed in [205], although here we have chosen to use the earlier proof because it is more transparent and elementary. The main result, including the -- ∈ ℒ , was first obtained by Connes, Sullinecessary and sufficient conditions for df d,∞ van, and Teleman [89]. The trace formula in Theorem 6.1.3 originates from Connes [68], who first proved the analogous assertion for compact manifolds. Commutators with the Riesz transform The study of the commutator [Rj , Mf ] : L2 (ℝd ) → L2 (ℝd ), where f ∈ L∞ (ℝd ) and Rj =

𝜕j √−Δ

is a Riesz transform, is a classical topic in harmonic

analysis. For the study of Dirac operators on ℝd and representations of the Clifford algebra, see [184].

6.7 Notes

� 387

The earliest results in this direction concern the one-dimensional case, where the commutator reduces to a direct sum of Hankel operators. In this one-dimensional situ-- are determined by the rate of approximation of f by ation, the singular numbers of df rational functions; this is the Adamyan–Arov–Krein theorem [220, Chapter 4, Section 1]. -- belongs to ℒ (L (ℝ)) if and only if f has vanishFor 0 < p < ∞, Peller has proved that df p 2 ing mean oscillation and coincides with a distribution in the homogeneous Besov space 1

p Ḃ p,p (ℝ) [220, Chapter 7, Section 7]. Similar results can be obtained for ℒp,∞ (L2 (ℝ)), p > 0, by interpolation (see [220, Theorem 4.4, Ch. 6, p. 255] and [237, 238]). Peller’s results for functions on the circle form a particularly precise expression of the general rule that -- ) should be related to the smoothness of f . the rate of decay of μ(df A more general study of commutators of pointwise multipliers with singular integral operators was embarked upon by Rochberg and Semmes [238] in 1989. Their results -- ∈ ℒ for all p ∈ [1, ∞) and q ∈ [1, ∞] in terms characterized the class of f such that df p,q of the oscillations of f on dyadic cubes.

Alternative methods The results of Chapter 6 can be proved by different methods. In particular, Lemma 6.3.8, which was proved here for m ∈ ℤ+ by induction, may be strengthened using results on an integral representation for the difference of complex powers [271]. This approach was adopted in [205].

�

Part III: Further applications

7 Connes character formula 7.1 Introduction In the minds of many experts, Dixmier traces – and, more generally, singular traces – are closely linked with the field of noncommutative geometry. Noncommutative geometry is a broad term, but in this specific context, it refers to the noncommutative differential geometry initiated by A. Connes in the 1980s. A key feature of this field is that the ordinary spaces we deal with in mathematics (topological spaces, measure spaces, manifolds, and so forth) are recontextualized as being the spectra of algebras of operators. In this way, spaces are traded for algebras of functions, and we can apply geometric language to algebras of operators, which may be noncommutative. This point of view is manifested throughout mathematics. The principal focus of the noncommutative geometry literature is index theory, and one of the leading achievements of this field is the local index theorem of Connes and Moscovici. The local index theorem is an algebraic reformulation of the index theorem of Atiyah and Singer for twisted Dirac operators in the setting of noncommutative geometry, and a key component is the Connes character formula. The Connes character formula predates the local index theorem and was originally proved by Connes in 1995. Connes’ formula involves singular traces and remains one of the well-known applications of singular traces. In the language of noncommutative geometry, the Connes character formula computes the Hochschild cohomology class of the cyclic cohomology Chern character of a spectral triple with integral spectral dimension. We will say little about noncommutative geometry, only recalling the key notions needed to formulate the Connes character formula.

7.1.1 Spectral triples and the Connes–Chern character The notion of a spectral triple is central in noncommutative geometry. The terminology is due to Connes, but the original concept can be traced back further and originates with the study of the algebraic underpinnings of index theory. Let H be a Hilbert space, and let D : dom(D) → H be a self-adjoint operator with dom(D) ⊂ H a dense linear subspace. The operator D admits a polar decomposition D = F|D|, where the phase F is a self-adjoint partial isometry defined using the Borel functional calculus and indicator functions by F := χ(0,∞) (D) − χ(−∞,0) (D), and |D| : dom(D) → H is a positive self-adjoint operator. Definition 7.1.1. A spectral triple (𝒜, H, D) consists of a Hilbert space H, a self-adjoint operator D : dom(D) → H, and a ∗-subalgebra 𝒜 of ℒ(H) such that: https://doi.org/10.1515/9783110700176-007

392 � 7 Connes character formula (a) a : dom(D) → dom(D) for all a ∈ 𝒜, (b) [D, a] : dom(D) → H extends to an operator 𝜕(a) ∈ ℒ(H) for all a ∈ 𝒜, and (c) (D + i)−1 is a compact operator. We will always assume that 𝒜 contains the identity operator of H. In what follows, if a : dom(D) → dom(D), then the (a priori unbounded) operator [|D|, a] : dom(D) → H is denoted by δ(a). Definition 7.1.2. A spectral triple is smooth if (a) a : dom(Dn ) → dom(Dn ) for all n ≥ 0 and a ∈ 𝒜, and (b) δn (a) : dom(Dn ) → H and δn (𝜕(a)) : dom(Dn+1 ) → H extend to bounded operators for all n ≥ 0 and a ∈ 𝒜. We will denote by δ(a), δn (𝜕(a)), n ∈ ℤ+ , and so on, the unique bounded extensions of these operators in ℒ(H). Definition 7.1.3. A spectral triple of parity ∙ ∈ {0, 1} is a 4-tuple (A, H, D, Γ), where Γ ∈ ℒ(H) is such that Γ = Γ∗ , Γ2 = 1, [Γ, a] = 0 for all a ∈ 𝒜, and ΓD + (−1)∙ DΓ = 0. If ∙ = 1, then we take Γ = 1 without loss of generality. Definition 7.1.4. For p > 0, a spectral triple (𝒜, H, D) is said to be ℒp,∞ -summable, or p-dimensional, if (D + i)−1 ∈ ℒp,∞ . The terminology “p-dimensional” for an ℒp,∞ -summable spectral triple (𝒜, H, D) should not lead to confusion with the dimension of the algebra 𝒜 or the Hilbert space H, which are typically both infinite dimensional. The following statement is proved in many places; see, for example, [52] and [224]. We prove a related statement in Lemma 7.4.3. Proposition 7.1.5. If (𝒜, H, D) is a smooth p-dimensional spectral triple with ker(D) = 0, then [F, a] and [F, δk (a)] lie in ℒp,∞ for all a ∈ 𝒜 and k ≥ 1. If (𝒜, H, D) is a smooth p-dimensional spectral triple with ker(D) = 0, then define the multilinear map ch : 𝒜⊗(p+1) → ℒ(H) by setting p

ch(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := ΓF ∏[F, ak ]. k=0

Since the spectral triple (𝒜, H, D) is p-dimensional, it follows from Proposition 7.1.5 and the Hölder inequality that ch(c) ∈ ℒ for all c ∈ 𝒜⊗(p+1) .

p ,∞ p+1

⊂ ℒ1

7.1 Introduction

� 393

Definition 7.1.6. If (𝒜, H, D) is a smooth p-dimensional spectral triple with ker(D) = 0, then the Connes–Chern character 𝒜⊗(p+1) → ℂ is defined as Ch(c) :=

1 Tr(ch(c)), 2

c ∈ 𝒜⊗(p+1) .

In fact, the Chern character is traditionally understood as the class of Ch in periodic cyclic cohomology, but we will further ignore this distinction. If (𝒜, H, D) is a spectral triple with ker(D) ≠ 0, then the definition of the Chern character in (7.1.6) is unsuitable. Instead, the definition should be modified by passing to another spectral triple (π(𝒜), H0 , D0 ), where H0 := H ⊕ H,

D D0 := ( 0

0 ), −D

π(a) := (

a 0

0 ). 0

For a tensor c ∈ 𝒜⊗(p+1) , we denote by π(c) the corresponding element of (π(𝒜))⊗(p+1) obtained by applying the map π ⊗(p+1) to c. If the spectral triple (𝒜, H, D) is equipped with grading Γ, then the spectral triple (π(𝒜), H0 , D0 ) is equipped with grading Γ0 := (

Γ 0

0 ). Γ

Define a self-adjoint unitary operator F0 on the Hilbert space H0 as F0 := (

F P

P ), −F

where P is the orthogonal projection onto the kernel of D. Define the multilinear mapping ch0 : 𝒜⊗(p+1) → ℒ(ℂ2 ⊗ H) by p

ch0 (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Γ0 F0 ∏[F0 , π(ak )]. k=0

Definition 7.1.7. If (𝒜, H, D) is a smooth p-dimensional spectral triple, then the Connes– Chern character 𝒜⊗(p+1) → ℂ is defined by setting 1 Ch(c) := (TrM2 (ℂ) ⊗ Tr)(ch0 (c)), 2

c ∈ 𝒜⊗(p+1) .

It is easy to see that the Definitions 7.1.6 and 7.1.7 give the same functional when ker(D) = 0.

394 � 7 Connes character formula 7.1.2 Hochschild homology and cohomology We now turn to Hochschild (co)homology. Hochschild homology and cohomology theories can be associated with any associative algebra, but we will only consider the algebra 𝒜 of a spectral triple. Definition 7.1.8. For n ≥ 1, the space C n (𝒜) of Hochschild chains is the algebraic tensor product ⊗n C n (𝒜) := ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝒜 ⊗ ⋅⋅⋅ ⊗ 𝒜 = 𝒜 . n times

For every n, the Hochschild boundary b : C n+1 (𝒜) → C n (𝒜) is defined by setting b(a0 ⊗ ⋅ ⋅ ⋅ ⊗ an ) := (a0 a1 ) ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ an n−1

+ ∑ (−1)k a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ak−1 ⊗ (ak ak+1 ) ⊗ ak+2 ⊗ ⋅ ⋅ ⋅ ⊗ an k=1

+ (−1)n (an a0 ) ⊗ a1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ an−1 , where a0 ⊗ ⋅ ⋅ ⋅ ⊗ an ∈ C n+1 (𝒜). Clearly, b2 = 0, that is, the composition b ∘ b : C n+2 (𝒜) → C n (𝒜) is zero. Thus we have the chain complex b

b

b

b

⋅⋅⋅ → 󳨀 C n+1 (𝒜) → 󳨀 C n (𝒜 ) → 󳨀 ⋅⋅⋅ → 󳨀 𝒜. This is called the Hochschild complex of 𝒜. Denoting by Cn (𝒜) the linear dual of C n (𝒜), we have the corresponding cochain complex b

b

b

b

⋅⋅⋅ ← 󳨀 Cn+1 (𝒜) ← 󳨀 C n (𝒜 ) ← 󳨀 ⋅⋅⋅ ← 󳨀 𝒜∗ , where b denotes the dual map, called the Hochschild coboundary. If θ : C n (𝒜) → ℂ is a multilinear functional, then the multilinear functional bθ : C n+1 (𝒜) → ℂ is defined by setting (bθ)(c) := θ(bc), Explicitly, (bθ)(a0 ⊗ ⋅ ⋅ ⋅ ⊗ an ) = θ(a0 a1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ an )

c ∈ C n+1 (𝒜).

7.1 Introduction

� 395

n−1

+ ∑ (−1)k θ(a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ak−1 ⊗ ak ak+1 ⊗ ak+2 ⊗ ⋅ ⋅ ⋅ ⊗ an ) k=1

+ (−1)n θ(an a0 ⊗ a1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ an−1 ),

a0 ⊗ ⋅ ⋅ ⋅ ⊗ an ∈ C n+1 (𝒜).

The corresponding homology and cohomology are termed Hochchild homology and Hochschild cohomology, respectively. Definition 7.1.9. For n ≥ 0, the Hochschild homology HH n (𝒜) of 𝒜 is the quotient space HH n (𝒜) :=

ker(b|C n (𝒜) ) . im(b|C n+1 (𝒜) )

A Hochschild chain c ∈ C n (𝒜) is a Hochschild cycle if bc = 0, whereas a Hochschild boundary is an element c ∈ C n (𝒜) such that there exists c′ ∈ C n+1 (𝒜) such that c = bc′ . Similarly, for n ≥ 1, the Hochschild cohomology is the quotient space HHn (𝒜) :=

ker(b|Cn (𝒜) )

im(b|Cn−1 (𝒜) )

.

A Hochschild cochain θ ∈ Cn (𝒜) is called a Hochschild cocycle if bθ = 0. A Hochschild cochain θ ∈ Cn (𝒜) is a Hochschild coboundary if there exists θ′ ∈ Cn+1 (𝒜) such that θ = bθ′ . Since (bθ)(c) = θ(bc), a Hochschild cocyle θ ∈ Cn (𝒜) vanishes on every Hochschild boundary bc for c ∈ C n+1 (𝒜). For example, if n = 1, then b(a0 ⊗ a1 ) = [a0 , a1 ]. Hence an elementary tensor a0 ⊗ a1 ∈ C 2 (𝒜) is a Hochschild cycle if and only if a0 and a1 commute. A linear functional θ ∈ C1 (𝒜) = 𝒜∗ is a Hochschild cocycle if and only if θ(a0 a1 ) = θ(a1 a0 ) for every a0 , a1 ∈ 𝒜, that is, if and only if θ is a trace on 𝒜. With this terminology, HH n (𝒜) is the linear space of all Hochschild cycles in C n (𝒜) modulo Hochschild boundaries in C n (𝒜). Similarly, HHn (𝒜) is the linear space of all Hochschild cocycles in Cn (𝒜) modulo Hochschild coboundaries in Cn (𝒜). A fundamental feature of the Connes–Chern character is the following: Proposition 7.1.10. The Connes–Chern character is a Hochschild cocycle. Far stronger statements about the Connes–Chern character are true. In fact, it represents a cocycle in cyclic cohomology. However, for our purposes, it suffices to know only that Ch is a Hochschild cocycle.

7.1.3 Statement of the Connes character formula We now have all the terminology needed to state the Connes character formula. Let p ∈ ℕ, and let (𝒜, H, D) be a smooth p-dimensional spectral triple with grading Γ. The

396 � 7 Connes character formula notion of “smoothness” that we introduce in Definition 7.1.2 is sometimes referred to as QC ∞ . Note that in principle the parity of the dimension p has no relation to the parity of the spectral triple (i. e., p can be an odd integer, whereas (𝒜, H, D) has a nontrivial grading Γ, and similarly p can be even, whereas (𝒜, H, D) has no grading). Nevertheless, as we will see in Lemma 7.5.1, the character formula reduces to a trivial statement when the parities of p and of (𝒜, H, D) do not match. Define the multilinear map Ω : 𝒜⊗(p+1) → ℒ(H) by setting p

Ω(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Γa0 ∏[D, ak ]. k=1

Now we state the main result of this chapter. Theorem 7.1.11 (Connes character formula). Let p ∈ ℕ, and let (𝒜, H, D) be a smooth p-dimensional spectral triple. If c ∈ 𝒜⊗(p+1) is a Hochschild cycle, then φ(Ω(c)(1 + D2 )

p

−2

) = Ch(c)

(7.1)

for every normalized trace φ on ℒ1,∞ . The purpose of the Connes character formula is to compute the Hochschild class of the Chern character by a “local” formula involving the derivations 𝜕 in the differential form Ω(c). A consequence of Theorem 7.1.11 stated for arbitrary normalized traces on ℒ1,∞ is that we can deduce the following statement on the behavior of the distribution of eigenp values of the operator Ω(c)(1 + D2 )− 2 : Corollary 7.1.12. Let (𝒜, H, D) be a smooth p-dimensional spectral triple, and let c ∈ p 𝒜⊗(p+1) be a Hochschild cycle. Then the sequence {λ(k, Ω(c)(1 + D2 )− 2 )}∞ k=0 of eigenvalues p 2 −2 of the operator Ω(c)(1 + D ) arranged in decreasing absolute value satisfies n

∑ λ(k, Ω(c)(1 + D2 )

p

−2

k=0

) = Ch(c) ⋅ log(n + 1) + O(1),

n ≥ 0.

The corollary is an immediate consequence of Theorem 7.1.11 and Theorem 9.1.2 in Volume I. This statement is particularly remarkable in that there is no assumption at the p outset that (1 + D2 )− 2 is uniquely traceable, that is, in general, it may not be the case that n

∑ λ(k, (1 + D2 )

k=0

p

−2

) = C ⋅ log(n + 1) + O(1),

n ≥ 0,

for some C ∈ ℝ. The unique traceability in Corollary 7.1.12 relies on the interplay of the algebraic properties of Hochschild homology and the analytic properties of the spectral triple.

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7.2 Exploiting Hochschild cohomology Our aim in this subsection is to prove Theorem 7.2.1 by refining the approach of [52, Section 3.5]. The following assumptions are made everywhere in this section. (a) (𝒜, H, D) is a smooth ℒp,∞ -summable spectral triple as in Definition 7.1.1; (b) ker(D) = 0 so that D is invertible; (c) the spectral triple and the integer p are either both odd or both even. To lighten the statements in this section, we will maintain these assumptions throughout this section without restating them. It follows from Definition 7.1.4 that (1 + D2 )

p

−2

− |D|−p ∈ ℒ1

when ker(D) = 0. Then the multilinear functional p

c 󳨃→ φ(Ω(c)(1 + D2 )

−2

)

in Theorem 7.1 becomes equivalent to c 󳨃→ φ(Ω(c)|D|−p ) for any Hochschild cycle c, since a trace φ on ℒ1,∞ vanishes on trace class operators. The title of this section refers to the fact that we will examine the spectral properties of the operator Ω(c)|D|−p , making essential use of the assumption that c is a Hochschild cycle. The first step in the proof of the Connes character formula in Theorem 7.1 is showing that the operator Ω(c)|D|−p can be replaced by some auxiliary multilinear mappings. For A ⊆ {1, . . . , p}, define the multilinear mapping 𝒲A : 𝒜⊗(p+1) → ℒ(H) by setting p

𝒲A (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Γa0 ∏[bk , ak ], k=1

a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ∈ 𝒜⊗(p+1) ,

where bk = |D| for k ∈ A and bk = F for k ∉ A . If A = {m}, then we denote 𝒲A by 𝒲m . Similarly, we define p

𝒱A (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Γa0 ∏[bk , ak ], k=2

398 � 7 Connes character formula where bk is either |D| or F depending on whether k ∈ A , as in the definition of 𝒲A . The definition of 𝒱A is the same as that of 𝒲A except that the product does not contain the k = 1 factor. It follows from Proposition 7.1.5 and the Hölder property that 𝒲A (a)D

−|A |

∈ ℒ1,∞ ,

A ⊆ {1, . . . , p},

where |A | is the cardinality of A . Recall that [ℒ1,∞ , ℒ(H)] denotes the linear span of commutators [A, B], A ∈ ℒ1,∞ , B ∈ ℒ(H). A linear functional φ : ℒ1,∞ → ℂ is a trace if and only if it vanishes on the linear space [ℒ1,∞ , ℒ(H)]. Theorem 7.2.1. We have Ω(c)|D|−p − p𝒲p (c)D−1 ∈ [ℒ1,∞ , ℒ(H)] for every Hochschild cycle c ∈ 𝒜⊗(p+1) . We prove Theorem 7.2.1 using a trace class approximation and then exploiting the fact that c is a Hochschild cycle. Approximation up to trace class operators The following proposition indicates how the mappings 𝒲A , A ⊆ {1, . . . , p}, substitute for the operator Ω(c). For every A ⊆ {1, . . . , p}, define the number 󵄨 󵄨 nA = 󵄨󵄨󵄨{(i, j) : i < j, i ∈ A , j ∉ A }󵄨󵄨󵄨. Proposition 7.2.2. For every c ∈ 𝒜⊗(p+1) , we have Ω(c)|D|−p −

∑

A ⊆{1,...,p}

(−1)nA 𝒲A (c)D−|A | ∈ ℒ1 .

