The d-bar Neumann Problem and Schrödinger Operators 9783111182926, 9783111182902

Table of contents :
Preface to the first edition
Preface to the second edition
Contents
1 Bergman spaces
2 The canonical solution operator to ᾱ
3 Spectral properties of the canonical solution operator to ᾱ
4 The ᾱ-complex
5 Density of smooth forms
6 The weighted ᾱ-complex
7 The twisted ᾱ-complex
8 Applications
9 Spectral analysis
10 Schrödinger operators and Witten Laplacians
11 Compactness
12 The ᾱ-Neumann operator and the Bergman projection
13 Compact resolvents
14 Spectrum of ◻ on the Fock space
15 Obstructions to compactness
16 The ᾱ-complex on the Segal–Bargmann space
Bibliography
Index

Citation preview

Friedrich Haslinger ̄ The 𝜕-Neumann Problem and Schrödinger Operators

De Gruyter Expositions in Mathematics

�

Edited by Lev Birbrair, Fortaleza, Brazil Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York City, New York, USA Markus J. Pflaum, Boulder, Colorado, USA Dierk Schleicher, Marseille, France Katrin Wendland, Freiburg, Germany

Volume 59

Friedrich Haslinger

̄ The 𝜕-Neumann Problem and Schrödinger Operators �

2nd edition

Mathematics Subject Classification 2020 Primary: 32W05, 32A25, 32A36; Secondary: 35P05, 35J10 Author Prof. Dr. Friedrich Haslinger Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 1090 Vienna Austria [email protected]

ISBN 978-3-11-118290-2 e-ISBN (PDF) 978-3-11-118292-6 e-ISBN (EPUB) 978-3-11-118467-8 ISSN 0938-6572 Library of Congress Control Number: 2023940864 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 to the first edition The subject of this book is complex analysis in several variables and its connections to partial differential equations and functional analysis. The first sections of each chapter contain prerequisites from functional analysis, Sobolev spaces, partial differential equations, and spectral analysis, which are used in the following sections devoted to the main topic of the book. In this way the book becomes self-contained, with only one exception, where we do not provide all details in the proof of the general spectral theorem for unbounded self-adjoint operators. We concentrate on the Cauchy–Riemann equation (𝜕-equation) and investigate the properties of the canonical solution operator to 𝜕, the solution with minimal L2 -norm and its relationship to the 𝜕-Neumann operator. The first chapter contains a discussion of Bergman spaces in one and several complex variables, including basic facts on Hilbert spaces. In the second chapter the solution operator to 𝜕 restricted to holomorphic L2 functions in one complex variable is investigated, pointing out that the Bergman kernel of the associated Hilbert space of holomorphic functions plays an important role. We investigate operator properties like compactness and Schatten class membership, also for the solution operator on weighted spaces of entire functions (Fock spaces). In the third chapter, we generalize the results to several complex variables and explain some new phenomena that do not appear in one variable. In the following, we consider the general 𝜕-complex and derive properties of the complex Laplacian on L2 -spaces of bounded pseudoconvex domains and on weighted L2 -spaces. For this purpose, we first concentrate on basic results about distributions, Sobolev spaces, and unbounded operators on Hilbert spaces. The key result in Kohn’s far-reaching method is the Kohn–Morrey formula, which is presented in different versions. Using this formula, the basic property of the 𝜕-Neumann operator – the bounded inverse of the complex Laplacian – is proved. In recent years, it has turned out to be useful to investigate an even more general situation, namely the twisted 𝜕-complex, where 𝜕 is composed with a positive twist factor. In this way, we obtain a rather general basic estimate, from which we get Hörmander’s L2 -estimates for the solution of the Cauchy– Riemann equation together with results on related weighted spaces of entire functions, such as that these spaces are infinite dimensional if the eigenvalues of the Levi matrix of the weight function show a certain behavior at infinity. In addition, it is pointed out that some L2 -estimates for 𝜕 can be interpreted in the sense of a general Brascamp–Lieb inequality. The next chapter contains a detailed account of the application of the 𝜕-methods to Schrödinger operators, Pauli and Dirac operators, and Witten Laplacians. In this context, spectral analysis plays an important role. Therefore an extensive chapter on spectral analysis was inserted to provide a better understanding for the operator theoretic aspects in the 𝜕-Neumann problem, which, in particular, is used to exactly describe the spectrum of complex Laplacian on the Fock space. Returning to the 𝜕-Neumann problem, we characterize the compactness of the 𝜕-Neumann operator using a description https://doi.org/10.1515/9783111182926-201

VI � Preface to the first edition of precompact subsets in L2 -spaces. The compactness of the 𝜕-Neumann operator is also related to properties of commutators of the Bergman projection and multiplication operators. In the last part, we use the 𝜕-methods and some spectral theory to settle the question whether certain Schrödinger operators with a magnetic field have a compact resolvent. It is also shown that a large class of Dirac operators fail to have a compact resolvent. Finally, we exhibit some situations where the 𝜕-Neumann operator is not compact. Numerous references for the topics of the text and for additional results are given in the notes at the end of each chapter. Most of the material of the book stems from various lectures of the author given at the University of Vienna, the Erwin Schrödinger International Institute for Mathematical Physics (ESI) in Vienna and at CIRM, Luminy, during programs on the 𝜕-Neumann operator in recent years. The author is indebted to both institutions, ESI and CIRM, for their help and hospitality. I would also like to thank my students Franz Berger, Damir Ferizović, and Tobias Preinerstorfer for their constructive criticisms of the manuscript and also for their help in eliminating a number of typos and minor errors. Vienna, March 2014

Friedrich Haslinger

Preface to the second edition We tried to correct some misprints of the first edition. Chapter 1 contains some new sections on the Segal–Bargmann space, the uncertainty principle, and nonradial weights. In Chapter 11, we included a complete treatment of magnetic Schrödinger operators with compact resolvent due to Iwatsuka [59]. In addition, we give a characterization of compact subsets in the Segal–Bargmann space, which is used in the new chapters. With the help of Muckenhoupt characteristics, we present a concise compactness criterion for ̄ the weighted 𝜕-operator on ℂ. This criterion shows the remarkable interplay between methods of real analysis (Schrödinger operators) and complex analysis (𝜕̄ equation). We also included a new chapter on the 𝜕-complex with the Segal–Bargmann space as an un̄ derlying Hilbert space. In many aspects, we can use methods of the 𝜕-operator for its analysis. The creation and annihilation operators of quantum mechanics play an important role. The last part is devoted to the generalized 𝜕-complex, where only partial results can be delivered. The main problem consists in analyzing the basic estimates, which can be seen as generalizations of the uncertainty principle, hence raising many interesting questions for further research. I was partially supported by the FWF-grant P 36884 of the Austrian Science Fund. Vienna, June 2023

https://doi.org/10.1515/9783111182926-202

Friedrich Haslinger

Contents Preface to the first edition � V Preface to the second edition � VII 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Bergman spaces � 1 Elementary properties � 1 Examples � 9 Biholomorphic maps � 12 The Segal–Bargmann space � 15 The uncertainty principle � 19 Nonradial weights � 22 Notes � 25

2 2.1 2.2 2.3

The canonical solution operator to 𝜕 � 26 Compact operators on Hilbert spaces � 26 The canonical solution operator to 𝜕 restricted to A2 (𝔻) � 37 Notes � 47

3 3.1 3.2 3.3 3.4

Spectral properties of the canonical solution operator to 𝜕̄ � 49 Complex differential forms � 49 (0, 1)-forms with holomorphic coefficients � 50 Compactness and Schatten class membership � 52 Notes � 63

4 4.1 4.2 4.3 4.4 4.5

The 𝜕-complex � 65 Unbounded operators on Hilbert spaces � 65 Distributions � 82 A finite-dimensional analog � 88 The 𝜕-Neumann operator � 89 Notes � 106

5 5.1 5.2 5.3

Density of smooth forms � 107 Friedrichs’ lemma and Sobolev spaces � 107 Density in the graph norm � 119 Notes � 125

6 6.1 6.2 6.3

The weighted 𝜕-complex � 126 The 𝜕-Neumann operator on (0, 1)-forms � 126 (0, q)-forms � 131 Notes � 134

X � Contents 7 7.1 7.2 7.3

The twisted 𝜕-complex � 136 An exact sequence of unbounded operators � 136 The twisted basic estimates � 137 Notes � 141

8 8.1 8.2 8.3

Applications � 142 Hörmander’s L2 -estimates � 142 Weighted spaces of entire functions � 145 Notes � 150

9 9.1 9.2 9.3 9.4 9.5 9.6

Spectral analysis � 151 Resolutions of the identity � 151 Spectral decomposition of bounded normal operators � 154 Spectral decomposition of unbounded self-adjoint operators � 161 Determination of the spectrum � 178 Variational characterization of the discrete spectrum � 190 Notes � 195

10 10.1 10.2 10.3 10.4 10.5 10.6

Schrödinger operators and Witten Laplacians � 196 Difference quotients � 196 Interior regularity � 198 Schrödinger operators with magnetic field � 202 Witten Laplacians � 209 Dirac and Pauli operators � 212 Notes � 213

11 11.1 11.2 11.3 11.4 11.5 11.6 11.7

Compactness � 215 Precompact sets in L2 -spaces � 215 Sobolev spaces and Gårding’s inequality � 218 Magnetic Schrödinger operators with compact resolvent � 223 Compactness in weighted spaces � 231 Compactness in the Segal–Bargmann space � 247 Bounded pseudoconvex domains � 249 Notes � 252

12 12.1 12.2 12.3

The 𝜕-Neumann operator and the Bergman projection � 254 The Stone–Weierstraß theorem � 254 Commutators of the Bergman projection � 257 Notes � 263

13 13.1

Compact resolvents � 264 Schrödinger operators � 264

Contents �

13.2 13.3

Dirac and Pauli operators � 271 Notes � 274

14 14.1 14.2 14.3

Spectrum of ◻ on the Fock space � 275 The general setting � 275 Determination of the spectrum � 277 Notes � 282

15 15.1 15.2 15.3

Obstructions to compactness � 283 The bidisc � 283 Weighted spaces � 284 Notes � 288

16 16.1 16.2 16.3 16.4 16.5

The 𝜕-complex on the Segal–Bargmann space � 290 Unbounded operators and the complex Laplacian � 291 Basic estimates and compactness � 293 Properties of the graph norm and compactness � 298 The generalized 𝜕-complex � 304 Notes � 315

Bibliography � 317 Index � 321

XI

1 Bergman spaces To investigate the solution to the inhomogeneous 𝜕-equation 𝜕u = g, we will first consider the case where the right-hand side g is a holomorphic function. Therefore we need an appropriate Hilbert space of holomorphic functions, the Bergman space. We will use standard basic facts about Hilbert spaces, such as the Riesz representation theorem for continuous linear functional and facts about orthogonal projections and complete orthonormal bases. Let Ω ⊆ ℂn be a domain. The Bergman space is defined as 󵄨 󵄨2 A2 (Ω) = {f : Ω 󳨀→ ℂ holomorphic : ‖f ‖2 = ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 dλ(z) < ∞}, Ω

where λ is the Lebesgue measure on ℂn . The inner product is given by (f , g) = ∫ f (z) g(z) dλ(z) Ω

for f , g ∈ A2 (Ω).

1.1 Elementary properties For simplicity, we first restrict ourselves to domains Ω ⊆ ℂ. We consider special continuous linear functionals on A2 (Ω), the point evaluations. Let f ∈ A2 (Ω) and fix z ∈ Ω. By Cauchy’s integral theorem we have f (z) =

f (ζ ) 1 dζ , ∫ 2πi ζ − z γs

where γs (t) = z + seit , t ∈ [0, 2π], 0 < s ≤ r, and D(z, r) = {w : |w − z| < r} ⊂ Ω. Using polar coordinates and integrating the above equality with respect to s between 0 and r, we get f (z) =

1 πr 2

∫ f (w) dλ(w). D(z,r)

Then, by the Cauchy–Schwarz inequality, 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ 2 ∫ 1 . 󵄨󵄨󵄨f (w)󵄨󵄨󵄨 dλ(w) πr D(z,r)

https://doi.org/10.1515/9783111182926-001

(1.1)

2 � 1 Bergman spaces 1/2

≤

1 ( ∫ 12 dλ(w)) πr 2 D(z,r)

≤

󵄨2 󵄨 ( ∫ 󵄨󵄨󵄨f (w)󵄨󵄨󵄨 dλ(w))

1 󵄨2 󵄨 (∫󵄨󵄨󵄨f (w)󵄨󵄨󵄨 dλ(w)) π 1/2 r Ω

1/2

D(z,r)

1/2

≤

1 ‖f ‖. π 1/2 r

If K is a compact subset of Ω, then there is r(K) > 0 such that for all z ∈ K, we have D(z, r(K)) ⊂ Ω, and we get 󵄨 󵄨 sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ z∈K

1 ‖f ‖. π 1/2 r(K)

If K ⊂ Ω ⊂ ℂn , then we can find a polycylinder P(z, r(K)) = {w ∈ ℂn : |wj − zj | < r(K), j = 1, . . . , n} such that for all z ∈ K, we have P(z, r(K)) ⊂ Ω. Hence by iterating the above Cauchy integrals we get the following: Proposition 1.1. Let K ⊂ Ω be a compact set. Then there exists a constant C(K), only depending on K, such that 󵄨 󵄨 sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ C(K) ‖f ‖

(1.2)

z∈K

for all f ∈ A2 (Ω). Proposition 1.2. A2 (Ω) is a Hilbert space. Proof. If (fk )k is a Cauchy sequence in A2 (Ω), by (1.2) it is also a Cauchy sequence with respect to uniform convergence on compact subsets of Ω. Hence the sequence (fk )k has a holomorphic limit f with respect to uniform convergence on compact subsets of Ω. On the other hand, the original L2 -Cauchy sequence has a subsequence that converges pointwise almost everywhere to the L2 -limit of the original L2 -Cauchy sequence (see, for instance, [83]), and so the L2 -limit coincides with the holomorphic function f . Therefore A2 (Ω) is a closed subspace of L2 (Ω) and itself a Hilbert space. In the sequel, we present basic facts about Hilbert spaces and their consequences for the Bergman spaces. Proposition 1.3. Let E be a nonempty, convex, and closed subset of the Hilbert space H, i. e., tx +(1−t)y ∈ E for all for x, y ∈ E and t ∈ [0, 1]. Then E contains a uniquely determined element of minimal norm. Proof. The parallelogram rule says that ‖x + y‖2 + ‖x − y‖2 = 2 ‖x‖2 + 2 ‖y‖2 ,

x, y ∈ H.

1.1 Elementary properties �

3

Let δ = inf{‖x‖ : x ∈ E}. For x, y ∈ E, we have 21 (x + y) ∈ E, and hence 󵄩 󵄩2 1/4 ‖x − y‖2 = 1/2 ‖x‖2 + 1/2 ‖y‖2 − 󵄩󵄩󵄩1/2(x + y)󵄩󵄩󵄩 , which implies that ‖x − y‖2 ≤ 2 ‖x‖2 + 2 ‖y‖2 − 4δ2 . So, if ‖x‖ = ‖y‖ = δ, then x = y (uniqueness). By the definition of δ there exists a sequence (yk )k in E such that ‖yk ‖ → δ as k → ∞. The estimate ‖yk − ym ‖2 ≤ 2 ‖yk ‖2 + 2 ‖ym ‖2 − 4δ2 implies that (yk )k is a Cauchy sequence in H. Since H is complete, there exists x0 ∈ H such that ‖yk − x0 ‖ → 0, and, as E is closed, we have x0 ∈ E; the mapping x 󳨃→ ‖x‖ is continuous, and therefore ‖x0 ‖ = limk→∞ ‖yk ‖ = δ. Theorem 1.4. Let M be a closed subspace of a Hilbert space H. Then there exist uniquely determined mappings P : H 󳨀→ M,

Q : H 󳨀→ M ⊥

such that (1) x = Px + Qx ∀x ∈ H. (2) for x ∈ M, we have Px = x, and hence P2 = P and Qx = 0; for x ∈ M ⊥ , we have Px = 0, Qx = x, and Q2 = Q. (3) The distance of x ∈ H to M is given by inf{‖x − y‖ : y ∈ M} = ‖x − Px‖. (4) For each x ∈ H, we have ‖x‖2 = ‖Px‖2 + ‖Qx‖2 . (5) P and Q are continuous, linear, and self-adjoint operators, and they are the orthogonal projections of H onto M and M ⊥ . Proof. For each x ∈ H, the set x +M = {x +y : y ∈ M} is convex. Hence, by Proposition 1.3, there exists a uniquely determined element of minimal norm in x + M, which is denoted by Qx. We set Px = x − Qx and see that Px ∈ M, since Qx ∈ x + M. Now we claim that Qx ∈ M ⊥ . We have to show that (Qx, y) = 0 for all y ∈ M. We can suppose that ‖y‖ = 1, and then we have

4 � 1 Bergman spaces (Qx, Qx) = ‖Qx‖2 ≤ ‖Qx − αy‖2 = (Qx − αy, Qx − αy)

∀α ∈ ℂ

by the minimality of Qx. Therefore we get 0 ≤ −α(y, Qx) − α(Qx, y) + |α|2 . Setting α = (Qx, y), we obtain 0 ≤ −|(Qx, y)|2 and (Qx, y) = 0; hence Q : H 󳨀→ M ⊥ . If x = x0 +x1 with x0 ∈ M and x1 ∈ M ⊥ , then x0 −Px = Qx −x1 , and since M ∩M ⊥ = {0}, we obtain x0 = Px and x1 = Qx, therefore P and Q are uniquely determined. In a similar way, we get that P(αx + βy) − αPx − βPy = αQx + βQy − Q(αx + βy). The left side belongs to M, the right side belongs to M ⊥ , and hence both sides are 0, which proves that P and Q are linear. Property 3 follows by the definition of Q, and property 4 by the fact that (Px, Qx) = 0

∀x ∈ H.

In addition, we have 󵄩󵄩 󵄩 󵄩󵄩Q(x − y)󵄩󵄩󵄩 = inf{‖x − y + m‖ : m ∈ M} ≤ ‖x − y‖, and hence Q and P = I − Q are continuous. For x, y ∈ H, we have (Px, y) = (Px, Py + Qy) = (Px, Py)

and (x, Py) = (Px + Qx, Py) = (Px, Py),

hence (Px, y) = (x, Py), and so P is self-adjoint. Corollary 1.5. If M ≠ H is a closed proper subspace of a Hilbert space H, then there exists an element y ≠ 0 such that y ⊥ M. Proof. Let x ∈ H be such that x ∉ M. Set y = Qx. Then x ≠ Px implies y ≠ 0. The next result is the Riesz representation theorem. Theorem 1.6. Let L be a continuous linear functional on a Hilbert space H. Then there exists a uniquely determined element y ∈ H such that Lx = (x, y) for all x ∈ H. Proof. If L(x) = 0 for all x ∈ H, then we set y = 0. Otherwise, we define M = {x : Lx = 0}. Then, by the continuity of L, the subspace M of H is closed. By Corollary 1.5 we have Lz ⊥ M ⊥ ≠ 0. Let z ∈ M ⊥ , z ≠ 0. Then Lz ≠ 0. Now set y = αz, where α = ‖z‖ 2 . Then y ∈ M and Ly = L(αz) =

Lz |Lz|2 Lz = = (y, y) = |α|2 (z, z). 2 ‖z‖ ‖z‖2

1.1 Elementary properties �

5

For x ∈ H, we define x′ = x −

Lx y (y, y)

and

x ′′ =

Lx y. (y, y)

Then we obtain Lx ′ = 0 and x ′ ∈ M; hence (x ′ , y) = 0, and (x, y) = (x ′′ , y) = (

Lx y, y) = Lx. (y, y)

If (x, y) = (x, y′ ) for all x ∈ H, then we get (x, y − y′ ) = 0 for all x ∈ H; in particular, (y − y′ , y − y′ ) = 0. Therefore y = y′ , which shows that y is uniquely determined. Corollary 1.7. Let H be a Hilbert space, and let L ∈ H ′ be a continuous linear functional. Then the dual norm ‖L‖ = sup{|Lx| : ‖x‖ ≤ 1} can be expressed in the form 󵄨 󵄨 ‖L‖ = sup{󵄨󵄨󵄨(x, y)󵄨󵄨󵄨 : ‖x‖ ≤ 1} = ‖y‖, where y ∈ H corresponds to L. For fixed z ∈ Ω, (1.2) also implies that the point evaluation f 󳨃→ f (z) is a continuous linear functional on A2 (Ω), and hence, by the Riesz representation, Theorem 1.6, there is a uniquely determined function kz ∈ A2 (Ω) such that f (z) = (f , kz ) = ∫ f (w) kz (w) dλ(w).

(1.3)

Ω

We set K(z, w) = kz (w). Then w 󳨃→ K(z, w) = kz (w) is an element of A2 (Ω), and hence the function w 󳨃→ K(z, w) is antiholomorphic on Ω, and we have f (z) = ∫ K(z, w)f (w) dλ(w),

f ∈ A2 (Ω).

Ω

The function of two complex variables (z, w) 󳨃→ K(z, w) is called the Bergman kernel of Ω, and the above identity represents the reproducing property of the Bergman kernel. Now we use the reproducing property for the holomorphic function z 󳨃→ ku (z), where u ∈ Ω is fixed: ku (z) = ∫ K(z, w)ku (w) dλ(w) = ∫ kz (w) K(u, w) dλ(w) Ω

Ω

6 � 1 Bergman spaces −

= (∫ K(u, w)kz (w) dλ(w)) = kz (u), Ω

and hence we have ku (z) = kz (u), or K(z, u) = K(u, z). It follows that the Bergman kernel is holomorphic in the first variable and antiholomorphic in the second variable. Proposition 1.8. The Bergman kernel is uniquely determined by the properties that it is an element of A2 (Ω) in z and that it is conjugate symmetric and reproduces A2 (Ω). Proof. To see this, let K ′ (z, w) be another kernel with these properties. Then we have K(z, w) = ∫ K ′ (z, u)K(u, w) dλ(u) Ω

= (∫ K(w, u)K ′ (u, z) dλ(u))

−

Ω

= K ′ (w, z)

= K ′ (z, w). Now let ϕ ∈ L2 (Ω). Since A2 (Ω) is a closed subspace of L2 (Ω), there exists a uniquely determined orthogonal projection P : L2 (Ω) 󳨀→ A2 (Ω); see Theorem 1.4. For the function Pϕ ∈ A2 (Ω), using the reproducing property, we obtain Pϕ(z) = ∫ K(z, w)Pϕ(w) dλ(w) = (Pϕ, kz ) = (ϕ, Pkz ) = (ϕ, kz ),

(1.4)

Ω

where we still have used that P is a self-adjoint operator and that Pkz = kz . Hence Pϕ(z) = ∫ K(z, w)ϕ(w) dλ(w).

(1.5)

Ω

The operator P is called the Bergman projection. To compute the Bergman kernel for some particular cases, we need the notion of a complete orthonormal basis. A subset {uα : α ∈ A} of a Hilbert space is called orthonormal if (uα , uβ ) = δαβ for each α, β ∈ A. If (xk )k is a linearly independent sequence in H, then there is a standard procedure, called the Gram–Schmidt process, for converting (xk )k into an orthonormal sequence (uk )k such that the linear span of (uk )Nk=1 equals the linear span of (xk )Nk=1 for all N ∈ ℕ. We start with u1 = x1 /‖x1 ‖. Having defined u1 , . . . , uN−1 , we set N−1

vN = xN − ∑ (xN , uj )uj . j=1

1.1 Elementary properties

� 7

The element vN is nonzero because xN is not in the linear span of x1 , . . . , xN−1 and hence not in the span of u1 , . . . , uN−1 . So we can set uN = vN /‖vN ‖. It is now clear that (uk )Nk=1 has the desired properties. Next we prove Bessel’s inequality. Proposition 1.9. If {uα : α ∈ A} is an orthonormal set in a Hilbert space H, then for all u ∈ H, 󵄨 󵄨2 ∑ 󵄨󵄨󵄨(u, uα )󵄨󵄨󵄨 ≤ ‖u‖2 .

(1.6)

α∈A

In particular, the set {α : (u, uα ) ≠ 0} is countable. Proof. It suffices to show that ∑α∈F |(u, uα )|2 ≤ ‖u‖2 for any finite set F ⊂ A. We use the property ‖uα + uβ ‖2 = ‖uα ‖2 + ‖uβ ‖2 for α, β ∈ A, α ≠ β: 󵄩󵄩 󵄩󵄩2 󵄩 󵄩 0 ≤ 󵄩󵄩󵄩u − ∑ (u, uα )uα 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 α∈F

󵄩󵄩 󵄩󵄩2 󵄩 󵄩 = ‖u‖2 − 2R(u, ∑ (u, uα )uα ) + 󵄩󵄩󵄩 ∑ (u, uα )uα 󵄩󵄩󵄩 󵄩󵄩α∈F 󵄩󵄩 α∈F 󵄨 󵄨2 󵄨 󵄨2 = ‖u‖2 − 2 ∑ 󵄨󵄨󵄨(u, uα )󵄨󵄨󵄨 + ∑ 󵄨󵄨󵄨(u, uα )󵄨󵄨󵄨 α∈F

󵄨 󵄨2 = ‖u‖ − ∑ 󵄨󵄨󵄨(u, uα )󵄨󵄨󵄨 .

α∈F

2

α∈F

Proposition 1.10. If {uα : α ∈ A} is an orthonormal set in a Hilbert space H, then the following conditions are equivalent: (1) Completeness: if (u, uα ) = 0 for all α ∈ A, then u = 0. (2) Parseval’s identity: ‖u‖2 = ∑α∈A |(u, uα )|2 for all u ∈ H. (3) u = ∑α∈A (u, uα )uα for each u ∈ H, where the sum has only countably many nonzero terms and converges in norm to u no matter how these terms are ordered. Proof. (1) implies (3): If u ∈ H, let α1 , α2 , . . . be any enumeration of those α for which 2 (u, uα ) ≠ 0. By Bessel’s inequality the series ∑∞ j=1 |(u, uαj )| converges, so 󵄩󵄩 M 󵄩󵄩2 M 󵄩󵄩 󵄩 󵄩󵄩 ∑ (u, uα )uα 󵄩󵄩󵄩 = ∑ 󵄨󵄨󵄨(u, uα )󵄨󵄨󵄨2 → 0 󵄩󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩j=m 󵄩󵄩 j=m

as m, M → ∞.

∞ Hence the series ∑∞ j=1 (u, uαj )uαj converges in H. If v = u − ∑j=1 (u, uαj )uαj , then (v, uα ) = 0 for all α ∈ A, so by (1), v = 0. (3) implies (2): As in the proof of Bessel’s inequality, we have

8 � 1 Bergman spaces 󵄩󵄩2 󵄩󵄩 m 󵄩󵄩 󵄨󵄨2 󵄩󵄩󵄩 󵄨󵄨 ‖u‖ − ∑󵄨󵄨(u, uαj )󵄨󵄨 = 󵄩󵄩u − ∑(u, uαj )uαj 󵄩󵄩󵄩 → 0 󵄩󵄩 󵄩󵄩 j=1 j=1 󵄩 󵄩 m

2

as m → ∞.