The following lemmas are used in the proof of Proposition 7.2.2. Lemma 7.2.3. Let ℬ denote the ∗-subalgebra of ℒ(H) generated by the operators {a, δ(a), 𝜕(a)}a∈𝒜 . For every x ∈ ℬ, we have |D|−1 x − x|D|−1 ∈ ℒ p ,∞ . 2

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Proof. The assumption that (𝒜, H, D) is smooth implies that δ(x) ∈ ℒ(H) for every x ∈ ℬ. We have |D|−1 x − x|D|−1 = [|D|−1 , x] = −|D|−1 δ(x)|D|−1 . Since |D|−1 ∈ ℒp,∞ , which follows from Definition 7.1.4 and the equality (D + i)−1 − D−1 = −i(D + i)−1 D−1 , the result follows from Hölder’s inequality. Lemma 7.2.4. For all a ∈ 𝒜, we have |D|−1 [F, a] − [F, a]|D|−1 ∈ ℒ p ,∞ . 3

Proof. By Proposition 7.1.5, [F, δ(a)] ∈ ℒp,∞ . We have [|D|−1 , [F, a]] = −|D|−1 [F, δ(a)]|D|−1 . Since |D|−1 ∈ ℒp,∞ , the latter summand belongs to ℒ p ,∞ by Hölder’s inequality. 3

Lemma 7.2.5. If a ∈ 𝒜, then [D, a]|D|−1 − (δ(a)D−1 + [F, a]) ∈ ℒ p ,∞ . 2

Proof. By the Leibniz rule we have [D, a] = [F|D|, a] = Fδ(a) + [F, a]|D| = [F, δ(a)] + δ(a)F + [F, a]|D|. Thus [D, a]|D|−1 = [F, δ(a)]|D|−1 + δ(a)D−1 + [F, a]. By Proposition 7.1.5 and the Hölder inequality we have [F, δ(a)]|D|−1 ∈ ℒp,∞ ⋅ ℒp,∞ ⊆ ℒ p ,∞ . 2

This completes the proof. Proof of Proposition 7.2.2. By linearity it suffices to prove the statement for an elementary tensor c = a0 ⊗⋅ ⋅ ⋅⊗ap . Noting that Γa0 is a common factor for Ω(c) and each 𝒲A (c), we may assume without loss of generality that Γ = 1 and a0 = 1. We will proceed by proving that for 1 ≤ q ≤ p and c = 1 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ aq , [D, a1 ] ⋅ ⋅ ⋅ [D, aq ]|D|−q −

∑

A ⊆{1,...,q}

(−1)nA 𝒲A (c)D−|A | ∈ ℒ

p ,∞ q+1

.

(7.2)

400 � 7 Connes character formula Here we slightly abuse notation by dropping the factors q < k ≤ p in the definition of 𝒲A . The base of induction (i. e., q = 1) follows from Lemma 7.2.5. Suppose then that we have proved the claim for some q < p. By Lemma 7.2.3 it follows that q+1

q+1

k=2

k=2

(∏[D, ak ])|D|−1 = |D|−1 (∏[D, ak ]) mod ℒ p ,∞ . 2

Therefore q+1

q+1

k=1

k=2

(∏[D, ak ])|D|−q−1 = [D, a1 ]((∏[D, ak ])|D|−1 )|D|−q q+1

= [D, a1 ](|D|−1 (∏[D, ak ] mod ℒ p ,∞ ))|D|−q 2

k=2 q+1

= [D, a1 ]|D|−1 ((∏[D, ak ])|D|−q ) mod ℒ k=2

p ,∞ q+2

By induction we have q+1

(∏[D, ak ])|D|−q = k=2

∑

A ⊂{2,...,q+1}

̂ −|A | mod ℒ (−1)nA 𝒱A (c)D

p ,∞ q+1

where ĉ = 1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ aq+1 . Thus q+1

(∏[D, ak ])|D|−q−1 k=1

= [D, a1 ]|D|−1 (

∑

A ⊂{2,...,q+1}

̂ −|A | ) mod ℒ (−1)nA 𝒱A (c)D

p ,∞ q+2

By Lemma 7.2.5 we have q+1

(∏[D, ak ])|D|−q−1 k=1

= δ(a1 )D−1 ( + [F, a1 ](

∑

̂ −|A | ) (−1)nA 𝒱A (c)D

∑

̂ −|A | ) mod ℒ (−1)nA 𝒱A (c)D

A ⊂{2,...,q+1}

A ⊂{2,...,q+1}

By repeated application of Lemmas 7.2.3 and 7.2.4 it follows that −1 ̂ |D|−1 𝒱A (c)̂ = 𝒱A (c)|D| mod ℒ

p ,∞ q+2−|A |

.

p ,∞ q+2

.

.

,

.

7.2 Exploiting Hochschild cohomology

� 401

Since [F, δ(a)] ∈ ℒp,∞ for all a ∈ 𝒜, it follows that ̂ mod ℒ F 𝒱A (c)̂ = (−1)q−|A | 𝒱A (c)F

p ,∞ q+1−|A |

.

Indeed, we have F[F, a] = −[F, a]F for every a ∈ 𝒜, and there are exactly q − |A | commutators [F, aj ] in 𝒲A . Therefore q+1

(∏[D, ak ])|D|−q−1 k=1

=

∑

A ⊂{2,...,q+1}

+

̂ −|A |−1 (−1)nA (−1)q−|A | δ(a1 )𝒱A (c)D

∑

A ⊂{2,...,q+1}

̂ −|A | mod ℒ (−1)nA [F, a1 ]𝒱A (c)D

p ,∞ q+2

.

Clearly, δ(a1 )𝒱A (c)̂ = 𝒲A ∪{1} (c),

[F, a1 ]𝒱A (c)̂ = 𝒲A (c),

where c = 1 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ aq+1 . Note also that (−1)nA (−1)q−|A | = (−1)nA ∪{1} . Thus q+1

(∏[D, ak ])|D|−q−1 k=1

=

∑

A ⊂{2,...,q+1}

+

∑

(−1)nA ∪{1} 𝒲A ∪{1} (c)D−|A ∪{1}|

A ⊂{2,...,q+1}

(−1)nA 𝒲A (c)D−|A | mod ℒ

p ,∞ q+2

.

Since every B ⊂ {1, . . . , q + 1} is of the form B=A

or B = A ∪ {1},

A ⊂ {2, . . . , q + 1},

it follows that q+1

(∏[D, ak ])|D|−q−1 = k=1

∑

B⊂{1,...,q+1}

(−1)nB 𝒲B (c)D−|B| mod ℒ

p ,∞ q+2

.

We have therefore established (7.2) for q + 1, thus proving the step of induction. This concludes the argument.

402 � 7 Connes character formula Hochschild coboundaries In this section, we describe the functional c 󳨃→ φ(𝒲A (c)D−|A | ),

c ∈ 𝒜⊗(p+1) ,

when A ⊆ {1, . . . , p} is not a singleton as a Hochschild coboundary. Applying the functional to Hochschild cycles allows us to improve the approximation in Proposition 7.2.2 on the commutator subspace and describe it in terms of a singleton. Lemma 7.2.6. Let A ⊂ {1, . . . , p} be such that m − 1, m ∈ A . Let φ be a trace on ℒ1,∞ . The mapping 𝒜⊗(p+1) → ℂ defined by c 󳨃→ φ(𝒲A (c)D−|A | ),

c ∈ 𝒜⊗(p+1) ,

is equal to bθ, where b is the Hochschild coboundary map, and θ : 𝒜⊗p → ℂ is the multilinear mapping θ : a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 󳨃→

p−1

m−2 (−1)m−1 φ(Γa0 ∏ [bk , ak ]δ2 (am−1 ) ∏ [bk+1 , ak ]D−|A | ), 2 k=1 k=m

where bk = |D| for k ∈ A and bk = F for k ∉ A . Proof. For brevity, we prove the statement for p = 2 as the proof in the general case is very similar. In this case, we automatically have m = 2 and A = {1, 2}. We have (bθ)(a0 ⊗ a1 ⊗ a2 ) = θ(a0 a1 ⊗ a2 ) − θ(a0 ⊗ a1 a2 ) + θ(a2 a0 ⊗ a1 )

1 1 1 = − φ(Γa0 a1 δ2 (a2 )D−2 ) + φ(Γa0 δ2 (a1 a2 )D−2 ) − φ(Γa2 a0 δ2 (a1 )D−2 ). 2 2 2

Since Γ commutes with a2 and since φ is a trace, it follows that φ(Γa2 a0 δ2 (a1 )D−2 ) = φ(Γa0 δ2 (a1 )D−2 a2 )

= φ(Γa0 δ2 (a1 )a2 D−2 ) + φ(Γa0 δ2 (a1 )[D−2 , a2 ]).

Since |D|2 = D2 , the Leibniz rule implies that [D−2 , a2 ] = [|D|−2 , a2 ] = −|D|−1 δ(a2 )|D|−2 − |D|−2 δ(a2 )|D|−1 ∈ ℒ 2 ,∞ ⊂ ℒ1 . 3

Therefore φ(Γa2 a0 δ2 (a1 )|D|−2 ) = φ(Γa0 δ2 (a1 )a2 |D|−2 ). Finally, we have 1 (bθ)(a0 ⊗ a1 ⊗ a2 ) = φ(Γa0 (δ2 (a1 a2 ) − a1 δ2 (a2 ) − δ2 (a1 )a2 )|D|−2 ). 2

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Since δ2 (a1 a2 ) − a1 δ2 (a2 ) − δ2 (a1 )a2 = 2δ(a1 )δ(a2 ), the statement follows. Lemma 7.2.7. Let c ∈ 𝒜⊗(p+1) be a Hochschild cycle. Suppose that |A | ≥ 2 and m − 1, m ∈ A for some m. We have 𝒲A (c)D

−|A |

∈ [ℒ1,∞ , ℒ(H)],

where [ℒ1,∞ , ℒ(H)] denotes the linear span of commutators in ℒ1,∞ . Proof. Let φ be a trace on ℒ1,∞ . The mapping on 𝒜⊗(p+1) given by c 󳨃→ φ(𝒲A (c)D−|A | ) is a Hochschild coboundary by Lemma 7.2.6. Recall that a Hochschild coboundary vanishes on every Hochschild cycle. Therefore, if c ∈ 𝒜⊗(p+1) is a Hochschild cycle, then φ(𝒲A (c)D−|A | ) = 0. Since φ is an arbitrary trace, the statement follows. Lemma 7.2.8. Let A1 , A2 ⊂ {1, . . . , p} be such that |A1 | = |A2 | ≥ 2 and the symmetric difference A1 ΔA2 = {m − 1, m}. Let φ be a trace on ℒ1,∞ . The mapping on 𝒜⊗(p+1) defined by c 󳨃→ φ(𝒲A1 (c)D−|A1 | ) + φ(𝒲A2 (c)D−|A2 | ) is equal to bθ, where b is the Hochschild coboundary map, and θ : 𝒜⊗p → ℂ is the multilinear mapping m−2

p−1

k=1

k=m

θ : a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 󳨃→ (−1)m−1 φ(Γa0 ∏ [bk , ak ][F, δ(am−1 )] ∏ [bk+1 , ak ]D−|A1 | ), where bk = |D| for k ∈ A and bk = F for k ∉ A . Proof. For brevity, we prove the statement for p = 2 as the proof in the general case is a slight extension of this argument. In this case, we automatically have m = 2 and A = {1, 2}. We have (bθ)(a0 ⊗ a1 ⊗ a2 ) = θ(a0 a1 ⊗ a2 ) − θ(a0 ⊗ a1 a2 ) + θ(a2 a0 ⊗ a1 ) = − φ(Γa0 a1 [F, δ(a2 )]D−1 ) + φ(Γa0 [F, δ(a1 a2 )]D−1 ) − φ(Γa2 a0 [F, δ(a1 )]D−1 ).

404 � 7 Connes character formula Since Γ commutes with a2 and since φ is a trace, it follows that φ(Γa2 a0 [F, δ(a1 )]D−1 ) = φ(Γa0 [F, δ(a1 )]D−1 a2 ) = φ(Γa0 [F, δ(a1 )]a2 D−1 ) + φ(Γa0 [F, δ(a1 )][D−1 , a2 ]). We have [D−1 , a2 ] = −D−1 𝜕(a2 )D−1 ∈ ℒ1,∞ and Γa0 [F, δ(a1 )][D−1 , a2 ] ∈ ℒ2,∞ ⋅ ℒ1,∞ = ℒ 2 ,∞ ⊂ ℒ1 . 3

Therefore φ(Γa2 a0 [F, δ(a1 )]D−1 ) = φ(Γa0 [F, δ(a1 )]a2 D−1 ). Finally, we have (bθ)(a0 ⊗ a1 ⊗ a2 ) = φ(Γa0 ([F, δ(a1 a2 )] − a1 [F, δ(a2 )] − [F, δ(a1 )]a2 )D−1 ). Since [F, δ(a1 a2 )] − a1 [F, δ(a2 )] − [F, δ(a1 )]a2 = [F, a1 ]δ(a2 ) + δ(a1 )[F, a2 ], the statement follows. Lemma 7.2.9. Let c ∈ 𝒜⊗(p+1) be a Hochschild cycle. Suppose that |A1 | = |A2 | ≥ 2 and symmetric difference A1 ΔA2 = {m − 1, m} for some m. We have 𝒲A1 (c)D

−|A1 |

+ 𝒲A2 (c)D−|A2 | ∈ [ℒ1,∞ , ℒ(H)],

where [ℒ1,∞ , ℒ(H)] denotes the linear span of commutators in ℒ1,∞ . Proof. Let φ be a trace on ℒ1,∞ . The mapping on 𝒜⊗(p+1) given by c 󳨃→ φ(𝒲A1 (c)D−|A1 | ) + φ(𝒲A2 (c)D−|A2 | ) is a Hochschild coboundary by Lemma 7.2.8. Therefore, if c ∈ 𝒜⊗(p+1) is a Hochschild cycle, then φ(𝒲A1 (c)D−|A1 | ) + φ(𝒲A2 (c)D−|A2 | ) = 0. Since φ is an arbitrary trace on ℒ1,∞ , the statement follows.

7.2 Exploiting Hochschild cohomology

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Corollary 7.2.10. Let c ∈ 𝒜⊗(p+1) be a Hochschild cycle. If |A | ≥ 2, then 𝒲A (c)D

−|A |

∈ [ℒ1,∞ , ℒ(H)],

where [ℒ1,∞ , ℒ(H)] denotes the linear span of commutators in ℒ1,∞ . Proof. Let n < m be such that n, m ∈ A . Without loss of generality, i + n ∉ A for all 0 < i < m − n. Set Ai := (A \{n}) ∪ {i + n},

0 ≤ i < m − n.

We have (a) |Ai | = |A | and |Ai ΔAi−1 | = 2 for all 1 ≤ i < m − n; (b) A0 = A , and m − 1, m ∈ Am−n−1 . It follows from Lemma 7.2.9 that 𝒲Am−n−1 (a)D−1 ∈ [ℒ1,∞ , ℒ(H)]. The statement follows by applying Lemma 7.2.7 m − n − 1 times. Lemma 7.2.11. For every c ∈ 𝒜⊗(p+1) , we have 𝒲⌀ (c) ∈ [ℒ1,∞ , ℒ(H)].

Proof. Let a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ∈ 𝒜⊗(p+1) . We have p

p

p

k=1

k=1

k=0

2Γa0 ∏[F, ak ] = [F, FΓa0 ∏[F, ak ]] + (−1)p−1 FΓ ∏[F, ak ],

(7.3)

2𝒲⌀ (c) = [F, F 𝒲⌀ (c)] + (−1)p−1 ch(c).

(7.4)

so that

Since 𝒲⌀ (c) ∈ ℒ1,∞ , it follows that [F, F 𝒲⌀ (c)] ∈ [ℒ1,∞ , ℒ(H)]. By Proposition 7.1.5 and the Hölder property we have ch(c) ∈ ℒ1 ⊂ [ℒ1,∞ , ℒ(H)]. Thus 𝒲⌀ (c) ∈ [ℒ1,∞ , ℒ(H)]. We are now ready to prove Theorem 7.2.1. Proof of Theorem 7.2.1. For every Hochschild cycle c ∈ 𝒜⊗(p+1) , it follows from Proposition 7.2.2 that Ω(c)|D|−p ∈

∑

A ⊆{1,...,p}

(−1)nA 𝒲A (c)D−|A | + ℒ1 .

Applying Corollary 7.2.10 to every summand in the sum ∑|A |≥2 and Lemma 7.2.11, we infer that Ω(c)|D|−p ∈ ∑ (−1)nA 𝒲A (c)D−1 + [ℒ1,∞ , ℒ(H)]. |A |=1

406 � 7 Connes character formula If A = {m}, then nA = p − m. Therefore p

Ω(c)|D|−p ∈ ∑ (−1)p−m 𝒲m (c)D−1 + [ℒ1,∞ , ℒ(H)]. m=1

Applying Lemma 7.2.9 p − m times, we obtain 𝒲m (c)D

−1

− (−1)p−m 𝒲p (c)D−1 ∈ [ℒ1,∞ , ℒ(H)],

1 ≤ m < p.

This suffices to conclude the argument.

7.3 Commutator estimates In this section, we prove some estimates related to the pseudodifferential calculus associated with a spectral triple. They will allow us to connect the auxiliary multilinear mappings of the last section to the Chern character. Following Section 1.5, the Fourier transform is defined by d

(ℱ f )(t) := (2π)− 2 ∫ e−i⟨t,ξ⟩ f (ξ) dξ,

f ∈ 𝒮 (ℝd ),

(7.5)

ℝd

and can be extended to tempered distributions. Lemma 7.3.1. Let f ∈ C 2 (ℝ) be such that ℱ (f ), ℱ (f ′ ), ℱ (f ′′ ) ∈ L1 (ℝ). If (𝒜, H, D) is a smooth spectral triple, then 󵄩󵄩 󵄩 󵄩 2 󵄩 ′ 2󵄩 ′′ 󵄩 󵄩󵄩[f (s|D|), a] − sf (s|D|)δ(a)󵄩󵄩󵄩ℒ(H) ≤ s 󵄩󵄩󵄩ℱ (f )󵄩󵄩󵄩L1 (ℝ) 󵄩󵄩󵄩δ (a)󵄩󵄩󵄩ℒ(H) and 󵄩󵄩 󵄩 󵄩 2 󵄩 ′ 2󵄩 ′′ 󵄩 󵄩󵄩[f (s|D|), a] − sδ(a)f (s|D|)󵄩󵄩󵄩ℒ(H) ≤ s 󵄩󵄩󵄩ℱ (f )󵄩󵄩󵄩L1 (ℝ) 󵄩󵄩󵄩δ (a)󵄩󵄩󵄩ℒ(H) for all s > 0 and l a ∈ 𝒜. Proof. We prove only the first inequality as the proof of the second inequality is similar. Following the proof of Lemma 1.4.12, the assumption on f ensures that f [1] ∈ BS, because 1 ∞

f (t) − f (s) 1 = ∫ ∫ (ℱ (f ′ ))(ξ)eiξ(1−θ)t eiξθs dξdθ, 1 t−s (2π) 2 0 −∞ By Theorem 1.4.11 we have [f (s|D|), a] = Tfs|D|,s|D| (sδ(a)), [1]

t ≠ s ∈ ℝ.

7.3 Commutator estimates

� 407

and Lemma 1.4.14 delivers [f (s|D|), a] = T |D|,|D| (δ(a)), [1] fs

where fs (t) := f (st). By (1.13) we have sf ′ (s|D|)δ(a) = Tϕ|D|,|D| (δ(a)), s

where ϕs (t, u) = sf ′ (st) = fs′ (t). We have from Theorem 1.4.8 that [f (s|D|), a] − sf ′ (s|D|)δ(a) = T |D|,|D| (δ(a)). [1] fs −ϕs

(7.6)

For s ∈ ℝ and t ≠ u ∈ ℝ, fs[1] (t, u) − ϕs (t, u) =

f (st) − f (su) f (st) − f (su) − s(t − u)f ′ (st) − sf ′ (st) = . t−u t−u

Therefore the symbol of the double operator integral on the right-hand side of (7.6) is given by the integral fs[1] (t, u)

2

1

− ϕs (t, u) = (t − u)s ∫ f ′′ (st(1 − θ) + suθ)(1 − θ) dθ. 0

Since ℱ (f ′′ ) ∈ L1 (ℝ), the Fourier inversion implies that fs[1] (t, u) − ϕs (t, u) = (t − u)s2 Gs (t, u), where Gs (t, u) :=

1

1 ∞

1

(2π) 2

∫ ∫ (1 − θ)ℱ (f ′′ )(ξ)eiξst(1−θ) eiξsuθ dξdθ. 0 −∞

This formula demonstrates that Gs ∈ BS and moreover that 1 󵄩 󵄩 ‖Gs ‖BS ≤ (2π)− 2 󵄩󵄩󵄩ℱ (f ′′ )󵄩󵄩󵄩1 ,

so that sups∈ℝ ‖Gs ‖BS < ∞. It follows from (7.6) that [f (s|D|), a] − sf ′ (s|D|)δ(a) = s2 TG|D|,|D| (δ2 (a)). s

By Theorem 1.4.9 and the BS-norm bound on Gs we have 󵄩󵄩 󵄩 󵄩 ′ − 1 2󵄩 ′′ 󵄩 󵄩 2 󵄩󵄩[f (s|D|), a] − sf (s|D|)δ(a)󵄩󵄩󵄩∞ ≤ (2π) 2 s 󵄩󵄩󵄩ℱ (f )󵄩󵄩󵄩1 󵄩󵄩󵄩δ (a)󵄩󵄩󵄩∞ .

408 � 7 Connes character formula This implies the first inequality, as claimed. The second inequality follows by taking the adjoints. From now on, we fix p+1

fp (s) := e−|s| ,

s ∈ ℝ,

where the integer p > 0 is defined by the summability of the spectral triple. Lemma 7.3.2. Let D be an invertible unbounded self-adjoint operator. If D−p ∈ ℒ1,∞ , then fp (s|D|) ∈ ℒ1 ,

s > 0,

and |D|−p−1 (1 − fp (s|D|)) ∈ ℒ1 ,

s > 0,

with the estimates Tr(fp (s|D|)) = O(s−p ),

s > 0,

and Tr(|D|−p−1 (1 − fp (s|D|))) = O(s),

s > 0.

Proof. Let {αk }k≥1 be the sequence of eigenvalues of D arranged so that {|αk |}k≥1 increases. Let {ξk }k≥1 be the corresponding sequence of eigenvectors of D. By assumption 1

we have |αk | ≥ (cD k) p for all k ≥ 1 and some constant cD > 0. Define the unbounded self-adjoint invertible operator D0 by setting 1

D0 ξk := (cD k) p ξk . To prove the first estimate, note that fp is decreasing on (0, ∞). It is immediate that 1 p

∞

1

Tr(fp (s|D|)) = ∑ fp (s|αk |) ≤ ∑ fp (s(cD k) ) ≤ ∫ fp (s(cD t) p )dt. k≥1

k≥1

0

Making the substitution t = s−p v, we rewrite the latter integral as ∞

1

s−p ⋅ ∫ fp ((cD v p ))dv. 0

To prove the second estimate, note that fp is decreasing on (0, ∞), and so is the function s 󳨃→ s−p−1 (1 − fp (s)). It is immediate that

7.3 Commutator estimates

�

409

Tr(|D|−p−1 (1 − fp (s|D|))) = ∑ |αk |−p−1 (1 − fp (s|αk |)) k≥1

1

≤ ∑ ((cD k) p ) k≥1 ∞

1

−p−1

−p−1

≤ ∫ ((cD t) p )

1

(1 − fp (s(cD k) p )) 1

(1 − fp (s(cD t) p ))dt.

0

Making the substitution t = s−p v, we rewrite the latter integral as ∞

1

s ⋅ ∫ ((cD v p ))

−p−1

1

(1 − fp ((cD v p )))dv.

0

Lemma 7.3.3. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. If a ∈ 𝒜, then [fp′ (s|D|), δ(a)] ∈ ℒ1 ,

s > 0,

and 󵄩󵄩 ′ 󵄩 1−p 󵄩󵄩[fp (s|D|), δ(a)]󵄩󵄩󵄩ℒ1 = O(s ),

s > 0.

Proof. Suppose first that p ≥ 4 or that p = 2. Define a positive function h by setting p

1

p+1

h(s) := |s| 2 e− 2 |s| ,

s ∈ ℝ.

We have h′ , h′′ ∈ L2 (ℝ). It follows now from the proof of Lemma 1.4.12 that ℱ (h′ ) ∈ L1 (ℝ). On the other hand, we have fp′ (s) = (p + 1) sgn(s) ⋅ h2 (s),

s ∈ ℝ.

fp′ (s|D|) = −(p + 1)h2 (s|D|),

s > 0.

In particular, we have

Applying Lemma 7.3.1 with h instead of f , we obtain 󵄩󵄩 󵄩 󵄩 󵄩 ′ 󵄩 󵄩 2 󵄩󵄩[h(s|D|), δ(a)]󵄩󵄩󵄩∞ ≤ s󵄩󵄩󵄩ℱ (h )󵄩󵄩󵄩1 󵄩󵄩󵄩δ (a)󵄩󵄩󵄩∞ . On the other hand, we have 1 [f ′ (s|D|), δ(a)] = [h2 (s|D|), δ(a)] p+1 p

= h(s|D|)[h(s|D|), δ(a)] + [h(s|D|), δ(a)]h(s|D|).

Therefore

410 � 7 Connes character formula 1 󵄩󵄩 ′ 󵄩 󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩[f (s|D|), δ(a)]󵄩󵄩󵄩1 ≤ 2󵄩󵄩󵄩h(s|D|)󵄩󵄩󵄩1 󵄩󵄩󵄩[h(s|D|), δ(a)]󵄩󵄩󵄩∞ = 󵄩󵄩󵄩h(s|D|)󵄩󵄩󵄩1 ⋅ O(s). p + 1󵄩 p Recall that h(s) ≤ cp fp ( 2s ) for all s ∈ ℝ. Since D−p ∈ ℒ1,∞ , it follows from Lemma 7.3.2 that 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 −p 󵄩󵄩h(s|D|)󵄩󵄩󵄩1 ≤ cp 󵄩󵄩󵄩fp ( ⋅ |D|)󵄩󵄩󵄩 = O(s ). 󵄩󵄩1 󵄩󵄩 2 This proves the statement for p ≥ 4 or p = 2. If p = 1 or p = 3, rather than to apply the estimate in the proof of Lemma 1.4.12, we instead proceed with a direct computation (which works for arbitrary odd p). Define the Schwartz functions h1 and h2 by setting h1 (s) := sp e−

sp+1 2

,

h2 (s) := e−

sp+1 2

,

s ∈ ℝ.

Obviously, h1 and h2 are Schwartz class functions on ℝ, and fp′ = −2h1 h2 on ℝ+ . Thus [fp′ (s|D|), δ(a)] = 2[h1 (s|D|)h2 (s|D|), δ(a)] = 2h1 (s|D|)[h2 (s|D|), δ(a)] + 2[h1 (s|D|), δ(a)]h2 (s|D|). It follows that 󵄩󵄩 ′ 󵄩 󵄩󵄩[fp (s|D|), δ(a)]󵄩󵄩󵄩1 󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩 󵄩 ≤ 2󵄩󵄩󵄩h1 (s|D|)󵄩󵄩󵄩1 󵄩󵄩󵄩[h2 (s|D|), δ(a)]󵄩󵄩󵄩∞ + 2󵄩󵄩󵄩h2 (s|D|)󵄩󵄩󵄩1 󵄩󵄩󵄩[h1 (s|D|), δ(a)]󵄩󵄩󵄩∞ . Applying Lemma 7.3.1 with h1 and h2 instead of f , we obtain 󵄩󵄩 ′ 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩[fp (s|D|), δ(a)]󵄩󵄩󵄩1 ≤ O(s) ⋅ (󵄩󵄩󵄩h1 (s|D|)󵄩󵄩󵄩1 + 󵄩󵄩󵄩h2 (s|D|)󵄩󵄩󵄩1 ). Recall that h1 (s), h2 (s) ≤ cp fp ( 2s ) for all s > 0. Since D−p ∈ ℒ1,∞ , it follows from Lemma 7.3.2 that 󵄩󵄩 s 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 −p 󵄩󵄩h1 (s|D|)󵄩󵄩󵄩1 , 󵄩󵄩󵄩h2 (s|D|)󵄩󵄩󵄩1 ≤ cp 󵄩󵄩󵄩fp ( |D|)󵄩󵄩󵄩 = O(s ). 󵄩󵄩 2 󵄩󵄩1 This proves the statement for p = 1 and p = 3. Lemma 7.3.4. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. If a ∈ 𝒜, then 󵄩󵄩 󵄩 ′ 2−p 󵄩󵄩[fp (s|D|), a] − sδ(a)fp (s|D|)󵄩󵄩󵄩ℒ = O(s ), 1

Proof. Let 1

p+1

h(s) := e− 2 |s| ,

s ∈ ℝ,

s > 0.

7.3 Commutator estimates

� 411

so that f = h2 . By the Leibniz rule we have s [fp (s|D|, a)] − {fp′ (s|D|), δ(a)} 2 = h(s|D|)([h(s|D|), a] − sh′ (s|D|)δ(a)) + ([h(s|D|), a] − sδ(a)h′ (s|D|))h(s|D|). It follows that 󵄩󵄩 󵄩󵄩 s ′ 󵄩󵄩 󵄩 󵄩󵄩[fp (s|D|, a)] − {fp (s|D|), δ(a)}󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩1 2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ≤ 󵄩󵄩h(s|D|)󵄩󵄩1 (󵄩󵄩[h(s|D|), a] − sh′ (s|D|)δ(a)󵄩󵄩󵄩∞ 󵄩 󵄩 + 󵄩󵄩󵄩[h(s|D|), a] − sδ(a)h′ (s|D|)󵄩󵄩󵄩∞ ). We infer from Lemma 7.3.1 that the expression in brackets is O(s2 ). It follows from Lemma 7.3.2 that ‖h(s|D|)‖1 = O(s−p ). Therefore 󵄩󵄩 󵄩󵄩 s ′ 󵄩󵄩 󵄩 2−p 󵄩󵄩[fp (s|D|, a)] − {fp (s|D|), δ(a)}󵄩󵄩󵄩 = O(s ), 󵄩󵄩 󵄩󵄩1 2

s > 0.

The statement now follows from Lemma 7.3.3. Recall from Example 1.1.6 the ideal of compact operators ℒp,1 := {A ∈ ℒ(H) : ‖A‖ℒp,1 < ∞},

where ∞

1

−1

‖A‖ℒp,1 := ∑ (k + 1) p μ(k, A). k=0

Proposition 7.3.5. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. If a ∈ 𝒜, then 󵄩󵄩 󵄩 ′ 󵄩󵄩[fp (s|D|), a] − sδ(a)fp (s|D|)󵄩󵄩󵄩ℒp,1 = O(s),

s > 0.

Proof. If p = 1, then the statement is proved in Lemma 7.3.4. Suppose p > 1 and set T = [fp (s|D|), a] − sδ(a)fp′ (s|D|). We infer from Lemma 7.3.1 that ‖T‖∞ = O(s2 ) and from Lemma 7.3.4 that ‖T‖1 = O(s2−p ) as s → 0. From the definition of the ℒp,1 norm or from (1.5) we have 1

1− 1

‖T‖p,1 ≤ ‖T‖1p ‖T‖∞ p = O(s

2−p p

⋅s

2⋅(1− p1 )

) = O(s).

412 � 7 Connes character formula

7.4 Asymptotics for the heat semigroup In Section 7.2, we introduced the multilinear mapping c 󳨃→ 𝒲p (c) for a Hochschild cycle c and a smooth p-dimensional spectral triple (𝒜, H, D). Theorem 7.2.1 states that, under the assumptions in Section 7.2, on Hochschild cycles the multilinear functional c 󳨃→ φ(Ω(c)|D|−p ) is equivalent to c 󳨃→ φ(p𝒲p (c)D−1 ) for any trace φ on ℒ1,∞ . In this section, we connect the value φ(p𝒲p (c)D−1 ) to the value of the Connes–Chern character of c using the heat semigroup formulation of uniquely traceable operators from Chapter 9 of Volume I. The central result is the following heat semigroup estimate. Theorem 7.4.1. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. Suppose ker(D) = 0. If the spectral triple and the integer p are both odd or both even, then p+1

Tr(𝒲p (c)D−1 e−(s|D|) ) = Ch(c) ⋅ log(s−1 ) + O(1),

s ∈ (0, 1),

for every Hochschild cycle c ∈ 𝒜⊗(p+1) , where Ch is the Connes–Chern character from Definition 7.1.7. Corollary 7.4.2. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. Suppose ker(D) = 0. If the spectral triple and the integer p are both odd or both even, then φ(Ω(c)|D|−p ) = Ch(c) for every Hochschild cycle c ∈ 𝒜⊗(p+1) and every normalized trace φ on ℒ1,∞ , where Ch is the Connes–Chern character. We further prove Theorem 7.4.1 and Corollary 7.4.2. Lemmas 7.4.3 and 7.4.4 are preliminary results for the proof of Theorem 7.4.1. Lemma 7.4.3. If (𝒜, H, D) is a smooth spectral triple with ker(D) = 0, then m

(∏[F, ak ])|D|m+1 ∈ ℒ(H), k=0

a k ∈ 𝒜,

0 ≤ k ≤ m.

7.4 Asymptotics for the heat semigroup

� 413

Proof. Define the set n

n

n

ℬ := {A ∈ ℒ(H) : A : ⋂ dom(D ) → ⋂ dom(D ), δ (A) ∈ ℒ(H) for all n ≥ 0}. n≥0

n≥0

By the Leibniz rule, ℬ is an algebra. By the definition of smoothness we have 𝒜, 𝜕(𝒜) ⊆ ℬ and every bounded function of D belongs to ℬ. By the Leibniz rule, for every a ∈ 𝒜, we have 𝜕(a) = Fδ(a) + [F, a]|D|, and therefore [F, a]|D| ∈ ℬ. For every A ∈ ℬ, we have δ(A) ∈ ℬ, and therefore for every n ≥ 0, n n |D|−n A|D|n = ∑ ( )(−1)k |D|−k δk (A) ∈ ℬ. k k=0

Hence, for all m ≥ 0 and a ∈ 𝒜, we have |D|−m−1 [F, a]|D|m+2 ∈ ℬ. In particular, |D|−m−1 [F, a]|D|m+2 ∈ ℒ(H). We will prove the lemma by induction on m. For m = 0, we have [F, a0 ]|D| = 𝜕(a0 ) − Fδ(a0 ) ∈ ℒ(H). Assume now that the statement is true for some m ≥ 0. We have m+1

m

k=0

k=0

( ∏ [F, ak ])|D|m+2 = (∏[F, ak ])|D|m+1 |D|−m−1 [F, am ]|D|m+2 . By the inductive hypothesis it follows that m

(∏[F, ak ])|D|m+1 ∈ ℒ(H). j=0

Lemma 7.4.4. Let T ∈ ℒ1 . Define the multilinear mapping θ : 𝒜⊗p → ℂ by the formula p−1

θ(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 ) := Tr(Γa0 (∏[F, ak ])T). k=1

We have p−1

(bθ)(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = (−1)p Tr(Γa0 (∏[F, ak ])[T, ap ]), k=0

414 � 7 Connes character formula where b is the Hochschild coboundary map. Proof. Denote for brevity k−1

p

l=1

l=k+1

Xk := Tr(Γa0 (∏[F, al ]) ⋅ ak ⋅ ( ∏ [F, al ]) ⋅ T),

1 ≤ k ≤ p.

Clearly, θ(a0 a1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = X1 . By the Leibniz rule we have θ(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ak−1 ⊗ ak ak+1 ⊗ ak+2 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = Xk + Xk+1 ,

1 ≤ k < p.

Also, p−1

θ(ap a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 ) = Tr(Γap a0 ⋅ (∏[F, ak ])T). k=1

Since ap commutes with Γ, it follows that p−1

θ(ap a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 ) = Tr(Γa0 (∏[F, ak ])Tap ) k=1 p−1

= Tr(Γa0 (∏[F, ak ])[T, ap ]) + Xp . k=1

It follows that p−1

(bθ)(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = X1 + ∑ (−1)k (Xk + Xk+1 ) k=1

p−1

+ (−1)p Xp + (−1)p Tr(Γa0 (∏[F, ak ])[T, ap ]). k=1

Clearly, p−1

X1 + (−1)p Xp + ∑ (−1)k (Xk + Xk+1 ) = 0. k=1

This completes the proof. Lemma 7.4.5 contains the central identity that connects the multilinear mapping

𝒲p (c) on a Hochschild cycle c to the Connes–Chern character.

7.4 Asymptotics for the heat semigroup

� 415

Lemma 7.4.5. Let (𝒜, H, D) be a smooth p-dimensional spectral triple, and let c ∈ 𝒜⊗(p+1) be a Hochschild cycle. Suppose that the spectral triple and p are both odd or both even. If ker(D) = 0, then s Tr(𝒲p (c)Ffp′ (s|D|)) = − Ch(c) + O(s),

s → 0+ ,

p+1

where fp (s) := e−|s| , s ∈ ℝ, and Ch(c) denotes the Connes–Chern character of the Hochschild cycle c. Proof. Let s > 0 and set f := fp for brevity. Define the multilinear mappings 𝒦s , ℋs : 𝒜⊗(p+1) → ℂ by setting p−1

𝒦s (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Tr(Γa0 (∏[F, ak ])[Ff (s|D|), ap ]), k=1 p−1

ℋs (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Tr(Γa0 (∏[F, ak ])F[f (s|D|), ap ]). k=1

By construction, for all c ∈ 𝒜⊗(p+1) , we have Tr(𝒲⌀ (c)f (s|D|)) = 𝒦s (c) − ℋs (c).