Finally, that (2) implies (1) is obvious. An orthonormal set having the properties of Proposition 1.10 is called an orthonormal basis of H. Remark 1.11. An application of Zorn’s lemma shows that the collection of orthonormal sets in a Hilbert space, ordered by inclusion, has a maximal element. Maximality is equivalent to (1) of Proposition 1.10, and hence for each orthonormal set, there exists an orthonormal basis that contains the given orthonormal set. Every separable Hilbert space H has a countable orthonormal basis. Proposition 1.12. Let K ⊂ Ω be a compact subset, and let {ϕj } be an orthonormal basis of A2 (Ω). Then the series ∞

∑ ϕj (z) ϕj (w) j=1

sums uniformly on K × K to the Bergman kernel K(z, w). Proof. For the proof of this statement, we use the duality for the sequence space l2 to get ∞

1/2

󵄨 󵄨2 sup(∑󵄨󵄨󵄨ϕj (z)󵄨󵄨󵄨 ) z∈K

j=1

󵄨󵄨 ∞ 󵄨󵄨 ∞ 󵄨󵄨 󵄨󵄨 = sup{󵄨󵄨󵄨∑ aj ϕj (z)󵄨󵄨󵄨 : ∑ |aj |2 = 1, z ∈ K} 󵄨󵄨 󵄨󵄨 󵄨j=1 󵄨 j=1 󵄨󵄨 󵄨󵄨 = sup{󵄨󵄨f (z)󵄨󵄨 : ‖f ‖ = 1, z ∈ K} ≤ C(K),

(1.7)

where we have used (1.2) in the last inequality. Now the Cauchy–Schwarz inequality gives ∞

∞

j=1

j=1

1/2

∞

1/2

󵄨 󵄨 󵄨2 󵄨 󵄨2 󵄨 ∑󵄨󵄨󵄨ϕj (z) ϕj (w)󵄨󵄨󵄨 ≤ (∑󵄨󵄨󵄨ϕj (z)󵄨󵄨󵄨 ) (∑󵄨󵄨󵄨ϕj (w)󵄨󵄨󵄨 ) j=1

with uniform convergence in z, w ∈ K. In addition, it follows that (ϕj (z))j ∈ l2 and the function ∞

w 󳨃→ ∑ ϕj (z) ϕj (w) j=1

belongs to A2 (Ω). Denote the sum of the series by K ′ (z, w). Notice that K ′ (z, w) is conjugate symmetric and that for f ∈ A2 (Ω), we get

1.2 Examples

� 9

∞

∫ K ′ (z, w)f (w) dλ(w) = ∑ ∫ f (w)ϕj (w) dλ(w) ϕj (z) = f (z) j=1 Ω

Ω

with convergence in the Hilbert space A2 (Ω). However, (1.7) implies uniform convergence on compact subsets of Ω, and hence f (z) = ∫ K ′ (z, w)f (w) dλ(w) Ω

for all f ∈ A2 (Ω), so K ′ (z, w) is a reproducing kernel. By the uniqueness of the Bergman kernel we obtain K ′ (z, w) = K(z, w). We note that (1.7) implies 󵄨 󵄨2 K(z, z) = sup{󵄨󵄨󵄨f (z)󵄨󵄨󵄨 : f ∈ A2 (Ω) , ‖f ‖ = 1}.

(1.8)

1.2 Examples (a) zn , n = 0, 1, 2, . . . , constitute an orthonormal basis in A2 (𝔻), The functions ϕn (z) = √ n+1 π 𝔻 = {z ∈ ℂ : |z| < 1}. This follows from 2π 1

2π δ . n + m + 2 n,m

∫ zn zm dλ(z) = ∫ ∫ r n einθ r m e−imθ r dr dθ = 0 0

𝔻

n For each f ∈ A2 (𝔻) with Taylor series expansion f (z) = ∑∞ n=0 an z , we get 1 2π

(f , zn ) = ∫ f (z)zn dλ(z) = ∫ ∫ f (reiθ )r n e−inθ r dr dθ 𝔻

1 2π

0 0

1

a f (reiθ ) = ∫ ∫ n+1 i(n+1)θ reiθ dθ r 2n+1 dr = 2πan ∫ r 2n+1 dr = π n , n +1 r e 0 0

0

where we used the fact that an =

f (z) 1 dz ∫ 2πi zn+1 γr

for γr (θ) = reiθ . Hence, by the uniqueness of the Taylor series expansion, we obtain that (f , ϕn ) = 0 for each n = 0, 1, 2, . . . implies f ≡ 0. This means that (ϕn )∞ n=0 constitutes an orthonormal basis for A2 (𝔻), and we get

10 � 1 Bergman spaces ∞

󵄨2 󵄨 ‖f ‖2 = ∑ 󵄨󵄨󵄨(f , ϕn )󵄨󵄨󵄨 , n=0

which is equivalent to |an |2 , n+1 n=0 ∞

‖f ‖2 = π ∑

∞

f (z) = ∑ an zn . n=0

Hence each f ∈ A2 (𝔻) can be written in the form f = ∑∞ n=0 cn ϕn , where the sum converges in A2 (𝔻) and also uniformly on compact subsets of 𝔻. For the coefficients cn , we have cn = (f , ϕn ). Now we compute an explicit formula for the Bergman kernel K(z, w) of 𝔻. The function z 󳨃→ K(z, w), with w ∈ 𝔻 fixed, belongs to A2 (𝔻). Hence we get from the above formula that ∞

K(z, w) = ∑ cn ϕn (z), n=0

where cn = (K(., w), ϕn ); in other words, cn = (ϕn , K(., w)) = ∫ ϕn (z)K(w, z) dλ(z) = ϕn (w) 𝔻

by the reproducing property of the Bergman kernel. This implies that the Bergman kernel is of the form ∞

K(z, w) = ∑ ϕn (z) ϕn (w), n=0

(1.9)

where the sum converges uniformly in z on all compact subsets of 𝔻. (This is true for any complete orthonormal system, as is shown above.) A simple computation now gives ∞

K(z, w) = ∑ ϕn (z) ϕn (w) = n=0

1 ∞ 1 1 . ∑ (n + 1)(zw)n = π n=0 π (1 − zw)2

Hence for each f ∈ A2 (𝔻), we have f (z) =

1 1 f (w) dλ(w). ∫ π (1 − zw)2 𝔻

If we fix z ∈ 𝔻 and set f (w) = 1/(1 − wz)2 , then we get the interesting formula 1 1 1 dλ(w) = . ∫ π |1 − zw|4 (1 − |z|2 )2 𝔻

(1.10)

1.2 Examples

� 11

(b) Next, we determine the Bergman kernel of the polycylinder. Proposition 1.13. Let Ωj ⊂ ℂnj , j = 1, 2, be two bounded domains with Bergman kernels KΩ1 and KΩ2 . Then the Bergman kernel KΩ of the product domain Ω = Ω1 × Ω2 is given by KΩ ((z1 , z2 ), (w1 , w2 )) = KΩ1 (z1 , w1 ) KΩ2 (z2 , w2 )

(1.11)

for (z1 , z2 ), (w1 , w2 ) ∈ Ω1 × Ω2 . Proof. To show this, let F denote the function on the right-hand side of (1.11). It is clear that (z1 , z2 ) 󳨃→ F((z1 , z2 ), (w1 , w2 )) belongs to A2 (Ω) for each fixed (w1 , w2 ) ∈ Ω and that F is antiholomorphic in the second variable. The reproducing property f (z1 , z2 ) = ∫ F((z1 , z2 ), (w1 , w2 ))f (w1 , w2 ) dλ(w1 , w2 ) Ω1 ×Ω2

is a consequence of Fubini’s theorem and the corresponding reproducing properties of KΩ1 and KΩ2 . Hence, by the uniqueness property of the Bergman kernel, Proposition 1.8, we obtain F = KΩ . From this we get that the Bergman kernel of the polycylinder 𝔻n is given by K𝔻n (z, w) =

1 1 n . ∏ π n j=1 (1 − zj wj )2

(1.12)

(c) For the computation of the Bergman kernel K𝔹n of the unit ball in ℂn , we use the beta and gamma functions 1

∫ x k (1 − x)m dx = B(k + 1, m + 1) = 0

Γ(k + 1)Γ(m + 1) , Γ(k + m + 2)

where k, m ∈ ℕ and that for 0 ≤ a < 1, √1−a2

∫ x 0

2k+1

m+1

x2 ) (1 − 1 − a2

1

1 k+1 dx = (1 − a2 ) ∫ yk (1 − y)m+1 dy 2 0

1 k+1 = (1 − a2 ) B(k + 1, m + 2) 2 1 k+1 Γ(k + 1)Γ(m + 2) . = (1 − a2 ) 2 Γ(k + m + 3) α

α

Now we can normalize the orthogonal basis {zα = z1 1 . . . znn } in A2 (𝔹n ) and obtain

12 � 1 Bergman spaces 󵄩󵄩 α 󵄩󵄩2 2α 2α 󵄩󵄩z 󵄩󵄩 = ∫ |z1 | 1 . . . |zn | n dλ(z) 𝔹n

=

π α +1 ∫ |z1 |2α1 . . . |zn−1 |2αn−1 (1 − |z1 |2 − ⋅ ⋅ ⋅ − |zn−1 |2 ) n dλ αn + 1 𝔹n−1

π α +1 = ∫ |z1 |2α1 . . . |zn−2 |2αn−2 (1 − |z1 |2 − ⋅ ⋅ ⋅ − |zn−2 |2 ) n αn + 1 𝔹n−1

αn +1

|zn−1 |2 ) 1 − |z1 |2 − ⋅ ⋅ ⋅ − |zn−2 |2 π πΓ(αn−1 + 1)Γ(αn + 2) = αn + 1 Γ(αn + αn−1 + 3) ⋅ |zn−1 |2αn−1 (1 −

dλ

⋅ ∫ |z1 |2α1 . . . |zn−2 |2αn−2 (1 − |z1 |2 − ⋅ ⋅ ⋅ − |zn−2 |2 )

αn +αn−1 +2

dλ

𝔹n−2

πΓ(α1 + 1)Γ(αn + ⋅ ⋅ ⋅ + α2 + n) π πΓ(αn−1 + 1)Γ(αn + 2) ... αn + 1 Γ(αn + αn−1 + 3) Γ(αn + ⋅ ⋅ ⋅ + α1 + n + 1) π n α1 ! . . . αn ! = . (αn + ⋅ ⋅ ⋅ + α1 + n)! =

Hence the Bergman kernel of the unit ball is given by K𝔹n (z, w) = ∑ α

= = =

(αn + ⋅ ⋅ ⋅ + α1 + n)! α α z w π n α1 ! . . . αn !

(αn + ⋅ ⋅ ⋅ + α1 + n)! α α 1 ∞ z w ∑ ∑ π n k=0 |α|=k α1 ! . . . αn !

1 ∞ ∑ (k + n)(k + n − 1) . . . (k + 1)(z1 w1 + ⋅ ⋅ ⋅ + zn wn )k π n k=0 1 n! . π n (1 − (z1 w1 + ⋅ ⋅ ⋅ + zn wn ))n+1

1.3 Biholomorphic maps Here we describe the behavior of the Bergman kernel under biholomorphic maps. Proposition 1.14. Let F : Ω1 󳨀→ Ω2 be a biholomorphic map between bounded domains in ℂn . Let f1 , . . . , fn be the components of F, and let F ′ (z) = ( Then

𝜕fj (z) n ) . 𝜕zk j,k=1

KΩ1 (z, w) = det F ′ (z) KΩ2 (F(z), F(w)) det F ′ (w) for all z, w ∈ Ω1 .

(1.13)

1.3 Biholomorphic maps

� 13

Proof. Notice that, by the Cauchy–Riemann equations for fj = uj + ivj , j = 1, . . . , n, the determinant of the Jacobian of the mapping F equals | det F ′ (z)|2 . The substitution formula for integrals implies that for g ∈ L2 (Ω2 ), we have 󵄨2 󵄨2 󵄨 󵄨2 󵄨 󵄨 ∫ 󵄨󵄨󵄨g(ζ )󵄨󵄨󵄨 dλ(ζ ) = ∫ 󵄨󵄨󵄨g(F(z))󵄨󵄨󵄨 󵄨󵄨󵄨det F ′ (z)󵄨󵄨󵄨 dλ(z).

(1.14)

Ω1

Ω2

Hence the map TF : g 󳨃→ (g ∘F) det F ′ establishes an isometric isomorphism from L2 (Ω2 ) to L2 (Ω1 ), with inverse map TF −1 , which restricts to an isomorphism between A2 (Ω1 ) and A2 (Ω2 ). Now let f ∈ A2 (Ω1 ) and apply the reproducing property of KΩ2 to the function TF −1 f = (f ∘ F −1 ) det(F −1 )′ ; setting F(z) = u, we get ∫ KΩ2 (u, v)TF −1 f (v) dλ(v) = TF −1 f (u) = f (z)(det F ′ (z)) . −1

(1.15)

Ω2

Since TF is an isometry, ∫ TF −1 f (v)[KΩ2 (v, u)] dλ(v) = ∫ f (w)[TF KΩ2 (., u)(w)] dλ(w). −

−

(1.16)

Ω1

Ω2

From (1.15) and (1.16) we obtain f (z) = ∫ det F ′ (z) KΩ2 (F(z), F(w)) det F ′ (w) f (w) dλ(w), Ω1

which means that the right-hand side of (1.13) has the required reproducing property, belongs to A2 (Ω1 ) in the variable z, is antiholomorphic in the variable w, and hence must agree with KΩ1 (z, w). Finally, we derive a useful formula for the corresponding orthogonal projections Pj : L2 (Ωj ) 󳨀→ A2 (Ωj ),

j = 1, 2.

Proposition 1.15. For all g ∈ L2 (Ω2 ), we have P1 (det F ′ g ∘ F) = det F ′ (P2 (g) ∘ F).

(1.17)

Proof. The left-hand side of (1.17) can be written in the form P1 (TF (g)), and hence, by (1.5), we obtain P1 (TF (g))(z) = ∫ KΩ1 (z, w)TF (g)(w) dλ(w), Ω1

Now (1.13), together with (1.16), implies that

z ∈ Ω1 .

14 � 1 Bergman spaces KΩ1 (w, z) = [TF (KΩ2 (., F(z)))(w)] det F ′ (z), so, since TF is an isometric isomorphism, we get P1 (TF (g))(z) = det F ′ (z) ∫ TF (g)(w) [TF (KΩ2 (., F(z)))(w)] dλ(w) −

Ω1

= det F (z) ∫ g(v) [KΩ2 (v, F(z))] dλ(v) −

′

Ω2

= det F (z) (P2 (g))(F(z)), ′

which proves (1.17). Remark 1.16. If n = 1 and Ω ⊊ ℂ is a simply connected domain, then there is an interesting connection between the Bergman kernel KΩ of Ω and the Riemann mapping F : Ω 󳨀→ 𝔻 with the uniqueness properties F(a) = 0 and F ′ (a) > 0 for some a ∈ Ω: F ′ (z) = √

π K (z, a), KΩ (a, a) Ω

z ∈ Ω.

(1.18)

By (1.14) the transformation TF establishes an isometry between L2 (Ω) and L2 (𝔻), which restricts to an isometry between A2 (Ω) and A2 (𝔻). Therefore we have (TF u, TF u)Ω = (u, u)𝔻 ,

u ∈ L2 (𝔻),

where (⋅, ⋅)Ω denotes the inner product in L2 (Ω), and (⋅, ⋅)𝔻 denotes the inner product of L2 (𝔻). Similarly, for v ∈ L2 (Ω) and G = F −1 , we have (TG v, TG v)𝔻 = (v, v)Ω . The polarization identity (u1 , u2 ) =

i 1 (‖u + u2 ‖2 − ‖u1 − u2 ‖2 ) − (‖u1 + iu2 ‖2 − ‖u1 − iu2 ‖2 ) 4 1 4

in an inner product space over ℂ yields (TF u1 , TF u2 )Ω = (u1 , u2 )𝔻 ,

u1 , u2 ∈ L2 (𝔻),

(1.19)

(TG v1 , TG v2 )𝔻 = (v1 , v2 )Ω ,

v1 , v2 ∈ L2 (Ω).

(1.20)

and

Since TF TG is the identity operator, from (1.19) and (1.20) we obtain (TF u, v)Ω = (TF u, TF (TG v))Ω = (u, TG v)𝔻 .

(1.21)

1.4 The Segal–Bargmann space

� 15

Now let h ∈ A2 (𝔻) and observe that, by (1.1), (h, 1)𝔻 = πh(0). By (1.21) we get that for f ∈ A2 (Ω), (f , F ′ )Ω = (G′ (f ∘ G), 1)𝔻 = πG′ (0)f (G(0)) =

π f (a). F ′ (a)

It follows that the function k(z, a) = F π(a) F ′ (z) has the reproducing property and belongs to A2 (Ω), so, by Proposition 1.8, k(z, a) = KΩ (z, a), and we get ′

F ′ (z) =

π K (z, a). F ′ (a) Ω

Setting z = a, we obtain F ′ (a)2 = πKΩ (a, a), which proves (1.18).

1.4 The Segal–Bargmann space 2

In the sequel, we will also consider the Segal–Bargmann space or Fock space A2 (ℂn , e−|z| ) consisting of all entire functions f such that 2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z) < ∞.

ℂn

It is clear that the Fock space is a Hilbert space with the inner product 2

(f , g) = ∫ f (z) g(z) e−|z| dλ(z). ℂn 2

Similarly as in the beginning of this chapter, setting n = 1, we obtain for f ∈ A2 (ℂ, e−|z| ) that 1 󵄨 󵄨󵄨 󵄨 −|w|2 /2 |w|2 /2 󵄨󵄨 dλ(w) 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ 2 ∫ e 󵄨󵄨f (w)󵄨󵄨󵄨 e πr D(z,r)

≤

1/2

2 1 ( ∫ e|w| dλ(w)) πr 2

D(z,r)

2 󵄨 󵄨2 ( ∫ 󵄨󵄨󵄨f (w)󵄨󵄨󵄨 e−|w| dλ(w))

1/2

D(z,r)

1/2

󵄨 󵄨2 ≤ C(∫󵄨󵄨󵄨f (w)󵄨󵄨󵄨 e−|w| dλ(w)) 2

ℂ

≤ C‖f ‖,

(1.22)

16 � 1 Bergman spaces 2

where C is a constant only depending on z. This implies that the Fock space A2 (ℂn , e−|z| ) has the reproducing property. In addition, for each compact subset L of ℂ, there exists a constant CL > 0 such that 󵄨 󵄨 sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ CL ‖f ‖

(1.23)

z∈L

2

for all f ∈ A2 (ℂ, e−|z| ). The monomials {zα } constitute an orthogonal basis, and the norms of the monomials are 󵄩󵄩 α 󵄩󵄩2 2α −|z |2 2α −|z |2 󵄩󵄩z 󵄩󵄩 = ∫ |z1 | 1 e 1 dλ(z1 )⋅ ⋅ ⋅ ∫ |zn | n e n dλ(zn ) ℂ ∞

ℂ ∞

2

2

= (2π)n ∫ r 2α1 +1 e−r dr⋅ ⋅ ⋅ ∫ r 2αn +1 e−r dr 0

0

= π n α1 ! . . . αn !. 2

It follows that each function f ∈ A2 (ℂn , e−|z| ) can be written in the form f = ∑ fα φα , α

where φα (z) =

zα √π n α!

and

∑ |fα |2 < ∞ α

(1.24)

with α! = α1 ! . . . αn !. 2 Hence the Bergman kernel of A2 (ℂn , e−|z| ) is of the form K(z, w) = ∑ α

α

α

zα w zα w 1 ∞ 1 = = exp(z1 w1 + ⋅ ⋅ ⋅ + zn wn ). ∑ ∑ n ‖zα ‖2 π k=0 |α|=k α1 ! . . . αn ! π n

(1.25)

In the following, we set ⟨z, w⟩ = z1 w1 + ⋅ ⋅ ⋅ + zn wn and |z|2 = |z1 |2 + ⋅ ⋅ ⋅ + |zn |2 . 1 exp⟨z, u⟩ for u, z ∈ ℂn . In simplicity, we set n = 1. Then Let ϕu (z) = π n/2 (ϕu , ϕv ) = =

1 ∞ ū k k ∞ v̄k k (∑ z ,∑ z ) π k=0 k! k! k=0 1 ∞ 1 (vu)̄ k (zk , zk ) ∑ π k=0 k!2

= exp⟨v, u⟩.

In particular, we have ‖ϕu ‖2 = exp(|u|2 ). The reproducing property implies that

(1.26)

1.4 The Segal–Bargmann space

f (u) =

1

(f , ϕu )

π 1/2

� 17

(1.27)

2

for f ∈ A2 (ℂ, e−|z| ), and hence 1 1 󵄨 1 |z|2 󵄨 󵄨 󵄨󵄨 )‖f ‖. 󵄨󵄨f (z)󵄨󵄨󵄨 = 1/2 󵄨󵄨󵄨(f , ϕz )󵄨󵄨󵄨 ≤ 1/2 ‖ϕz ‖‖f ‖ = 1/2 exp( 2 π π π

(1.28)

In a similar way, 1 |z|2 /2 󵄨󵄨 󵄨 ‖f ‖ 󵄨󵄨f (z)󵄨󵄨󵄨 ≤ n/2 e π

(1.29)

2

for each f ∈ A2 (ℂn , e−|z| ). 2 We point out that the space A2 (ℂ, e−|z| ) serves for a representation of the states in quantum mechanics (see [30]), where a(f ) =

df dz

is the annihilation operator, and a∗ (f ) = zf is the creation operator, both of them being densely defined unbounded operators 2 on A2 (ℂ, e−|z| ). The span of the finite linear combinations of the basis functions φα is 2 dense in A2 (ℂ, e−|z| ). Hence both operators a and a∗ are densely defined. The function φk (z)

∞

F(z) = ∑

k=2

√k(k − 1)

2

∈ A2 (ℂ, e−|z| ),

but 2 φk (z) ∉ A2 (ℂ, e−|z| ), √ k k=1

∞

F ′ (z) = ∑ and

2 φk (z) ∈ A2 (ℂ, e−|z| ), k + 1 k=0

∞

G(z) = ∑ but

2 φk (z) ∉ A2 (ℂ, e−|z| ), √ k k=1

∞

zG(z) = ∑

18 � 1 Bergman spaces 2

and hence both operators a and a∗ are unbounded operators on A2 (ℂ, e−|z| ). 2 However, taking a primitive of a function f ∈ A2 (ℂ, e−|z| ) yields a bounded operator 2

2

T : A2 (ℂ, e−|z| ) 󳨀→ A2 (ℂ, e−|z| ); let ∞

f (z) = ∑ fk k=0

zk , √π √k!

∞

∑ |fk |2 < ∞.

k=0

Then the function ∞

h(z) = ∑ fk k=0

zk+1 √π √k + 1√(k + 1)!

defines a primitive of f , and we can write 1 (f , φ̃ k )φk , √ k=1 k! ∞

T(f ) = h = ∑

where φ̃ k = φk−1 , and the constant term in the Fourier series expansion of the primitive is always 0. This implies immediately that T is even a compact operator. This is also a special result from the theory of Volterra-type integration operators on the Segal–Bargmann space of the form z

Tg f (z) = ∫ fg ′ dζ 0

(see [19]). We define the domain of the operator a as 2

2

dom(a) = {f ∈ A2 (ℂ, e−|z| ) : f ′ ∈ A2 (ℂ, e−|z| )}. 2

Then dom(a∗ ) consists of all functions g ∈ A2 (ℂn , e−|z| ) such that the densely defined linear functional L(f ) = (a(f ), g) is continuous on dom(a). This implies that there exists 2 a function h ∈ A2 (ℂ, e−|z| ) such that L(f ) = (a(f ), g) = (f , h). Next, we show that 2

2

dom(a∗ ) = {g ∈ A2 (ℂ, e−|z| ) : zg ∈ A2 (ℂ, e−|z| )}. For this purpose, we use Fourier series expansion: let ∞

f (z) = ∑ fk φk (z) and k=0

∞

g(z) = ∑ gk φk (z), k=0

1.5 The uncertainty principle

� 19

where (fk )k , (gk )k ∈ l2 . Then we have ∞

(a(f ), g) = ∑ √k + 1 fk+1 gk = (f , a∗ (g)). k=0

Notice that the commutator satisfies [a, a∗ ] = I,

(1.30)

which is important in quantum mechanics; see [30]. For f ∈ dom(a) ∩ dom(a∗ ) with Fourier expansion f (z) = ∑∞ k=0 fk φk (z), we have ∞

󵄩󵄩 ∗ 󵄩󵄩2 2 󵄩󵄩a (f )󵄩󵄩 = ∑ (k + 1)|fk | k=0 ∞

∞

k=0

k=0

(1.31)

= ∑ k|fk |2 + ∑ |fk |2 󵄩 󵄩2 = 󵄩󵄩󵄩a(f )󵄩󵄩󵄩 + ‖f ‖2 , which implies that dom(a) = dom(a∗ ).

1.5 The uncertainty principle If we consider entire functions of several variables, then we get that the operators aj (f ) = 𝜕f 𝜕zj

2

and aj∗ (f ) = zj f are densely defined unbounded operators on A2 (ℂn , e−|z| ). They are

adjoint to each other, [aj , aj∗ ] = I, and dom(aj ) = dom(aj∗ ). Using (1.31), we obtain 󵄩 󵄩2 󵄩 󵄩2 ‖f ‖2 ≤ 󵄩󵄩󵄩aj (f )󵄩󵄩󵄩 + 󵄩󵄩󵄩aj∗ (f )󵄩󵄩󵄩

(1.32)

for f ∈ dom(aj ) and j = 1, . . . , n. Now we consider the position operators Xj (u) = xj u and the momentum operators 𝜕u Dj u = −i 𝜕x on L2 (ℝn ). Recall that j

󵄩󵄩 󵄩2 󵄩 󵄩2 2 󵄩󵄩Xj (u)󵄩󵄩󵄩 + 󵄩󵄩󵄩Dj (u)󵄩󵄩󵄩 ≥ ‖u‖ ,

u ∈ dom(Xj ) ∩ dom(Dj ),

(1.33)

which is a variant of the uncertainty principle; see [30]. We indicate that (1.33) follows from inequality (1.32). To show this, we use the 2 Bargmann transform ℬ : L2 (ℝn ) 󳨀→ A2 (ℂn , e−|z| ) given by ℬ(f )(z) = π

−n/4

∫ f (x) exp[−z2 /2 + √2zx − x 2 /2] dx, ℝn

f ∈ L2 (ℝn ),

20 � 1 Bergman spaces where z2 = ∑nj=1 z2j and xz = ∑nj=1 xj zj .