(7.7)

By Lemma 7.4.4, 𝒦s is a Hochschild coboundary. If c ∈ 𝒜⊗(p+1) is a Hochschild cycle, then 𝒦s (c) = 0. The vanishing of the functional 𝒦s (c) is the only place where we use the fact that c is a Hochschild cycle. Thus Tr(𝒲⌀ (c)f (s|D|)) = −ℋs (c).

(7.8)

On the other hand, for c′ = a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap , we have ′

′

′

ℋs (c ) = s Tr(𝒲p (c )Ff (s|D|)) p−1

+ Tr(Γa0 (∏[F, ak ])F([f (s|D|), ap ] − sδ(ap )f ′ (s|D|))) k=1

p−1

+ s Tr(Γa0 (∏[F, ak ])[F, δ(ap )]f ′ (s|D|)). k=1

Using Proposition 7.3.5, we can estimate the asymptotic behavior of the second term: 󵄨󵄨 󵄨󵄨 p−1 󵄨󵄨 󵄨 󵄨󵄨Tr(Γa0 (∏[F, ak ])F([f (s|D|), ap ] − sδ(ap )f ′ (s|D|)))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 k=1 󵄩󵄩 󵄩󵄩 p−1 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 ′ ≤ 󵄩󵄩󵄩Γa0 ∏[F, ak ]F 󵄩󵄩󵄩 󵄩[f (s|D|), ap ] − sδ(ap )f (s|D|)󵄩󵄩󵄩p,1 = O(s), 󵄩󵄩 󵄩󵄩 ∗ 󵄩 k=1 󵄩 󵄩p ,∞

s ∈ (0, 1),

416 � 7 Connes character formula where p∗ is the Hölder conjugate of p. The inequality follows from Hölder’s theorem for Lorentz spaces, Theorem 1.2.6. The asymptotic behavior of the third term is similar: 󵄨󵄨 󵄨󵄨 p−1 󵄨󵄨 󵄨󵄨 s󵄨󵄨󵄨Tr(Γa0 (∏[F, ak ])[F, δ(ap )]f ′ (s|D|))󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 k=1 󵄨 󵄨 󵄩󵄩 󵄩󵄩 p−1 󵄩󵄩 󵄩󵄩 󵄩 ′ 󵄩 ≤ s󵄩󵄩󵄩Γa0 (∏[F, ak ])[F, δ(ap )]󵄩󵄩󵄩 󵄩󵄩󵄩f 󵄩󵄩󵄩∞ = O(s), 󵄩󵄩󵄩 󵄩󵄩󵄩1 k=1

s ∈ (0, 1).

Combining these estimates, we obtain ′

′

′

ℋs (c ) = s Tr(𝒲p (c )Ff (s|D|)) + O(s),

s ∈ (0, 1),

(7.9)

for every c′ ∈ 𝒜⊗(p+1) . Combining equalities (7.8) and (7.9), we infer that Tr(𝒲⌀ (c)f (s|D|)) = −s Tr(𝒲p (c)Ff ′ (s|D|)) + O(s),

s ∈ (0, 1),

(7.10)

s ∈ (0, 1),

(7.11)

for every Hochschild cycle c ∈ 𝒜⊗(p+1) . Note that ch(c) = 𝒲⌀ (c) + F 𝒲⌀ (c)F, where ch(c) is introduced before Definition 7.1.7, so that Ch(c) = Tr(ch(c)). Thus Tr(ch(c)f (s|D|)) = −2s Tr(𝒲p (c)Ff ′ (s|D|)) + O(s), for every Hochschild cycle c ∈ 𝒜⊗(p+1) . Finally, 󵄨󵄨 󵄨 󵄨󵄨Tr(ch(c)f (s|D|)) − Tr(ch(c))󵄨󵄨󵄨 󵄨 󵄨 = 󵄨󵄨󵄨Tr(ch(c)|D|p+1 ⋅ |D|−p−1 (1 − f (s|D|)))󵄨󵄨󵄨 󵄩 󵄩 ≤ 󵄩󵄩󵄩ch(c)|D|p+1 󵄩󵄩󵄩∞ ⋅ Tr(|D|−p−1 (1 − f (s|D|))) = O(s),

s ∈ (0, 1).

The last bound follows from Lemmas 7.4.3 and 7.3.2. Combining this estimate with (7.11), we complete the proof. Proof of Theorem 7.4.1. By Lemma 7.4.5 we have p+1

Tr(𝒲p (c)F|D|p e−(s|D|) ) = Setting u := sp+1 , we obtain

Ch(c) −p−1 ⋅s + O(s−p ), (p + 1)

s ∈ (0, 1).

7.4 Asymptotics for the heat semigroup

p+1

Tr(𝒲p (c)D−1 ⋅ |D|p+1 e−u|D| ) =

p Ch(c) − + O(u p+1 ), (p + 1)u

� 417

u ∈ (0, 1).

Integrating over u ∈ [t, 1], we obtain Tr(𝒲p (c)D−1 (e−t|D|

p+1

p+1

− e−|D| )) =

Ch(c) ⋅ log(t −1 ) + O(1), (p + 1)

Taking into account that D−1 ∈ ℒp,∞ , we get that 𝒲p (c)D−1 e−|D| sp+1 , we conclude the argument.

p+1

t ∈ (0, 1).

∈ ℒ1 . Replacing t with

Proof of Corollary 7.4.2. Set A := 𝒲p (c)D−1 ⋅ |D|p .

V := |D|−p , By construction 𝒲p (c)D

−1

p−1

⋅ |D|p = Γa0 (∏[F, ak ])[|D|, ap ]|D|p−1 F. k=1

Using the Leibniz rule, we get [|D|, ap ]|D|p−1 = |D|[|D|, ap ]|D|p−2 + [|D|, [|D|, a]]|D|p−2 . Repeating p − 2 times and using the fact that the spectral triple is smooth, we get [|D|, ap ]|D|p−1 = |D|p−1 ([|D|, ap ] + B) for a bounded operator B ∈ ℒ(H). Then 𝒲p (c)D

−1

p−1

⋅ |D|p = Γa0 (∏[F, ak ])|D|p−1 ⋅ ([|D|, ap ]F + BF). k=1

The second term in the product is a bounded operator, and the first term in the product is a bounded operator by Lemma 7.4.3. Hence A ∈ ℒ(H), and V ∈ ℒ1,∞ by assumption. The statement of Theorem 7.4.1 now reads as (here, α = p+1 and t = sp ) p −α

Tr(AVe−(tV ) ) =

Ch(c) ⋅ log(t −1 ) + O(1), p

By Theorem 9.1.4 in Volume I we have φ(AV ) =

Ch(c) p

for every normalized trace φ on ℒ1,∞ . In other words,

t ∈ (0, 1).

418 � 7 Connes character formula φ(𝒲p (c)D−1 ) =

Ch(c) p

for every normalized trace φ on ℒ1,∞ . The statement now follows from Theorem 7.2.1.

7.5 Proof of the Connes character formula In this section, we complete the proof of the Connes character formula. The formula was proved in Corollary 7.4.2 under the assumptions that the operator D has trivial kernel and that the parity and the dimension of the spectral triple are either both even or both odd. Triviality when parities are mismatched We start by removing the parity assumptions. The following three lemmas show that if the parity of the dimension p does not match that of (𝒜, H, D), then the statement of (7.1) becomes trivial. Lemma 7.5.1. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. Suppose that ker(D) = 0. Suppose that (a) p is odd, but (𝒜, H, D) is even, or (b) p is even, but (𝒜, H, D) is odd. Then c ∈ 𝒜⊗(p+1) .

Tr(ch(c)) = 0,

(7.12)

Proof. First, consider (a). Note that Γ[F, a] = −[F, a]Γ for every a ∈ 𝒜. Thus since p + 1 is even, p

p

p

k=0

k=0

k=0

ch(c) = ΓF ∏[F, ak ] = −FΓ ∏[F, ak ] = −F ⋅ ∏[F, ak ]Γ. Therefore p

Tr(ch(c)) = − Tr(ΓF ∏[F, ak ]) = − Tr(ch(c)). k=0

This proves (7.12). Now consider (b). For all a ∈ 𝒜, we have F[F, a] = −[F, a]F. Since p + 1 is odd, it follows that p

p

k=0

k=0

F ⋅ ∏[F, ak ] = −(∏[F, ak ])F.

7.5 Proof of the Connes character formula

� 419

Hence p

p

k=0

k=0

Tr(ch(c)) = Tr(F ∏[F, ak ]) = − Tr(F ∏[F, ak ]) = − Tr(ch(c)). This proves (7.12). Lemma 7.5.2. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. Suppose that ker(D) = 0. If p is odd and (𝒜, H, D) is even, then φ(Ω(c)|D|−p ) = 0,

c ∈ 𝒜⊗(p+1) .

Proof. Since ΓD = −DΓ and Γ commutes with a ∈ 𝒜, we have Γ[D, a] = −[D, a]Γ on H∞ . Hence Γ𝜕(a) = −𝜕(a)Γ for all a ∈ 𝒜. Since p is odd, it follows that p

p

k=1

k=1

Γa0 ∏ 𝜕(ak ) = −a0 (∏ 𝜕(ak ))Γ,

a0 , . . . , ap ∈ 𝒜.

Thus Ω(c) = −ΓΩ(c)Γ,

c ∈ 𝒜⊗(p+1) .

However, ΓD2 = D2 Γ, so by the spectral theorem we have Γ|D|−p = |D|−p Γ. Thus Ω(c)|D|−p = −ΓΩ(c)Γ|D|−p = −ΓΩ(c)|D|−p Γ. Applying the trace φ to both sides, we obtain φ(Ω(c)|D|−p ) = −φ(ΓΩ(c)|D|−p Γ) = −φ(Γ2 Ω(c)|D|−p ) = −φ(Ω(c)|D|−p ). This proves the statement. Now we deal with the other case, where the parity of (𝒜, H, D) does not match that of p. Lemma 7.5.3. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. Suppose ker(D) = 0. If (𝒜, H, D) is odd but p is even, then φ(Ω(c)|D|−p ) = 0 for every Hochschild cycle c ∈ 𝒜⊗(p+1) .

420 � 7 Connes character formula Proof. Consider the multilinear mapping θ : 𝒜⊗p → ℂ defined by the formula p−1

θ(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 ) = φ((∏ 𝜕(ak ))|D|−p ). k=0

We claim that (bθ)(c) = 2φ(Ω(c)|D|−p ),

c ∈ 𝒜⊗(p+1) .

To see this, let c = a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap and denote for brevity k−1

p

l=0

l=k+1

Xk := φ((∏ 𝜕(al )) ⋅ ak ⋅ ( ∏ 𝜕(ak )) ⋅ |D|−p ). With k = 0, the initial product is assumed to be 1. By the Leibniz rule we have θ(a0 a1 ⊗ a2 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = X0 + X1 . Similarly, θ(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ak−1 ⊗ ak ak+1 ⊗ ak+2 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = Xk + Xk+1 . Also from the Leibniz rule we have p−1

θ(ap a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ap−1 ) = φ(𝜕(ap a0 ) ⋅ (∏ 𝜕(ak ))|D|−p ) k=1 p−1

p−1

= φ(𝜕(ap )a0 ⋅ (∏ 𝜕(ak ))|D|−p ) + φ(ap (∏ 𝜕(ak ))|D|−p ) p−1

k=1

k=0

p−1

= φ(a0 ⋅ (∏ 𝜕(ak ))|D|−p 𝜕(ap )) + φ((∏ 𝜕(ak ))|D|−p ap ). k=1

k=0

By the definition of X0 , p−1

p−1

φ(a0 ⋅ (∏ 𝜕(ak ))|D|−p 𝜕(ap )) = φ(a0 ⋅ (∏ 𝜕(ak ))[|D|−p , 𝜕(ap )]) + X0 , k=1

p−1

p−1

k=0

k=0

k=1

φ((∏ 𝜕(ak ))|D|−p ap ) = φ((∏ 𝜕(ak ))[|D|−p , ap ]) + Xp . Since [|D|−p , ap ] ∈ ℒ1 , and since φ vanishes on ℒ1 , it follows that

[|D|−p , 𝜕(ap )] ∈ ℒ1 ,

7.5 Proof of the Connes character formula

� 421

θ(ap a0 ⊗ a1 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = X0 + Xp . Thus p−1

(bθ)(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = (X0 + X1 ) − ∑ (−1)k (Xk + Xk+1 ) + (−1)p (X0 + Xp ). k=1

Since p is even, it follows that (bθ)(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) = 2X0 = 2φ(Ω(a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap )|D|−p ). This proves the claim. Since c 󳨃→ φ(Ω(c)|D|−p ) is a Hochschild coboundary, it must vanish on every Hochschild cycle c. This completes the proof. The preceding two lemmas show the statement of Theorem 7.1.11 if the parities of p and (𝒜, H, D) do not match. Nontriviality of the kernel of D To complete the proof of Theorem 7.1.11, we remove the assumption that ker(D) = 0. For this purpose, we use the “doubling trick”, which in this form follows [52, Definition 6]. Let μ > 0. We define another spectral triple (π(𝒜), H0 , Dμ ), where H0 := ℂ2 ⊗ H,

Dμ := (

D μ

μ ), −D

π(a) := (

a 0

0 ). 0

For a tensor c ∈ 𝒜⊗(p+1) , we denote by π(c) for the corresponding element of (π(𝒜))⊗(p+1) obtained by applying the map π ⊗(p+1) to c. If the spectral triple (𝒜, H, D) is equipped with the grading Γ, then the spectral triple (π(𝒜), H0 , Dμ ) is equipped with grading Γ0 := (

Γ 0

0 ). Γ

In accordance with this, we write D

Fμ := sgn(Dμ ) =

1

(D2 +μ2 ) 2 ( μ 1 (D2 +μ2 ) 2

μ

1

(D2 +μ2 ) 2 D

−

1 (D2 +μ2 ) 2

).

As μ → 0, Fμ does not converge to F ⊕ (−F) unless D has trivial kernel. In general, Fμ converges to the operator F0 := (

F χ{0} (D)

χ{0} (D) ). −F

422 � 7 Connes character formula We establish the proper sense of this convergence in the following lemma. Lemma 7.5.4. Let D : dom(D) → H be a self-adjoint operator such that (D + i)−1 ∈ ℒp,∞ . We have Fμ − F0 → 0,

μ → 0+ ,

in ℒp,∞ . Proof. By definition we have D

Fμ =

1

(D2 +μ2 ) 2 ( μ 1 (D2 +μ2 ) 2

μ

1

(D2 +μ2 ) 2 D

−

1 (D2 +μ2 ) 2

)

and F0 = (

sgn(D) χ{0} (D)

χ{0} (D) ). − sgn(D)

By the functional calculus we have 󵄨󵄨 󵄨󵄨 D 󵄨󵄨 󵄨 − sgn(D)󵄨󵄨󵄨 ≤ μ2 |D|−2 (1 − χ{0} (D)) 󵄨󵄨 1 󵄨󵄨 (D2 + μ2 ) 2 󵄨󵄨 and 󵄨󵄨 󵄨󵄨 μ 󵄨󵄨 󵄨 − χ{0} (D)󵄨󵄨󵄨 ≤ μ|D|−1 (1 − χ{0} (D)). 󵄨󵄨 1 󵄨󵄨 (D2 + μ2 ) 2 󵄨󵄨 Since D has compact resolvent, |D|−k (1 − χ{0} (D)) ∈ ℒp,∞ for k = 1, 2. Thus sgn(Dμ ) − F0 ∈ ℒp,∞ and 󵄩󵄩 󵄩 󵄩󵄩sgn(Dμ ) − F0 󵄩󵄩󵄩p,∞ ≤ c ⋅ μ for a constant c > 0 independent of μ. This completes the proof. Let Ωμ and chμ be the multilinear mappings Ω and ch as in Definition 7.1.6 for the spectral triple (π(𝒜), H0 , Dμ ).

7.5 Proof of the Connes character formula

� 423

Lemma 7.5.5. Let (𝒜, H, D) be a smooth p-dimensional spectral triple. If c ∈ 𝒜⊗(p+1) , then μ → 0+ ,

chμ (π(c)) → ch0 (c), in ℒ1 .

Proof. It suffices to prove the statement for c = a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap . Then we have p

chμ (π(c)) = Γ0 sgn(Dμ ) ∏[sgn(Dμ ), π(ak )]. k=0

By Lemma 7.5.4 we have Γ0 sgn(Dμ ) → Γ0 F0 in ℒ(H). Also by Lemma 7.5.4 we have [sgn(Dμ ), π(ak )] → [F0 , π(ak )] in ℒp,∞ and therefore in ℒp+1 . The statement follows now from the continuity of the product mapping. We are now ready to prove the main result of this chapter. Proof of Theorem 7.1.11. For μ > 0, the spectral triple (π(𝒜), H0 , Dμ ) is again smooth and p-dimensional. We note that since π is an algebra homomorphism, if c ∈ 𝒜⊗(p+1) is a Hochschild cycle, then so is π(c). Suppose that the parities of p and (𝒜, H, D) match. In that case the parities of p and (π(𝒜), H0 , Dμ ) also match. Clearly, ker(Dμ ) = 0. This allows us to apply Corollary 7.4.2 to the Hochschild cycle π(c) ∈ (π(𝒜))⊗(p+1) . We have (Tr2 ⊗φ)(Ωμ (π(c))|Dμ |−p ) = Chμ (π(c)).

(7.13)

Suppose now that the parities of p and (𝒜, H, D) do not match. By Lemma 7.5.1 the righthand side in (7.13) is 0. By Lemma 7.5.2 or Lemma 7.5.3 the left-hand side in (7.13) is 0. Thus (7.13) also holds in this case. Since |Dμ |−p = (1 ⊗ (μ2 + D2 )

p

−2

),

we have |Dμ |−p − (1 ⊗ (1 + D2 )

p

−2

Thus

) ∈ ℒ1 .

424 � 7 Connes character formula (Tr2 ⊗φ)(Ωμ (π(c))(1 ⊗ (1 + D2 )

p

−2

)) = Chμ (π(c)).

For A ⊆ {1, . . . , p}, we define the multilinear functional on 𝒜⊗(p+1) by p

𝒯A (a0 ⊗ ⋅ ⋅ ⋅ ⊗ ap ) := Γ0 π(a0 ) ∏ yk (ak ), k=1

where 𝜕(a) 0 { { { ( ), { { { 0 0 yk (a) := { { 0 −a { { { ), {( { a 0

k ∉A, k ∈A.

For c ∈ 𝒜⊗(p+1) , we have Ωμ (π(c)) =

∑

A ⊆{1,...,p}

μ|A | 𝒯A (c).

Denote for brevity 𝒮k (c) :=

∑

A ⊆{1,...,p} |A |=k

𝒯A (c).

We have p

Ωμ (c) = ∑ μk 𝒮k (c), k=0

c ∈ 𝒜⊗(p+1) .

This allows us to rewrite (7.14) as follows: p

∑ μk (Tr2 ⊗φ)(𝒮k (c)(1 ⊗ (1 + D2 )

p

−2

k=0

)) = Chμ (π(c)).

Passing to the limit as μ → 0+ , we obtain (Tr2 ⊗φ)(𝒮0 (c)(1 ⊗ (1 + D2 )

p

−2

)) = lim Chμ (π(c)) = Ch(c), μ→0

where the last inequality follows from Lemma 7.5.5. Since 𝒮0 (c) = 𝒯0 (c) = (

the statement follows.

Ω(c) 0

0 ), 0

c ∈ 𝒜⊗(p+1) ,

(7.14)

7.6 Notes

� 425

7.6 Notes Spectral triples in noncommutative geometry The term spectral triple was introduced by Connes [73]. They have also been called K-cycles over a C ∗ -algebra 𝒩 [72, p. 546] and unbounded Fredholm modules over the C ∗ -algebra 𝒩 [19, 70]. The notion of a spectral triple can be found in the reformulation by S. Baaj and P. Julg of the external Kasparov product on KK-theory. The operator D in a spectral triple (𝒜, H, D) is a generalization of a Dirac-type operator [184]. The prototype spectral triple is defined in terms of a compact Riemannian spin manifold M with spinor bundle S and a Dirac-type operator D acting on sections of S. In this case, 𝒜 = C ∞ (M), acting on H = L2 (M, S) by pointwise multiplication [139, Part III]. The Connes–Moscovici approach to the pseudodifferential calculus of spectral triples was introduced in [86] and [73]. The character formula The formula for the Hochschild class of the Chern character for p-dimensional spectral triples was first proved by Connes [72, Section IV.2.γ]. Later presentations include [139, Section 10.4] and [50]. The version of the theorem given here is closest to [54]. The extension of the result to nonunital spectral triples is nontrivial; this was achieved in [49]. The necessity of replacing D by D0 in the even case (as in Definition 7.1.7) was unnoticed in [54]. The Chern character is more fully understood as the class of ch in periodic cyclic cohomology; see, for example, [49, Section 2.6]. For further details on cyclic cohomology, see [188].

8 Density of states 8.1 Introduction In this chapter, we describe recent applications of singular traces to the density of states (or DOS) measure. Originating from solid state physics, the DOS is a Borel measure on ℝ associated with a Schrödinger operator on Euclidean space. After briefly describing the physical background, we state the main result that the DOS measure can be recovered from the Dixmier trace of a smooth compactly supported function of the Schrödinger operator. We describe some applications of the main result. Physical background Let d ≥ 2, and let V ∈ L∞ (ℝd ) be real valued. The Schrödinger operator with potential V is the linear operator HV := −Δ + MV . This can be understood as a densely defined symmetric operator on the domain H 2 (ℝd ). In fact, HV is a self-adjoint operator on L2 (ℝd ) with domain H 2 (ℝd ) [249, Theorem 8.8]. Since V is bounded, HV is a lower bounded operator whose spectrum is a closed subset of [−‖V ‖∞ , ∞). In quantum mechanics, HV is the Hamiltonian operator for a particle moving in the d-dimensional space under the potential V . The spectrum of HV , a closed lower bounded subset of ℝ, corresponds to the observable energy of the particle. In general, the spectrum can be very difficult to describe. The most physically relevant potentials in solid state physics are those that have some regular predictable behavior on large scales, such as periodic or almost periodic potentials, or certain classes of random potentials. When the DOS measure νHV associated with HV exists, it is supported on the essential spectrum of HV and encodes important information about the physics associated with HV . The DOS νHV is a substitute for the spectral counting function t 󳨃→ Tr(χ(−∞,t) (HV )) when the spectrum of HV is not discrete. Definition of the DOS For R > 0, denote by ΔR the Laplace operator on the open ball B(0, R) := {x ∈ ℝd : |x| < R} with Dirichlet boundary conditions. Weyl’s law for a bounded domain implies that the spectrum of ΔR consists of a discrete set of eigenvalues of finite multiplicity, and furthermore we have (1 − ΔR )−1 ∈ ℒ d ,∞ (L2 (B(0, R))). 2

https://doi.org/10.1515/9783110700176-008

8.1 Introduction

� 427

As in the ℝd case described before, the Schrödinger operator on B(0, R) with potential V is the linear operator HV , R := −ΔR + MV , R . Here V ∈ L∞ (ℝd ) is real valued, and MV , R is the restriction to L2 (B(0, R)) of the product operator MV on L2 (ℝd ). As with ΔR , the spectrum of HV , R is a discrete lower bounded subset of ℝ consisting of eigenvalues of finite multiplicity. If I ⊂ ℝ is a bounded open interval, then HV , R has only finitely many eigenvalues contained in I, and it is meaningful to define νHV , R (I) := TrL2 (B(0,R)) (χI (HV , R )), that is, νHV , R (I) is the number of eigenvalues of HV , R contained in I, counted according to multiplicity. The density of states measure of HV is a Borel regular measure νHV on ℝ such that for every open interval I ⊂ ℝ, we have νHV (I) = lim

R→∞

νHV , R (I)

Vol(B(0, R))

,

that is, νHV (I) is the asymptotic number of eigenvalues of HV , R contained in I per unit volume as R → ∞. This limit does not necessarily exist, and even if does, it is not obvious that νHV really does extend to a Borel measure on ℝ. In general, the DOS does not always exist; see, for example, [259, p. 513]. However, it is well known to exist for certain classes of Hamiltonians important for solid state physics such as those corresponding to periodic, almost periodic, and ergodic potentials; see, for example, [4, 25, 219, 230, 259]. Remark 8.1.1. The preceding definition of νHV is not universally followed. Alternative definitions may be based on either changing the boundary conditions for ΔR , for example, using Neumann boundary conditions, or by changing the family of domains {B(0, R)}R>0 into some other exhausting family for ℝd , such as a family of boxes {[−R, R]d }R>0 . It is known that the choice of boundary conditions for ΔR is irrelevant [119]. However, the choice of domain can in principle change the limit; see [17]. Should it exist, the DOS measure νHV possesses the following properties: (i) Invariance under isometries. If O is an orthogonal d × d matrix, then νHV ∘O = νHV , where (V ∘ O)(t) := V (Ot),

t ∈ ℝd .

428 � 8 Density of states To see this, note that the Laplace operator is invariant under rotations; that is, the unitary UO,R on L2 (B(0, R)) given by (UO,R h)(x) := h(O(x)), h ∈ L2 (B(0, R)), commutes with ΔR . Thus −1 UO,R (−ΔR + MV , R )UO,R = −ΔR + MV ∘O,R .

The invariance of νHV under isometries follows now from the unitary invariance of the trace Tr. (ii) Insensitivity to localized perturbations. If V0 ∈ L∞ (ℝd ) is real valued and compactly supported, then νHV +V = νHV . 0

This statement will be proved in Section 8.4 as a consequence of the Dixmier trace formula for the DOS given below. Example 8.1.2. We can explicitly compute the density of states for H0 = −Δ: dνH0 (λ) =

d Vol(𝕊d−1 ) max{0, λ} 2 −1 dλ. d 2(2π)

See [259, Theorem C.7.7]. Adding a scalar α ∈ ℝ to H0 corresponds to a shift of the spectrum by α, and hence for every α ∈ ℝ, we have dνHα⋅1 (λ) =

d Vol(𝕊d−1 ) max{0, λ − α} 2 −1 dλ. d 2(2π)

Here Hα⋅1 is the Hamiltonian with potential V being a constant function equal to α. DOS and Connes’ integral formula The main result of this chapter is a formula for the DOS in terms of a Dixmier trace on the two-sided ideal ℒ1,∞ . We recall Theorem 3.4.1, the integration formula of Connes. Let d ≥ 2. For all normalized traces φ on ℒ1,∞ (L2 (ℝd )), we have d

φ(Mf (1 − Δ)− 2 ) =

Vol(𝕊d−1 ) ⋅ ∫ f (t) dt, d(2π)d

f ∈ Cc (ℝd ).

ℝd

Since the Fourier transform ℱ from Section 1.5.1 is a unitary operator on L2 (ℝd ) and exchanges Fourier multipliers g(∇) with product operators Mg , we have d

d

−d φ(Mf (1 − Δ)− 2 ) = φ(ℱ −1 Mf (1 − Δ)− 2 ℱ ) = φ(f (∇)M⟨x⟩ ),

8.1 Introduction

� 429

where the product operator M⟨x⟩ is the notation we adopt from Section 1.5 for the operator of pointwise multiplication by the function 1

x 󳨃→ ⟨x⟩ := (1 + |x|2 ) 2 ,

x ∈ ℝd .

Technically, M⟨x⟩ is an unbounded positive operator on L2 (ℝd ) when given the domain ℱ H 1 (ℝd ), M⟨x⟩ : ℱ H 1 (ℝd ) → L2 (ℝd ). Here H 1 (ℝd ) is the Sobolev Hilbert space defined in Section 1.6.3 by Bessel potentials. −m Clearly, M⟨x⟩ has a bounded inverse, and for any m ≥ 0, M⟨x⟩ is bounded on L2 (ℝd ), and −m M⟨x⟩ = M⟨x⟩−m : L2 (ℝd ) → ℱ H m (ℝd ) −d is continuous. In particular, M⟨x⟩ is the operator − d2

−d (M⟨x⟩ u)(x) = (1 + |x|2 )

u(x),

x ∈ ℝd ,

u ∈ L2 (ℝd ).

Considering the case where f is a radial compactly supported function, which implies that there exists g ∈ Cc ([0, ∞)) such that f (t) = g(|t|2 ), Connes’ integration formula specializes to ∞

−d φ(g(−Δ)M⟨x⟩ )=

d Vol(𝕊d−1 ) Vol(𝕊d−1 )2 ⋅ ∫ g(|x|2 ) dx = ⋅ ∫ g(λ)λ 2 −1 dλ, d d d(2π) 2d(2π)

0

ℝd

where the latter equality arises from switching to polar coordinates. Example 8.1.2 provides the formula for the DOS of H0 = −Δ, and we see that −d φ(g(H0 )M⟨x⟩ )=

Vol(𝕊d−1 ) ⋅ ∫ g(λ)dνH0 (λ), d

g ∈ Cc ([0, ∞)),

(8.1)

ℝ

where g is identified with a function on ℝ by setting it to be identically zero on (−∞, 0). The idea behind the singular trace formula for the DOS is replacing H0 in (8.1) with HV = −Δ + MV for some real-valued V ∈ L∞ (ℝd ). As a consequence of the Riesz representation theorem and a product-convolution compactness estimate proved below, if φ : ℒ1,∞ → ℂ is a continuous trace, then for every HV , there exists a Borel measure μHV ,φ on ℝ such that −d φ(g(HV )M⟨x⟩ ) = ∫ g dμHV ,φ , ℝ

g ∈ Cc (ℝ).

430 � 8 Density of states The measure μHV ,φ is associated with the Schrödinger operator in a natural way and shares some features with the density of states measure νHV . In particular, μHV ,φ has the following features: (i) Invariance under isometries: If O is an orthogonal d × d matrix, then μHV ∘O ,φ = μHV ,φ . (ii) Insensitivity to localized perturbations: If V0 ∈ L∞ (ℝd ) is compactly supported, then μHV +V

0

,φ

= μHV ,φ .

The invariance of μHV ,φ under isometries follows from the same argument as before in Example 8.1.2. The statement on localized perturbations will be proved in Section 8.4. In addition to these similarities between νHV and μHV ,φ , when V = 0, we have μ−Δ,φ =

Vol(𝕊d−1 ) ⋅ ν−Δ d

from (8.1). DOS trace formula All this leads to the conjecture that μHV ,φ is in fact a constant multiple of the DOS measure νHV for HV , at least whenever the latter measure exists. The main result of this chapter confirms that conjecture when φ is a Dixmier trace. Theorem 8.1.3. Let d ≥ 2. For all Dixmier traces Trω , ω an extended limit on l∞ , and all real-valued V ∈ L∞ (ℝd ) such that the DOS measure νHV exists, we have μHV ,Trω =

Vol(𝕊d−1 ) ⋅ νHV , d

that is, −d Trω (f (HV )M⟨x⟩ )=

Vol(𝕊d−1 ) ⋅ ∫ f dνHV , d

f ∈ Cc (ℝ).

(8.2)

ℝ

Theorem 8.1.3 is proved in Section 8.3. Remark 8.1.4. The DOS measure νHV is only defined for some potentials V on ℝd , d ≥ 2. On the other hand, the measure μHV ,Trω exists for every real-valued V ∈ L∞ (ℝd ) and Dixmier trace Trω . Theorem 8.1.3 indicates that μHV ,Trω extends the DOS construction to define a measure for every bounded real-valued potential that coincides with the DOS

8.1 Introduction

� 431

when the latter exists and retains some of the essential properties of the DOS in the general case. The Dixmier trace is not a normal semifinite trace on ℒ(H), and therefore it is not clear that formula (8.2) extends to bounded Borel functions on ℝd using an approximation by continuous functions of compact support. An order argument provides the following for indicator functions on intervals. Corollary 8.1.5. If the density of states of HV exists and I = (a, b) ⊂ ℝ is a bounded open interval with endpoints being points of continuity of the distribution function t 󳨃→ νHV (−∞, t),

t ∈ ℝ,

then for all extended limits ω on l∞ , we have −d Trω (χI (HV )M⟨x⟩ )=

Vol(𝕊d−1 ) ⋅ νHV (I), d

where χI is the indicator function of I. Proof. If f , g ∈ Cc (ℝ) are functions such that f (t) ≤ χI (t) ≤ g(t),

t ∈ ℝ,

then −d

−d

−d

−d

−d

−d

M⟨x⟩2 f (HV )M⟨x⟩2 ≤ M⟨x⟩2 χI (HV )M⟨x⟩2 ≤ M⟨x⟩2 g(HV )M⟨x⟩2 . Since the Dixmier trace Trω is positive, if 0 ≤ A ≤ B ≤ C ∈ ℒ1,∞ , then Trω (A) ≤ Trω (B) ≤ Trω (C). Thus −d

−d

−d

−d

−d

−d

Trω (M⟨x⟩2 f (HV )M⟨x⟩2 ) ≤ Trω (M⟨x⟩2 χI (HV )M⟨x⟩2 ) ≤ Trω (M⟨x⟩2 g(HV )M⟨x⟩2 ). Since Trω is a trace, it follows that −d −d −d Trω (f (HV )M⟨x⟩ ) ≤ Trω (χI (HV )M⟨x⟩ ) ≤ Trω (g(HV )M⟨x⟩ ).

Since f and g are continuous, Theorem 8.1.3 applies to the upper and lower bounds, and therefore Vol(𝕊d−1 ) Vol(𝕊d−1 ) −d )≤ ∫ f dνH ≤ Trω (χI (HV )M⟨x⟩ ∫ g dνH . d d ℝ

ℝ

432 � 8 Density of states Since νH is a regular positive measure, we have νH (a, b) = sup ∫ f dνH , f ≤χI

νH [a, b] = inf ∫ g dνH . χI ≤g

ℝ

ℝ

Therefore Vol(𝕊d−1 ) Vol(𝕊d−1 ) −d νH (a, b) ≤ Trω (χI (HV )M⟨x⟩ )≤ νH [a, b]. d d If a and b are points of continuity of the distribution function, then νH ({a}) = limt→0 νH ((a − t, a + t)) = 0, and similarly νH ({b}) = 0. Therefore −d Trω (χI (HV )M⟨x⟩ )=

Vol(𝕊d−1 ) νH (I). d

Strategy of the proof of the trace formula Sections 8.2 and 8.3 are devoted to the proof of Theorem 8.1.3. The argument relies on the following theorem, which greatly simplifies the computation of the DOS measure. Theorem 8.1.6 ([259, Proposition C.7.2]). Let V ∈ L∞ (ℝd ) be a real-valued potential. Then the following statements are equivalent: (i) The density of states measure νHV for HV = −Δ + MV exists; (ii) For all s > 0, the limit lim

R→∞

1 Tr(e−sHV MχB(0,R) ) Vol(B(0, R))

exists. If either of the above holds, then we have lim

R→∞

1 Tr(e−sHV MχB(0,R) ) = ∫ e−st dνHV (t), Vol(B(0, R))

s > 0.

ℝ

In view of Theorem 8.1.6, the proof of Theorem 8.1.3 is reduced to demonstrating that for all s > 0, we have −d Trω (e−sHV M⟨x⟩ )=

Vol(𝕊d−1 ) 1 lim Tr(e−sHV MχB(0,R) ) R→∞ Vol(B(0, R)) d

(8.3)

whenever the limit on the right exists and Trω is a Dixmier trace associated with an extended limit ω on l∞ . The proof of Theorem 8.1.3 then has three steps:

8.2 Compactness estimate for the density of states

� 433

−d (a) In Section 8.2, we show that e−sHV M⟨x⟩ ∈ ℒ1,∞ and related estimates, so that the left-hand side (8.3) is well defined. (b) Using the estimates from Section 8.2, we verify the conditions necessary for Theorem 1.3.20 to compute the Dixmier trace via a ζ -function residue. (c) In Section 8.3, we use the Laplace transform of measures to transfer equality (8.3) to the statement of Theorem 8.1.3.

8.2 Compactness estimate for the density of states In this section, we prove a convolution-product-type compactness estimate, which can be compared with Theorems 1.5.20 and 1.5.22. The important difference is that we replace radial functions of ∇ (or a function of H0 = −Δ) with functions of HV = −Δ + MV . Recall that V is a bounded real-valued function on ℝd , and HV is self-adjoint as an unbounded operator on the Hilbert space L2 (ℝd ) with domain H 2 (ℝd ). Theorem 8.2.1. Let d ≥ 2. Suppose V = V ∗ ∈ L∞ (ℝd ) is bounded from below by a strictly positive constant. For any n ∈ ℕ, we have −n HV−n M⟨x⟩ ∈ ℒ d ,∞ . n

In the case V = 0, Theorem 8.2.1 reduces to a particular case of either Theorem 1.5.20 or Theorem 1.5.22, depending on dn < 2 or dn > 2. However, when dn = 2 or V ≠ 0, Theorem 8.2.1 is not covered by any of the estimates in Section 1.5. Theorem 8.2.1 has the following corollary. −n Corollary 8.2.2. For every s > 0, we have that e−sHV M⟨x⟩ ∈ ℒ d ,∞ . n

Proof. Write −n −n e−sHV M⟨x⟩ = e−sHV HVn ⋅ HV−n M⟨x⟩ .