We claim that the Bargmann transform is a unitary operator between L2 (ℝn ) and 2 A2 (ℂn , e−|z| ). To show this, we first mention that ℬ(f ) is an entire function and set again n = 1: ℬ(f )(z) = π

−1/4

zk ∫ f (x)(√2x)k exp[−x 2 /2] dx, k! k=0 ∞

exp[−z2 /2] ∑

(1.34)

ℝ

and 1/2 1/2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨2 k 2 2k −x 2 󵄨󵄨∫ f (x)(√2x) exp(−x /2) dx 󵄨󵄨󵄨 ≤ (∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 dx) (∫ x e dx) 󵄨󵄨 󵄨󵄨 ℝ

ℝ

ℝ

∞

= ‖f ‖L2 (ℝ) ( ∫ r

k−1/2 −r

e

1/2

dr)

0

≤ ‖f ‖L2 (ℝ) √k!, which gives that ℬ(f ) is an entire function. Let Φz (x) = π −n/4 exp[−z̄2 /2 + √2zx̄ − x 2 /2] dx. Then ℬf (z) = (f , Φz )L2 (ℝn ) .

Lemma 1.17.

(Φu , Φv )L2 (ℝn ) = exp⟨v, u⟩

for all u, v ∈ ℂn . Proof. (Φu , Φv )L2 (ℝn ) ̄ − x 2 /2] exp[−v2 /2 + √2vx − x 2 /2] dx = π −n/2 ∫ exp[−ū 2 /2 + √2ux ℝn

1 = π −n/2 exp(vu)̄ ∫ exp[− (v + u)̄ 2 + √2x(v + u)̄ − x 2 ] dx 2 ℝn

2

̄ √2) ] dx = π −n/2 exp(vu)̄ ∫ exp[−(x − (v + u)/ ℝn

1 = (2π)−n/2 exp(vu)̄ ∫ exp[− y2 ] dy 2 = exp⟨v, u⟩.

ℝn

(1.35)

1.5 The uncertainty principle

� 21

Using (1.26), we see now that (Φu , Φv )L2 (ℝn ) = (ϕu , ϕv )

(1.36)

for all u, v ∈ ℂn . The system {Φz : z ∈ ℂn } is total in L2 (ℝn ), i. e., (f , Φz )L2 (ℝn ) = 0 for all z ∈ ℂn implies f = 0: by (1.34) we have ℬf (−it/√2) = π

−1/4 −t 2 /2

e

∫ f (x)e−itx e−x

2

/2

dx = 0

ℝ 2

for all t ∈ ℝ. Hence the Fourier transform of the function f (x)e−x /2 is 0, so f = 0 almost everywhere. The linear combinations of vectors Φz are dense in L2 (ℝn ), which is a consequence of the Hahn–Banach theorem: we have to show that if a continuous linear functional L has the property that L(Φz ) = 0 for all z ∈ ℂn , then L = 0, which follows again from the fact that each continuous linear functional is given by a function f in L2 (ℝn ) and we have 2 L(Φz ) = (Φz , f )L2 (ℝn ) . The same is also true for the system {ϕz : z ∈ ℂn } in A2 (ℂn , e−|z| ), where we use the reproducing property (1.27). 2 Now we define the unitary operator U : L2 (ℝn ) 󳨀→ A2 (ℂn , e−|z| ) by U(Φz ) = ϕz ,

z ∈ ℂn .

We indicate that for linear combinations, we have by (1.26) ‖a1 ϕZ1 + ⋅ ⋅ ⋅ + am ϕZm ‖2 = ‖a1 ΦZ1 + ⋅ ⋅ ⋅ + am ΦZm ‖2L2 (ℝn ) , and hence we can extend U by continuity and obtain Uf (z) = (Uf , ϕz ) = (f , U −1 ϕz )L2 (ℝn ) = (f , Φz )L2 (ℝn ) = ℬf (z) by (1.35). Concerning the operators aj and aj∗ , we have ℬ aj ℬ = ∗

1 (X + iDj ) √2 j

and ℬ∗ aj∗ ℬ =

1 (X − iDj ), √2 j

which follows from integration by parts and 𝜕 exp[−z2 /2 + √2zx − x 2 /2] = (√2zj − xj ) exp[−z2 /2 + √2zx − x 2 /2]; 𝜕xj see also [30]. So we get 󵄩󵄩 󵄩2 1 󵄩 ∗ 󵄩2 ∗ ∗ 󵄩󵄩Xj (u)󵄩󵄩󵄩 = 󵄩󵄩󵄩ℬ aj ℬ(u) + ℬ aj ℬ(u)󵄩󵄩󵄩 , 2

22 � 1 Bergman spaces and 󵄩2 1 󵄩 ∗ 󵄩󵄩 󵄩2 ∗ ∗ 󵄩󵄩Dj (u)󵄩󵄩󵄩 = 󵄩󵄩󵄩ℬ aj ℬ(u) − ℬ aj ℬ(u)󵄩󵄩󵄩 . 2 Hence 󵄩2 󵄩2 󵄩 ∗ ∗ 󵄩2 󵄩 ∗ 󵄩2 󵄩 󵄩󵄩 󵄩󵄩Xj (u)󵄩󵄩󵄩 + 󵄩󵄩󵄩Dj (u)󵄩󵄩󵄩 = 󵄩󵄩󵄩ℬ aj ℬ(u)󵄩󵄩󵄩 + 󵄩󵄩󵄩ℬ aj ℬ(u)󵄩󵄩󵄩 󵄩 󵄩2 󵄩 󵄩2 = 󵄩󵄩󵄩aj ℬ(u)󵄩󵄩󵄩 + 󵄩󵄩󵄩aj∗ ℬ(u)󵄩󵄩󵄩

󵄩 󵄩2 ≥ 󵄩󵄩󵄩ℬ(u)󵄩󵄩󵄩

= ‖u‖2 ,

where we used (1.32) and that ℬ is an isometry. The last computation also shows that (1.33) implies (1.32).

1.6 Nonradial weights We consider spaces of entire functions with nonradial weights and determine their Bergman kernels using properties of the Segal–Bargmann space (Section 1.4). For 0 ≤ β ≤ 1, define 2 2 󵄨 󵄨2 Hβ = {f : ℂ 󳨀→ ℂ entire : ‖f ‖2 = (f , f ) = ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−((1+β)x +(1−β)y ) dλ(z) < ∞},

ℂ

where z = x + iy. In general, we cannot expect that the monomials in z constitute a biorthogonal system for Hβ . We define β ψk (z) = zk exp( z2 ) 2 for k = 0, 1, . . . . Since 2 󵄨󵄨 β 2 󵄨󵄨󵄨 󵄨󵄨 2 2 󵄨󵄨exp( z )󵄨󵄨󵄨 = exp(β(x − y )), 󵄨󵄨 󵄨󵄨 2

we have ψk ∈ Hβ . Let k ≠ m. For the inner product in Hβ , we have 2 2 β β (ψk , ψm ) = ∫ zk exp( z2 )(zm exp( z2 )) e−((1+β)x +(1−β)y ) dλ(z). 2 2

−

ℂ

The exponential terms deliver

1.6 Nonradial weights

2 2 β β exp( z2 )(exp( z2 )) e−((1+β)x +(1−β)y ) 2 2

� 23

−

= exp(β(x 2 − y2 )) e−((1+β)x

2

+(1−β)y2 )

2

= e−|z| .

(1.37)

Hence 2

(ψk , ψm ) = ∫ zk z̄m e−|z| dλ(z) = 0. ℂ

Now let f ∈ Hβ be such that (f , ψk ) = 0 for each k = 0, 1, . . . . Then 2 2 β β β (f , ψk ) = ∫ f (z) exp(− z2 ) exp( z2 ) (zk exp( z2 )) e−((1+β)x +(1−β)y ) dλ(z); 2 2 2

−

ℂ

using (1.37), we get 2 β (f , ψk ) = ∫ f (z) exp(− z2 ) z̄k e−|z| dλ(z), 2

ℂ

which implies that f (z) exp(− β2 z2 ) = 0, and hence f = 0. Therefore (ψk )k is a complete biorthogonal system in Hβ . 2

The mapping Ψ : Hβ 󳨀→ A2 (ℂ, e−|z| ) defined by

β Ψ(f )(z) = f (z) exp(− z2 ) 2 is an isometry: f ∈ Hβ .

󵄩󵄩 󵄩 󵄩󵄩Ψ(f )󵄩󵄩󵄩A2 (ℂ,e−|z|2 ) = ‖f ‖Hβ ,

Hence the norms of ψk in Hβ are ‖ψk ‖2 = πk!, and the Bergman kernel of Hβ has the form Kβ (z, w) = =

β 2 β 2 − k k 1 ∞ z exp( 2 z ) (w exp( 2 w )) ∑ π k=0 k!

β 1 ̄ exp( (z2 + w̄ 2 )) exp(zw). π 2

Next, we compute the adjoint of a(f ) = df on Hβ . dz For this purpose, we use the orthonormal basis χk (z) =

β zk exp( z2 ), √πk! 2

k = 0, 1, . . . ,

24 � 1 Bergman spaces ∞ ∗ and write f = ∑∞ k=0 fk χk and g = ∑k=0 gk χk for f ∈ dom(a) and g ∈ dom(a ), where 2 (fk )k , (gk )k ∈ l . Then we get ∞

a(f ) = f1 χ0 + ∑ (fk+1 √k + 1 + βfk−1 √k)χk . k=1

Hence ∞

(a(f ), g)H = f1 ḡ0 + ∑ (fk+1 √k + 1 + βfk−1 √k)ḡk β

k=1 ∞

= f0 βḡ1 + ∑ fk (ḡk−1 √k + βḡk √k) k=1

= (f , a (g))H . ∗

β

This implies that a∗ (g) = β

dg + (1 − β2 )zg. dz

For β = 0, we have the Segal–Bargmann space; for β = 1, we get a∗ = a, so the operator a is self-adjoint on H1 . In addition, we use Parseval’s identity to compute the norms ‖a(f )‖2Hβ and ‖a∗ (f )‖2Hβ : ∞

󵄩󵄩 󵄩2 2 2 󵄩󵄩a(f )󵄩󵄩󵄩H = |f1 | + ∑ |β√k + 1fk + √k + 2fk+2 | β

k=0

(1.38)

and ∞

󵄩󵄩 ∗ 󵄩󵄩2 2 2 2 󵄩󵄩a (f )󵄩󵄩Hβ = β |f1 | + ∑ |√k + 1fk + β√k + 2fk+2 | . k=0

A simple computation shows that 󵄩󵄩 ∗ 󵄩󵄩2 󵄩 󵄩2 2 2 󵄩󵄩a (f )󵄩󵄩Hβ − 󵄩󵄩󵄩a(f )󵄩󵄩󵄩Hβ = (1 − β )‖f ‖Hβ ; it follows that dom(a) = dom(a∗ ), and for 0 ≤ β < 1, we obtain ‖f ‖2Hβ ≤ compare with (1.32).

1 󵄩 󵄩2 󵄩 󵄩2 (󵄩󵄩a(f )󵄩󵄩󵄩H + 󵄩󵄩󵄩a∗ (f )󵄩󵄩󵄩H ), β β 1 − β2 󵄩

(1.39)

1.7 Notes

�

25

1.7 Notes The basics on Hilbert spaces can be found, for instance, in [93]. There are numerous good texts on different aspects of Bergman spaces and their reproducing kernels; the reader may consult the books by Chen and Shaw [16], Krantz [69], and Jarnicki and Pflug [60]. The remark about the Riemann mapping and the Bergman kernel is contained in Bell’s book [4]. Further results regarding the Bergman kernel and the solution to 𝜕 will be discussed in the next chapters. Folland’s book [30] yields an excellent approach to mathematical concepts of quantum mechanics, like creation and annihilation operators and the uncertainty principle. The fundamentals of the Fock (Segal–Bargmann) space can be found in Zhu’s book [97]. We will come back to these notions at the end of this book; see Chapter 16. An even more general nonradial form of the weight is investigated in [80].

2 The canonical solution operator to 𝜕 In this chapter, we will use properties of the Bergman kernel to solve the inhomogeneous Cauchy–Riemann equation 𝜕u =g 𝜕z

or

𝜕u = g,

where 𝜕 1 𝜕 𝜕 = ( + i ), 𝜕z 2 𝜕x 𝜕y

z = x + iy,

(2.1)

and g ∈ A2 (𝔻). Before we proceed, we recall some basic facts from operator theory.

2.1 Compact operators on Hilbert spaces Let H1 and H2 be separable Hilbert spaces, and let A : H1 󳨀→ H2 be a bounded linear operator. The operator A is compact if the image A(B1 ) of the unit ball B1 in H1 is a relatively compact subset of H2 ; since H2 is complete, this is equivalent to the notion of a totally bounded set, i. e., for each ϵ > 0, there exists a finite number of elements v1 , . . . , vm ∈ H2 such that m

A(B1 ) ⊂ ⋃ B(vj , ϵ), j=1

where B(vj , ϵ) = {v ∈ H2 : ‖v − vj ‖ < ϵ}. Another equivalent definition of compactness is as follows: for each bounded sequence (uk )k in H1 , the image sequence (A(uk ))k has a convergent subsequence in H2 . Let ℒ(H1 , H2 ) denote the space of all bounded linear operators from H1 to H2 endowed with the topology generated by the operator norm ‖A‖ = sup{‖Au‖ : ‖u‖ ≤ 1}. In this way, ℒ(H1 , H2 ) becomes a Banach space. Let 𝒦(H1 , H2 ) denote the subspace of all compact operators from H1 to H2 . The following characterization of compactness is useful for the special operators in the text; see, for instance, [23]: Proposition 2.1. Let H1 and H2 be Hilbert spaces, and let S: H1 → H2 be a bounded linear operator. The following three statements are equivalent:

https://doi.org/10.1515/9783111182926-002

2.1 Compact operators on Hilbert spaces

– –

S is compact. For every ϵ > 0, there are C = Cϵ > 0 and a compact operator T = Tϵ : H1 → H2 such that ‖Sv‖ ≤ C‖Tv‖ + ϵ‖v‖.

–

� 27

(2.2)

For every ϵ > 0, there are C = Cϵ > 0 and a compact operator T = Tϵ : H1 → H2 such that ‖Sv‖2 ≤ C‖Tv‖2 + ϵ‖v‖2 .

(2.3)

Proof. First, we show that (2.2) and (2.3) are equivalent. Suppose that (2.3) holds. Write (2.3) with ϵ and C replaced by their squares to obtain 2

‖Sv‖2 ≤ C 2 ‖Tv‖2 + ϵ2 ‖v‖2 ≤ (C‖Tv‖ + ϵ‖v‖) , which implies (2.2). Now suppose that (2.2) holds. Choose η such that ϵ = 2η2 and apply (2.2) with ϵ replaced by η to get ‖Sv‖2 ≤ C 2 ‖Tv‖2 + 2ηC‖v‖‖Tv‖ + η2 ‖v‖2 . It is easily seen (small–large constant trick) that there is C ′ > 0 such that 2ηC‖v‖‖Tv‖ ≤ η2 ‖v‖2 + C ′ ‖Tv‖2 , and hence ‖Sv‖2 ≤ (C 2 + C ′ )‖Tv‖2 + 2η2 ‖v‖2 = C ′′ ‖Tv‖2 + ϵ‖v‖2 . To prove the proposition, it therefore suffices to show that (2.2) is equivalent to compactness. When S is known to be compact, we choose T = S and C = 1, and then (2.2) holds for every positive ϵ. For the converse, let (vn )n be a bounded sequence in H1 . We want to extract a Cauchy subsequence from (Svn )n . From (2.2) we have ‖Svn − Svm ‖ ≤ C‖Tvn − Tvm ‖ + ϵ‖vn − vm ‖.

(2.4)

Given a positive integer N, we may choose ϵ sufficiently small in (2.4) so that the second term on the right-hand side is at most 1/(2N). The first term can be made smaller than 1/(2N) by extracting a subsequence of (vn )n (still labeled the same) for which (Tvn )n converges and then choosing n and m large enough.

28 � 2 The canonical solution operator to 𝜕 Let (v(0) n )n denote the original bounded sequence. The above argument shows that (N) for each positive integer N, there is a sequence (v(N) n )n satisfying: (vn )n is a subsequence (N−1) (N) of (vn )n , and for any pair v and w in (vn )n , we have ‖Sv − Sw‖ ≤ 1/N. . Then (wk )k is a subseLet (wk )k be the diagonal sequence defined by wk = v(k) k quence of (v(0) ) , and the image sequence under S of (w ) is a Cauchy sequence. Since n k k n H2 is complete, the image sequence converges, and S is compact. Proposition 2.2. 𝒦(H1 , H2 ) is a closed subspace of ℒ(H1 , H2 ) endowed with the operator norm. Proof. Let A ∈ ℒ(H1 , H2 ). Suppose that for each ϵ > 0, there is a compact operator Aϵ such that ‖A − Aϵ ‖ ≤ ϵ. Then for each u ∈ H1 , we have ‖Au − Aϵ u‖ ≤ ϵ‖u‖. Now we get ‖Au‖ = ‖Au − Aϵ u + Aϵ u‖

≤ ‖Au − Aϵ u‖ + ‖Aϵ u‖

≤ ϵ‖u‖ + ‖Aϵ u‖. Proposition 2.1 implies that A is compact.

Proposition 2.3. Suppose that A ∈ 𝒦(H1 , H2 ), S ∈ ℒ(H1 , H1 ), and T ∈ ℒ(H2 , H2 ). Then both AS and TA are compact. Proof. If (uk )k is a bounded sequence in H1 , then (S(uk ))k is also bounded, because S is a bounded operator. Since A is compact, (A(S(uk )))k has a convergent subsequence. Thus AS is compact. To show that TA is compact, we use Proposition 2.1: ‖TAu‖ ≤ ‖T‖ ‖Au‖ ≤ ‖T‖(ϵ‖u‖ + C‖Au‖) ≤ ϵ ‖T‖ ‖u‖ + C ‖T‖ ‖Au‖. Corollary 2.4. Let H be a Hilbert space. Then 𝒦(H, H) forms a two-sided closed ideal in

ℒ(H, H).

Theorem 2.5. Let A : H1 󳨀→ H2 be a bounded linear operator. The following properties are equivalent: (i) A is compact; (ii) the adjoint operator A∗ : H2 󳨀→ H1 is compact; (iii) A∗ A : H1 󳨀→ H1 is compact.

2.1 Compact operators on Hilbert spaces

� 29

Proof. Suppose that A is compact. Then, by Proposition 2.3, AA∗ is also compact. Given ϵ > 0, it follows that there is a constant C such that 󵄩󵄩 ∗ 󵄩󵄩2 ∗ ∗ ∗ 󵄩󵄩A u󵄩󵄩 = (A u, A u) = (u, AA u) 󵄩 󵄩 ≤ ‖u‖ 󵄩󵄩󵄩AA∗ u󵄩󵄩󵄩 󵄩2 󵄩 ≤ ϵ‖u‖2 + C 󵄩󵄩󵄩AA∗ u󵄩󵄩󵄩 . Therefore, by Proposition 2.1, A∗ is compact. Since A∗∗ = A, (i) and (ii) are equivalent. (i) implies (iii) (Proposition 2.3), so it remains to show that (iii) implies (i). Let A∗ A be compact. Given ϵ > 0, there is again a constant C such that 󵄩 󵄩2 ‖Au‖2 = (Au, Au) = (u, A∗ Au) ≤ ϵ‖u‖2 + C 󵄩󵄩󵄩A∗ Au󵄩󵄩󵄩 , so Proposition 2.1 implies that A is compact. Proposition 2.6. A bounded operator A : H 󳨀→ H is compact if and only if there exists a sequence (Ak )k of linear operators with finite-dimensional range such that ‖A − Ak ‖ → 0 as k → ∞. Proof. An operator with finite-dimensional range is compact, because any bounded sequence in a finite-dimensional Hilbert space has a convergent subsequence. The limit of compact operators in the operator norm is again compact, so we proved one direction. The converse will follow from the next two results. The following theorem is the spectral theorem for compact self-adjoint operators: Theorem 2.7. Let A : H 󳨀→ H be a compact self-adjoint operator on a separable Hilbert space H. Then there exist a real zero sequence (μn )n and an orthonormal system (en )n in H such that for x ∈ H, ∞

Ax = ∑ μn (x, en )en , n=0

where the sum converges in the operator norm, i. e., 󵄩󵄩 󵄩󵄩 N 󵄩󵄩 󵄩󵄩 sup 󵄩󵄩󵄩Ax − ∑ μn (x, en )en 󵄩󵄩󵄩 → 0 󵄩 󵄩󵄩 ‖x‖≤1󵄩 n=0 󵄩 󵄩 as N → ∞. Proof. First, we collect some elementary properties of self-adjoint operators: Claim (a): Let A ∈ ℒ(H, H) = ℒ(H) and suppose that (Ax, x) = 0 for all x ∈ H. Then A = 0. We have

30 � 2 The canonical solution operator to 𝜕 0 = (A(x + y), x + y) − (A(x − y), x − y) = 2[(Ax, y) + (Ay, x)], and hence (Ax, y) + (Ay, x) = 0. Now replace x by ix. Then i(Ax, y) − i(Ay, x) = 0, and therefore (Ax, y) = 0 for all x, y ∈ H, which implies A = 0. Claim (b): A = A∗ if and only if (Ax, x) ∈ ℝ for all x ∈ H. The equality A = A∗ implies (Ax, x)− = (x, Ax) = (Ax, x) ∈ ℝ. If (Ax, x) = (Ax, x)− for all x ∈ H, then (Ax, x) = (x, A∗ x)− = (A∗ x, x), and so ((A − A∗ )x, x) = 0 for all x ∈ H. By (a) we have A = A∗ . Now we show that for a self-adjoint operator A ∈ ℒ(H), the norm of A is given by 󵄨 󵄨 ‖A‖ = sup 󵄨󵄨󵄨(Ax, x)󵄨󵄨󵄨. ‖x‖=1

(2.5)

For this aim, let NA = sup‖x‖=1 |(Ax, x)|. Then NA ≤ ‖A‖. We also have 2[(Ax, y) + (Ay, x)] = (A(x + y), x + y) − (A(x − y), x − y) as A∗ = A and, by the parallelogram rule, 󵄨󵄨 󵄨 2 2 2 2 󵄨󵄨4R(Ax, y)󵄨󵄨󵄨 ≤ NA (‖x + y‖ + ‖x − y‖ ) = 2NA (‖x‖ + ‖y‖ ). There is θ ∈ ℝ such that e−iθ (Ax, y) = |(Ax, y)|. Now replace y by e−iθ y. Then 󵄨 󵄨 2󵄨󵄨󵄨(Ax, y)󵄨󵄨󵄨 ≤ NA (‖x‖2 + ‖y‖2 ). For t > 0, we replace x by tx and y by y/t. Then we obtain 1 󵄨 󵄨 2󵄨󵄨󵄨(Ax, y)󵄨󵄨󵄨 ≤ NA (t 2 ‖x‖2 + 2 ‖y‖2 ). t We consider the right side of this inequality as a function in t, and after differentiation with respect to t, we get 2t‖x‖2 −

2 ‖y‖2 , t3

so the right side of the inequality is minimal if t 2 = ‖y‖/‖x‖. Hence we obtain 󵄨 󵄨 2󵄨󵄨󵄨(Ax, y)󵄨󵄨󵄨 ≤ 2NA ‖x‖ ‖y‖ and ‖A‖ ≤ NA , which proves (2.5). Next we show that if A ≠ 0, then there exists an eigenvector x0 ∈ H of A such that ‖x0 ‖ = 1 and 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨(Ax0 , x0 )󵄨󵄨󵄨 = ‖A‖ = sup 󵄨󵄨󵄨(Ax, x)󵄨󵄨󵄨, ‖x‖=1

(2.6)

2.1 Compact operators on Hilbert spaces

� 31

and we show that the corresponding eigenvalue λ0 is real and satisfies |λ0 | = ‖A‖. By (2.5) we have ‖A‖ = sup‖x‖=1 |(Ax, x)|. Hence there is a sequence (xn )n in H such that ‖xn ‖ = 1 and limn→∞ |(Axn , xn )| = ‖A‖. The inner products (Axn , xn ) are real, so there exists a subsequence, again denoted by (xn )n , such that limn→∞ (Axn , xn ) = λ0 . Then we have λ0 = ‖A‖ or λ0 = −‖A‖. Now we get 0 ≤ ‖Axn − λ0 xn ‖2 = ‖Axn ‖2 − 2λ0 (Axn , xn ) + λ20 ‖xn ‖2 ≤ ‖A‖2 − 2λ0 (Axn , xn ) + ‖A‖2

and, letting n → ∞, lim ‖Axn − λ0 xn ‖ = 0.

n→∞

Since A is compact and (xn )n is a bounded sequence, there exists a subsequence (xnk )k such that limk→∞ Axnk = x. Hence limk→∞ λ0 xnk = x, and as λ0 ≠ 0, we also have limk→∞ xnk = x0 with λ0 x0 = x. Since A is continuous, this implies lim Axnk = Ax0 ,

k→∞

and hence x = λ0 x0 = Ax0 . So x0 is an eigenvector of A with eigenvalue λ0 . This proves (2.6). Now each eigenvalue of A is real: Ax = λx and x ≠ 0 imply λ(x, x) = (Ax, x) = (x, Ax) = (x, λx) = λ(x, x). Therefore λ ∈ ℝ. Let Hλ = {x ∈ H : Ax = λx} be the eigenspace of the eigenvalue λ. Since A is continuous, Hλ is closed. There exists an orthonormal basis in Hλ consisting of eigenvectors of A. If λ1 ≠ λ2 are eigenvalues, then Hλ1 ⊥Hλ2 , because for x ∈ Hλ1 and y ∈ Hλ2 , we have λ1 (x, y) = (Ax, y) = (x, Ay) = (x, λ2 y) = λ2 (x, y), and hence (x, y) = 0. We claim that the eigenspace Hλ of an eigenvalue λ ≠ 0 is always finite-dimensional: if Hλ is of infinite dimension, we would have an infinite orthonormal system {xα }α∈A such that Axα = λxα , and since the restriction of A to Hλ is also compact, we could find a convergent subsequence of (Axα )α , which contradicts to ‖λxα1 − λxα2 ‖2 = 2|λ|2 for α1 ≠ α2 . Now we have two possible cases: (i) there are finitely many eigenvalues ≠ 0; (ii) there are infinitely many eigenvalues ≠ 0. (i) In this case, we have H = Hλ0 ⊕ ⋅ ⋅ ⋅ ⊕ Hλk−1 ⊕ M, where M = (Hλ0 ⊕ ⋅ ⋅ ⋅ ⊕ Hλk−1 )⊥ . We claim that A(M) ⊆ M: for y ∈ M, we have (Ay, x) = (y, Ax) = 0 for all x ∈ Hλ0 ⊕ ⋅ ⋅ ⋅ ⊕ Hλk−1 , because Ax ∈ Hλ0 ⊕ ⋅ ⋅ ⋅ ⊕ Hλk−1 , and hence Ay ∈ M. The restriction AM of A to M is again compact and self-adjoint: for x, y ∈ M, we have (AM x, y) = (Ax, y) = (x, Ay) = (x, AM y), because A(M) ⊆ M. We claim that AM = 0.