The first factor has a bounded extension. The second factor is in ℒ d ,∞ by Theorem 8.2.1. n

We begin the proof of Theorem 8.2.1 with a lemma of elementary operator theory. Lemma 8.2.3. Let A, B, C be bounded operators such that A = B − AC. If B ∈ ℒp0 ,∞ and C ∈ ℒp1 ,∞ for 0 < p0 , p1 < ∞, then A ∈ ℒp0 ,∞ . Proof. Writing B = A(1 + C), we can see that for each n ≥ 1, n−1

n−1

k=0

k=0

B ⋅ ∑ (−C)k = A ⋅ (1 + C) ⋅ ∑ (−C)k = A + (−1)n−1 AC n . Hence

434 � 8 Density of states n−1

A = ∑ (−1)k BC k + (−1)n AC n . k=0

Since ℒp0 ,∞ is an ideal, we have BC k ∈ ℒp0 ,∞ for all k ≥ 0. Choose n sufficiently large such that p1 < np0 . From Hölder’s inequality (1.7) it follows that AC n ∈ ℒ p1 ,∞ ⊂ ℒp0 ,∞ . n

Hence A ∈ ℒp0 ,∞ . The following lemma contains a central piece of the proof of Theorem 8.2.1. For uniformity of notation, we set H0 := −Δ. Lemma 8.2.4. For all p ≥ 0, we have p

−p

M⟨x⟩ [H0 , M⟨x⟩ ](H0 + 1)−1 ∈ ℒ2d,∞ . Proof. By linearity it suffices to verify that for all 1 ≤ j ≤ d, p

M⟨x⟩ [𝜕j2 , M⟨x⟩ ](H0 + 1)−1 ∈ ℒ2d,∞ . −p

The commutator [𝜕j2 , M⟨x⟩ ] is equal to −p

2M𝜕j ⟨x⟩−p 𝜕j + M𝜕2 ⟨x⟩−p . j

However, it is easily verified that for all α ∈ ℤd+ , 󵄨 󵄨 ⟨x⟩p ⋅ 󵄨󵄨󵄨𝜕xα (⟨x⟩−p )󵄨󵄨󵄨 ≤ Cα ⟨x⟩−|α| . Indeed, this was already noted in Section 1.6.3. It follows that the operator p

−p

M⟨x⟩ [H0 , M⟨x⟩ ](H0 + 1)−1 is a linear combination of operators of the form Mf1 (1 − Δ)−1 and Mf2 𝜕j (1 − Δ)−1 , where |f1 (x)| ≤ C⟨x⟩−2 in the first case and |f2 (x)| ≤ C⟨x⟩−1 in the second case, so that both f1 and f2 belong to L2d (ℝd ). It follows from Theorem 1.5.20 that these operators belong to ℒ2d,∞ by taking g(ξ) = (1 + |ξ|2 )−1 ∈ L2d (ℝd ) in the first case and g(ξ) = ξj (1 + |ξ|2 )−1 ∈ L2d (ℝd ) in the second case.

8.2 Compactness estimate for the density of states

� 435

The next lemma may be seen as a particular case of Theorem 1.5.20 when the dimension d is greater than 2, but the two-dimensional case is not covered in Section 1.5. Lemma 8.2.5. We have −1 (H0 + 1)−1 M⟨x⟩ ∈ ℒd,∞ .

Proof. Let us introduce the functions f , g : ℝd → ℝ by setting f : x 󳨃→ ⟨x⟩−1 ,

g : x 󳨃→ ⟨x⟩−2 ,

x ∈ ℝd .

We have −1 (H0 + 1)−1 M⟨x⟩ = g(∇)Mf .

We consider separately d > 2 and d = 2. Consider first d > 2. Note that f ∈ Ld,∞ (ℝd ) and g ∈ (L d ,∞ ∩ L∞ )(ℝd ) ⊂ Ld (ℝd ). The “Fourier dual” of Theorem 1.5.20 yields 2

g(∇)Mf ∈ ℒd,∞ . This proves the statement for d > 2. Now we prove the statement for d = 2. Recall that the Dirac operator D on ℂ2 ⊗L2 (ℝ2 ) is defined by D := γ1 ⊗ 𝜕1 + γ2 ⊗ 𝜕2 , where γ1 γ2 = −γ1 γ2 and γ12 = γ22 = 1. We have D2 = −1 ⊗ Δ. Therefore 1 ⊗ g(∇)Mf = (1 + D2 )

−1

⋅ (1 ⊗ Mf ) = (D + i)−1 ⋅ (D − i)−1 ⋅ (1 ⊗ Mf )

= (D + i)−1 (1 ⊗ Mf )(D − i)−1 + (D + i)−1 ⋅ [(D − i)−1 , 1 ⊗ Mf ]. We prove that the two summands belong to ℒ2,∞ . In fact, the first summand belongs to ℒ2,∞ , and the second summand belongs to ℒ2 . 1

Indeed, for the first summand, note that (D + i)(D2 + 1)− 2 is unitary. The unitary invariance of the ℒ2,∞ norm implies that −1 −1 󵄩 󵄩󵄩 󵄩 2 −1 −1 󵄩 2 󵄩󵄩(D + i) (1 ⊗ Mf )(D − i) 󵄩󵄩󵄩2,∞ = 󵄩󵄩󵄩(D + 1) 2 (1 ⊗ Mf )(D + 1) 2 󵄩󵄩󵄩2,∞ 1 1 󵄩 󵄩 = 󵄩󵄩󵄩(1 − Δ)− 2 Mf (1 − Δ)− 2 󵄩󵄩󵄩2,∞ .

436 � 8 Density of states By Hölder’s inequality (1.7) there is a constant c such that 󵄩󵄩 󵄩 −1 −1 󵄩 − 1 󵄩2 󵄩󵄩(1 − Δ) 2 Mf (1 − Δ) 2 󵄩󵄩󵄩2,∞ ≤ c󵄩󵄩󵄩M 21 (1 − Δ) 2 󵄩󵄩󵄩4,∞ . f The “Fourier dual” of Theorem 1.5.20 yields 1

M 1 (1 − Δ)− 2 ∈ ℒ4,∞ , f

1

2

1

as f 2 ∈ L4,∞ (ℝ2 ) and g(ξ) = (1 + |ξ|2 )− 2 ∈ L4 (ℝ2 ). Hence the first summand indeed belongs to ℒ2,∞ , as it is a product of two operators in ℒ4,∞ . Now we treat the second summand. Clearly, [D, 1 ⊗ Mf ] = γ1 ⊗ M𝜕1 f + γ2 ⊗ M𝜕2 f . Therefore [(D − i)−1 , 1 ⊗ Mf ] = −(D − i)−1 (γ1 ⊗ M𝜕1 f + γ2 ⊗ M𝜕2 f )(D − i)−1 , and (D + i)−1 [(D − i)−1 , 1 ⊗ Mf ] = (γ1 ⊗ 1) ⋅ (D2 + 1) (1 ⊗ M𝜕1 f )(D − i)−1 −1

+ (γ2 ⊗ 1) ⋅ (D2 + 1) (1 ⊗ M𝜕2 f )(D − i)−1 . −1

The triangle inequality for ℒ2 yields 󵄩󵄩 󵄩 −1 −1 󵄩󵄩(D + i) [(D − i) , 1 ⊗ Mf ]󵄩󵄩󵄩2 −1 󵄩 󵄩 ≤ 󵄩󵄩󵄩(γ1 ⊗ 1) ⋅ (D2 + 1) (1 ⊗ M𝜕1 f )(D − i)−1 󵄩󵄩󵄩2 −1 󵄩 󵄩 + 󵄩󵄩󵄩(γ2 ⊗ 1) ⋅ (D2 + 1) (1 ⊗ M𝜕2 f )(D − i)−1 󵄩󵄩󵄩2 1 1 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩(1 − Δ)−1 M𝜕1 f (1 − Δ)− 2 󵄩󵄩󵄩2 + 󵄩󵄩󵄩(1 − Δ)−1 M𝜕2 f (1 − Δ)− 2 󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩 ≤ 󵄩󵄩󵄩(1 − Δ)−1 M𝜕1 f 󵄩󵄩󵄩2 + 󵄩󵄩󵄩(1 − Δ)−1 M𝜕2 f 󵄩󵄩󵄩2 . Theorem 1.5.20 yields (1 − Δ)−1 M𝜕1 f , (1 − Δ)−1 M𝜕2 f ∈ ℒ2 . Hence the second summand indeed belongs to ℒ2 . These considerations yield 1 ⊗ g(∇)Mf ∈ ℒ2,∞ . This completes the proof in the two-dimensional case.

8.2 Compactness estimate for the density of states

�

437

Proof of Theorem 8.2.1. Denote by An , Bn , and Cn the following operators: −n An := HV−n M⟨x⟩ ,

−n −1 Bn := HV1−n M⟨x⟩ HV ,

n −n Cn := M⟨x⟩ [H0 , M⟨x⟩ ]HV−1 .

The theorem states that An ∈ ℒ d ,∞ for all n ∈ ℕ, which we will prove by induction on n. n

Observe that since MV and HV−1 are bounded by the condition on V , we have HV−1 (H0 + 1) = 1 + HV−1 (1 − MV ) ∈ ℒ(L2 (ℝd )).

(8.4)

By Lemma 8.2.5 we have −1 A1 = HV−1 (H0 + 1) ⋅ (H0 + 1)−1 M⟨x⟩ ∈ ℒ∞ ⋅ ℒd,∞ = ℒd,∞ .

This proves the n = 1 case. Suppose the estimate holds for n ≥ 1. Let us prove it for n + 1. If X and Y are linear operators where Y and X −1 are bounded and Y preserves the domain of X, then we have the identity [X −1 , Y ] = −X −1 [X, Y ]X −1 .

(8.5)

−n−1 This identity applies with Y = M⟨x⟩ and X = HV , since the domain of X is equal to the domain of H0 and, clearly, Y preserves H 2 (ℝd ), the domain of H0 . Thus we may write −n−1 −n−1 An+1 − Bn+1 = HV−n ⋅ [HV−1 , M⟨x⟩ ] = −HV−n−1 [H0 , M⟨x⟩ ]HV−1 = −An+1 Cn+1 .

We now verify that the operators An+1 , Bn+1 , and Cn+1 satisfy the assumptions in Lemma 8.2.3. With the identity Bn+1 = An A∗1 , Hölder’s inequality (1.7) and the inductive assumption yield Bn+1 ∈ ℒ d ,∞ ⋅ ℒd,∞ ⊂ ℒ n

d ,∞ n+1

.

Also, n+1 −n−1 Cn+1 = M⟨x⟩ [H0 , M⟨x⟩ ](H0 + 1)−1 ⋅ (H0 + 1)HV−1 .

Lemma 8.2.4 states that n+1 −n−1 M⟨x⟩ [H0 , M⟨x⟩ ](H0 + 1)−1 ∈ ℒ2d,∞ .

438 � 8 Density of states Since (H0 + 1)HV−1 = 1 + (1 − MV )HV−1 ∈ ℒ∞ (L2 (ℝd )), it follows that Cn+1 ∈ ℒ2d,∞ ⋅ ℒ∞ = ℒ2d,∞ ; that is, Bn+1 ∈ ℒ

d ,∞ n+1

and Cn+1 ∈ ℒ2d,∞ , so applying Lemma 8.2.3 to the operators An+1 ,

Bn+1 , and Cn+1 yields An+1 ∈ ℒ

d ,∞ n+1

, so the statement follows by induction on n.

As a useful corollary to Theorem 8.2.1, we also include the following: Corollary 8.2.6. Let d ≥ 2. Suppose V = V ∗ ∈ L∞ (ℝd ) is bounded from below by a strictly positive constant. If s > 0, then −d −sHV [M⟨x⟩ ,e ] ∈ ℒ1 .

Proof. Define the function ϕs ∈ C 1 (ℝd ) by setting s

ϕs (t) := {

e− 2t , 0,

t > 0, t ≤ 0.

It is immediate that ϕ′s ∈ H 1 (ℝ) with the norm estimate 󵄩󵄩 ′ 󵄩󵄩 −1 −3 󵄩󵄩ϕs 󵄩󵄩H 1 ≤ c ⋅ (s 2 + s 2 ),

s > 0,

(8.6)

for a constant c > 0. Since ϕ′s ∈ H 1 (ℝ), it follows from Theorem 1.4.9 and Lemma 1.4.12 that the transformer T Moreover, we have

HV−1 ,HV−1

ϕ[1] s

is bounded from ℒ1 to ℒ1 and also from ℒ

d ,1 d−1

to ℒ

󵄩󵄩 HV−1 ,HV−1 󵄩󵄩 󵄩 ′󵄩 󵄩󵄩T [1] 󵄩󵄩ℒ1 →ℒ1 ≤ b ⋅ 󵄩󵄩󵄩ϕs 󵄩󵄩󵄩H 1 ϕs

s

s

s

−d −sHV −d − 2 HV −d − 2 HV − 2 HV [M⟨x⟩ ,e ] = e− 2 HV [M⟨x⟩ ,e ] + [M⟨x⟩ ,e ]e . s

Since e− 2 HV = ϕs (HV−1 ), it follows that from (1.15) in Section 1.4.3 that s

−d − 2 HV [M⟨x⟩ ,e ]=T

HV−1 ,HV−1

ϕ[1] s

.

(8.7)

for some constant b > 0. Using the semigroup property of s 󳨃→ e−sHV and the Leibniz rule, we have s

d ,1 d−1

−d ([M⟨x⟩ , HV−1 ]).

Combining the preceding two displays, (1.13) and (1.14) from Section 1.4.3 lead us to

8.2 Compactness estimate for the density of states

−d −sHV [M⟨x⟩ ,e ]=T

HV−1 ,HV−1

ϕ[1] s

s

� 439

s

−d −d (e− 2 HV [M⟨x⟩ , HV−1 ] + [M⟨x⟩ , HV−1 ]e− 2 HV ).

Now (8.7) implies that 󵄩󵄩 −d −sHV 󵄩󵄩 󵄩 󵄩 󵄩 s 󵄩 󵄩 −d −1 − 2s HV 󵄩󵄩 −d ]󵄩󵄩1 ≤ b ⋅ 󵄩󵄩󵄩ϕ′s 󵄩󵄩󵄩W 1 (󵄩󵄩󵄩e− 2 HV [M⟨x⟩ , HV−1 ]󵄩󵄩󵄩1 + 󵄩󵄩󵄩[M⟨x⟩ , HV ]e 󵄩󵄩[M⟨x⟩ , e 󵄩󵄩1 ). 2

(8.8)

We write −d −d −d [M⟨x⟩ , HV−1 ] = −HV−1 [M⟨x⟩ , HV ]HV−1 = −HV−1 [M⟨x⟩ , H0 ]HV−1

−d−1 d+1 −d = −HV−1 M⟨x⟩ ⋅ M⟨x⟩ [M⟨x⟩ , H0 ](H0 + 1)−1 ⋅ (H0 + 1)HV−1 .

Therefore s

−d e− 2 HV [M⟨x⟩ , HV−1 ] = X1 X2 X3 X4 ,

where s

X1 = e− 2 HV HVd ,

−d−1 X2 = HV−d−1 M⟨x⟩ ,

d+1 −d X3 = M⟨x⟩ [M⟨x⟩ , H0 ](H0 + 1)−1 ,

X4 = (H0 + 1)HV−1 .

Clearly, X1 ∈ ℒ(H). Also, X3 ∈ ℒ(H), as can be established by writing H0 = − ∑dj=1 𝜕j2 and applying the Leibniz rule similarly to the proof of Lemma 8.2.4. By Theorem 8.2.1 we have X2 ∈ ℒ1 . From (8.4), X4 ∈ ℒ(H). Combining these estimates, we obtain s

−d e− 2 HV [M⟨x⟩ , HV−1 ] ∈ ℒ1 .

(8.9)

Similarly, we have s

−d [M⟨x⟩ , HV−1 ]e− 2 HV ∈ ℒ1 .

(8.10)

Substituting (8.9), (8.10), and (8.6) into (8.8), we complete the proof. Corollary 8.2.7. Let d ≥ 2. Suppose V = V ∗ ∈ L∞ (ℝd ) is bounded from below by a strictly positive constant. If s > 0, then 󵄩 −pd 󵄩 (p − 1)󵄩󵄩󵄩M⟨x⟩ e−sHV 󵄩󵄩󵄩ℒ ≤ cV ,s,d , 1

1 < p ≤ 2,

for a constant cV ,s,d > 0 depending on V , d, and s > 0. Proof. Let 1 < p ≤ 2. If p < p′ < p2 , then p − 1 < p′ − 1 < p2 − 1 = (p + 1)(p − 1) ≤ 3(p − 1) and

440 � 8 Density of states −p′ d

(p−p′ )d

M⟨x⟩ = M⟨x⟩ (p−p′ )d

where M⟨x⟩

−pd

M⟨x⟩ ,

is a bounded operator with 󵄩󵄩 (p−p′ )d 󵄩󵄩 󵄩󵄩M⟨x⟩ 󵄩󵄩∞ ≤ 1.

Hence, when p < p′ < p2 , 󵄩 −p′ d 󵄩 󵄩 (p−p′ )d 󵄩󵄩 󵄩󵄩 −pd −sHV 󵄩󵄩 (p′ − 1)󵄩󵄩󵄩M⟨x⟩ e−sHV 󵄩󵄩󵄩1 ≤ 3(p − 1)󵄩󵄩󵄩M⟨x⟩ 󵄩󵄩∞ 󵄩󵄩M⟨x⟩ e 󵄩󵄩1 󵄩󵄩 −pd −sHV 󵄩󵄩 ≤ 3 ⋅ (p − 1)󵄩󵄩M⟨x⟩ e 󵄩󵄩1 . It is therefore sufficient to prove the statement for 1 < p ≤ p0 for some p0 > 1. We take 1 < p ≤ 2d+1 . We write 2d −pd

−(p−1)d

M⟨x⟩ e−sHV = M⟨x⟩

s

s

−(p−1)d − 2s HV

−d − 2 HV ⋅ [M⟨x⟩ ,e ] ⋅ e− 2 HV + M⟨x⟩

e

s

−d − 2 HV ⋅ M⟨x⟩ e .

By the triangle inequality and (1.5) we have 󵄩󵄩 −pd −sHV 󵄩󵄩 󵄩󵄩 −d − 2s HV 󵄩󵄩 󵄩󵄩 −(p−1)d − 2s HV 󵄩󵄩 󵄩󵄩 −d − 2s HV 󵄩󵄩 ]󵄩󵄩1 + 󵄩󵄩M⟨x⟩ e 󵄩󵄩M⟨x⟩ e 󵄩󵄩1 ≤ 󵄩󵄩[M⟨x⟩ , e 󵄩󵄩∞,1 󵄩󵄩M⟨x⟩ e 󵄩󵄩1,∞ . The first summand and the second term in the second summand are independent of p, so it suffices to estimate the first term in the second summand. We write −(p−1)d − 2s HV

M⟨x⟩

e

−(p−1)d

= M⟨x⟩

s

(H0 + 1)−1 ⋅ (H0 + 1)HV−1 ⋅ HV e− 2 HV .

s

Clearly, the operator HV e− 2 HV is bounded. It is established in the proof of Theorem 8.2.1 that (H0 + 1)HV−1 is bounded. Therefore 󵄩󵄩 −(p−1)d − 2s HV 󵄩󵄩 󵄩 −(p−1)d 󵄩 e (H0 + 1)−1 󵄩󵄩󵄩∞,1 , 󵄩󵄩M⟨x⟩ 󵄩󵄩∞,1 ≤ bV ,s 󵄩󵄩󵄩M⟨x⟩ where s 󵄩 󵄩 bV ,s := 󵄩󵄩󵄩(H0 + 1)HV−1 ⋅ HV e− 2 HV 󵄩󵄩󵄩∞ .