32 � 2 The canonical solution operator to 𝜕 Suppose AM ≠ 0. Then, by (2.5), there is λ ≠ 0 such that AM x = λx for some x ≠ 0, and hence we would get x ∈ Hλ , which is a contradiction. Let Pj : H 󳨀→ Hλj be the orthogonal projection. Then we obtain x = P0 x + P1 x + ⋅ ⋅ ⋅ + Pk−1 x + y, where Ay = 0, and therefore Ax = λ0 P0 x + λ1 P1 x + ⋅ ⋅ ⋅ + λk−1 Pk−1 x. (ii) For each ϵ > 0, the set {λi : λi eigenvalue of A with |λi | ≥ ϵ} is finite; otherwise, we would get an infinite orthonormal system {xi } such that Axi = λi xi , and the sequence (Axi )i would have to contain a convergent subsequence. However, ‖Axk − Axl ‖2 = ‖λk xk − λl xl ‖2 = |λk |2 + |λl |2 ≥ 2ϵ2 , which is a contradiction. Let λ0 , . . . , λk−1 be eigenvalues with |λi | ≥ ϵ, and let M = (Hλ0 ⊕ ⋅ ⋅ ⋅ ⊕ Hλk−1 )⊥ . The restriction AM of A to M is self-adjoint and compact; hence, by (2.5), there is an eigenvector x such that AM x = λx and |λ| = ‖AM ‖, and thus Ax = λx. Therefore x ∈ (Hλi )⊥ and |λ| = ‖AM ‖ < ϵ. So we have Ax = Ax1 + AM x2 , x1 ∈ M ⊥ ,

x2 ∈ M.

Since Ax1 = λ0 P0 x + ⋅ ⋅ ⋅ + λk−1 Pk−1 x, we obtain ‖AM x2 ‖ = ‖Ax − λ0 P0 x − ⋅ ⋅ ⋅ − λk−1 Pk−1 x‖ < ϵ‖x‖. Letting ϵ → 0, we get ∞

Ax = ∑ λj Pj x. j=0

Now let {ψ1 , . . . , ψk } be an orthonormal basis of Hλi . Then k

APi (x) = λi Pi (x) = λi ∑(x, ψj )ψj . j=1

The sequence of the eigenvalues can be ordered in the following way: |λ0 | ≥ |λ1 | ≥ ⋅ ⋅ ⋅ , because for each ϵ > 0, there are only finitely many λi with |λi | ≥ ϵ. Finally, the orthogonality of the eigenvectors xk and Bessel’s inequality (1.6) give

2.1 Compact operators on Hilbert spaces

� 33

󵄩󵄩2 󵄩󵄩 ∞ n 2 󵄩 󵄩󵄩 󵄩󵄩Ax − ∑ λk (x, xk )xk 󵄩󵄩󵄩 = ∑ 󵄨󵄨󵄨λk (x, xk )󵄨󵄨󵄨2 ≤ (‖x‖ sup |λk |) , 󵄨 󵄨 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 k>n k=n+1 k=1 and since (λk )k≥1 is a zero sequence, the above series converges to A in the operator norm. Now we drop the assumption of self-adjointness and obtain the following: Proposition 2.8. Let A : H1 󳨀→ H2 be a compact operator. There exist a decreasing zero sequence (sn )n in ℝ+ and orthonormal systems (en )n≥0 in H1 and (fn )n≥0 in H2 such that ∞

Ax = ∑ sn (x, en )fn n=0

∀x ∈ H1 ,

where the sum converges again in the operator norm. Proof. To show this, we apply the spectral theorem for the self-adjoint compact operator A∗ A : H1 󳨀→ H1 and get ∞

A∗ Ax = ∑ sn2 (x, en )en , n=0

where sn2 are the eigenvalues of A∗ A. If sn > 0, then we set fn = sn−1 Aen and get (fn , fm ) =

s2 1 1 (Aen , Aem ) = (A∗ Aen , em ) = n (en , em ) = δn,m . sn sm sn sm sn sm

For y ∈ H1 such that y ⊥ en for each n ∈ ℕ0 , we have by (2.7) that ‖Ay‖2 = (Ay, Ay) = (A∗ Ay, y) = 0. Hence we have ∞

∞

Ax = A(x − ∑ (x, en )en ) + A( ∑ (x, en )en ) n=0

n=0

∞

∞

n=0

n=0

= ∑ (x, en )Aen = ∑ sn (x, en )fn . Similarly to the last theorem, we get 󵄩󵄩 󵄩󵄩2 n ∞ 2 󵄩󵄩 󵄩 󵄩󵄩Ax − ∑ sk (x, ek )fk 󵄩󵄩󵄩 = ∑ 󵄨󵄨󵄨sk (x, ek )󵄨󵄨󵄨2 ≤ (‖x‖ sup |sk |) , 󵄩󵄩 󵄩󵄩 󵄨 󵄨 󵄩󵄩 󵄩󵄩 k>n k=1 k=n+1 which implies that the series converges in the operator norm.

(2.7)

34 � 2 The canonical solution operator to 𝜕 Remark 2.9. From the last statement we get the missing direction in the proof of Proposition 2.6. The numbers sn are called the s-numbers of A. They are uniquely determined by the operator A, since they are the square roots of the eigenvalues of A∗ A. Let 0 < p < ∞. The operator A belongs to the Schatten class Sp if its sequence (sn )n of s-numbers belongs to lp . The elements of the Schatten class S2 are called Hilbert–Schmidt operators. Proposition 2.10. Let A : H1 󳨀→ H2 be a bounded linear operator between Hilbert spaces. The following conditions are equivalent: (i) there is an orthonormal basis (ei )i∈I of H1 , such that ∑i∈I ‖Aei ‖2 < ∞; (ii) for each orthonormal basis (fj )j∈J of H1 , we have ∑j∈J ‖Afj ‖2 < ∞; (iii) A is a Hilbert–Schmidt operator. Proof. (i) implies (ii): Let (ei )i∈I be as in (i), and let (gl )l∈L be an orthonormal basis of H2 . Then by Parseval’s equality (Proposition 1.10) 󵄩 󵄩2 󵄨 󵄨2 󵄨 󵄨2 ∑󵄩󵄩󵄩A∗ gl 󵄩󵄩󵄩 = ∑ ∑󵄨󵄨󵄨(A∗ gl , ei )󵄨󵄨󵄨 = ∑ ∑󵄨󵄨󵄨(gl , Aei )󵄨󵄨󵄨 = ∑ ‖Aei ‖2 < ∞.

l∈L

l∈L i∈I

i∈I l∈L

i∈I

If (fj )j∈J is an arbitrary orthonormal basis of H1 , then we get again 󵄩 󵄩2 ∑ ‖Afj ‖2 = ∑󵄩󵄩󵄩A∗ gl 󵄩󵄩󵄩 = ∑ ‖Aei ‖2 < ∞. j∈J

i∈I

l∈L

(ii) implies (iii): Let (ei )i∈I be an orthonormal basis of H1 , and let M be a finite subset of I. We set PM x = ∑i∈M (x, ei )ei . Then 󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩 󵄩 󵄩󵄩(A − APM )x 󵄩󵄩󵄩 = 󵄩󵄩󵄩A(I − PM )x 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ (x, ei )Aei 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 i∈I\M 1/2

1/2

󵄨 󵄨2 ≤ ( ∑ ‖Aei ‖2 ) ( ∑ 󵄨󵄨󵄨(x, ei )󵄨󵄨󵄨 ) i∈I\M

i∈I\M

1/2

≤ ( ∑ ‖Aei ‖2 ) ‖x‖, i∈I\M

where we used Bessel’s inequality (1.6). By assumption, ∑i∈I ‖Aei ‖2 < ∞, and hence we can approximate A in the operator norm by operators of finite range. Therefore A is compact and, by Proposition 2.8, can be written as ∞

Ax = ∑ sn (x, xn )fn . n=0

By Remark 1.11 there exists an orthonormal basis (ξj )j∈J that contains the orthonormal system (xn )n≥0 . Then we have

2.1 Compact operators on Hilbert spaces

� 35

󵄩󵄩2 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 󵄩 ∑ ‖Aξj ‖ = ∑󵄩󵄩 ∑ sn (ξj , xn )fn 󵄩󵄩󵄩 = ∑ sn2 < ∞, 󵄩󵄩 󵄩󵄩 n=0 j∈J j∈J 󵄩n=0 󵄩 2

so A is a Hilbert–Schmidt operator. (iii) implies (i): If A is a Hilbert–Schmidt operator, then it can be written as above, and we obtain (i). Corollary 2.11. Let A : H1 󳨀→ H2 be a Hilbert–Schmidt operator. Then for each orthonormal basis (ei )i∈I of H1 , we have ∞

ν2 (A)2 := ∑ sn2 = ∑ ‖Aei ‖2 ≥ ‖A‖2 . n=0

i∈I

In particular, ν2 is a norm in the space S2 (H1 , H2 ) of all Hilbert–Schmidt operators between H1 and H2 . Proof. We have 1/2 󵄩󵄩 󵄩󵄩 󵄩 󵄩 ‖Ax‖ = 󵄩󵄩󵄩∑(x, ei )Aei 󵄩󵄩󵄩 ≤ (∑ ‖Aei ‖2 ) ‖x‖, 󵄩󵄩 󵄩󵄩 i∈I i∈I

and hence ν2 (A) ≥ ‖A‖. It is easily seen that A 󳨃→ (∑i∈I ‖Aei ‖2 )1/2 is a norm. If H1 and H2 are separable Hilbert spaces, A ∈ ℒ(H1 , H2 ), (en )n≥1 is an orthonormal basis of H1 , and (fn )n≥1 is an orthonormal basis of H2 , then for each x ∈ H1 , we have ∞

Ax = ∑ (x, en )Aen , n=1

and hence for all j ∈ ℕ, ∞

(Ax, fj ) = ∑ (x, en )(Aen , fj ). n=1

The coefficients of Ax with respect to (fj )j≥1 can be computed from the coefficients of x with respect to (en )n≥1 by means of the infinite matrix aj,n = (Aen , fj ),

j, n ∈ ℕ.

Parseval’s equation implies ∞

∞

∞

j,n=1

j,n=1

n=1

󵄨 󵄨2 ∑ |aj,n |2 = ∑ 󵄨󵄨󵄨(Aen , fj )󵄨󵄨󵄨 = ∑ ‖Aen ‖2 ,

and we obtain the following:

36 � 2 The canonical solution operator to 𝜕 Corollary 2.12. An operator A ∈ ℒ(H1 , H2 ) is a Hilbert–Schmidt operator if and only if the matrix (aj,n )j,n≥1 of A with respect to arbitrary orthonormal bases satisfies ∞

∑ |aj,n |2 < ∞.

j,n=1

In this case, ∞

ν2 (A)2 = ∑ |aj,n |2 . j,n=1

Corollary 2.13. A linear operator A : l2 󳨀→ l2 is a Hilbert–Schmidt operator if and only 2 2 if there exists a matrix (aj,n )j,n≥1 with ∑∞ j,n=1 |aj,n | < ∞ such that for all x = (xn )n≥1 ∈ l , we have ∞

Ax = ( ∑ aj,n xn ) n=1

. j≥1

Proof. If A has the given form, we can apply the Cauchy–Schwarz inequality to get ∞

‖Ax‖22 ≤ ( ∑ |aj,n |2 )‖x‖22 , j,n=1

and hence everything follows from Corollary 2.12. Proposition 2.14. Let S ⊆ ℝn and T ⊆ ℝm be open sets, and let A : L2 (T) 󳨀→ L2 (S) be a bounded linear operator. Then A is a Hilbert–Schmidt operator if and only if there exists K ∈ L2 (S × T) such that Af (s) = ∫ K(s, t)f (t) dt,

f ∈ L2 (T).

T

In this case, 1/2

󵄨 󵄨2 ν2 (A) = ( ∫ 󵄨󵄨󵄨K(s, t)󵄨󵄨󵄨 ds dt) . S×T

Proof. Let (gk )k≥1 and (fj )j≥1 be orthonormal bases of L2 (T) and L2 (S), respectively. Then (hj,k )j,k≥1 defined by hj,k (s, t) := fj (s)gk (t),

(s, t) ∈ S × T,

is an orthonormal basis of L2 (S × T). This can be shown as follows: if F ∈ L2 (S × T), then t 󳨃→ F(s, t)fj (s) is in L2 (T) for almost every s ∈ S. If F ⊥ hj,k for all j, k ∈ ℕ, then it follows

2.2 The canonical solution operator to 𝜕 restricted to A2 (𝔻)

�

37

that the functions t 󳨃→ F(s, t)fj (s) vanish in L2 (T) for all j ∈ ℕ almost everywhere with respect to s. This implies F = 0 and (hj,k )j,k≥1 is an orthonormal basis. If A is a Hilbert–Schmidt operator and (aj,k )j,k≥1 is its matrix with respect to (gk )k≥1 and (fj )j≥1 , then, by Corollary 2.12, K(s, t) := ∑ aj,k hj,k (s, t) = ∑ aj,k fj (s)gk (t) j,k≥1

j,k≥1

belongs to L2 (S × T), and ‖K‖22 = ∑j,k≥1 |aj,k |2 = ν2 (A)2 . For f ∈ L2 (T), we have Af (s) = ∑ aj,k (f , gk )fj (s) = ∫ ∑ aj,k fj (s)gk (t)f (t) dt = ∫ K(s, t)f (t) dt. j,k≥1

T j,k≥1

T

If A is given by the kernel function K ∈ L2 (S × T), then A is continuous, because 󵄨 󵄨2 󵄨 󵄨2 ‖Af ‖2 ≤ ( ∫ 󵄨󵄨󵄨K(s, t)󵄨󵄨󵄨 ds dt) ∫󵄨󵄨󵄨f (t)󵄨󵄨󵄨 dt. S×T

T

For the matrix (aj,k )j,k≥1 of A, we get aj,k = (Agk , fj ) = ∫(∫ K(s, t)gk (t) dt)fj (s) ds = (K, hj,k ) S

T

for all j, k ∈ ℕ. Bessel’s inequality and Corollary 2.12 imply that A is a Hilbert–Schmidt operator.

2.2 The canonical solution operator to 𝜕 restricted to A2 (𝔻) We return to the inhomogeneous Cauchy–Riemann equation on the disc 𝔻 and use the notations of Chapter 1. Let S(g)(z) = ∫ K(z, w)g(w)(z − w)− dλ(w). 𝔻

Using the Bergman projection P : L2 (𝔻) 󳨀→ A2 (𝔻) and (1.5), we get ̃ S(g)(z) = zg(z) − P(g)(z),

(2.8)

38 � 2 The canonical solution operator to 𝜕 ̃ where g(w) = wg(w). We claim that S(g) is a solution of the inhomogeneous Cauchy– Riemann equation 𝜕g 𝜕P(g)̃ 𝜕 𝜕z S(g)(z) = g(z) + z + = g(z), 𝜕z 𝜕z 𝜕z 𝜕z because g and P(g)̃ are holomorphic functions, and therefore 𝜕S(g) = g. In addition, we have S(g) ⊥ A2 (𝔻), because for arbitrary f ∈ A2 (𝔻), we get ̃ f ) = (g,̃ f ) − (P(g), ̃ f ) = (g,̃ f ) − (g,̃ Pf ) = (g,̃ f ) − (g,̃ f ) = 0. (Sg, f ) = (g̃ − P(g), The operator S : A2 (𝔻) 󳨀→ L2 (𝔻) is called the canonical solution operator to 𝜕. Now we want to show that S is a compact operator. For this purpose, we consider the adjoint operator S ∗ and prove that S ∗ S is compact, which implies that S is compact (Theorem 2.5). For g ∈ A2 (𝔻) and f ∈ L2 (𝔻), we have (Sg, f ) = ∫(∫ K(z, w)g(w)(z − w)− dλ(w) )f (z) dλ(z) 𝔻 𝔻 −

= ∫(∫ K(w, z)(z − w)f (z) dλ(z)) g(w) dλ(w) = (g, S ∗ f ), 𝔻 𝔻

and hence S ∗ (f )(w) = ∫ K(w, z)(z − w)f (z) dλ(z).

(2.9)

𝔻

Now set cn2 = ∫ |z|2n dλ(z) = 𝔻

π n+1

and ϕn (z) = zn /cn , n ∈ ℕ0 . Then the Bergman kernel K(z, w) can be expressed in the form k

zk w . 2 k=0 ck ∞

K(z, w) = ∑ Next, we compute

k ∞ k k ∞ k−1 c zn−1 z w wn z w wn P(ϕ̃ n )(z) = ∫ ∑ 2 w dλ(w) = ∑ 2 ∫ dλ(w) = n 2 , cn cn cn−1 k=0 ck k=1 ck−1 𝔻

and hence we have

𝔻

2.2 The canonical solution operator to 𝜕 restricted to A2 (𝔻)

S(ϕn )(z) = z ϕn (z) −

cn zn−1 , 2 cn−1

� 39

n ∈ ℕ.

Now we apply S ∗ and get w k zk zzn cn zn−1 (z − w)( − 2 ) dλ(z). 2 cn cn−1 k=0 ck ∞

S ∗ S(ϕn )(w) = ∫ ∑ 𝔻

The last integral is computed in two steps: first, by the multiplication by z, zn+1 ∞ wk zk+1 wk zk zzn+1 cn zn − ) dλ(z) = ( dλ(z) ∑ ∫ 2 2 cn cn k=0 ck2 cn−1 k=0 ck ∞

∫∑ 𝔻

𝔻

−

∞ cn wk zk n z dλ(z) ∑ ∫ 2 2 cn−1 k=0 ck 𝔻

n

=

w wn ∫ |z|2n+2 dλ(z) − 2 ∫ |z|2n dλ(z) 3 cn cn−1 cn 𝔻

c2 c = ( n+1 − 2 n ) wn . 3 cn cn−1

𝔻

Next, by the multiplication by w, wk zk zzn cn zn−1 zn ∞ wk zk+1 ( − 2 ) dλ(z) = w ∫ dλ(z) ∑ 2 cn cn k=0 ck2 cn−1 k=0 ck ∞

w∫ ∑ 𝔻

𝔻

−w∫ 𝔻

= w( = 0.

cn zn−1 ∞ wk zk ∑ 2 dλ(z) 2 cn−1 k=0 ck

cn wn−1 cn wn−1 − 2 ) 2 cn−1 cn−1

From this it follows that S ∗ S(ϕn )(w) = (

2 cn+1 cn2 − )ϕn (w), 2 cn2 cn−1

for n = 0, an analogous computation shows S ∗ S(ϕ0 )(w) = Finally, we get the following:

c12 ϕ0 (w). c02

n = 1, 2, . . . ;

40 � 2 The canonical solution operator to 𝜕 Proposition 2.15. Let S : A2 (𝔻) 󳨀→ L2 (𝔻) be the canonical solution operator for 𝜕, and let (ϕk )k be the normalized monomials. Then S ∗ Sϕ =

2 ∞ cn+1 cn2 c12 (ϕ, ϕ )ϕ + ( ) (ϕ, ϕn )ϕn − ∑ 0 0 2 cn2 c02 cn−1 n=1

(2.10)

for each ϕ ∈ A2 (𝔻). Since 2 cn+1 cn2 1 − = →0 2 (n + 2)(n + 1) cn2 cn−1

it follows that S ∗ S and S are compact.

We have also shown that the s-numbers of S are ( ∞

∑(

n=1

as n → ∞,

2 cn+1 cn2

−

cn2 1/2 , 2 ) cn−1

and since

2 cn+1 cn2 ) < ∞, − 2 cn2 cn−1

it follows that S is Hilbert–Schmidt. This can also be shown directly using Proposition 2.14. For this purpose, we claim that the function (z, w) 󳨃→ K(z, w)(z − w)− belongs to L2 (𝔻 × 𝔻). We have to prove that ∫∫ 𝔻𝔻

|z − w|2 dλ(z) dλ(w) < ∞. |1 − zw|4

An easy estimate gives |z − w| ≤ |1 − zw| for z, w ∈ 𝔻. Hence ∫∫ 𝔻𝔻

|z − w|2 1 dλ(z) dλ(w). dλ(z) dλ(w) ≤ ∫ ∫ 4 |1 − zw| |1 − zw|2 𝔻𝔻

Introducing the polar coordinates z = r eiθ and w = s eiϕ , we can write the last integral in the form 1 1 2π 2π

r s dθ dϕ dr ds 1 dλ(z) dλ(w) = ∫ ∫ ∫ ∫ ∫∫ |1 − zw|2 1 − 2 r s cos(θ − ϕ) + r 2 s2

𝔻𝔻

0 0 0 0

1 1 2π 2π

= ∫∫∫ ∫ 0 0 0 0

1 − r 2 s2 rs dθ dϕ dr ds. 1 − 2 r s cos(θ − ϕ) + r 2 s2 1 − r 2 s2

Integration of the Poisson kernel with respect to θ yields

2.2 The canonical solution operator to 𝜕 restricted to A2 (𝔻) 2π

∫ 0

1 − ρ2 dθ = 2π, 1 − 2ρ cos(θ − ϕ) + ρ2

�

41

0 < ρ < 1.

Therefore we have 1 1 2π 2π

∫∫∫ ∫ 0 0 0 0

rs 1 − r 2 s2 dθ dϕ dr ds 2 2 1 − 2 r s cos(θ − ϕ) + r s 1 − r 2 s2 1 1

1

0 0

0

log(1 − s2 ) rs 2 = (2π) ∫ ∫ dr ds = − (2π) ds < ∞. ∫ 2s 1 − r 2 s2 2

In the following, we consider the weighted spaces of entire functions m m 󵄨 󵄨2 A2 (ℂ, e−|z| ) = {f : ℂ 󳨀→ ℂ : ‖f ‖2m := ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z) < ∞},

ℂ

where m > 0. Let m

ck2 = ∫ |z|2k e−|z| dλ(z). ℂ

Then zk w 2 k=0 ck ∞

Km (z, w) = ∑

k

m

is the reproducing kernel for A2 (ℂ, e−|z| ). In the sequel the expression 2 ck+1

ck2

−

ck2

2 ck−1

will be important. Using the integral representation of the Γ-function, we easily see that the above expression is equal to Γ( 2k+4 ) m Γ( 2k+2 ) m

−

Γ( 2k+2 ) m Γ( 2k ) m

.

For m = 2, this expression equals 1 for each k = 1, 2, . . . . We will be interested in the limit behavior as k → ∞. By Stirling’s formula the limit behavior is equivalent to the limit behavior of the expression 2/m

(

2k + 2 ) m

2/m

−(

2k ) m

42 � 2 The canonical solution operator to 𝜕 as k → ∞. Hence we have shown the following: Lemma 2.16. The expression ) Γ( 2k+4 m

Γ( 2k+2 ) m

−

) Γ( 2k+2 m

) Γ( 2k m

tends to ∞ for 0 < m < 2, is equal to 1 for m = 2, and tends to zero for m > 2 as k tends to ∞. m For 0 < ρ < 1, define fρ (z) := f (ρz) and fρ̃ (z) = zfρ (z) for f ∈ A2 (ℂ, e−|z| ). Then m m we easily see that f ̃ ∈ L2 (ℂ, e−|z| ) but there are functions g ∈ A2 (ℂ, e−|z| ) such that

ρ

m

zg ∈ ̸ L2 (ℂ, e−|z| ). m m Let Pm : L2 (ℂ, e−|z| ) 󳨀→ A2 (ℂ, e−|z| ) denote the orthogonal projection. Then Pm can be written in the form m

Pm (f )(z) = ∫ Km (z, w)f (w)e−|w| dλ(w),

m

f ∈ L2 (ℂ, e−|z| ).

ℂ

Proposition 2.17. Let m ≥ 2. Then there is a constant Cm > 0, depending only on m, such that m m 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨fρ̃ (z) − Pm (fρ̃ )(z)󵄨󵄨󵄨 e−|z| dλ(z) ≤ Cm ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z)

ℂ

ℂ m

for each 0 < ρ < 1 and for each f ∈ A2 (ℂ, e−|z| ). k Proof. First, we observe that for the Taylor expansion of f (z) = ∑∞ k=0 ak z , we have k

∞ ∞ k m z w Pm (fρ̃ )(z) = ∫ ∑ 2 (w ∑ aj ρj wj ) e−|w| dλ(w) c j=0 k=0 k ℂ ∞

= ∑ ak k=1

ck2

2 ck−1

ρk zk−1 .

Now we obtain ∞ ∞ m c2 󵄨 󵄨2 ∫󵄨󵄨󵄨fρ̃ (z) − Pm (fρ̃ )(z)󵄨󵄨󵄨 e−|z| dλ(z) = ∫(z ∑ ak ρk zk − ∑ ak 2k ρk zk−1 ) ck−1 k=0 k=1

ℂ

ℂ

∞

∞

× (z ∑ ak ρk zk − ∑ ak k=0

k=1

ck2

2 ck−1

∞

∞

k=0

k=1

= ∫( ∑ |ak |2 ρ2k |z|2k+2 − 2 ∑ |ak |2 ℂ

m

ρk zk−1 ) e−|z| dλ(z) ck2

2 ck−1

ρ2k |z|2k

2.2 The canonical solution operator to 𝜕 restricted to A2 (𝔻)

� 43

ck4 2k 2k−2 −|z|m ρ |z| )e dλ(z) 4 ck−1

∞

+ ∑ |ak |2 k=1

∞

|a0 |2 c12 + ∑ |ak |2 ck2 ρ2k ( k=1

2 ck+1

ck2

−

ck2

2 ck−1

).

Now the result follows from the facts that ∞

m 󵄨 󵄨2 ∫󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z) = ∑ |ak |2 ck2

k=0

ℂ

and that the sequence (

2 ck+1 ck2

−

ck2 2 )k ck−1

is bounded.

Remark 2.18. Already in the last proposition, the sequence (

2 ck+1 ck2

c2

− c2k )k plays an impork−1

tant role, and it will turn out that this sequence is the sequence of eigenvalues of the ∗ operator Sm Sm (see below). m

Proposition 2.19. Let m ≥ 2 and consider an entire function f ∈ A2 (ℂ, e−|z| ) with Taylor k series expansion f (z) = ∑∞ k=0 ak z . Let ck2

∞

∞

F(z) := z ∑ ak zk − ∑ ak k=1

k=0

2 ck−1

zk−1 m

m

and define Sm (f ) := F. Then Sm : A2 (ℂ, e−|z| ) 󳨀→ L2 (ℂ, e−|z| ) is a continuous linear m operator representing the canonical solution operator to 𝜕 restricted to A2 (ℂ, e−|z| ), i. e., m 𝜕Sm (f ) = f and Sm (f ) ⊥ A2 (ℂ, e−|z| ). Proof. By the proof of Proposition 2.17 and by Abel’s and Fatou’s theorems (see, for instance, [26]) we have m m 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨F(z)󵄨󵄨󵄨 e−|z| dλ(z) = ∫ lim󵄨󵄨󵄨fρ̃ (z) − Pm (fρ̃ )(z)󵄨󵄨󵄨 e−|z| dλ(z) ρ→1

ℂ

ℂ

m 󵄨 󵄨2 ≤ sup ∫󵄨󵄨󵄨fρ̃ (z) − Pm (fρ̃ )(z)󵄨󵄨󵄨 e−|z| dλ(z)

00

for all multi-indices γ with γj = −1. In the case of a rotation-invariant measure μ, we write md = ∫ |z|2d dμ ℂn

and claim that cγ =

(n − 1 + |γ|)! , (n − 1)! γ! m|γ|

(3.5)

where |γ| = γ1 + ⋅ ⋅ ⋅ + γn and γ! = γ1 ! . . . γn !. To show this, let 𝒰 be the unitary group consisting of all n × n unitary matrices, and let dU denote the Haar probability measure on 𝒰 . Let σ be the rotation-invariant probability measure on the unit sphere 𝕊 in ℂn . For a multi-index α = (α1 , . . . , αn ), we α α define zα = z1 1 . . . znn , α! = α1 ! . . . αn !, and |α| = α1 + ⋅ ⋅ ⋅ + αn . Due to the invariance of μ, it follows by Fubini’s theorem that ∫ zα zβ dμ(z) = ∫ ∫ (Uz)α (Uz)β dμ(z) dU ℂn

𝒰 ℂn

= ∫ ∫(Uz)α (Uz)β dU dμ(z) ℂn 𝒰

56 � 3 Spectral properties of the canonical solution operator to 𝜕̄ β

= ∫ |z||α|+|β| ∫ ζ α ζ dσ(ζ ) dμ(z), ℂn

(3.6)

𝕊

where we used the fact that for a continuous function f ∈ 𝒞 (𝕊), we have ∫ f (ζ ) dσ(ζ ) = ∫ f (Uη) dU 𝒰

𝕊

for all η ∈ 𝕊 (see [84, Proposition 1.4.7]). It is clear that for α ≠ β, we have β

∫ ζ α ζ dσ(ζ ) = 0. 𝕊

Next, we claim that for any multi-index γ, (n − 1)! γ! . (n − 1 + |γ|)!