If x ∈ ℒq,∞ , where q =

1 p−1

∈ [2d, ∞), then ∞

μ(k, x) k+1 k=0

‖x‖∞,1 := ∑ ∞

1

− q1 −1

= ∑ (k + 1) q μ(k, x) ⋅ (k + 1) k=0

8.2 Compactness estimate for the density of states

�

441

∞

1

≤ sup(k + 1) q μ(k, x) ⋅ ∑ k −p k≥0

k=1

∞

≤ ‖x‖q,∞ (1 + ∫ u−p du) 1

p = ‖x‖q,∞ . p−1 Since p ≤

2d+1 , 2d

‖x‖∞,1 ≤ cd (p − 1)−1 ‖x‖q,∞ for a constant cd > 0. Thus 󵄩󵄩 −(p−1)d − 2s HV 󵄩󵄩 󵄩 −1 󵄩 −(p−1)d e (H0 + 1)−1 󵄩󵄩󵄩q,∞ . 󵄩󵄩M⟨x⟩ 󵄩󵄩∞,1 ≤ cd bV ,s ⋅ (p − 1) 󵄩󵄩󵄩M⟨x⟩ By Theorem 1.5.20 we have 󵄩󵄩 −(p−1)d 󵄩 󵄩 󵄩 󵄩 󵄩 (H0 + 1)−1 󵄩󵄩󵄩q,∞ ≤ cq,d ⋅ 󵄩󵄩󵄩⟨x⟩−d(p−1) 󵄩󵄩󵄩q,∞ 󵄩󵄩󵄩⟨x⟩−2 󵄩󵄩󵄩q , 󵄩󵄩M⟨x⟩

11

t>0

Recalling that there is a constant ad ≥ 1 such that μ(t, |x|−d ) ≤ ad ⋅ t −1 ,

t > 1,

we have 1

󵄩󵄩 −d(p−1) 󵄩󵄩 q 󵄩󵄩⟨x⟩ 󵄩󵄩q,∞ ≤ 1 + ad ≤ 1 + ad . Let Vd be the volume of the unit ball in ℝd . Then 1

q 󵄩󵄩 −2 󵄩󵄩 −2q 󵄩󵄩⟨x⟩ 󵄩󵄩q ≤ (Vd + ∫ |x| dx)

|x|>1

1

d−1

≤ (Vd + Vol(𝕊

q 1 Vol(𝕊d−1 ) )⋅ ) ≤ Vd + , 2q − d 3d 1

since 2q − d ≥ 3d by assumption on q, Vd ≥ 1, and r q ≤ r for r ≥ 1.

442 � 8 Density of states

8.3 The density of states trace formula Corollary 8.2.2 in Section 8.2 ensures that −d e−sHV M⟨x⟩ ∈ ℒ1,∞ ,

s > 0,

where HV := −Δ + MV is the Hamiltonian associated with a real-valued potential V ∈ L∞ (ℝd ), d ≥ 2, and u ∈ L2 (ℝd ),

−d (M⟨x⟩ u)(x) := ⟨x⟩−d u(x),

x ∈ ℝd ,

is the product operator associated with the (−d)th power of the function 1

⟨x⟩ := (1 + |x|2 ) 2 ,

x ∈ ℝd .

This section continues the proof of Theorem 8.1.3 by proving equation (8.3), −d Trω (e−sHV M⟨x⟩ )=

Vol(𝕊d−1 ) 1 ⋅ lim Tr(e−sHV MχB(0,R) ), R→∞ Vol(B(0, R)) d

s > 0,

using ζ -function residues. Here Trω is a Dixmier trace on ℒ1,∞ associated with an extended limit ω on l∞ , as described in Section 1.1.2. Equality of measures by equality of their Laplace transforms will complete the proof of Theorem 8.1.3. Lemma 8.3.1. Let d ≥ 2. For any Dixmier trace Trω and all s > 0, we have −d −r Trω (e−sHV M⟨x⟩ ) = d −1 ⋅ lim+ (r − d) Tr(e−sHV M⟨x⟩ ), r→d

provided that the limit on the right-hand side exists. −d Proof. Let A = e−sHV and B = M⟨x⟩ . Corollaries 8.2.6 and 8.2.7 prove that A and B satisfy the assumptions in Theorem 1.3.20. Applying Theorem 1.3.20, we infer the statement.

As justified by Lemma 8.3.1, formula (8.3) is equivalent to the identity −dr lim+ (r − 1) Tr(e−sHV M⟨x⟩ )=

r→1

Vol(𝕊d−1 ) 1 lim Tr(e−sHV MχB(0,R) ) R→∞ Vol(B(0, R)) d

(8.11)

whenever the limit on the right exists. This can be made a little clearer by writing the traces in terms of the integral kernels. For each s > 0, the operator e−sHV is an integral operator [258, Corollary 25.9], and we denote its kernel by Ks,V .

8.3 The density of states trace formula

� 443

We will compute the traces in (8.11) by the formulas Tr(e−sHV MχB(0,R) ) = ∫ Ks,V (t, t) dt B(0,R)

and −dr Tr(e−sHV M⟨x⟩ ) = ∫ Ks,V (t, t)⟨t⟩−dr dt. ℝd

Although the kernel Ks,V is only a priori defined pointwise-almost everywhere, which means that the restriction to the diagonal t 󳨃→ Ks,V (t, t) is ambiguous, these identities are correct if we understand Ks,V (t, t) as the Lebesgue mean value of Ks,V at the point (t, t). For further details, see Brislawn’s version of Mercer’s theorem [42, Theorem 3.1]. Then (8.11) is equivalent to the claim that lim (r − 1) ∫ Ks,V (t, t)⟨t⟩−dr dt = lim R−d ∫ Ks,V (t, t) dt,

r→1+

R→∞

ℝd

B(0,R)

which is immediately plausible and can be justified by the following Abelian theorem. Lemma 8.3.2. Let F be a locally integrable function on ℝd . Then 1 1 lim+ (r − 1) ∫ ⟨t⟩−dr F(t) dt = lim ∫ F(t) dt R→∞ Vol(B(0, R)) Vol(B(0, 1)) r→1 B(0,R)

ℝd

whenever the right-hand side limit exists. Lemma 8.3.2 may be proved from the following single-variable statement. Lemma 8.3.3. If f ∈ Lloc 1 [0, ∞), then R

∞

lim+ (r − 1) ∫ ⟨u⟩−dr f (u) du = lim R−d ∫ f (u) du

r→1

0

R→∞

(8.12)

0

whenever the limit on the right exists. Proof. Assume that f ∈ Lloc 1 [0, ∞) is a function such that there exists c ∈ ℂ such that R

∫(f (u) − cdud−1 )du = ρ(R), 0

where ρ is a function such that limR→∞ R−d ρ(R) = 0. Replacing f by the function u 󳨃→ f (u) − cdud−1 if necessary, we may assume that c = 0. Then the assumption becomes

444 � 8 Density of states R

∫ f (u) du = ρ(R), 0

where ρ(R) = o(Rd ) as R → ∞. By the fundamental theorem of calculus we have f = ρ′ . Using integration by parts, we write ∞

∞

∞

∫ ⟨u⟩−dr f (u) du = ∫ ⟨u⟩−dr ρ′ (u) du = dr ∫ u⟨u⟩−dr−2 ρ(u)du. 0

0

0

Fix ε > 0 and let |ρ(u)| < εud for u > u(ε) > 1. It is immediate that ∞ 󵄨󵄨 ∞ 󵄨󵄨 u(ε) 󵄨󵄨 󵄨 󵄨󵄨 ∫ u⟨u⟩−dr−2 ρ(u)du󵄨󵄨󵄨 ≤ ∫ 󵄨󵄨󵄨ρ(u)󵄨󵄨󵄨du + ε ∫ ud+1 ⟨u⟩−dr−2 du 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 0 0 u(ε) u(ε)

∞

󵄨 󵄨 ≤ ∫ 󵄨󵄨󵄨ρ(u)󵄨󵄨󵄨du + ε ∫ u−d(r−1)−1 du 0

u(ε)

1

ε 󵄨 󵄨 = ∫ 󵄨󵄨󵄨ρ(u)󵄨󵄨󵄨du + . d(r − 1) 0

Consequently, 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim sup(r − 1)󵄨󵄨󵄨 ∫ u⟨u⟩−dr−2 ρ(u)du󵄨󵄨󵄨 ≤ ε. 󵄨 󵄨󵄨 󵄨󵄨 r→1 󵄨 0 Since ε > 0 is arbitrary, it follows that 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 󵄨󵄨 lim sup(r − 1)󵄨󵄨󵄨 ∫ u⟨u⟩−dr−2 ρ(u)du󵄨󵄨󵄨 = 0. 󵄨 󵄨󵄨 󵄨󵄨 r→1 󵄨 0 Therefore ∞

lim(r − 1) ∫ u⟨u⟩−dr−2 ρ(u)du = 0, r→1

0

and ∞

lim(r − 1) ∫ ⟨u⟩−dr f (u) du = 0. r→1

0

8.3 The density of states trace formula

� 445

Proof of Lemma 8.3.2. Set G(t) =

1 ∫ F(tθ)dθ, Vol(𝕊d−1 )

t ∈ ℝd .

𝕊d−1

It is immediate that ∫ F(t)dt = ∫ G(t)dt, B(0,R) −dr

∫ ⟨t⟩

R > 0,

B(0,R)

F(t)dt = ∫ ⟨t⟩−dr G(t)dt,

ℝd

r > 1.

ℝd

It therefore suffices to prove the statement for G instead of F. Clearly, G is radially symmetric. Therefore it suffices to prove the statement for a radially symmetric function F. If F is radially symmetric, then write F(t) = f (|t|), where f ∈ L∞ [0, ∞). Then ∫ F(t) dt = Vol(𝕊

d−1

R

) ∫ ud−1 f (u) du, 0

B(0,R)

and ∞

∫ F(t)⟨t⟩−dr dt = Vol(𝕊d−1 ) ∫ ⟨u⟩−dr ud−1 f (u) du. 0

ℝd

The claim of the lemma is hence reduced to that of Lemma 8.3.3 (applied with the function u 󳨃→ ud−1 f (u)). Corollary 8.3.4. Let d ≥ 2. If the density of states measure νHV exists, then ∫ e−sλ dνHV (λ) = ℝ

1 −dr lim (r − 1) Tr(e−sHV M⟨x⟩ ) |B(0, 1)| r→1+

for all s > 0. Proof. By Theorem 8.1.6 we have ∫ e−sλ dνHV (λ) = lim ℝ

R→∞

= lim

R→∞

By Lemma 8.3.2 we have

1 Tr(MχB(0,R) e−sHV ) |B(0, R)| 1 ∫ Ks,V (x, x) dx. |B(0, R)| B(0,R)

446 � 8 Density of states ∫ e−sλ dνHV (λ) =

1 lim (r − 1) ∫ ⟨x⟩−dr Ks,V (x, x) dx. |B(0, 1)| r→1+ ℝd

ℝ

The statement now follows from the equality −dr ). ∫ ⟨x⟩−dr Ks,V (x, x) dx = Tr(e−sHV M⟨x⟩ ℝd

The final step in the proof of Theorem 8.1.3 crucially relies on the following wellknown property of the Laplace transform of measures. Lemma 8.3.5. Let ν1 and ν2 be two finite Borel measures on [0, ∞) such that ∫ e−st dν1 (t) = ∫ e−st dν2 (t), ℝ

s ≥ 0,

ℝ

that is, the Laplace transforms of ν1 and ν2 coincide. Then ν1 = ν2 . Proof. Putting s = 0, we infer ν1 ([0, ∞)) = ν2 ([0, ∞)), where both numbers are finite. Hence we may assume without loss of generality that ν1 and ν2 are probability measures. The statement now follows from, for example, [131, Theorem 1, p. 430]. Finally, we conclude with the proof of Theorem 8.1.3. Proof of Theorem 8.1.3. Note that adding a constant to HV induces a shift in the measure νHV , and hence we may assume without loss of generality that V ≥ 0, and hence νHV is supported in ℝ+ . Corollary 8.3.4 yields ∫ e−sλ dνHV (λ) = ℝ

1 −dr lim (r − 1) Tr(e−sHV M⟨x⟩ ). |B(0, 1)| r→1+

Theorem 8.3.1 identifies the limit above as 1 −r −d lim (r − d) Tr(e−sHV M⟨x⟩ ) = Trω (e−sHV M⟨x⟩ ). d r→d+ Therefore, for every s > 0 and every extended limit ω, we have ∫ e−sλ dνHV (λ) = ℝ

1 −d Tr (e−sHV M⟨x⟩ ). |B(0, 1)| ω

(8.13)

Theorem 8.2.1 implies that if f ∈ C0 (ℝ+ ), then 󵄨󵄨 −d −d 󵄨 󵄨󵄨Trω (f (HV )⟨HV ⟩ M⟨x⟩ )󵄨󵄨󵄨 ≤ CV ⋅ ‖f ‖∞ for a constant CV > 0.

(8.14)

8.3 The density of states trace formula

� 447

Since V ≥ 0, it follows that HV ≥ 0. Hence the mapping −d f 󳨃→ Trω (f (HV )⟨HV ⟩−d M⟨x⟩ ),

f ∈ C0 (ℝ+ ),

defines a positive bounded linear functional on C0 (ℝ+ ). By the Riesz theorem there exists a Borel measure ρHV on ℝ+ such that −d Trω (f (HV )⟨HV ⟩−d M⟨x⟩ ) = ∫ f dρHV ,

f ∈ C0 (ℝ+ ).

ℝ+

Setting f (t) = e−st ⟨t⟩d , we obtain −d Trω (e−sHV M⟨x⟩ ) = ∫ e−st ⟨t⟩d dρHV (t),

s > 0.

(8.15)

ℝ+

Combining (8.15) with (8.13), we obtain ∫ e−sλ dνHV (λ) = ℝ

1 ∫ e−st ⟨t⟩d dρHV (t), |B(0, 1)|

s > 0.

ℝ+

Fixing ε > 0, we write ∫ e−sλ ⋅ e−ελ dνHV (λ) = ℝ

1 ∫ e−st ⋅ e−εt ⟨t⟩d dρHV (t), |B(0, 1)|

s > 0.

ℝ+

The measures A 󳨃→ ∫ e−ελ dνHV (λ), A

A 󳨃→

1 ∫ e−εt ⟨t⟩d dρHV (t) |B(0, 1)| A

are positive and finite, and their Laplace transforms coincide. By Lemma 8.3.5 these measures coincide. Passing to the limit as ε → 0, we obtain by the monotone convergence theorem that νHV (A) =

1 ∫⟨t⟩d dρHV (t). |B(0, 1)| A

Now let f ∈ Cc (ℝ+ ). Set f1 (t) := f (t)⟨t⟩d ∈ C0 (ℝ+ ). By the above arguments, −d −d Trω (f (HV )M⟨x⟩ ) = Trω (f1 (HV )⟨HV ⟩−d M⟨x⟩ ) = ∫ f1 dρHV = ∫ f dνHV , ℝ+

and the statement follows.

ℝ+

448 � 8 Density of states

8.4 Invariance under perturbation Theorem 8.1.3 makes certain features of the density of states more transparent. One feature of the DOS measure is the invariance under “small” variations of the potential. There are various ways of making this precise (see [259, Theorem C.7.7]), but one possible approach is the following. Let d ≥ 2. Let V ∈ L∞ (ℝd ) be real valued, and let HV := −Δ + MV be the corresponding Schrödinger operator. We are interested in perturbations HV +V0 = HV + MV0 , where V0 ∈ L∞ (ℝd ) is real valued such that MV0 esΔ is a compact operator on L2 (ℝd ) for any s > 0. We have the following: Theorem 8.4.1. Let d ≥ 2. Let V , V0 ∈ L∞ (ℝd ) be real-valued functions such that MV0 esΔ ∈ 𝒞0 (L2 (ℝd )) for all s > 0. If the density of states measure exists for both HV and HV +V0 , then the two measures are equal. This theorem reflects the stability of the density of states under “localized” perturbations. For example, if V0 is compactly supported, then MV0 esΔ is compact for every s > 0 by Theorem 1.5.6. More generally, there is the following simple criterion. Lemma 8.4.2. Let V0 ∈ L∞ (ℝd ) be a potential such that for all t > 0, 󵄨󵄨 󵄨 󵄨 d 󵄨 󵄨󵄨{x ∈ ℝ : 󵄨󵄨󵄨V0 (x)󵄨󵄨󵄨 ≥ t}󵄨󵄨󵄨 < ∞. Then MV0 esΔ is compact for all s > 0. Proof. Let ε > 0, and set 󵄨 󵄨 Aε := {x ∈ ℝd : 󵄨󵄨󵄨V0 (x)󵄨󵄨󵄨 ≥ ε}. Then ‖(1 − χAε )V0 ‖∞ ≤ ε, and hence 󵄩󵄩 󵄩 󵄩 sΔ 󵄩 sΔ sΔ 󵄩 sΔ 󵄩 󵄩󵄩MV0 e − MχAε V0 e 󵄩󵄩󵄩∞ = 󵄩󵄩󵄩M(1−χAε )V e 󵄩󵄩󵄩∞ ≤ ε󵄩󵄩󵄩e 󵄩󵄩󵄩∞ = ε. Therefore in the operator norm we have

8.4 Invariance under perturbation

� 449

lim MχA MV0 esΔ = MV0 esΔ .

ε→0

ε

On the other hand, Aε has finite measure, and V0 is bounded, so it follows that the function χAε V0 is square integrable. By Theorem 1.5.6 the operator MχA V0 esΔ is Hilbert– ε

Schmidt and, in particular, compact. Thus MV0 esΔ is the limit in the operator norm of a sequence of compact operators, and so it is itself compact. We now prove Theorem 8.4.1. Let the real-valued potential V ∈ L∞ (ℝd ) be fixed. Lemma 8.4.3. If V0 ∈ L∞ (ℝd ) is such that MV0 esΔ is compact for all s > 0, then MV0 e−sHV is also compact for all s > 0. Proof. By taking the adjoint it suffices to check that e−sHV MV0 is compact. Using Duhamel’s formula e

−sHV

sΔ

1

= e − s ∫ e−sθHV MV es(1−θ)Δ dθ, 0

it follows that 1

e−sHV MV0 − esΔ MV0 = −s ∫ e−sθHV MV es(1−θ)Δ MV0 dθ. 0

We will complete the proof by showing that the integral on the right-hand side is a convergent 𝒞0 (L2 (ℝd ))-valued Bochner integral and in particular is an element of 𝒞0 (L2 (ℝd )). Note that the integrand belongs to 𝒞0 (L2 (ℝd )) for each θ ∈ (0, 1) and has a uniformly bounded operator norm. Since 𝒞0 (L2 (ℝd )) is separable, the Bochner theorem implies that to ensure Bochner integrability, it suffices to show that θ 󳨃→ e−sθHV MV e−s(1−θ)Δ MV0 is weakly measurable. Since the semigroups e−sθHV and es(1−θ)Δ are strongly continuous and uniformly bounded, it follows that the integrand is weakly continuous and hence weakly measurable. Thus the integral defines an element of 𝒞0 (L2 (ℝd )), which completes the proof. Lemma 8.4.4. If V0 ∈ L∞ (ℝd ) is such that MV0 esΔ is compact for all s > 0, then the difference e−s(HV +MV0 ) − e−sHV is compact for all s > 0. Proof. Using Duhamel’s formula e

−s(HV +MV0 )

=e

−sHV

1

− s ∫ e−sθ(HV +MV0 ) MV0 e−s(1−θ)HV dθ, 0

450 � 8 Density of states Lemma 8.4.3 implies that the integrand e−sθ(HV +MV0 ) MV0 e−s(1−θ)HV is compact for all θ ∈ [0, 1] and s > 0. A reasoning similar to the proof of Lemma 8.4.3 implies that the integral converges as a 𝒞0 (L2 (ℝd ))-valued Bochner integral, and hence the difference e−s(HV +MV0 ) − e−sHV is compact. To prove Theorem 8.4.1, we will use (8.15) from the proof of Theorem 8.1.3, which states that −d Trω (e−sHV M⟨x⟩ )=

Vol(𝕊d−1 ) ⋅ ∫ e−sλ dνHV (λ), d

s > 0.

ℝ

The proof of Theorem 8.4.1 now follows from Theorem 8.1.3, the uniqueness of the Laplace transform (see Remark 8.3.5), and the following statement. Proposition 8.4.5. Suppose that HV = −Δ + MV is a Schrödinger operator with bounded real-valued potential V , and let V0 ∈ L∞ (ℝd ) be a real-valued potential such that MV0 esΔ is compact for all s > 0. Then for all Dixmier traces Trω , where ω is an extended limit on l∞ , and all s > 0, we have −d −d Trω (e−sHV M⟨x⟩ ) = Trω (e−s(HV +MV0 ) M⟨x⟩ ).

Proof. For brevity, set H := HV . It follows from Corollary 8.2.2 that the operators −d −d e−sH M⟨x⟩ and e−s(H+MV0 ) M⟨x⟩ individually belong to ℒ1,∞ for each s > 0. So each side of the equality is meaningful. Using the cyclic property of the Dixmier trace, we have s

s

−d −d − 2 H Trω (e−sH M⟨x⟩ ) = Trω (e− 2 H M⟨x⟩ e ),

and similarly s

s

−d −d − 2 (H+MV0 ) Trω (e−s(H+MV0 ) M⟨x⟩ ) = Trω (e− 2 (H+MV0 ) M⟨x⟩ e ).

For each s > 0, we have the identity s

s

s

s

−d − 2 (H+MV0 ) −d − 2 H e− 2 (H+MV0 ) M⟨x⟩ e − e− 2 H M⟨x⟩ e s

s

s

−d − 2 (H+MV0 ) = (e− 2 (H+MV0 ) − e− 2 H )M⟨x⟩ e s

s

s

−d − 2 (H+MV0 ) + e− 2 H M⟨x⟩ (e − e− 2 H ). s

s

−d −d − 2 (H+MV0 ) The operators e− 2 H M⟨x⟩ and M⟨x⟩ e belong to ℒ1,∞ , and Lemma 8.4.4 implies s

s

that the difference e− 2 (H+MV0 ) − e− 2 H is compact. Hence s

s

s

s

−d − 2 (H+MV0 ) −d − 2 H e− 2 (H+MV0 ) M⟨x⟩ e − e− 2 H M⟨x⟩ e ∈ 𝒞0 ⋅ ℒ1,∞ + ℒ1,∞ ⋅ 𝒞0 .

The product of a compact operator and a weak trace class operator belongs to (ℒ1,∞ )0 and in particular is in the kernel of every Dixmier trace. Thus

8.5 Density of states for homogeneous potentials

s

s

s

�

451

s

−d − 2 (H+MV0 ) −d − 2 H Trω (e− 2 (H+MV0 ) M⟨x⟩ e ) = Trω (e− 2 H M⟨x⟩ e ).

Once again using the cyclic property of the trace, we immediately obtain the result.

8.5 Density of states for homogeneous potentials Theorem 8.1.3 can be used in combination with the trace theorem, Theorem 5.3.5 in Section 5.3, to compute the density of states for radially homogeneous potentials. A potential V ∈ L∞ (ℝd ) is said to be radially homogeneous if V (tx) = V (x),

x ∈ ℝd , t > 0.

The density of states for the Schrödinger operator HV = −Δ + MV is known to exist when V is radially homogeneous [17]. The following theorem computes νHV explicitly since νHα⋅1 can be computed for any scalar multiple α ⋅ 1, α ∈ ℝ, of the identity function 1 in L∞ (Example 8.1.2). Theorem 8.5.1. Let d ≥ 2. Denote H0 = −Δ and let HV = H0 + MV , where V ∈ L∞ (ℝd ) is real valued, radially homogeneous, and continuous on the unit sphere. Then the density of states of HV is the average over ξ ∈ 𝕊d−1 of the density of states of H0 + V (ξ) ⋅ 1; in other words, νHV =

1 ⋅ ∫ νH0 +V (ξ)⋅1 dξ. Vol(𝕊d−1 ) 𝕊d−1

Here V (ξ) ⋅ 1, ξ ∈ 𝕊d−1 , is a scalar multiple of the identity function in L∞ (ℝd ). The integral in Theorem 8.5.1 is understood in the sense that if I ⊆ ℝ is a Borel set, then νHV (I) =

1 ⋅ ∫ νH0 +V (ξ)⋅1 (I) dξ. Vol(𝕊d−1 ) 𝕊d−1

The integrand is indeed integrable due to the explicit expression for νH0 +V (ξ)⋅1 given in Example 8.1.2. Lemma 8.5.2. Let g be a compactly supported smooth function on ℝ with g(0) = 1. Then −d −d lim Trω (e−sHV M⟨x⟩ g(εH0 )) = Trω (e−sHV M⟨x⟩ ),

ε→0

s

s > 0.

Proof. Note that the operator (HV + i)e− 2 HV is bounded. The resolvent identity implies s that (H0 + i)(HV + i)−1 is bounded, so it follows that (H0 + i)e− 2 HV is also bounded. By the functional calculus for self-adjoint operators we have

452 � 8 Density of states lim g(εH0 )(H0 + i)−1 = (H0 + i)−1

ε→0

in the uniform norm. Hence as ε → 0, we have s

s

g(εH0 )e− 2 HV = g(εH0 )(H0 + i)−1 ⋅ (H0 + i)e− 2 HV s

s

→ (H0 + i)−1 ⋅ (H0 + i)e− 2 HV = e− 2 HV in the uniform norm. Therefore, as ε → 0, s

s

s

s

−d −d − 2 HV e− 2 HV M⟨x⟩ g(εH0 )e− 2 HV → e− 2 HV M⟨x⟩ e

in the ℒ1,∞ topology. s s We now write e−sHV = e− 2 HV e− 2 HV and use the cyclic property of Trω : s

s

−d −d Trω (e−sHV M⟨x⟩ g(sεH0 )) = Trω (e− 2 HV M⟨x⟩ g(sεH0 )e− 2 HV ).

The statement now follows from the preceding paragraph and the continuity of Trω . Lemma 8.5.3. If a measure ν is such that ∫ e−sλ dν(λ) =

d 1 ⋅ ∫ (4πs)− 2 e−sV (ξ) dξ, d−1 Vol(𝕊 )

s > 0,

𝕊d−1

ℝ

then ν(I) =

1 ⋅ ∫ νH0 +V (ξ)⋅1 (I) dξ. Vol(𝕊d−1 ) 𝕊d−1

Proof. Recall from Example 8.1.2 that the Laplace transform of the density of states for H0 is given by d

∫ e−sλ dνH0 (λ) = (4πs)− 2 ,

s > 0,

ℝ

and for the constant potential V = c, where c ∈ ℝ, we have d

∫ e−sλ dνH0 +c (λ) = (4πs)− 2 e−sc . ℝ

Therefore ∫ e−sλ dν(λ) = ℝ

1 ∫ ∫ e−sλ dνH0 +V (ξ) (λ)dξ. Vol(𝕊d−1 ) 𝕊d−1 ℝ

Finally, Fubini’s theorem and the uniqueness of the Laplace transform yield

8.5 Density of states for homogeneous potentials

ν=

�

453

1 ∫ νH0 +V (ξ) dξ Vol(𝕊d−1 ) 𝕊d−1

in the sense that if I ⊆ ℝ is a Borel set, then ν(I) =

1 ∫ νH0 +V (ξ) (I) dξ. Vol(𝕊d−1 ) 𝕊d−1

Proof of Theorem 8.5.1. Let s > 0. For each N > 0, performing N-fold iteration of Duhamel’s formula gives the Dyson expansion with remainder term, N

(−s)n ∫ e−θ0 sH0 MV e−θ1 sH0 MV ⋅ ⋅ ⋅ MV e−θn sH0 dθ n! n=1

e−sHV = e−sH0 + ∑

Δn

(−s)N+1 + ∫ e−θ0 sH0 MV ⋅ ⋅ ⋅ e−θN sH0 MV e−θN+1 sHV dθ, (N + 1)! ΔN+1

where Δn = {(θ0 , . . . , θn ): θ0 + ⋅ ⋅ ⋅ + θn = 1} is the n-simplex, and dθ is the normalized Lebesgue measure (so ∫Δ dθ = 1). For each fixed s > 0, because V is bounded, it is easy n to see that the remainder term tends to zero in the operator norm as N → ∞, so we have the operator-norm convergent Dyson series (−s)n ∫ e−θ0 sH0 MV e−θ1 sH0 MV ⋅ ⋅ ⋅ MV e−θn sH0 dθ. n! n=1 ∞

e−sHV = e−sH0 + ∑

Δn

Taking the Fourier transform, we have (−s)n ∫ e−θ0 sM|x|2 V (∇)e−θ1 sM|x|2 ⋅ ⋅ ⋅ V (∇)e−θn sM|x|2 dθ. n! n=1 ∞

2

e−s(M|x| +V (∇)) = e−sM|x|2 + ∑

Δn

For each n ≥ 1, the integral In defined by In := ∫ e−sθ0 M|x|2 V (∇)e−sθ1 M|x|2 V (∇) ⋅ ⋅ ⋅ e−sθn M|x|2 dθ Δn

converges in the C ∗ -algebra Π(C0 (ℝd ) + ℂ, C(𝕊d−1 )) (as defined in terms of the representations in Theorem 5.3.1). Indeed, at each θ ∈ Δn the integrand belongs to Π(C0 (ℝd ) + ℂ, C(𝕊d−1 )) and is norm continuous as a function of θ. Since the principal symbol function sym stated in Theorem 5.3.1 is norm continuous, we have 2

sym(In )(x, ξ) = ∫ e−s(θ0 +⋅⋅⋅+θn )|x| V (ξ)n dθ Δn

2

= e−s|x| V (ξ)n ,

x ∈ ℝd ,

ξ ∈ 𝕊d−1 .

454 � 8 Density of states The norm convergence of the Dyson series and the norm continuity of the principal symbol mapping imply that e−s(M|x|2 +V (∇)) ∈ Π(C0 (ℝd ), C(𝕊d−1 )) and (−s)n sym(In )(x, ξ) n! n=1 ∞

2

sym(e−s(M|x|2 +V (∇)) )(x, ξ) = e−s|x| + ∑

(−sV (ξ))n n! n=0 ∞

2

= e−s|x| ∑ 2

= e−s(|x| +V (ξ)) . Let g be a compactly supported smooth function on ℝ with g(0) = 1, and let ε > 0. Applying Theorem 5.3.5 (and the unitary invariance of the Dixmier trace), we have −d

d

Trω (e−sHV M⟨x⟩2 g(εH0 )) = Trω (g(εM|x|2 )e−s(M|x|2 +V (∇)) (1 − Δ)− 2 ) =

1

2

d(2π)d s

d 2

∫ e−sV (ξ) dξ ∫ g(ε|x|2 )e−|x| dx. 𝕊d−1

ℝd d

2

As ε → 0, the integral ∫ℝd e−|x| g(ε|x|2 ) dx converges to π 2 by the dominated convergence theorem. By Lemma 8.5.2 we have, for all s > 0, −d Trω (e−sHV M⟨x⟩ )=

1 d(4π)

d

d 2

∫ s− 2 e−sV (ξ) dξ. 𝕊d−1

The statement now follows from Lemma 8.5.3 and Theorem 8.1.3.

8.6 Notes The density of states The density of states is a notion of long-standing interest in the mathematical physics literature. Monographs that include details about this topic are [4, 25, 56, 219, 240, 280]. The definition we use here is identical to that of [259], but as indicated, some variations in the literature do exist. Dixmier trace formula for the DOS The content of this chapter is primarily based on the paper [16], although the proof given here benefits from a number of substantial simplifications. We have used the uniqueness property of the Laplace transform, with a short argument given in Theorem 8.3.5. This property is well known; see [285, Theorem II.6.3] or [13, Theorem 1.7.3] for a more modern treatment. The Dixmier trace formula in this chapter has also been proven to hold in certain discrete metric spaces [15].

8.6 Notes

� 455

Insensitivity to localized perturbations The fact that the density of states of H = −Δ + MV depends only on the asymptotics of V at infinity is very prominent in the theory. Some related facts are outlined in [259, Section C]. For example, in [259, Proposition C.7.7], it is proved that if V belongs to the closure of Cc∞ (ℝd ) in the Kato class Kd , then the density of states exists for −Δ + MV and is the same as the density of states for V = 0. An even stronger statement is proved in [166]. Radially homogeneous potentials Schrödinger operators with radially homogeneous potentials are studied in [152–154]. The formula in Theorem 8.5.1 was first proved in [16], although at that time the authors were unaware if the density of states exists for these potentials. An alternative proof of Theorem 8.5.1 based on semiclassical Weyl asymptotics is given in [17].

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Index Banach limit 8 – Factorizable 9 Beals’ theorem 93 Bessel potential 77, 88 Birman–Solomyak function space 47, 123 C ∗ -algebra – Tensor product 282 Calkin correspondence 5, 17, 118 Canonical commutation relations 214, 276 Cesaro mean 8 Commutator XVII Compact operator 3, 5 Connes character formula 396, 425 Connes–Chern character 393 Decreasing rearrangement – Function 15 – Sequence 5 Density of states 426 Dirac operator 350 Divided difference function 50, 123 Double operator integral 41, 49, 123 Eigenvalue sequence 4, 118 Extended limit XXXVIII, 8 Fan inequalities 32 Fourier transform 52, 54, 83 – Discrete 78 Fubini’s theorem 33 Fully symmetric – Function space 17, 22, 122 – Ideal see Ideal, Fully symmetric – Operator space 22, 68 – Sequence space 5, 72 Hausdorff–Young inequality 123 – Noncommutative torus 192 Hilbert space 3 – Direct product 33 – Direct sum 29 – Of square integrable functions 21 Hochschild (co)homology 394 Hölder inequality – Lorentz function spaces 16 https://doi.org/10.1515/9783110700176-010

– Lorentz ideals 32 Hörmander symbol class 83, 126 Ideal 5 – Banach 7 – Fully symmetric 12, 23, 52, 68, 118 – Lorentz 6, 32, 34, 68, 74 – of compact operators 5 – Quasi-Banach 7, 25 – Weak trace class XL, 7, 33 Integration formula for noncommutative plane see Trace formula, Noncommutative plane Interpolation 12, 18, 24, 49, 122 – Exact-space 14 – Pair 20 Lorentz function spaces 16, 119 Majorization 26, 30, 121 Modulated function 132 – Square- 134, 142 Moyal plane 210 Moyal product 124, 230, 277 Noncommutative geometry XVII Noncommutative integral 167, 197, 198 – Closed manifold 199 – Compactly supported square integrable functions XXIV, 171, 185 – Curved plane 179, 199 – Elliptic differential operators XXIV, 171 – Noncommutative torus 196, 204 – Torus 181, 204 Noncommutative Lorentz spaces 22, 70, 118 Noncommutative plane 210, 214 – Characterization 216 – Product-convolution estimate 213, 241 – Smooth product-convolution estimate 251 – Symplectic group 240 Noncommutative residue 116, 128 Noncommutative torus 190 – C ∗ -algebra 191 – Von Neumann algebra 190 Norm – Birman–Solomyak 47 – Laplacian modulated operator 129, 140 – Lorentz 6, 16, 22

470 � Index

– Modulated functions 132 – Modulus of concavity 72 – Operator 3 – Quasi- 7 – Square modulated functions 134 Operator – τ-measurable 22 – Absolute value 4, 119 – Adjoint 3 – Affiliated 22 – Bounded 3 – Compact 3, 5 – Compactly supported 143 – Contraction 12 – Convolution 55 – Diagonal 4 – Disjoint from the left/right 30 – Dixmier measurable 11 – Finite-rank 3 – Gradient 55 – Hilbert–Schmidt 7, 57, 125, 137 – Integral kernel 57 – Integral symbol 57 – Laplace–Beltrami 177 – Laplacian 77, 129 – Laplacian modulated 129, 136, 163 – Measurable 11 – Norm 3 – Pairwise orthogonal 64 – Positive 3 – Product 56 – Product-convolution 56, 57, 60, 74, 78, 124 – Pseudodifferential see Pseudodifferential operator – Rank 3 – Schrödinger 426 – Self-adjoint 3 – Smoothing 85 – Trace class 7, 43, 58, 126 – Unitary 3 – Unitary equivalence 3 – Universally measurable 11 Pseudodifferential operator 83, 85, 126, 163 – Adjoint 86 – Asymptotic expansion 110 – Classical 112 – Compactly supported 107, 127

– Complex powers of 96 – Elliptic 105, 127, 170 – m-homogeneous symbol 112 – Principal symbol 112 – Principal symbol class 115 – Symbol 83, 85 Quantized differential XXXV, 349, 351 Schur’s test 89 Schwartz function 82 – Noncommutative plane 222, 277 – Twisted convolution 228 Shift operator 8 Singular value – Function 5, 15, 22, 23, 64, 118 – Sequence 4, 118 Singular value estimate – Product-convolution operators 74 – Product-convolution operators on tori 79 Sobolev space 88, 128, 352 – Noncommutative plane 213, 234 Spectral triple 391, 425 – Dimension 392 – Parity 392 – Smooth 392 Submajorization 12, 16, 17, 24, 46, 49, 64, 89, 122 – Logarithmic 25 Symbol see Pseudodifferential operator, Symbol Symbol map – Abstract 280, 281, 284 – Noncommutative plane 315 – Noncommutative torus 303 – SU(2) 335 Tempered distribution 82 Trace XVIII, 7, 121 – Continuous 8 – Diagonal formula 9 – Dixmier XL, XVII, 8 – Normalized 8 – Positive 8 – Semifinite trace 20 – Singular XL, XVII, 10, 121 Trace formula – Abstract 280, 286 – Connes’ 131, 160, 162, 278 – Density of states 430, 446 – Integral operators 58, 129

Index

– Laplacian modulated operators 130, 152 – Noncommutative plane 214, 265, 275, 321 – Noncommutative torus 310 – Product-convolution operators 77, 297 – Pseudodifferential operators 109, 153 – Quantized differential 353, 371, 383 – SU(2) 344 – Zero order pseudodifferential operators 297 Transformer 49 Unique trace theorem 131

von Neumann algebra 20, 63 – Commutant 21 – Lattice of projections 20 – Projection 20 – Spatial tensor product 63 Weak operator topology – Integrable 42 – Integral 43, 47 – Measurable 42 Weyl transform 222, 277

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