∫ |ζ γ |2 dσ(ζ ) = 𝕊

(3.7)

To prove (3.7), we use the integral n

I = ∫ |zγ |2 exp(−|z|2 ) dλ2n (z) = ∏ ∫ |w|2γj exp(−|w|2 ) dλ2 (w), j=1 ℂ

ℂn

where λ2n is the Lebesgue measure on ℝ2n . It follows easily that I = π n γ!. Now applying integration in polar coordinates to I, we get ∞

2

π γ! = 2n cn ∫ r 2|γ|+2n−1 e−r dr ∫ |ζ γ |2 dσ(ζ ), n

0

𝕊

where cn is the volume of the unit ball in ℂn . Hence ∫ |ζ γ |2 dσ(ζ ) = 𝕊

π n γ! , (n − 1 + |γ|)! ncn

and taking γ = 0, we get cn = π n /n!, which proves (3.7). For d ∈ ℕ, we set md = ∫ |z|2d dμ ℂn

and from (3.6) and (3.7) obtain

and

cγ−1 = ∫ |zγ |2 dμ ℂn

3.3 Compactness and Schatten class membership

cγ =

� 57

(n − 1 + |γ|)! . (n − 1)! γ! m|γ|

To express the conditions of our theorems, we compute (setting d = |γ| + 1)

∑( j

cγ+ej

cγ+2ej

−

m

d+2n−1 d+1 { d+n md − )={ 1 md+1 cγ+ej { d+n md

cγ

md , md−1

γj ≠ −1 for all j, else.

(3.8)

Note that the Cauchy–Schwarz inequality implies that the first case in (3.8) always dominates the second case for n ≥ 2; for n = 1, we observe that the second case in (3.8) reduces m1 ; compare with Proposition 2.15. to m 0

ies.

Using this observation and some trivial inequalities, we get the following corollar-

Corollary 3.9. Let μ be a rotation-invariant measure on ℂn . Then the canonical solution operator to 𝜕̄ is bounded on A2(0,1) (dμ) if and only if sup( d∈ℕ

(2n + d − 1)md+1 md − ) < ∞. (n + d)md md−1

(3.9)

Corollary 3.10. Let μ be a rotation-invariant measure on ℂn . Then the canonical solution operator to 𝜕̄ is compact on A2(0,1) (dμ) if and only if lim (

d→∞

(2n + d − 1)md+1 md − ) = 0. (n + d)md md−1

(3.10)

Corollary 3.11. Let μ be a rotation-invariant measure on ℂn . Then the canonical solution operator to 𝜕̄ is a Hilbert–Schmidt operator on A2(0,1) (dμ) if and only if n + d − 2 md+1 ) < ∞. n−1 md

lim (

d→∞

(3.11)

Remark 3.12. It follows that the canonical solution operator to 𝜕̄ is a Hilbert–Schmidt operator on A2 (𝔻) but fails to be Hilbert–Schmidt on A2 (𝔹n ), where 𝔹n is the unit ball in ℂn for n ≥ 2. Corollary 3.13. Let μ be a rotation-invariant measure on ℂn , and let p > 0. Then the canonical solution operator to 𝜕̄ is in the Schatten class Sp as an operator from A2(0,1) (dμ) to L2 (dμ) if and only if p

md 2 n + d − 2 (2n + d − 1)md+1 )( − ) < ∞. ∑( n−1 (n + d)md md−1 d=1 ∞

Now we prove the theorems of this section. In what follows, we denote by

(3.12)

58 � 3 Spectral properties of the canonical solution operator to 𝜕̄ uα = √cα zα the orthonormal basis of monomials for the space A2 (dμ) and by Uα,j = uα d z̄j

(3.13)

the corresponding basis of A2(0,1) (dμ). We first note that it is always possible to solve the ̄ 𝜕-equation for the elements of this basis; indeed, 𝜕̄ z̄j uα = Uα,j . The canonical solution operator is also easily determined for forms with monomial coefficients: Lemma 3.14. The canonical solution Szα d z̄j for monomial forms is given by Szα d z̄j = z̄j zα −

cα−ej cα

zα−ej ,

α ∈ ℕn0 .

(3.14)

Proof. We have ⟨z̄j zα , zβ ⟩ = ⟨zα , zβ+ej ⟩; so this expression is nonzero only if β = α − ej (in particular, this implies (3.14) for multi-indices α with αj = 0; recall our convention that cγ = 0 if one of the entries of γ is negative). Thus Szα d z̄j = z̄j zα + czα−ej , and c is computed by −1 0 = ⟨z̄j zα + czα−ej , zα−ej ⟩ = cα−1 + ccα−e , j

which gives c = −cα−ej /cα . We are going to introduce an orthogonal decomposition A2(0,1) (dμ) = ⨁ Eγ γ∈Γ

of A2(0,1) (dμ) into at most n-dimensional subspaces Eγ indexed by multi-indices γ ∈ Γ (we will describe the index set below) and a corresponding sequence of mutually orthogonal finite-dimensional subspaces Fγ ⊂ L2 (dμ), which diagonalizes S (by this we mean that SEγ = Fγ ). To motivate the definition of Eγ , note that {0, ⟨Szα d z̄k , Szβ d z̄ℓ ⟩ = { 1 cα ( − { cα cα+eℓ

cα−ek

cα+eℓ −ek

β ≠ α + eℓ − ek ,

),

β = α + eℓ − ek ,

(3.15)

so that ⟨Szα d z̄k , Szβ d z̄ℓ ⟩ ≠ 0 if and only if there exists a multi-index γ such that α = γ + ek and β = γ + eℓ . Recall (3.13) and define Eγ = span{Uγ+ej ,j : 1 ≤ j ≤ n} = span{zγ+ej d z̄j : 1 ≤ j ≤ n}, and likewise Fγ = SEγ . Recall that Γ is defined to be the set of all multi-indices whose entries are greater than or equal to −1 and has at most one negative entry. Note that Eγ

3.3 Compactness and Schatten class membership

� 59

is one-dimensional if exactly one entry in γ equals −1 and n-dimensional otherwise. We have already observed that Fγ are mutually orthogonal subspaces of L2 (dμ) (see (3.15)). Whenever we use multi-indices γ and integers p ∈ {1, . . . , n} as indices, we use the convention that p runs over all p such that γ + ep ≥ 0; that is, for a fixed multi-index γ ∈ Γ, either the indices are all p ∈ {1, . . . , n}, or there is exactly one p such that γp = −1, in which case the index is exactly this one p. We next observe that we can find an orthonormal basis of Eγ and an orthonormal basis of Fγ such that Sγ = S|Eγ : Eγ → Fγ acts diagonally in these bases. First, note that it suffices to do this if dim Eγ = n (since an operator between one-dimensional spaces is automatically diagonal). Fixing γ, the functions Uj := Uγ+ej ,j are an orthonormal basis of Eγ . The operator Sγ is clearly nonsingular on this space, so the functions SUj = Ψj j

j

constitute a basis of Fγ . For a basis B of vectors vj = (v1 , . . . , vn ), j = 1, . . . , n, of ℂn , we consider the new basis n

j

Vk = ∑ vk Uj . j=1

Since the basis given by the Uj is orthonormal, the basis given by the Vk is also orthonormal, provided that the vectors vk = (v1k , . . . , vnk ) constitute an orthonormal basis for ℂn with the standard Hermitian product. Let us write j

Φk = SVk = ∑ vk SUj . j

j

The inner product ⟨Φp , Φq ⟩ is then given by ∑j,k vp v̄kq ⟨SUj , SUk ⟩. We therefore have ⟨Φ1 , Φ1 ⟩ .. ( . ⟨Φn , Φ1 ⟩ v11

. = ( .. v1n

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅

⟨Φ1 , Φn ⟩ .. ) . ⟨Φn , Φn ⟩

vn1 ⟨Ψ1 , Ψ1 ⟩ .. .. . )( . vnn ⟨Ψn , Ψ1 ⟩

⟨Ψ1 , Ψn ⟩ v̄11 .. . ) ( .. . ⟨Ψn , Ψn ⟩ v̄n1

⋅⋅⋅ ⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅

v̄1n .. . ). v̄nn

(3.16)

Since the matrix (⟨Ψj , Ψk ⟩)j,k is Hermitian, we can unitarily diagonalize it; that is, we can choose an orthonormal basis B of ℂn such that with this choice of B, the vectors j

φγ,k = Vk = ∑ vk Uγ+ej ,j j

(3.17)

of Eγ are orthonormal, and their images Φk = SVk are orthogonal in Fγ . Therefore Φk /‖Φk ‖ is an orthonormal basis of Fγ such that Sγ : Eγ → Fγ is diagonal when expressed in terms of the bases {V1 , . . . , Vn } ⊂ Eγ and {Φ1 , . . . , Φn } ⊂ Fγ with entries ‖Φk ‖.

60 � 3 Spectral properties of the canonical solution operator to 𝜕̄ Furthermore, the ‖Φk ‖ are exactly the square roots of the eigenvalues of the matrix (⟨Ψp , Ψq ⟩)p,q , which by (3.15) is given by ⟨Ψp , Ψq ⟩ = ⟨SUγ+ep ,p , SUγ+eq ,q ⟩

= √cγ+ep √cγ+eq ⟨S zγ+ep d z̄p , S zγ+eq d z̄q ⟩

= √cγ+ep cγ+eq

1

(

cγ+ep

cγ+ep cγ+ep +eq cγ+ep cγ+eq − cγ cγ+ep +eq = . cγ+ep +eq √cγ+ep cγ+eq

−

cγ

cγ+eq

)

(3.18)

Summarizing, we have the following proposition. Proposition 3.15. With μ as above, the canonical solution operator S: A2(0,1) (dμ) → L2(0,1) (dμ) admits a diagonalization by orthonormal bases. In fact, we have a decomposition A2(0,1) = ⨁γ Eγ into mutually orthogonal finite-dimensional subspaces Eγ indexed by the multiindices γ with at most one negative entry (equal to −1), which are of dimension 1 or n, and orthonormal bases φγ,j of Eγ such that Sφγ,j is a set of mutually orthogonal vectors in L2 (dμ). For fixed γ, the norms ‖Sφγ,j ‖ are the square roots of the eigenvalues of the matrix Cγ = (Cγ,p,q )p,q given by Cγ,p,q =

cγ+ep cγ+eq − cγ cγ+ep +eq cγ+ep +eq √cγ+ep cγ+eq

(3.19)

.

In particular, we have that n

n

∑ ‖Sφγ,j ‖2 = tr(Cγ,p,q )p,q = ∑ ( p=1

j=1

cγ+ep

cγ+2ep

−

cγ

cγ+ep

).

(3.20)

To prove Theorem 3.4, we use Proposition 3.15. We have seen that we have an orthonormal basis φγ,j , γ ∈ Γ, j ∈ {1, . . . , n}, such that the images Sφγ,j are mutually orthogonal. Thus S is bounded if and only if there exists a constant C such that ‖Sφγ,j ‖2 ≤ C for all γ ∈ Γ and j ∈ {1, . . . , dim Eγ }. If dim Eγ = 1, then γ has exactly one entry (say the jth one) equal to −1; in that case, let us write φγ = Uγ+ej d z̄j . We have Sφγ = √cγ+ej z̄j zγ+ej , and so

‖Sφγ ‖2 =

cγ+ej

cγ+2ej

.

3.3 Compactness and Schatten class membership

� 61

On the other hand, if dim Eγ = n, then we argue as follows. Writing ‖Sφγ,j ‖2 = λ2γ,j with λγ,j > 0, from (3.20) we find that n

n

∑ λ2γ,j = ∑( j=1

j=1

cγ+ej

cγ

−

cγ+2ej

cγ+ej

).

The last two equations complete the proof of Theorem 3.4. To prove Theorem 3.5, we use Proposition 2.1. Proof of Theorem 3.5. We first show that (3.2) implies compactness. We will use the notation already used in the proof of Theorem 3.4; that is, we write ‖Sφγ,j ‖2 = λ2γ,j . Let ε > 0. There exists a finite set Aε of multi-indices γ ∈ Γ such that for all γ ∉ Aε , n

n

∑ λ2γ,j = ∑( j=1

j=1

cγ+ej

cγ+2ej

−

cγ

cγ+ej

) < ε.

Hence, if we consider the finite-dimensional (and thus compact) operator Tε defined by Tε (∑ aγ,j φγ,j ) = ∑ aγ,j Sφγ,j γ∈Aε

for every v = ∑ aγ,j φγ,j ∈ A2(0,1) (dμ), then we obtain 󵄩󵄩 󵄩󵄩2 󵄩 󵄩 ‖Sv‖2 = ‖Tε v‖2 + 󵄩󵄩󵄩S ∑ aγ,j φγ,j 󵄩󵄩󵄩 󵄩󵄩 γ∉A 󵄩󵄩 ε

= ‖Tε v‖2 + ∑ |aγ,j |2 ‖Sφγ,j ‖2 2

γ∉Aε

= ‖Tε v‖ + ∑ |aγ,j |2 λ2γ,j 2

γ∉Aε

≤ ‖Tε v‖ + ε ∑ |aγ,j |2 2

γ∉Aε

≤ ‖Tε v‖ + ε‖v‖2 . Hence (2.2) holds, and we have proved the first implication in Theorem 3.5. We now turn to the other direction. Assume that (3.2) is not satisfied. Then there exist K > 0 and an infinite family A of multi-indices γ such that for all γ ∈ A, n

n

∑ λ2γ,j = ∑( j=1

j=1

cγ+ej

cγ+2ej

−

cγ

cγ+ej

) > nK.

In particular, for each γ ∈ A, there exists jγ such that λ2γ,jγ > K. Thus we have an infinite orthonormal family {φγ,jγ : γ ∈ A} of vectors such that their images Sφγ,jγ are orthogonal and have the norms bounded from below by √K, which contradicts compactness.

62 � 3 Spectral properties of the canonical solution operator to 𝜕̄ In the following, we will also need to introduce the usual grading on the index set Γ, that is, we write Γk = {γ ∈ Γ: |γ| = k},

k ≥ −1.

(3.21)

To study the membership in the Schatten class, we need the following elementary lemma. Lemma 3.16. Assume that p and q are continuous real-valued functions on ℝN that are homogeneous of degree 1 (i. e., p(tx) = tp(x) and q(tx) = tq(x) for t ∈ ℝ) and that q(x) = 0 and p(x) = 0 implies x = 0. Then there exists a constant C such that 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨q(x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨p(x)󵄨󵄨󵄨 ≤ C 󵄨󵄨󵄨q(x)󵄨󵄨󵄨. C󵄨

(3.22)

Proof. Note that the set Bq = {x: |q(x)| = 1} is compact: it is closed since q is continuous, and since |q| is bounded from below on the unit sphere of ℝN by some m > 0, it is necessarily contained in the closed ball of radius 1/m. Now the function |p| is bounded on the compact set Bq , say, by 1/C from below and C from above. Thus for all x ∈ ℝN , 1 󵄨󵄨󵄨󵄨 x 󵄨󵄨󵄨󵄨 ≤ 󵄨󵄨p( )󵄨 ≤ C, C 󵄨󵄨 q(x) 󵄨󵄨󵄨 which proves (3.22). Proof of Theorem 3.6. Note that S is in the Schatten class Sp if and only if p

∑ λγ,j < ∞.

(3.23)

γ∈Γ, j

We rewrite this sum as p

∑ (∑ λγ,j ) =: M ∈ ℝ ∪ {∞}.

γ∈Γ

j

Lemma 3.16 implies that there exists a constant C such that for every γ ∈ Γ, p/2

1 (∑ λ2 ) C j γ,j

p/2

p

≤ ∑ λγ,j ≤ C(∑ λ2γ,j ) j

j

.

Hence M < ∞ if and only if p/2

∑(∑ λ2γ,j ) γ

j

< ∞,

which after applying (3.20) becomes condition (3.6) claimed in Theorem 3.6.

3.4 Notes

� 63

Proof of Theorem 3.7. S is in the Hilbert–Schmidt class if and only if ∑ λ2γ,j < ∞.

(3.24)

γ∈Γ,j

We will prove that k

∑ ∑ λ2γ,j =

ℓ=−1 γ∈Γℓ ,j

cα , c α+e p ,|α|=k+1

(3.25)

∑

n

α∈ℕ 1≤p≤n

which immediately implies Theorem 3.7. The proof is by induction over k. For k = −1, the left-hand side of (3.25) is n

n

n

j=1

j=1

j=1

∑ λ2−ej ,j = ∑ ‖zj ‖2 c0 = ∑

c0 , cep

which is equal to the right-hand side of (3.25). Now assume that (3.25) holds for k = K − 1. We will show that this implies that it holds for k = K. We write K

∑ ∑ λ2γ,j =

ℓ=−1 γ∈Γℓ ,j

=

∑

cα

c α∈ℕn ,|α|=K−1 α+ep 1≤p≤n ∑ n

α∈ℕ ,|α|=K 1≤p≤n

+ ∑ ( γ∈ΓK ,j

cγ+ej

cγ+2ej

−

cγ

cγ+ej

)

cα . cα+ep

This finishes the proof of Theorem 3.7.

3.4 Notes Most of the material in this chapter is taken from [47], which can be viewed as a generalization of results from [38, 39] and [71]. Corollary 3.13 improves Theorem C of [71] in the sense that it also covers the case 0 < p < 2. We would like to note that our techniques can be adapted to the setting of [71] by considering the canonical solution operator on a Hilbert space ℋ of holomorphic functions endowed with a norm comparable to the L2 -norm on each subspace generated by monomials of a fixed degree d if in addition to the requirements in [71], we also assume that the monomials belong to ℋ; this introduces the additional weights found by [71] in the formulas, as the reader can check. In our setting the formulas are somewhat “cleaner” by working with A2 (dμ) (in particular, Corollary 3.11 only holds in this setting). The restriction of the canonical solution operator to 𝜕 can be seen as a Hankel operator. Additional contributions were made by Knirsch and Schneider [63, 88]. Later, in

64 � 3 Spectral properties of the canonical solution operator to 𝜕̄ Chapter 12, we will investigate more general properties of this restriction, which are related to commutators between multiplication operators and the Bergman projection. In Chapter 13, we will study the canonical solution operator to 𝜕 on L2 -spaces without restriction.

4 The 𝜕-complex Our main task will be to solve the inhomogeneous Cauchy–Riemann equation 𝜕u = f , where the right-hand side f is given and satisfies the necessary condition 𝜕f = 0. For n > 1, this is an overdetermined system of partial differential equations, which will be reduced to a system with equal numbers of unknowns and equations. We also need some background from the theory of unbounded operators on Hilbert spaces and the basics of the theory of distributions. Afterwards, we introduce the complex Laplacian (box operator). We show that under suitable assumptions this operator has a bounded inverse, the 𝜕-Neumann operator, and we discuss important properties of the latter.

4.1 Unbounded operators on Hilbert spaces In the sequel, we develop elements of unbounded self-adjoint operators, which are used for the 𝜕-complex. Definition 4.1. Let H1 , H2 be Hilbert spaces, and let T : dom(T) 󳨀→ H2 be a densely defined linear operator, i. e., dom(T) is a dense linear subspace of H1 . Let dom(T ∗ ) be the space of all y ∈ H2 such that x 󳨃→ (Tx, y)2 defines a continuous linear functional on dom(T). Since dom(T) is dense in H1 , there exists a uniquely determined element T ∗ y ∈ H1 such that (Tx, y)2 = (x, T ∗ y)1 (Riesz representation Theorem 1.6). The map y 󳨃→ T ∗ y is linear, and T ∗ : dom(T ∗ ) 󳨀→ H1 is the adjoint operator to T. An operator T is said to be closed if the graph 𝒢 (T) = {(f , Tf ) ∈ H1 × H2 : f ∈ dom(T)}

is a closed subspace of H1 × H2 . The inner product in H1 × H2 is ((x, y), (u, v)) = (x, u)1 + (y, v)2 . If Ṽ is a linear subspace of H1 containing dom(T) and an operator T̃ defined on Ṽ is ̃ = Tx for all x ∈ dom(T), then we say that T̃ is an extension of T. such that Tx An operator T with domain dom(T) is said to be closable if it has a closed extension T.̃ Lemma 4.2. Let T be a densely defined closable operator. Then there is a closed extension T, called its closure, whose domain is the smallest among all closed extensions. Proof. Let 𝒱 be the set of x ∈ H1 for which there exist xk ∈ dom(T) and y ∈ H2 such that limk→∞ xk = x and limk→∞ Txk = y. Since T̃ is a closed extension of T, it follows ̃ = y. Therefore y is uniquely determined by x. We define Tx = y that x ∈ dom(T)̃ and Tx with dom(T) = 𝒱 . Then T is an extension of T, and every closed extension of T is also https://doi.org/10.1515/9783111182926-004

66 � 4 The 𝜕-complex an extension of T. The graph of T is the closure of the graph of T in H1 × H2 . Hence T is a closed operator. Lemma 4.3. Let T1 : dom(T1 ) 󳨀→ H2 be a densely defined operator, and let T2 : H2 󳨀→ H3 be a bounded operator. Then (T2 T1 )∗ = T1∗ T2∗ , which includes that dom((T2 T1 )∗ ) = dom(T1∗ T2∗ ). Proof. Note that dom(T1∗ T2∗ ) = {f ∈ dom(T2∗ ) : T2∗ (f ) ∈ dom(T1∗ )}. Let f ∈ dom(T1∗ T2∗ ) and g ∈ dom(T2 T1 ). Then (T1∗ T2∗ f , g) = (T2∗ f , T1 g) = (f , T2 T1 g), and hence dom(T1∗ T2∗ ) ⊆ dom((T2 T1 )∗ ). Now let f ∈ dom((T2 T1 )∗ ). As T2∗ is bounded and everywhere defined on H3 , we have for all g ∈ dom(T2 T1 ) = dom(T1 ) that ((T2 T1 )∗ f , g) = (f , T2 T1 g) = (T2∗ f , T1 g). Hence T2∗ f ∈ dom(T1∗ ) and f ∈ dom(T1∗ T2∗ ). Lemma 4.4. Let T be a densely defined operator on H, and let S be a bounded operator on H. Then (T + S)∗ = T ∗ + S ∗ . Proof. Let f ∈ dom(T ∗ + S ∗ ) = dom(T ∗ ). Then for all g ∈ dom(T + S) = dom(T), we have ((T ∗ + S ∗ )f , g) = (T ∗ f , g) + (S ∗ f , g) = (f , Tg) + (f , Sg) = (f , (T + S)g), and hence f ∈ dom((T + S)∗ ) and (T + S)∗ f = T ∗ f + S ∗ f . If f ∈ dom((T + S)∗ ), then for all g ∈ dom(T + S) = dom(T), we have ([(T + S)∗ − S ∗ ]f , g) = (f , (T + S)g) − (f , Sg) = (f , Tg), and therefore f ∈ dom(T ∗ ) and dom((T + S)∗ ) = dom(T ∗ + S ∗ ) = dom(T ∗ ). Lemma 4.5. Let T : dom(T) 󳨀→ H2 be a densely defined linear operator and define V : H1 × H2 󳨀→ H2 × H1 by V ((x, y)) = (y, −x). Then ∗

⊥

⊥

𝒢 (T ) = [V (𝒢 (T))] = V (𝒢 (T) );

in particular, T ∗ is always closed. Proof. (y, z) ∈ 𝒢 (T ∗ ) ⇔ (Tx, y)2 = (x, z)1 for each x ∈ dom(T) ⇔ ((x, Tx), (−z, y)) = 0 for each x ∈ dom(T) ⇔ V −1 ((y, z)) = (−z, y) ∈ 𝒢 (T)⊥ . Hence 𝒢 (T ∗ ) = V (𝒢 (T)⊥ ), and since V is unitary, we have V ∗ = V −1 and [V (𝒢 (T))]⊥ = V (𝒢 (T)⊥ ).

4.1 Unbounded operators on Hilbert spaces

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Lemma 4.6. Let T : dom(T) 󳨀→ H2 be a densely defined closed linear operator. Then H2 × H1 = V (𝒢 (T)) ⊕ 𝒢 (T ∗ ). Proof. Since 𝒢 (T) is closed, 𝒢 (T ∗ )⊥ = V (𝒢 (T)) by Lemma 4.5. Lemma 4.7. Let T : dom(T) 󳨀→ H2 be a densely defined closed linear operator. Then dom(T ∗ ) is dense in H2 and T ∗∗ = T. Proof. Let z⊥ dom(T ∗ ). Hence (z, y)2 = 0 for each y ∈ dom(T ∗ ). We have V −1 : H2 × H1 󳨀→ H1 × H2 , where V −1 ((y, x)) = (−x, y), and V −1 V = Id. Now by Lemma 4.6 we have H1 × H2 ≅ V −1 (H2 × H1 ) = V −1 (V (𝒢 (T)) ⊕ 𝒢 (T ∗ )) ≅ 𝒢 (T) ⊕ V −1 (𝒢 (T ∗ )). Hence (z, y)2 = 0 ⇔ ((0, z), (−T ∗ y, y)) = 0 for each y ∈ dom(T ∗ ). This implies (0, z) ∈ 𝒢 (T), and therefore z = T(0) = 0, which means that dom(T ∗ ) is dense in H2 . Since T and T ∗ are densely defined and closed, by Lemma 4.5 we have ⊥⊥

𝒢 (T) = 𝒢 (T)

⊥

= [V −1 𝒢 (T ∗ )] = 𝒢 (T ∗∗ ),

where −V −1 corresponds to V in considering operators from H2 to H1 . Lemma 4.8. Let T : dom(T) 󳨀→ H2 be a densely defined linear operator. Then ker T ∗ = (im T)⊥ , which means that ker T ∗ is closed. Proof. Let v ∈ ker T ∗ and y ∈ im T, which means that there exists u ∈ dom(T) such that Tu = y. Hence (v, y)2 = (v, Tu)2 = (T ∗ v, u)1 = 0, and ker T ∗ ⊆ (im T)⊥ . If y ∈ (im T)⊥ , then (y, Tu)2 = 0 for each u ∈ dom(T), which implies that y ∈ dom(T ∗ ) and (y, Tu)2 = (T ∗ y, u)1 for each u ∈ dom(T). Since dom(T) is dense in H1 , we obtain T ∗ y = 0 and (im T)⊥ ⊆ ker T ∗ . Lemma 4.9. Let T : dom(T) 󳨀→ H2 be a densely defined closed linear operator. Then ker T is a closed linear subspace of H1 . Proof. We use Lemma 4.8 for T ∗ and get ker T ∗∗ = (im T ∗ )⊥ . Since, by Lemma 4.7, T ∗∗ = T, we obtain that ker T = (im T ∗ )⊥ and ker T is a closed linear subspace of H1 . For our applications to the 𝜕-equation, it will be important to know whether the differential operators involved have a closed range or are even surjective. In the fol-

68 � 4 The 𝜕-complex lowing propositions, we will explain the functional analysis background and show how inequalities correspond to the closed range property. First, we collect some corresponding results about bounded operators and begin with the Baire category theorem and the open-mapping theorem. Theorem 4.10. Let X be a complete metric space. (i) If {Un }n is a sequence of open dense subsets of X, then ⋂∞ n=1 Un is dense in X. (ii) X is not a countable union of nowhere dense sets, i. e., of sets En such that E n has an empty interior. Proof. (i) Let W be a nonempty open set in X. We have to show that ∞

( ⋂ Un ) ∩ W ≠ 0. n=1

Since U1 ∩ W is open and nonempty, it contains a ball B(r1 , x1 ), and we can assume that 0 < r1 < 1. We choose sequences (xn )n in X and (rn )n in ℝ+ inductively as follows: having chosen xj and rj for j < n, we observe that Un ∩ B(rn−1 , xn−1 ) is open and nonempty, so we can choose xn and rn such that 0 < rn < 2−n and B(rn , xn ) ⊂ Un ∩ B(rn−1 , xn−1 ). Then, for n, m ≥ N, we see that xn , xm ∈ B(rN , xN ), and since rn → 0, the sequence (xn )n is a Cauchy sequence. As X is complete, the limit x = limn→∞ xn exists. Since xn ∈ B(rN , xN ) for all n ≥ N, we have x ∈ B(rN , xN ) ⊂ UN ∩ B(r1 , x1 ) ⊂ UN ∩ W for all N, which proves (i). (ii) If {En }n is a sequence of nowhere dense sets, then {(E n )c }n is a sequence of open c dense sets. Since, by (i), ⋂∞ n=1 (E n ) ≠ 0, we have ∞

∞

n=1

n=1

⋃ En ⊂ ⋃ E n ≠ X.

A set E ⊂ X is called meager (or of the first category) if it is a countable union of nowhere dense sets; otherwise, E is of the second category. Now we apply the Baire category theorem to linear maps between Banach spaces to prove the open-mapping the closed graph theorems. Theorem 4.11. Let X and Y be Banach spaces, and let T : X 󳨀→ Y be a bounded linear operator that is surjective. Then T is open. Proof. Let Br denote the open ball of radius r and center 0 in X. We have to show that T(Br ) contains an open ball about 0 in Y . Since X = ⋃∞ n=1 Bn and T is surjective, we have Y = ⋃∞ T(B ). As Y is complete and the map y → 󳨃 ny is a homeomorphism of Y that n n=1 maps T(B1 ) to T(Bn ), applying Theorem 4.10, we obtain that T(B1 ) cannot be nowhere

4.1 Unbounded operators on Hilbert spaces

� 69

dense in Y . Hence there exist y0 ∈ Y and r > 0 such that the ball B(4r, y0 ) = {y ∈ Y : ‖y − y0 ‖ < 4r} is contained in T(B1 ). Pick y1 = Tx1 ∈ T(B1 ) such that ‖y1 − y0 ‖ < 2r. Then B(2r, y1 ) ⊂ T(B1 ), so if ‖y‖ < 2r, then we have y = Tx1 + (y − y1 ) ∈ T(x1 + B1 ) ⊂ T(B2 ). Dividing both sides by 2, we obtain that there exists r > 0 such that if ‖y‖ < r, and then y ∈ T(B1 ). If we could replace T(B1 ) by T(B1 ), the proof would be complete. Dilation invariance implies that if ‖y‖ < r2−n , then y ∈ T(B2−n ). Suppose that ‖y‖ < r/2; we can find x1 ∈ B1/2 such that ‖y − Tx1 ‖ < r/4, and proceeding inductively, we can find xn ∈ B2−n such that 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 󵄩 󵄩󵄩y − ∑ Txj 󵄩󵄩󵄩 < r2−n−1 . 󵄩󵄩󵄩 󵄩󵄩󵄩 j=1 󵄩 󵄩

(4.1)

∞ −n We have ∑∞ = 1, and since X is complete, the absolutely convergent n=1 ‖xn ‖ < ∑n=1 2 ∞ series ∑n=1 xn is also convergent, say to x = ∑∞ n=1 xn ∈ X, and by (4.1) we have y = Tx. This means that T(B1 ) contains all y with ‖y‖ < r/2.

Corollary 4.12. Let T : X 󳨀→ Y be a bijective bounded linear operator between Banach spaces. Then T −1 is also bounded. Proof. By Theorem 4.11, T is open, and hence T −1 is bounded. Now it is easy to prove the closed graph theorem. Theorem 4.13. Let X and Y be Banach spaces. Let T : X 󳨀→ Y be a closed operator with dom(T) = X. Then T is bounded. Proof. Let π1 and π2 be the projections of 𝒢 (T) onto X and Y . They are both continuous. Since X × Y is also complete, 𝒢 (T) is a closed subspace of X × Y and therefore also complete. Since π1 is a bijection from 𝒢 (T) to X, π1−1 is bounded by Corollary 4.12. Therefore T = π2 ∘ π1−1 is bounded. Before we proceed in the theory for unbounded operators on Hilbert spaces, we still prove the uniform boundedness principle again as an application of the Baire category theorem. Theorem 4.14. Suppose X and Y are normed spaces and 𝒜 is a subset of ℒ(X, Y ). (i) If supT∈𝒜 ‖Tx‖ < ∞ for all x in a nonmeager subset of X, then sup ‖T‖ < ∞.

T∈𝒜

(ii) If X is a Banach space and supT∈𝒜 ‖Tx‖ < ∞ for all x ∈ X, then sup ‖T‖ < ∞.

T∈𝒜

70 � 4 The 𝜕-complex Proof. Let En = {x ∈ X : sup ‖Tx‖ ≤ n} = ⋂ {x ∈ X : ‖Tx‖ ≤ n}. T∈𝒜

T∈𝒜

Then the sets En are closed, so under the hypothesis of (i), some En must contain a nontrivial closed ball B(r, x0 ). Then E2n ⊃ B(r, 0), for if ‖x‖ ≤ r, then x − x0 ∈ En and 󵄩 󵄩 ‖Tx‖ ≤ 󵄩󵄩󵄩T(x − x0 )󵄩󵄩󵄩 + ‖Tx0 ‖ ≤ 2n. In other words, ‖Tx‖ ≤ 2n whenever T ∈ 𝒜 and ‖x‖ ≤ r, so sup ‖T‖ ≤ 2n/r.

T∈𝒜

(ii) follows from (i) by Theorem 4.10. The following characterization of compactness uses the uniform boundedness principle and the notion of weak convergence. Definition 4.15. A sequence (xk )k in a Hilbert space H is a weak null-sequence if (xk , x) → 0 for each x ∈ H. A sequence (xk )k converges weakly to x0 if (xk − x0 )k is a weak null sequence. Remark 4.16. A weakly convergent sequence (xk )k in a Hilbert space is always bounded: we have 󵄨 󵄨 sup󵄨󵄨󵄨(xk − x0 , x)󵄨󵄨󵄨 < ∞ k

for all x ∈ H. Then, by Theorem 4.14, 󵄨 󵄨 sup ‖xk − x0 ‖ = sup sup 󵄨󵄨󵄨(xk − x0 , x)󵄨󵄨󵄨 < ∞, k

k

‖x‖≤1

and therefore ‖xk ‖ ≤ ‖xk − x0 ‖ + ‖x0 ‖ < ∞ for all k ∈ ℕ. In the same way, we can show that each weak Cauchy sequence is bounded. If A ∈ ℒ(H1 , H2 ) and (xk )k is a weakly convergent sequence in H1 , then (Axk )k converges weakly in H2 , which follows from (Axk − Ax0 , y)2 = (xk − x0 , A∗ y)1 , where y ∈ H2 . Proposition 4.17. Let A ∈ ℒ(H1 , H2 ) be a bounded operator between Hilbert spaces. Then A is compact if and only if (Axk )k converges to 0 in H2 for each weak null sequence (xk )k in H1 .

4.1 Unbounded operators on Hilbert spaces

� 71

Proof. Let A be compact, and let (xk )k be a weak null sequence in H1 . Then (Axk )k is a weak null sequence in H2 . Suppose that (‖Axk ‖)k does not converge to 0. Then there exist ϵ > 0 and a subsequence (yk )k of (xk )k such that ‖Ayk ‖ ≥ ϵ for each k ∈ ℕ. By Remark 4.16 the sequence (‖yk ‖)k is bounded. As A is compact, there exists a subsequence (zk )k of (yk )k such that (Azk )k is convergent in H2 . Since (Azk )k converges weakly to 0, we would have 󵄨 󵄨 ‖Azk ‖ = sup 󵄨󵄨󵄨(Azk , y)󵄨󵄨󵄨 → 0, ‖y‖≤1

contradicting ‖Ayk ‖ ≥ ϵ. For the opposite direction, we need a number of prerequisites. First, we claim the following statement: if (xk )k is a bounded sequence in a Hilbert space H and if ((xk , y))k is a Cauchy sequence in ℂ for each y in a dense subset M of H, then (xk )k is a weak Cauchy sequence in H. To show this, let x ∈ H be an arbitrary element and choose y ∈ M such that ‖x − y‖ < ϵ. Then 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨(xk − xm , x)󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(xk , x − y)󵄨󵄨󵄨 + 󵄨󵄨󵄨(xk − xm , y)󵄨󵄨󵄨 + 󵄨󵄨󵄨(xm , y − x)󵄨󵄨󵄨. Since (xk )k is a bounded sequence, we get that ((xk , x))k is a Cauchy sequence in ℂ for each x ∈ H. Next, we claim that each weak Cauchy sequence in H is also weakly convergent: a weak Cauchy sequence is bounded, and therefore we have |(xk , x)| ≤ C‖x‖ for some constant C > 0 and for each k ∈ ℕ. Hence F(x) := lim (x, xk ) k→∞

defines a continuous linear functional for which there exists x0 ∈ H such that (x, x0 ) = lim (x, xk ), k→∞

which means that (xk )k converges weakly to x0 . Next, we show that each bounded sequence (xk )k contains a weakly convergent subsequence: the sequence ((xk , x))k is bounded in ℂ, and hence we can find subsequences (xk(m) )k of (xk(m−1) )k for m ∈ ℕ with (xk )k = (xk(0) )k such that ((xk(m) , xm ))k converges. The diagonal sequence (xk(k) )k has the property that the sequence ((xk(k) , xm ))k is convergent for each m ∈ ℕ. This implies that ((xk(k) , x))k converges for each x in the linear span L of the elements {xℓ : ℓ ∈ ℕ}. Since the sequence (xk(k) )k is bounded, we have from above that ((xk(k) , x))k is convergent for each x ∈ L. It is clear that ((xk(k) , x)) = 0 for each x ∈ L⊥ , and therefore we get that (xk(k) )k is a weak Cauchy sequence, which converges weakly by our prerequisites. Now we can finish the proof of the opposite direction: let (xk )k be a bounded sequence in H1 . We know that it contains a weakly convergent subsequence xkℓ → x0 , and

72 � 4 The 𝜕-complex hence the sequence (xkℓ − x0 )ℓ is a weak null sequence. By assumption we obtain that Axkℓ → Ax0 in H2 . So A is a compact operator. Now we continue to derive further important properties of unbounded operators on Hilbert spaces, which will later be used for the 𝜕-operator. Lemma 4.18. Let T : H1 󳨀→ H2 be a bounded linear operator. Then T(H1 ) is closed if and only if T|(ker T)⊥ is bounded from below, i. e., ‖Tf ‖ ≥ C‖f ‖

∀f ∈ (ker T)⊥ .

Proof. If T(H1 ) is closed, then the mapping T : (ker T)⊥ 󳨀→ T(H1 ) is bijective and continuous and, by the open-mapping Theorem 4.11, also open. This implies the desired inequality. To prove the other direction, let (fn )n be a sequence in H1 such that Tfn → y in H2 . We have to show that there exists h ∈ H1 such that Th = y. Decompose fn = gn + hn , where gn ∈ ker T and hn ∈ (ker T)⊥ . By assumption we have ‖hn − hm ‖ ≤ C‖Thn − Thm ‖ = C‖Tfn − Tfm ‖ < ϵ for all sufficiently large n and m. Hence (hn )n is a Cauchy sequence. Let h = limn→∞ hn . Then we have ‖Tfn − Th‖ = ‖Thn − Th‖ ≤ ‖T‖ ‖hn − h‖, and therefore y = lim Tfn = Th. n→∞

Lemma 4.19. Let T be as before. Then T(H1 ) is closed if and only if T ∗ (H2 ) is closed. Proof. Since T ∗∗ = T, it suffices to show one direction. We will show that the closedness of T(H1 ) implies that (ker T)⊥ = im T ∗ ; since (ker T)⊥ is closed, we will be finished. Let x ∈ im T ∗ . Then there exists y ∈ H2 such that x = T ∗ y. Now we get for x ′ ∈ ker T that (x, x ′ ) = (T ∗ y, x ′ ) = (y, Tx ′ ) = 0, and hence im T ∗ ⊆ (ker T)⊥ . For x ′ ∈ (ker T)⊥ , we define the linear functional λ(Tx) = (x, x ′ )

4.1 Unbounded operators on Hilbert spaces

� 73

on the closed subspace T(H1 ) of H2 . We remark that λ is well defined, since Tx = T x̃ implies that x − x̃ ∈ ker T, and hence (x − x,̃ x ′ ) = 0 and (x, x ′ ) = (x,̃ x ′ ). The operator T : H1 󳨀→ T(H1 ) is continuous and surjective. Since T(H1 ) is closed, the open-mapping theorem implies ‖v‖ ≤ C‖Tv‖ for all v ∈ (ker T)⊥ , where C > 0 is a constant. Set y = Tx and write x = u + v, where u ∈ ker T and v ∈ (ker T)⊥ . Then we obtain 󵄨 󵄨 󵄨󵄨 ′ 󵄨 󵄨 ′ 󵄨 󵄨󵄨λ(y)󵄨󵄨󵄨 = 󵄨󵄨󵄨(x, x )󵄨󵄨󵄨 = 󵄨󵄨󵄨(v, x )󵄨󵄨󵄨 󵄩 󵄩 ≤ ‖v‖󵄩󵄩󵄩x ′ 󵄩󵄩󵄩 󵄩 󵄩 ≤ C‖Tv‖󵄩󵄩󵄩x ′ 󵄩󵄩󵄩 󵄩 󵄩 = C‖Tx‖󵄩󵄩󵄩x ′ 󵄩󵄩󵄩 󵄩 󵄩 = C‖y‖󵄩󵄩󵄩x ′ 󵄩󵄩󵄩. Hence λ is continuous on im T. By Theorem 1.6 there exists a uniquely determined element z ∈ im T such that λ(y) = (y, z)2 = (x, x ′ )1 . This implies (y, z)2 = (Tx, z)2 = (x, T ∗ z)1 = (x, x ′ )1 for all x ∈ H1 , and hence x ′ = T ∗ z ∈ im T ∗ . Lemma 4.20. Let T : H1 󳨀→ H2 be a densely defined closed operator. Then im T is closed in H2 if and only if T|dom(T)∩(ker T)⊥ is bounded from below, i. e., ‖Tf ‖ ≥ C‖f ‖

∀f ∈ dom(T) ∩ (ker T)⊥ .

̃ , Tf )) = Tf and get a bounded Proof. On the graph 𝒢 (T), we define the operator T((f linear operator T̃ : 𝒢 (T) 󳨀→ H2 , since 󵄩󵄩 ̃ 󵄩 󵄩 󵄩 2 2 1/2 󵄩󵄩T((f , Tf ))󵄩󵄩󵄩 = ‖Tf ‖ ≤ (‖f ‖ + ‖Tf ‖ ) = 󵄩󵄩󵄩(f , Tf )󵄩󵄩󵄩, and im T̃ = im T. By Lemma 4.18, im T is closed if and only if T|̃ (ker T)̃ ⊥ is bounded from below. We have ker T̃ = ker T ⊕ {0}, and it remains to show that T|̃ (ker T)̃ ⊥ is bounded from below if and only if T|dom(T)∩(ker T)⊥ is bounded from below. This follows from 󵄩󵄩 ̃ 󵄩 2 2 1/2 󵄩󵄩T((f , Tf ))󵄩󵄩󵄩 = ‖Tf ‖ ≥ C(‖f ‖ + ‖Tf ‖ ) , and hence, for 0 < C < 1,

74 � 4 The 𝜕-complex

‖Tf ‖2 ≥

C2 ‖f ‖2 . 1 − C2

Lemma 4.21. Let P, Q : H 󳨀→ H be orthogonal projections on the Hilbert space H. Then the following statements are equivalent: (i) im(PQ) is closed; (ii) im(QP) is closed; (iii) im(I − P)(I − Q) is closed; (iv) P(H) + (I − Q)(H) is closed. Proof. (i) and (ii) are equivalent by Lemma 4.19 and the fact that QP = Q∗ P∗ = (PQ)∗ . Suppose (ii) holds and let (fn )n and (gn )n be sequences in H such that Pfn + (I − Q)gn → h. Then Q(Pfn + (I − Q)gn ) = QPfn → Qh. By assumption, im(QP) is closed, and hence there exists f ∈ H such that QPf = Qh; it follows that Qh = Pf − (I − Q)(Pf ) and h = Qh + (I − Q)h = Pf − (I − Q)(Pf ) + (I − Q)h = Pf + (I − Q)(h − Pf ) ∈ P(H) + (I − Q)(H), which yields (iv). If (iv) holds and (fn )n is a sequence in H such that QPfn → h, then we get QPfn = Pfn − (I − Q)Pfn ∈ P(H) + (I − Q)(H). Hence there exist f , g ∈ H such that h = Pf + (I − Q)g, and it follows that Qh = Q( lim QPfn ) = lim Q2 Pfn = h n→∞

n→∞

and h = Pf + (I − Q)g = Qh = QPf ∈ im(QP), and therefore (ii) holds. Finally, replace P by I − P and Q by I − Q. Then, using the statements proved so far, we obtain the equivalence im(I − P)(I − Q) closed ⇔ (I − P)(H) + Q(H) closed, which proves the remaining statements. At this point, we are able to prove Lemma 4.19 for densely defined closed operators.

4.1 Unbounded operators on Hilbert spaces

� 75

Proposition 4.22. Let T : H1 󳨀→ H2 be a densely defined closed operator. Then im T is closed if and only if im T ∗ is closed. Proof. Let P : H1 × H2 󳨀→ 𝒢 (T) be the orthogonal projection of H1 × H2 onto the closed subspace 𝒢 (T) of H1 × H2 , and let Q : H1 × H2 󳨀→ {0} × H2 be the canonical orthogonal projection. Then im T ≅ im QP, and since I − Q : H1 × H2 󳨀→ H1 × {0} and I − P : H1 × H2 󳨀→ 𝒢 (T)⊥ = V (𝒢 (T ∗ )) ≅ 𝒢 (T ∗ ), we obtain the desired result from Lemma 4.21. Proposition 4.23. Let T : H1 󳨀→ H2 be a densely defined closed operator, and let G be a closed subspace of H2 such that G ⊇ im T. Suppose that T ∗ |dom(T ∗ )∩G is bounded from below, i. e., ‖f ‖ ≤ C‖T ∗ f ‖ for all f ∈ dom(T ∗ ) ∩ G, where C > 0 is a constant. Then G = im T. Proof. We have ker T ∗ = (im T)⊥ . Since im T ⊆ G, it follows that ker T ∗ ⊇ G⊥ . If G⊥ is a proper subspace of ker T ∗ , then G ∩ ker T ∗ ≠ {0}, which is a contradiction to the assumption that T ∗ |dom(T ∗ )∩G is bounded from below. Hence ker T ∗ = G⊥ , and ⊥

G = G⊥⊥ = (ker T ∗ ) = im T ⊥⊥ = (im T). In addition, we have 󵄨 󵄨 T ∗ 󵄨󵄨󵄨dom(T ∗ )∩G = T ∗ 󵄨󵄨󵄨dom(T ∗ )∩(ker T ∗ )⊥ , and by Lemma 4.20 we obtain that im T ∗ is closed. By Proposition 4.22, im T is also closed, and we get that G = im T. Remark 4.24. The last proposition also holds in the other direction: if T : H1 󳨀→ H2 is a densely defined closed operator and G is a closed subspace of H2 with G = im T, then T ∗ |dom(T ∗ )∩G is bounded from below. Since in this case G = im T, we have that im T is closed, and hence, by Lemma 4.22, im T ∗ is also closed. Therefore Lemma 4.20 and the fact that G = (ker T ∗ )⊥ imply that T ∗ |dom(T ∗ )∩G is bounded from below. Proposition 4.25. Let T : H1 󳨀→ H2 be a densely defined closed operator, and let G be a closed subspace of H2 such that G ⊇ im T. Suppose that T ∗ |dom(T ∗ )∩G is bounded from below. Then for each v ∈ H1 such that v ⊥ ker T, there exists f ∈ dom(T ∗ ) ∩ G such that T ∗ f = v and ‖f ‖ ≤ C‖v‖. Proof. We have ker T = (im T ∗ )⊥ , and hence v ∈ (ker T)⊥ = im T ∗ . In addition, G⊥ ⊆ (im T)⊥ = ker T ∗ , and therefore 󵄨 im T ∗ 󵄨󵄨󵄨dom(T ∗ )∩G = im T ∗ ,

76 � 4 The 𝜕-complex which means that im T ∗ is closed and that for v ∈ (ker T)⊥ = im T ∗ , there exists f ∈ dom(T ∗ ) ∩ G such that T ∗ f = v. The desired norm-inequality follows from the assumption that T ∗ |dom(T ∗ )∩G is bounded from below. In the following, we introduce the fundamental concept of an unbounded selfadjoint operator, which will be crucial for both spectral theory and its applications to complex analysis. Definition 4.26. A densely defined linear operator T : dom(T) 󳨀→ H is symmetric if (Tx, y) = (x, Ty) for all x, y ∈ dom(T). We say that T is self-adjoint if T is symmetric and dom(T) = dom(T ∗ ). This is equivalent to requiring that T = T ∗ and implies that T is closed. We say that T is essentially self-adjoint if it is symmetric and its closure T is self-adjoint. Remark 4.27. (a) If T is a symmetric operator, then there are two natural closed extensions. We have dom(T) ⊆ dom(T ∗ ) and T ∗ = T on dom(T). Since T ∗ is closed (Lemma 4.5), T ∗ is a closed extension of T, and it is the maximal closed extension. Since T is also closable by Lemma 4.2, T exists and is the minimal closed extension. (b) If T is essentially self-adjoint, then its self-adjoint extension is unique. To prove this, let S be a self-adjoint extension of T. Then S is closed, and, being an extension of T, it is also an extension of its smallest extension T. Hence T ⊂ S = S ∗ ⊂ (T)∗ = T, and S = T. Lemma 4.28. Let T be a densely defined symmetric operator. (i) If dom(T) = H, then T is self-adjoint and bounded. (ii) If T is self-adjoint and injective, then im T is dense in H, and T −1 is self-adjoint. (iii) If im T is dense in H, then T is injective. (iv) If im T = H, then T is self-adjoint, and T −1 is bounded. Proof. (i) By assumption dom(T) ⊆ dom(T ∗ ). If dom(T) = H, then it follows that T is self-adjoint and therefore also closed (Lemma 4.5) and continuous by the closed graph theorem. (ii) Suppose y⊥ im T. Then x 󳨃→ (Tx, y) = 0 is continuous on dom(T), and hence y ∈ dom(T ∗ ) = dom(T), and (x, Ty) = (Tx, y) = 0 for all x ∈ dom(T). Thus Ty = 0, and since T is assumed to be injective, it follows that y = 0. This proves that im T is dense in H. Therefore T −1 is densely defined with dom(T −1 ) = im T, and (T −1 )∗ exists. Now let U : H × H 󳨀→ H × H be defined by U((x, y)) = (−y, x). It easily follows that U 2 = −I and U 2 (M) = M for any subspace M of H × H, and we get 𝒢 (T −1 ) = U(𝒢 (−T)) and

4.1 Unbounded operators on Hilbert spaces

� 77

U(𝒢 (T −1 )) = 𝒢 (−T). Being self-adjoint, T is closed; hence −T is closed, and T −1 is closed. By Lemma 4.6 applied to T −1 and to −T we get the orthogonal decompositions ∗

H × H = U(𝒢 (T −1 )) ⊕ 𝒢 ((T −1 ) ) and H × H = U(𝒢 (−T))⊕𝒢 (−T) = 𝒢 (T −1 ) ⊕ U(𝒢 (T −1 )). Consequently, −1 ∗

−1

⊥

−1

𝒢 ((T ) ) = [U(𝒢 (T ))] = 𝒢 (T ),

which shows that (T −1 )∗ = T −1 . (iii) Suppose Tx = 0. Then (x, Ty) = (Tx, y) = 0 for each y ∈ dom(T). Thus x⊥ im(T), and therefore x = 0. (iv) Since im(T) = H, (iii) implies that T is injective and dom(T −1 ) = H. If x, y ∈ H, then x = Tz and y = Tw for some z ∈ dom(T) and w ∈ dom(T), so that (T −1 x, y) = (z, Tw) = (Tz, w) = (x, T −1 y). Hence T −1 is symmetric. (i) implies that T −1 is self-adjoint (and bounded), and now it follows from (ii) that T = (T −1 )−1 is also self-adjoint. Lemma 4.29. Let T be a densely defined closed operator, dom(T) ⊆ H1 , and T : dom(T)󳨀→ H2 . Then B = (I + T ∗ T)−1 and C = T(I + T ∗ T)−1 are everywhere defined and bounded with ‖B‖ ≤ 1 and ‖C‖ ≤ 1; in addition, B is self-adjoint and positive. Proof. Let h ∈ H1 be an arbitrary element and consider (h, 0) ∈ H1 × H2 . From the proof of Lemma 4.7 we get H1 × H2 = 𝒢 (T) ⊕ V −1 (𝒢 (T ∗ )),

(4.2)

which implies that (h, 0) can be written in a unique way as (h, 0) = (f , Tf ) + (−T ∗ (−g), −g) for f ∈ dom(T) and g ∈ dom(T ∗ ), which gives h = f + T ∗ g and 0 = Tf − g. We set Bh := f and Ch := g. In this way, we get two linear operators B and C everywhere defined on H1 . The two equations from above can now be written as I = B + T ∗ C, which gives

0 = TB − C,

78 � 4 The 𝜕-complex C = TB

and

I = B + T ∗ TB = (I + T ∗ T)B.

(4.3)

Decomposition in (4.2) is orthogonal, and therefore we obtain 󵄩2 󵄩 󵄩2 󵄩2 󵄩 󵄩2 󵄩 󵄩 ‖h‖2 = 󵄩󵄩󵄩(h, 0)󵄩󵄩󵄩 = 󵄩󵄩󵄩(f , Tf )󵄩󵄩󵄩 + 󵄩󵄩󵄩(T ∗ g, −g)󵄩󵄩󵄩 = ‖f ‖2 + ‖Tf ‖2 + 󵄩󵄩󵄩T ∗ g 󵄩󵄩󵄩 + ‖g‖2 , and hence ‖Bh‖2 + ‖Ch‖2 = ‖f ‖2 + ‖g‖2 ≤ ‖h‖2 , which implies ‖B‖ ≤ 1 and ‖C‖ ≤ 1. For each u ∈ dom(T ∗ T), we get ((I + T ∗ T)u, u) = (u, u) + (Tu, Tu) ≥ (u, u), and hence if (I + T ∗ T)u = 0, then u = 0. Therefore (I + T ∗ T)−1 exists, and (4.3) implies that (I + T ∗ T)−1 is defined everywhere and B = (I + T ∗ T)−1 . Finally, let u, v ∈ H1 . Then (Bu, v) = (Bu, (I + T ∗ T)Bv) = (Bu, Bv) + (Bu, T ∗ TBv) = (Bu, Bv) + (T ∗ TBu, Bv) = ((I + T ∗ T)Bu, Bv) = (u, Bv), and (Bu, u) = (Bu, (I + T ∗ T)Bu) = (Bu, Bu) + (TBu, TBu) ≥ 0, which proves the lemma. At this point, we can describe the notion of the core of an operator, which will be very useful later for spectral analysis. Definition 4.30. Let T be a closable operator with domain dom(T). A subspace D ⊂ dom(T) is called a core of the operator T if the closure of the restriction T|D is an extension of T. Remark 4.31. If T is a closed operator, then T|D = T. Lemma 4.32. Let T be a densely defined closed operator, dom(T) ⊆ H1 , and T : dom(T)󳨀→ H2 . Then dom(T ∗ T) is a core of the operator T. Proof. We have to show that 𝒢 (T) = 𝒢 (T|dom(T ∗ T) ). For this purpose, we consider elements (x, Tx) in the graph of T. We suppose that (x, Tx) ⊥ (y, Ty) for each y ∈ dom(T ∗ T). Then (x, (I + T ∗ T)y) = (x, y) + (Tx, Ty) = ((x, Tx), (y, Ty)) = 0, and as im(I + T ∗ T) = H1 (Lemma 4.29), we conclude that x = 0, which means that 𝒢 (T|dom(T ∗ T) ) is dense in 𝒢 (T).

4.1 Unbounded operators on Hilbert spaces

� 79

Finally, we describe a general method to construct self-adjoint operators associated with Hermitian sesquilinear forms. This leads to a self-adjoint extension of an unbounded operator, which is known as the Friedrichs extension. Definition 4.33. Let (𝒱 , ‖ ⋅ ‖𝒱 ) and (H, ‖ ⋅ ‖H ) be Hilbert spaces such that 𝒱 ⊂ H,

(4.4)

and suppose that there exists a constant C > 0 such that ‖u‖H ≤ C ‖u‖𝒱

(4.5)

for all u ∈ 𝒱 . We also assume that 𝒱 is dense in H. In this situation the space H can be imbedded into the dual space 𝒱 ′ : for h ∈ H, the mapping L(u) = (u, h)H ,

u ∈ 𝒱,

is continuous on 𝒱 , which follows from (4.5): 󵄨󵄨 󵄨 󵄨󵄨L(u)󵄨󵄨󵄨 ≤ ‖u‖H ‖h‖H ≤ C‖h‖H ‖u‖𝒱 . Hence there exists a uniquely determined vh ∈ 𝒱 ′ such that vh (u) = (u, h)H ,

u ∈ 𝒱,

and the mapping h 󳨃→ vh is injective, as 𝒱 is dense in H. Definition 4.34. A form a : 𝒱 × 𝒱 󳨀→ ℂ is sesquilinear if it is linear in the first component and antilinear in the second component. The form a is continuous if there exists a constant C > 0 such that 󵄨󵄨 󵄨 󵄨󵄨a(u, v)󵄨󵄨󵄨 ≤ C‖u‖𝒱 ‖v‖𝒱

(4.6)

for all u, v ∈ 𝒱 , and it is Hermitian if a(u, v) = a(v, u) for all u, v ∈ 𝒱 . The form a is called 𝒱 -elliptic if there exists a constant α > 0 such that 󵄨󵄨 󵄨 2 󵄨󵄨a(u, u)󵄨󵄨󵄨 ≥ α‖u‖𝒱 for all u ∈ 𝒱 .

(4.7)

80 � 4 The 𝜕-complex Proposition 4.35. Let a be a continuous 𝒱 -elliptic form on 𝒱 × 𝒱 . Using (4.6) and the Riesz representation Theorem 1.6, we can define a linear operator A : 𝒱 󳨀→ 𝒱 such that a(u, v) = (Au, v)𝒱 .

(4.8)

This operator A is a topological isomorphism from 𝒱 onto 𝒱 . Proof. First, we show that A is injective: (4.8) and (4.7) imply that for u ∈ 𝒱 , we have 󵄨 󵄨 ‖Au‖𝒱 ‖u‖𝒱 ≥ 󵄨󵄨󵄨(Au, u)𝒱 󵄨󵄨󵄨 ≥ α‖u‖2𝒱 , and hence ‖Au‖𝒱 ≥ α‖u‖𝒱 ,

(4.9)

which implies that A is injective. Now we claim that A(𝒱 ) is dense in 𝒱 . Let u ∈ 𝒱 be such that (Av, u)𝒱 = 0 for each v ∈ 𝒱 . Taking v = u, we get a(u, u) = 0, and, by (4.7), u = 0, which proves the claim. Next, we observe that (4.8) implies a(u, Au) = ‖Au‖2𝒱 . Therefore, using (4.6), we obtain ‖A(u)‖𝒱 ≤ C‖u‖𝒱 , and hence A ∈ ℒ(𝒱 ). If (vn )n is a Cauchy sequence in A(𝒱 ) and Aun = vn , then we derive from (4.9) that (un )n is also a Cauchy sequence. Let u = limn→∞ un . We know already that A is continuous, and therefore limn→∞ Aun = Au, which shows that limn→∞ vn = v = Au and A(𝒱 ) is closed. As we have already shown that A(𝒱 ) is dense in 𝒱 , we conclude that A is surjective. Finally, (4.9) yields that A−1 is continuous. Proposition 4.36. Let a be a Hermitian continuous 𝒱 -elliptic form on 𝒱 × 𝒱 and suppose that (4.4) and (4.5) hold. Let dom(S) be the set of all u ∈ 𝒱 such that the mapping v 󳨃→ a(u, v) is continuous on 𝒱 for the topology induced by H. For each u ∈ dom(S), there exists a uniquely determined element Su ∈ H such that a(u, v) = (Su, v)H

(4.10)

for each v ∈ 𝒱 (by the Riesz representation Theorem 1.6). Then S : dom(S) 󳨀→ H is a bijective densely defined self-adjoint operator, and S −1 ∈ ℒ(H). Moreover, dom(S) is also dense in 𝒱 . Proof. First, we show that S is injective. For each u ∈ dom(S), we get from (4.7) and (4.5) that

4.1 Unbounded operators on Hilbert spaces

� 81

󵄨 󵄨 α‖u‖2H ≤ Cα‖u‖2𝒱 ≤ C 󵄨󵄨󵄨a(u, u)󵄨󵄨󵄨 󵄨 󵄨 = C 󵄨󵄨󵄨(Su, u)H 󵄨󵄨󵄨 ≤ C‖Su‖H ‖u‖H , which implies that α‖u‖H ≤ C‖Su‖H

(4.11)

for all u ∈ dom(S), and therefore S is injective. Now let h ∈ H and consider the mapping v 󳨃→ (h, v)H for v ∈ 𝒱 . Then by (4.5) we obtain 󵄨󵄨 󵄨 󵄨󵄨(h, v)H 󵄨󵄨󵄨 ≤ ‖h‖H ‖v‖H ≤ C‖h‖H ‖v‖𝒱 , which implies that there exists a uniquely determined w ∈ 𝒱 such that (h, v)H = (w, v)𝒱 for all v ∈ 𝒱 . Now we apply Proposition 4.35 and get from (4.8) that a(u, v) = (w, v)𝒱 , where u = A−1 w. Since a(u, v) = (h, v)H for each v ∈ 𝒱 , we conclude that u ∈ dom(S) and that Su = h, which shows that S is surjective. Suppose that (u, h)H = 0 for each u ∈ dom(S). As S is surjective, there is v ∈ dom(S) such that Sv = h, and we get that (u, Sv)H = 0 for each u ∈ dom(S). Using the 𝒱 -ellipticity (4.7), we get for u = v that 0 = (Sv, v)H = a(v, v) ≥ α‖v‖2𝒱 , which implies that v = 0 and consequently h = 0. Therefore we have shown that dom(S) is dense in H. As a(u, v) is Hermitian, we get for u, v ∈ dom(S) that (Su, v)H = a(u, v) = a(v, u) = (Sv, u)H = (u, Sv)H . Hence S is symmetric, and dom(S) ⊂ dom(S ∗ ). Let v ∈ dom(S ∗ ). Since S is surjective, there exists v0 ∈ dom(S) such that Sv0 = S ∗ v. This implies (Su, v0 )H = (u, Sv0 )H = (u, S ∗ v)H = (Su, v)H for all u ∈ dom(S). Using again the surjectivity of S, we derive that v = v0 ∈ dom(S). This implies that dom(S) = dom(S ∗ ) and that S is self-adjoint. Finally, we show that dom(S) is dense in 𝒱 . Let h ∈ 𝒱 be such that (u, h)𝒱 = 0 for all u ∈ dom(S). By Proposition 4.35 there exists f ∈ 𝒱 such that Af = h. Then 0 = (u, h)𝒱 = (u, Af )𝒱 = (Af , u)𝒱 = a(f , u) = a(u, f ) = (Su, f )H . Since S is surjective, we obtain f = 0 and h = Af = 0.

82 � 4 The 𝜕-complex

4.2 Distributions In our context the unbounded operator T will mainly be the 𝜕-operator. To achieve an appropriate closed extension, we will have to consider the derivatives 𝜕z𝜕 in the sense of k distributions. Therefore we will now briefly summarize elementary definitions and results of distribution theory. Definition 4.37. Let Ω ⊆ ℝn be an open subset, and let 𝒟(Ω) = 𝒞0∞ (Ω) be the space of 𝒞 ∞ -functions with compact support (test functions). A sequence (ϕj )j tends to 0 in 𝒟(Ω) if there exists a compact set K ⊂ Ω such that supp(ϕj ) ⊂ K for every j and 𝜕|α| ϕj

α

α

𝜕x1 1 . . . 𝜕xn n

→0

uniformly on K for each α = (α1 , . . . , αn ). A distribution is a linear functional u on 𝒟(Ω) such that for every compact subset K ⊂ Ω, there exist k ∈ ℕ0 = ℕ ∪ {0} and a constant C > 0 such that 󵄨󵄨 𝜕|α| ϕ(x) 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨󵄨u(ϕ)󵄨󵄨󵄨 ≤ C ∑ sup󵄨󵄨󵄨 α1 αn 󵄨󵄨󵄨 󵄨 |α|≤k x∈K 󵄨 𝜕x1 . . . 𝜕xn 󵄨 for each ϕ ∈ 𝒟(Ω) with support in K. We denote the space of distributions on Ω by 𝒟′ (Ω). It is easily seen that u ∈ 𝒟′ (Ω) if and only if u(ϕj ) → 0 for every sequence (ϕj )j in 𝒟(Ω) converging to 0 in 𝒟(Ω). Example 4.38. (1) Let f ∈ L1loc (Ω), where L1loc (Ω) = {f : Ω 󳨀→ ℂ measurable : f |K ∈ L1 (K) ∀K ⊂ Ω, K compact}. The mapping Tf (ϕ) = ∫Ω f (x)ϕ(x) dλ(x), ϕ ∈ 𝒟(Ω), is a distribution. (2) Let a ∈ Ω and δa (ϕ) := ϕ(a), which is the point evaluation in a. The distribution δa is called the Dirac delta distribution. In the sequel, certain operations for ordinary functions, such as multiplication of functions and differentiation, are generalized to distributions. Definition 4.39. Let f ∈ 𝒞 ∞ (Ω) and u ∈ 𝒟′ (Ω). The multiplication of u with f is defined by (fu)(ϕ) := u(fϕ) for ϕ ∈ 𝒟(Ω). Notice that fϕ ∈ 𝒟(Ω). For u ∈ 𝒟′ (ℝn ) and f ∈ 𝒟(ℝn ), the convolution of u and f is defined by (u ∗ f )(x) := u(y 󳨃→ f (x − y)).

4.2 Distributions

� 83

If u = Tg for some locally integrable function g, then it is the usual convolution of functions (Tg ∗ f )(x) = ∫ g(y)f (x − y) dλ(y) = (g ∗ f )(x). Ω

Let Dk =

𝜕 𝜕xk

and

Dα =

𝜕|α| α , α 𝜕x1 1 . . . 𝜕xn n

where α = (α1 , . . . , αn ) is a multi-index. The partial derivative of a distribution u ∈ 𝒟′ (Ω) is defined by (Dk u)(ϕ) := −u(Dk ϕ),

ϕ ∈ 𝒟(Ω);

higher-order mixed derivatives are defined as (Dα u)(ϕ) := (−1)|α| u(Dα ϕ),

ϕ ∈ 𝒟(Ω).

This definition stems from integrating by parts: ∫(Dk f )ϕ dλ = − ∫ f (Dk ϕ) dλ, Ω

Ω

where f ∈ 𝒞 1 (Ω) and ϕ ∈ 𝒟(Ω). Definition 4.40. Let Ω1 ⊆ Ω be an open subset, and let u ∈ 𝒟′ (Ω). The restriction of u to Ω1 is the distribution u|Ω1 defined by u|Ω1 (ϕ) = u(ϕ) for ϕ ∈ 𝒟(Ω1 ) ⊆ 𝒟(Ω). The support supp(u) of u ∈ 𝒟′ (Ω) consists of x ∈ Ω such that there is no open neighborhood U of x with u|U = 0. We denote the space of distributions with compact support by ℰ ′ (Ω). A fundamental solution of a differential operator P(D) is a distribution E ∈ ℰ ′ (ℝn ) with P(D)E = δ0 . In what follows, we will find a fundamental solution for 𝜕, and we will investigate regularity properties of the 𝜕-operator. For this purpose, we recall Stokes’ theorem in the following way: let Ω ⊂ ℂ be a bounded domain with piecewise 𝒞 1 -boundary γ (positively oriented), let f ∈ 𝒞 1 (Ω) and ζ ∈ Ω; for 0 < ϵ < dist(ζ , Ωc ), let Ωϵ := {z ∈ Ω : |z − ζ | > ϵ},

84 � 4 The 𝜕-complex where the boundary of Ωϵ consists of γ and the negatively oriented circle ρϵ = {z : (z) dz. Then by Stokes’ theorem |z − ζ | = ϵ}. Consider the differential ω = fz−ζ 2π

f (z) dz − ∫ if (ζ + ϵeiθ ) dθ. ∫ dω = ∫ ω + ∫ ω = ∫ z−ζ γ

Ωϵ

ρϵ

γ

(4.12)

0

We have 𝜕f 1 𝜕 f (z) ( ) dz ∧ dz = − dz ∧ dz, 𝜕z z − ζ 𝜕z z − ζ

dω = 2π

and by continuity ∫0 if (ζ + ϵeiθ ) dθ → 2πif (ζ ) as ϵ → 0. Hence we obtain f (z) 𝜕f 1 1 1 dz + dz ∧ dz. ∫ ∫ 2πi z − ζ 2πi 𝜕z z − ζ

f (ζ ) =

γ

(4.13)

Ω

In particular, if ϕ ∈ 𝒟(Ω), then ϕ(ζ ) = −

1 𝜕ϕ 1 dλ(z), ∫ π 𝜕z z − ζ

(4.14)

Ω

where we used the fact that dz ∧ dz = (−2i) dx ∧ dy = (−2i) dλ(z). This can be interpreted in the sense of distributions as δζ (ϕ) = ϕ(ζ ) = −

1 1 𝜕ϕ 1 𝜕 dλ(z) = ∫ (z 󳨃→ )ϕ(z) dλ(z), ∫ π 𝜕z z − ζ 𝜕z π(z − ζ ) Ω

which means that z 󳨃→

1 πz

(4.15)

Ω

is a fundamental solution to 𝜕.

Proposition 4.41. Let Ω ⊆ ℂ be a domain, and let u ∈ 𝒟′ (Ω) be a distribution satisfying 𝜕u = 0. Then there exists a holomorphic function f on Ω such that u = Tf . Proof. Let E be the fundamental solution to 𝜕 given in (4.15). We have 𝜕E = δ0 . Now let Ω′ ⊂⊂ Ω be a relatively compact subset and choose g ∈ 𝒟(Ω) such that g = 1 in some neighborhood of Ω′ . Let S := 𝜕(gu). Then S ∈ ℰ ′ (Ω) with supp(S) ⊂ ℂ \ Ω′ , and hence gu = δ0 ∗ (gu) = (𝜕E) ∗ (gu) = E ∗ 𝜕(gu) = E ∗ S.

(4.16)

Now pick z0 ∈ Ω′ and ϵ > 0 such that Vϵ = {z : dist(z, ℂ \ Ω′ ) > ϵ} is a neighborhood of z0 . Let ψϵ ∈ 𝒟(ℂ) be such that ψϵ (z) = 1 for |z| ≤ ϵ/2 and ϕϵ (z) = 0 for |z| ≥ ϵ. We can write (4.16) in the form gu = (ψϵ E) ∗ S + ((1 − ψϵ )E) ∗ S.

(4.17)

4.2 Distributions

� 85

Since supp((ψϵ E) ∗ S) ⊆ supp(ψϵ E) + supp(S) ⊆ {z : dist(z, supp(S)) ≤ ϵ}, we have that (ψϵ E) ∗ S has to vanish in Vϵ . This implies 󵄨 u|Vϵ = (gu)|Vϵ = (((1 − ψϵ )E) ∗ S)󵄨󵄨󵄨V . ϵ

(4.18)

The function (1 − ψϵ )E is in 𝒞 ∞ (ℂ), and supp(S) is compact; therefore ((1 − ψϵ )E) ∗ S is smooth. Hence, using (4.18), we obtain that u is smooth in a neighborhood of z0 , and z0 being arbitrary, u is smooth in Ω′ and therefore also in Ω. This implies that u is holomorphic, since it also solves the Cauchy–Riemann equation. For an appropriate description of the appearing phenomena, we will further use Hilbert spaces of differentiable functions, the Sobolev spaces. Definition 4.42. If Ω is a bounded open set in ℝn , and k is a nonnegative integer, then we define the Sobolev space W k (Ω) = {f ∈ L2 (Ω) : 𝜕α f ∈ L2 (Ω), |α| ≤ k}, where the derivatives are taken in the sense of distributions, and endow the space with the norm 1/2

󵄨 󵄨2 ‖f ‖k,Ω = [ ∑ ∫󵄨󵄨󵄨𝜕α f 󵄨󵄨󵄨 dλ] , |α|≤k Ω

(4.19)

where α = (α1 , . . . , αn ) is a multi-index, |α| = ∑nj=1 αj , and 𝜕α f =

𝜕|α| f α . α 𝜕x1 1 . . . 𝜕xn n

By W0k (Ω) we denote the completion of 𝒞0∞ (Ω) under W k (Ω)-norm. Since 𝒞0∞ (Ω) is dense in L2 (Ω), it follows that W00 (Ω) = W 0 (Ω) = L2 (Ω). Using the Fourier transform, it is also possible to introduce Sobolev spaces of noninteger exponent. (See [1, 31].) In general, a function can belong to a Sobolev space and yet be discontinuous and unbounded. Example 4.43. Take Ω = 𝔹 the open unit ball in ℝn and u(x) = |x|−α ,

x ∈ 𝔹, x ≠ 0.

. We claim that u ∈ W 1 (𝔹) if and only if α < n−2 2 First note that u is smooth away from 0 and that uxj (x) =

−αxj

|x|α+2

,

x ≠ 0.

86 � 4 The 𝜕-complex Hence |α| 󵄨 󵄨󵄨 󵄨󵄨∇u(x)󵄨󵄨󵄨 = α+1 , |x|

x ≠ 0.

Now recall the Gauß–Green theorem: for a smoothly bounded ω ⊆ ℝn , we have ∫ ∇ . F(x) dλ(x) = ∫ (F(x), ν(x)) dσ(x), ω

bω

where ν(x) = ∇r(x) is the normal to bω at x, F is a 𝒞 1 vector field on ω, and n

∇ . F(x) = ∑ j=1

𝜕Fj 𝜕xj

(see [29]). Let ϕ ∈ 𝒞0∞ (𝔹), and let 𝔹ϵ be the open ball around 0 with radius ϵ > 0. Take ω = 𝔹 \ 𝔹ϵ and F(x) = (0, . . . , 0, uϕ, 0, . . . , 0), where uϕ appears at the jth component. Then ∫ u(x) ϕxj (x) dλ(x) = − ∫ uxj (x) ϕ(x) dλ(x) + ∫ u(x)ϕ(x)νj (x) dσ(x), 𝔹\𝔹ϵ

b𝔹ϵ

𝔹\𝔹ϵ

where ν(x) = (ν1 (x), . . . , νn (x)) denotes the inward pointing normal on b𝔹ϵ . If α < n − 1, then |∇u(x)| ∈ L1 (𝔹), and we obtain 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 −α 󵄨󵄨 ∫ u(x)ϕ(x)νj (x) dσ(x)󵄨󵄨󵄨 ≤ ‖ϕ‖∞ ∫ ϵ dσ(x) 󵄨󵄨 󵄨󵄨 b𝔹ϵ

b𝔹ϵ

≤Cϵ

n−1−α

→0

as ϵ → 0. Thus ∫ u(x) ϕxj (x) dλ(x) = − ∫ uxj (x) ϕ(x) dλ(x) 𝔹

𝔹

for all ϕ ∈ 𝒞0∞ (𝔹). As α 󵄨󵄨 󵄨 2 󵄨󵄨∇u(x)󵄨󵄨󵄨 = α+1 ∈ L (𝔹) |x| if and only if 2(α + 1) < n, we get that u ∈ W 1 (𝔹) if and only if α
0. Let f ∈ L2 (ℝn ), and for x ∈ ℝn , define fϵ (x) = (f ∗ χϵ )(x) = ∫ f (x ′ )χϵ (x − x ′ ) dλ(x ′ ) ℝn

= ∫ f (x − x ′ )χϵ (x ′ ) dλ(x ′ ) ℝn

= ∫ f (x − ϵx ′ )χ(x ′ ) dλ(x ′ ). ℝn

In the first integral, we can differentiate under the integral sign to show that fϵ ∈

𝒞 ∞ (ℝn ).

The family of functions (χϵ )ϵ is called an approximation to the identity.

Lemma 5.3. ‖fϵ − f ‖2 → 0 as ϵ → 0. Proof. fϵ (x) − f (x) = ∫ [f (x − ϵx ′ ) − f (x)]χ(x ′ ) dλ(x ′ ). ℝn

We use Minkowski’s inequality (5.1) to get

5.1 Friedrichs’ lemma and Sobolev spaces

� 109

󵄨 󵄨 ‖fϵ − f ‖2 ≤ ∫ ‖f−ϵx ′ − f ‖2 󵄨󵄨󵄨χ(x ′ )󵄨󵄨󵄨 dλ(x ′ ). ℝn

However, ‖f−ϵx ′ − f ‖2 is bounded by 2‖f ‖2 and tends to 0 as ϵ → 0 by Lemma 5.3. Now set Fϵ (x ′ ) = ‖f−ϵx ′ − f ‖2 χ(x ′ ). Then Fϵ (x ′ ) → 0 as ϵ → 0, and 󵄨󵄨 ′ 󵄨 ′ 󵄨󵄨Fϵ (x )󵄨󵄨󵄨 ≤ 2‖f ‖2 χ(x ), and we can apply the dominated convergence theorem to get the desired result. If u ∈ 𝒞0∞ (ℝn ), then we have Dj (u ∗ χϵ ) = (Dj u) ∗ χϵ , where Dj = 𝜕/𝜕xj . This is also true if u ∈ L2 (ℝn ) and Dj u is defined in the sense of distributions. We will show even more using these methods for approximating a function in a Sobolev space by smooth functions. Let Ω ⊆ ℝn be an open subset, and let Ωϵ = {x ∈ Ω : dist(x, bΩ) > ϵ}. Lemma 5.4. Let u ∈ W k (Ω) and set uϵ = u ∗ χϵ in Ωϵ . Then (i) uϵ ∈ 𝒞 ∞ (Ωϵ ) for each ϵ > 0, (ii) Dα uϵ = Dα u ∗ χϵ in Ωϵ for |α| ≤ k. Proof. (i) has already been shown. (ii) means that the ordinary αth partial derivative of the smooth functions uϵ is the ϵ-mollification of the αth weak partial derivative of u. To see this, we take x ∈ Ωϵ and compute Dα uϵ (x) = Dα ∫ u(y)χϵ (x − y) dλ(y) Ω

= ∫ Dαx χϵ (x − y)u(y) dλ(y) Ω

= (−1)|α| ∫ Dαy χϵ (x − y)u(y) dλ(y). Ω

For a fixed x ∈ Ωϵ the function ϕ(y) := χϵ (x − y) belongs to 𝒞 ∞ (Ω). The definition of the αth weak partial derivative implies ∫ Dαy χϵ (x − y)u(y) dλ(y) = (−1)|α| ∫ χϵ (x − y) Dα u(y) dλ(y).

Ω

Ω

110 � 5 Density of smooth forms Thus Dα uϵ (x) = (−1)|α|+|α| ∫ χϵ (x − y) Dα u(y) dλ(y) α

Ω

= (D u ∗ χϵ )(x), which proves (ii). We are now ready to prove the following: Lemma 5.5 (Friedrichs’ lemma). If v ∈ L2 (ℝn ) is a function with compact support and a is a 𝒞 1 -function in a neighborhood of the support of v, then 󵄩󵄩 󵄩 󵄩󵄩aDj (v ∗ χϵ ) − (aDj v) ∗ χϵ 󵄩󵄩󵄩2 → 0

as ϵ → 0,

where Dj = 𝜕/𝜕xj and aDj v is defined in the sense of distributions. Proof. If v ∈ 𝒞0∞ (ℝn ), then we have Dj (v ∗ χϵ ) = (Dj v) ∗ χϵ → Dj v,

(aDj v) ∗ χϵ → aDj v,

with uniform convergence. We claim that 󵄩󵄩 󵄩 󵄩󵄩aDj (v ∗ χϵ ) − (aDj v) ∗ χϵ 󵄩󵄩󵄩2 ≤ C‖v‖2 ,

(5.3)

where v ∈ L2 (ℝn ), and C is some positive constant independent of ϵ and v. Since 𝒞0∞ (ℝn ) is dense in L2 (ℝn ), the lemma will follow as in the proof of Lemma 5.3 from (5.3) and the dominated convergence theorem. To show (5.3), we may assume that a ∈ 𝒞01 (ℝn ), since v has a compact support. For v ∈ 𝒞0∞ (ℝn ), we have a(x)Dj (v ∗ χϵ )(x) − ((aDj v) ∗ χϵ )(x) = a(x) Dj ∫ v(x − y)χϵ (y) dλ(y) − ∫ a(x − y)

𝜕v (x − y)χϵ (y) dλ(y) 𝜕xj

𝜕v (x − y)χϵ (y) dλ(y) 𝜕xj 𝜕v = − ∫(a(x) − a(x − y)) (x − y)χϵ (y) dλ(y) 𝜕yj 𝜕 = ∫(a(x) − a(x − y))v(x − y) χ (y) dλ(y) 𝜕yj ϵ 𝜕 − ∫( a(x − y))v(x − y)χϵ (y) dλ(y). 𝜕yj

= ∫(a(x) − a(x − y))

Let M be the Lipschitz constant for a, that is, |a(x) − a(x − y)| ≤ M|y| for all x, y ∈ ℝn . Then

5.1 Friedrichs’ lemma and Sobolev spaces

� 111

󵄨󵄨 𝜕 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨a(x)Dj (v ∗ χϵ )(x) − ((aDj v) ∗ χϵ )(x)󵄨󵄨󵄨 ≤ M ∫󵄨󵄨󵄨v(x − y)󵄨󵄨󵄨(χϵ (y) + 󵄨󵄨󵄨 y χϵ (y)󵄨󵄨󵄨) dλ(y). 󵄨󵄨 󵄨󵄨 𝜕yj By Minkowski’s inequality (5.1) we obtain 󵄨󵄨 𝜕 󵄨󵄨 󵄨 󵄩 󵄩󵄩 󵄨 󵄩󵄩aDj (v ∗ χϵ ) − (aDj v) ∗ χϵ 󵄩󵄩󵄩2 ≤ M ‖v‖2 ∫(χϵ (y) + 󵄨󵄨󵄨 y χϵ (y)󵄨󵄨󵄨) dλ(y) 󵄨󵄨 󵄨󵄨 𝜕yj = M(1 + mj )‖v‖2 , where 󵄨󵄨 𝜕 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕 󵄨 󵄨 󵄨 󵄨 mj = ∫󵄨󵄨󵄨 y χϵ (y)󵄨󵄨󵄨 dλ(y) = ∫󵄨󵄨󵄨 y χ(y)󵄨󵄨󵄨 dλ(y). 󵄨󵄨 𝜕yj 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕yj This shows (5.3) when v ∈ 𝒞0∞ (ℝn ). Since 𝒞0∞ (ℝn ) is dense in L2 (ℝn ), we have proved (5.3) and the lemma. Lemma 5.6. Let n

L = ∑ aj Dj + a0 j=1

be a first-order differential operator with variable coefficients aj ∈ 𝒞 1 (ℝn ) and a0 ∈ 𝒞 (ℝn ). If v ∈ L2 (ℝn ) is a function with compact support and Lv = f ∈ L2 (ℝn ), where Lv is defined in the distribution sense, then the convolution vϵ = v ∗ χϵ is in 𝒞0∞ (ℝn ), vϵ → v, and Lvϵ → f in L2 (ℝn ) as ϵ → 0. Proof. Since a0 v ∈ L2 (ℝn ), we have lim a0 (v ∗ χϵ ) = lim(a0 v ∗ χϵ ) = a0 v

ϵ→0

ϵ→0

in L2 (ℝn ). Using Friedrichs’ Lemma 5.5, we have Lvϵ − Lv ∗ χϵ = Lvϵ − f ∗ χϵ → 0 in L2 (ℝn ) as ϵ → 0. The lemma follows easily since f ∗ χϵ → f in L2 (ℝn ). Before we proceed with results about Sobolev spaces, we prove an important inequality for the sgn-function. Let z ∈ ℂ. Define z/|z|, z ≠ 0, 0, z = 0.

sgn z = {

Proposition 5.7. Suppose that f ∈ L1loc (ℝn ) with ∇f ∈ L1loc (ℝn ). Then

112 � 5 Density of smooth forms ∇|f | ∈ L1loc (ℝn ), and (5.4)

∇|f |(x) = R[sgn(f (x)) ∇f (x)] almost everywhere. In particular, we have 󵄨󵄨 󵄨 󵄨󵄨∇|f |󵄨󵄨󵄨 ≤ |∇f |

(5.5)

almost everywhere. Proof. Let z ∈ ℂ and ϵ > 0. We define |z|ϵ := √|z|2 + ϵ2 − ϵ and observe that 0 ≤ |z|ϵ ≤ |z| and

lim |z|ϵ = |z|.

ϵ→0

If u ∈ 𝒞 ∞ (ℝn ), then |u|ϵ ∈ 𝒞 ∞ (ℝn ), and as |u|2 = u u, we get ∇|u|ϵ =

R(u ∇u) . √|u|2 + ϵ2

(5.6)

Now let f be as assumed, take an approximation to the identity (χδ )δ , and define fδ = f ∗ χδ . By Lemmas 5.2, 5.3, and 5.4 we obtain that fδ → f , |fδ | → |f |, and ∇fδ → ∇f in L1loc (ℝn ) as δ → 0. Let ϕ ∈ 𝒞0∞ (ℝn ) be a test function. There exists a subsequence δk → 0 such that fδk (x) → f (x) for almost every x ∈ supp(ϕ). For simplicity, we now omit the index k. Using the dominated convergence theorem and (5.6), we get ∫(∇ϕ) |f | dλ = lim ∫(∇ϕ) |f |ϵ dλ ϵ→0

= lim lim ∫(∇ϕ) |fδ |ϵ dλ ϵ→0 δ→0

= − lim lim ∫ ϕ ϵ→0 δ→0

R(f δ ∇fδ ) √|fδ |2 + ϵ2

Since ∇fδ → ∇f in L1loc (ℝn ), taking the limit as δ → 0, we get

dλ.

5.1 Friedrichs’ lemma and Sobolev spaces

∫(∇ϕ) |f | dλ = − lim ∫ ϕ ϵ→0

R(f ∇f ) √|f |2 + ϵ2

� 113

dλ,

and since ϕ∇f ∈ L1 (ℝn ) and f /√|f |2 + ϵ2 → sgnf as ϵ → 0, we get the desired result by applying once more dominated convergence. In the sequel, we still use the methods from above for approximating a function in a Sobolev space by smooth functions. In a similar way, as in the last lemma, we get the following: k Lemma 5.8. If u ∈ W k (Ω), then uϵ → u in Wloc (Ω) as ϵ → 0, which means that this k happens in each space W (ω), where ω is an open subset such that ω ⋐ Ω.

Using a smooth partition of unity, we still show that we can find smooth functions k that approximate in the W k (Ω)-norm, and not just in Wloc (Ω). Lemma 5.9. Let Ω ⊂ ℝn be a bounded open set, and let u ∈ W k (Ω). Then there exist functions um ∈ 𝒞 ∞ (Ω) ∩ W k (Ω) such that um → u in W k (Ω). Note that we do not assume that um ∈ 𝒞 ∞ (Ω). Proof. We write Ω = ⋃∞ j=1 ωj , where ωj := {x ∈ Ω : dist(x, bΩ) > 1/j},

j = 1, 2, . . . .

Set Uj := ωj+3 \ ωj+1 and choose any open set U0 ⋐ Ω so that Ω = ⋃∞ j=0 Uj . Let (ϕj )j be a smooth partition of unity subordinate to the open sets (Uj )j , that is, 0 ≤ ϕj ≤ 1, ϕj ∈ 𝒞0∞ (Uj ), and ∑∞ j=0 ϕj = 1 on Ω.

According to Proposition 4.44, ϕj u ∈ W k (Ω), and the support of ϕj u is contained in Uj . Now we use Lemma 5.8: fix ϵ > 0 and choose ϵj > 0 so small that uj := (ϕj u) ∗ χϵj satisfies ‖uj − ϕj u‖W k (Ω) ≤

ϵ , 2j+1

j = 0, 1, . . . ,

and uj has support in Vj := ωj+4 \ ωj ⊃ Uj for j = 1, 2, . . . . ∞ Now define v := ∑∞ j=0 uj . This function belongs to 𝒞 (Ω), since for each open set ω ⋐ Ω, there are at most finitely many nonzero terms in the sum. Since u = ∑∞ j=0 ϕj u, for each ω ⋐ Ω, we have ∞

∞

‖v − u‖W k (ω) ≤ ∑ ‖uj − ϕj u‖W k (Ω) ≤ ϵ ∑ j=0

j=0

1

2j+1

Finally, take the supremum over all sets ω ⋐ Ω to conclude that ‖v − u‖W k (Ω) ≤ ϵ.

= ϵ.

114 � 5 Density of smooth forms Before we proceed to prove the density result for the 𝜕-setting, we show that a function u ∈ W k (Ω) can be approximated by functions in 𝒞 ∞ (Ω), where all derivatives extend continuously to Ω. This of course requires some conditions on the boundary bΩ. Proposition 5.10. Let Ω be a bounded open set in ℝn and assume that bΩ is 𝒞 1 . Let u ∈ W k (Ω). Then there exist functions um ∈ 𝒞 ∞ (Ω) such that um → u in W k (Ω). Proof. Let x0 ∈ bΩ. As bΩ is 𝒞 1 , there exist a radius r > 0 and a 𝒞 1 -function γ : ℝn−1 󳨀→ ℝ such that Ω ∩ B(x0 , r) = {x ∈ B(x0 , r) : xn > γ(x1 , . . . , xn−1 )}. We set V := Ω ∩ B(x0 , r/2) and define the shifted point xϵ := x + μϵen for x ∈ V and ϵ > 0. We see that for some fixed, sufficiently large number μ > 0, the ball B(xϵ , ϵ) lies in Ω ∩ B(x0 , r) for all x ∈ V and all small ϵ > 0. Now we define uϵ (x) := u(xϵ ) for x ∈ V ; this is the function u translated a distance μϵ in the en -direction. Next, we write vϵ = uϵ ∗ χϵ . The idea is that we have moved up enough so that there is room to mollify within Ω. We have vϵ ∈ 𝒞 ∞ (V ). We now claim that vϵ → u in W k (V ) as ϵ → 0. Let α be a multi-index with |α| ≤ k. Then 󵄩󵄩 α 󵄩 α 󵄩 α α 󵄩 α 󵄩 α 󵄩 󵄩󵄩D vϵ − D u󵄩󵄩󵄩L2 (V ) ≤ 󵄩󵄩󵄩D vϵ − D uϵ 󵄩󵄩󵄩L2 (V ) + 󵄩󵄩󵄩D uϵ − D u󵄩󵄩󵄩L2 (V ) . The second term on the right-hand side goes to zero with ϵ, since by Lemma 5.2 translation is continuous in the L2 -norm. The first term also vanishes as ϵ → 0 by a similar reasoning as in Lemma 5.6. Let δ > 0. Since bΩ is compact, we can find finitely many points xj ∈ bΩ, radii rj > 0, corresponding sets Vj = Ω ∩ B(xj , rj /2), and functions vj ∈ 𝒞 ∞ (V j ), j = 1, . . . , N, such that N

bΩ ⊂ ⋃ B(xj , rj /2) and j=1

‖vj − u‖W k (Vj ) ≤ δ.

(5.7)

Now we take an open set V0 ⋐ Ω such that N

Ω ⊂ ⋃ Vj j=0

and select, using Lemma 5.8, a function v0 ∈ 𝒞 ∞ (V 0 ) satisfying ‖v0 − u‖W k (V0 ) ≤ δ.

(5.8)

Finally, we take a smooth partition (ϕj )j of unity subordinate to the open sets (Vj )j in Ω for j = 0, . . . , N. Define v := ∑Nj=0 ϕj vj . Then v ∈ 𝒞 ∞ (Ω). Since u = ∑Nj=0 ϕj u, we see that for each |α| ≤ k,

5.1 Friedrichs’ lemma and Sobolev spaces

� 115

N

󵄩 󵄩 α 󵄩󵄩 α α α 󵄩 󵄩󵄩D v − D u󵄩󵄩󵄩L2 (Ω) ≤ ∑󵄩󵄩󵄩D (ϕj vj ) − D (ϕj u)󵄩󵄩󵄩L2 (Vj ) j=0 N

≤ ∑ ‖vj − u‖W k (Vj ) j=0

≤ (N + 1)δ, where we used (5.7) and (5.8). In the next step, we show that functions in the Sobolev space W 1 (Ω) can be extended to functions in W 1 (ℝn ), provided that Ω is a bounded domain with 𝒞 1 -boundary. Proposition 5.11. Assume that Ω is a bounded domain with 𝒞 1 -boundary. Select a bounded open set V such that Ω ⋐ V . Then there exists a bounded linear operator E : W 1 (Ω) 󳨀→ W 1 (ℝn ) such that for each u ∈ W 1 (Ω): (i) Eu = u almost everywhere in Ω, (ii) Eu has a support within V , (iii) there exists a constant C, depending only on Ω and V , such that ‖Eu‖W 1 (ℝn ) ≤ C‖u‖W 1 (Ω) . Proof. We use the method of a higher-order reflection for the extension. Let x0 ∈ bΩ and suppose first that bΩ is flat near x0 , lying in the plane {xn = 0}. Then we may assume that there exists an open ball B centered in x0 with radius r such that B+ := B ∩ {xn ≥ 0} ⊂ Ω,

B− := B ∩ {xn ≤ 0} ⊂ ℝn \ Ω.

Temporarily, we suppose that u ∈ 𝒞 ∞ (Ω). We define then u(x)

̃ u(x) ={

x

−3u(x1 , . . . , xn−1 , −xn ) + 4u(x1 , . . . , xn−1 , − 2n )

if x ∈ B+ , if x ∈ B− .

This is called a higher-order reflection of u from B+ to B− . First, we show that ũ ∈ 𝒞 1 (B). To check this, we write u− := u|̃ B−

and

u+ := u|̃ B+ .

By definition we have x 𝜕u− 𝜕u 𝜕u (x) = 3 (x , . . . , xn−1 , −xn ) − 2 (x , . . . , xn−1 , − n ), 𝜕xn 𝜕xn 1 𝜕xn 1 2

116 � 5 Density of smooth forms and so ux−n |{xn =0} = ux+n |{xn =0} . Now since u+ = u− on {xn = 0}, we see that also ux−j |{xn =0} = ux+j |{xn =0} for j = 1, . . . , n − 1. Hence we have Dα u− |{xn =0} = Dα u+ |{xn =0} for each |α| ≤ 1, which implies ũ ∈ 𝒞 1 (B). Using these computations, we readily see that ‖u‖̃ W 1 (B) ≤ C‖u‖W 1 (B+ )

(5.9)

for some constant C > 0 that does not depend on u. If bΩ is not flat near x0 , then we can find a 𝒞 1 -mapping Φ, with inverse Ψ, which straightens out bΩ near x0 . We write y = Φ(x) and x = Ψ(y) and define u∗ (y) := u(Ψ(y)). We choose a small ball B and use the same reasoning as before to extend u∗ from B+ to a function ũ ∗ defined on all of B and such that ũ ∗ ∈ 𝒞 1 (B), and as in (5.9), we get 󵄩󵄩 ̃ ∗ 󵄩󵄩 󵄩 ∗󵄩 󵄩󵄩u 󵄩󵄩W 1 (B) ≤ C 󵄩󵄩󵄩u 󵄩󵄩󵄩W 1 (B+ ) .

(5.10)

Let W := Ψ(B). Then converting back to the x-variables, we obtain an extension ũ of u to W such that ‖u‖̃ W 1 (W ) ≤ C‖u‖W 1 (Ω) .

(5.11) j

Since bΩ is compact, there exist finitely many points x0 ∈ bΩ, open sets Wj , and extensions ũ j of u to Wj for j = 1, . . . , N such that bΩ ⊂ ⋃Nj=1 Wj . Take W0 ⋐ Ω such that

Ω ⊂ ⋃Nj=0 Wj , and let (ϕj )j be an associated partition of unity. Write N

ũ = ∑ ϕj ũ j , j=0

where ũ 0 = u. Then by (5.11) we obtain the estimate ‖u‖̃ W 1 (ℝn ) ≤ C‖u‖W 1 (Ω)

(5.12)

for some constant C > 0 independent of u. In addition, we arrange the support of ũ to lie within V ⋑ Ω.

5.1 Friedrichs’ lemma and Sobolev spaces

� 117

We define Eu := ũ and observe that the mapping u 󳨃→ Eu is linear. So far we have assumed that u ∈ 𝒞 ∞ (Ω). Now take u ∈ W 1 (Ω), and choose a sequence um ∈ 𝒞 ∞ (Ω) converging to u in W 1 (Ω) (see Proposition 5.10). Estimate (5.12) implies ‖Eum − Euℓ ‖W 1 (ℝn ) ≤ C‖um − uℓ ‖W 1 (Ω) . Hence (Eum )m is a Cauchy sequence and so converges to ũ =: Eu. This extension does not depend on the particular choice of the approximating sequence (um )m . In a similar way, we treat the problem how to assign boundary values along bΩ to a function u ∈ W 1 (Ω) assuming that bΩ is 𝒞 1 . Proposition 5.12. Let Ω be a bounded domain with 𝒞 1 -boundary. Then there exists a bounded linear operator T : W 1 (Ω) 󳨀→ L2 (bΩ) such that (i) Tu = u|bΩ if u ∈ W 1 (Ω) ∩ 𝒞 (Ω) and (ii) ‖Tu‖L2 (bΩ) ≤ C‖u‖W 1 (Ω) for each u ∈ W 1 (Ω), with the constant C depending only on Ω. We call Tu the trace of u on bΩ. Proof. First, we assume that u ∈ 𝒞 1 (Ω) and proceed as in the proof of Proposition 5.11. We suppose that x0 ∈ bΩ and that bΩ is flat near x0 , lying in the plane {xn = 0}. We choose an open ball B as in the previous proof and let B̃ denote the concentric ball of radius r/2. Select a function χ ∈ 𝒞0∞ (B) such that χ ≥ 0 in B and χ = 1 on B.̃ Denote by Γ the portion of bΩ within B.̃ Set x ′ = (x1 , . . . , xn−1 ) ∈ ℝn−1 = {xn = 0}. Then we have ∫ |u|2 dλ(x ′ ) ≤ ∫ χ|u|2 dλ(x ′ ) = − ∫ (χ|u|2 )x dλ(x) Γ

B+

{xn =0}

n

= − ∫ (|u|2 χxn + |u|(sgnu) uxn χ) dλ(x) B+

≤ C ∫ (|u|2 + |∇u|2 ) dλ(x), B+

where we used Proposition 5.7 and the inequality ab ≤ a2 /2 + b2 /2 for a, b ≥ 0. After this, we straighten out the boundary near x0 to get the estimate ∫ |u|2 dλ(x ′ ) ≤ C ∫(|u|2 + |∇u|2 ) dλ(x), Γ

Ω

118 � 5 Density of smooth forms where Γ is some open subset of bΩ containing x0 . Since bΩ is compact, there exist finitely many points x0,k ∈ bΩ and open subsets Γk ⊂ bΩ (k = 1, . . . , N) such that N

bΩ = ⋃ Γk k=1

and ‖u‖L2 (Γk ) ≤ C‖u‖W 1 (Ω) for k = 1, . . . , N. Hence, writing Tu := u|bΩ , we get ‖Tu‖L2 (bΩ) ≤ C‖u‖W 1 (Ω)

(5.13)

for some constant C, which does not depend on u. Inequality (5.13) holds for u ∈ 𝒞 1 (Ω). Assume now that u ∈ W 1 (Ω). Then by Proposition 5.10 there exist functions um ∈ 𝒞 ∞ (Ω) converging to u in W 1 (Ω). By (5.13) we have ‖Tum − Tuℓ ‖L2 (bΩ) ≤ C‖um − uℓ ‖W 1 (Ω) ,

(5.14)

and hence (Tum )m is a Cauchy sequence in L2 (bΩ). Set Tu := lim Tum , m→∞

where the limit is taken in L2 (bΩ). By (5.14) this definition does not depend on the particular choice of the smooth functions approximating u. If u ∈ W 1 (Ω) ∩ 𝒞 (Ω), then we can use the fact that the functions um ∈ 𝒞 ∞ (Ω) constructed in the proof of Proposition 5.10 converge uniformly to u on Ω. This implies Tu = u|bΩ . Proposition 5.13. Let Ω be a bounded domain with 𝒞 1 -boundary. Then u ∈ W01 (Ω) if and only if Tu = 0 on bΩ. Proof. If u ∈ W01 (Ω), then there exist functions um ∈ 𝒞0∞ (Ω) such that um → u in W 1 (Ω). As Tum = 0 on bΩ and T : W 1 (Ω) 󳨀→ L2 (bΩ) is a continuous linear operator (Proposition 5.12), we obtain Tu = 0 on bΩ.

5.2 Density in the graph norm

� 119

The converse statement is more difficult, but it will not be important in the following, so we omit the proof; see, for instance, [27].

5.2 Density in the graph norm We are now able to prove the density result for the basic estimates. ∗

∗

k Proposition 5.14. If bΩ is 𝒞 k+1 , then 𝒞(0,q) (Ω) ∩ dom(𝜕 ) is dense in dom(𝜕) ∩ dom(𝜕 ) in ∗

the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 . The statement also holds with k + 1 and k replaced by ∞.

Before we begin with the proof of this important approximation result, we gather ∗ some observations about dom(𝜕) and dom(𝜕 ). Remark 5.15. ∗ (a) It is useful to know that dom(𝜕 ) is preserved under multiplication by a function in ∗ 𝒞 1 (Ω). Indeed, let u ∈ dom(𝜕 ), v ∈ dom(𝜕), and ψ ∈ 𝒞 1 (Ω). Then ∗

(𝜕v, ψu) = (ψ 𝜕v, u) = (𝜕(ψv), u) − (𝜕 ψ ∧ v, u) = (ψv, 𝜕 u) − (𝜕 ψ ∧ v, u). ∗

The right-hand side is bounded by ‖v‖, and hence ψu ∈ dom(𝜕 ) (see [91]). ∗ (b) Compactly supported forms are not dense in dom(𝜕) ∩ dom(𝜕 ) in the graph norm: for compactly supported forms, Proposition 4.56 gives n 󵄩 󵄩󵄩 𝜕uj 󵄩󵄩󵄩2 󵄩 ∗ 󵄩2 󵄩󵄩 , ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 = ∑ 󵄩󵄩󵄩 󵄩 𝜕z 󵄩󵄩 j,k=1󵄩 k 󵄩

and integration by parts also shows that in this case 󵄩󵄩 𝜕uj 󵄩󵄩2 󵄩󵄩 𝜕uj 󵄩󵄩2 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩 󵄩. 󵄩󵄩 󵄩󵄩 𝜕zk 󵄩󵄩󵄩 󵄩󵄩 𝜕zk 󵄩󵄩󵄩 Hence 󵄩 ∗ 󵄩2 ‖u‖21 ≤ 2 (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ), where ‖u‖21 denotes the standard Sobolev-1 norm of u on Ω. Therefore the closure of ∗ the compactly supported forms in dom(𝜕) ∩ dom(𝜕 ) in the graph norm is contained in the Sobolev space W01 (Ω) for forms that are 𝒞 ∞ on Ω, which means that they are zero on the boundary (Proposition 5.13), which is stronger than the condition n

∑ j=1

on bΩ from Proposition 4.53.

𝜕r u =0 𝜕zj j

120 � 5 Density of smooth forms ∗

(c) If Ω is a smoothly bounded pseudoconvex domain, then dom(𝜕) ∩ dom(𝜕 ) is ∗ a Hilbert space in the graph norm u 󳨃→ (‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 . This follows from (4.38). We carry out the proof of Proposition 5.14 in several steps. ∗

∞ Lemma 5.16. Let Ω be as in Proposition 5.14. Then 𝒞(0,q) (Ω) is dense in dom(𝜕) ∩ dom(𝜕 ) ∗

in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 .

∗

Proof. By this we mean that if u ∈ dom(𝜕) ∩ dom(𝜕 ), then we can construct a sequence ∞ um ∈ 𝒞(0,q) (Ω) such that um → u, 𝜕um → 𝜕u, and ϑum → ϑu in L2 (Ω); recall (4.28). We use a method closely related to Friedrichs’ Lemma 5.5 and use the notation from there. Let (χϵ )ϵ be an approximation of the identity, let (δν )ν be a sequence of small positive numbers such that δν → 0, and let Ωδν = {z ∈ Ω : r(z) < −δν }. Then (Ωδν )ν is a sequence of relatively compact open subsets of Ω with union equal to Ω. ∞ The forms uϵ = u ∗ χϵ belong to 𝒞(0,q) (Ωδν ), and uϵ → u, 𝜕uϵ → 𝜕u, and ϑuϵ → ϑu in

L2 (Ωδν ); see Lemmas 5.3 and 5.5. To see that this can be done up to the boundary, we first assume that the domain Ω is star-shaped and 0 ∈ Ω is a center. Let Ωϵ = {(1 + ϵ)z : z ∈ Ω} and uϵ (z) = u(

z ), 1+ϵ

where the dilation is performed for each coefficient of u. Then Ω ⋐ Ωϵ and uϵ ∈ L2 (Ωϵ ). Also, by the dominated convergence theorem, uϵ → u, 𝜕uϵ → 𝜕u, and ϑuϵ → ϑu in L2 (Ω). Now we regularize uϵ defining u(ϵ) = uϵ ∗ χδϵ ,

(5.15)

∞ where δϵ → 0 as ϵ → 0, and δϵ are chosen sufficiently small. Then u(ϵ) ∈ 𝒞(0,q) (Ω), and

∞ (Ω) is dense in the graph norm u(ϵ) → u, 𝜕u(ϵ) → 𝜕u, and ϑu(ϵ) → ϑu in L2 (Ω). Thus 𝒞(0,q) when Ω is star-shaped. The general case follows by using a partition of unity since we assume that our domain has at least 𝒞 2 boundary.

Lemma 5.17. Let Ω be as in Proposition 5.14. Then compactly supported smooth forms ∗ ∗ are dense in dom(𝜕 ) in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕 u‖2 )1/2 . ∗

Proof. We remark that if u ∈ dom(𝜕 ) and if we extend u to ũ on the whole space ℂn by setting ũ to be zero outside of Ω, then ϑ ũ ∈ L2 (ℂn ) in the distribution sense; in fact, for ∗ u ∈ dom(𝜕 ), we have ̃ ϑ ũ = ϑu,

5.2 Density in the graph norm

� 121

̃ = ϑu in Ω and ϑu ̃ = 0 outside of Ω. This can be checked from the definition of where ϑu ∗ ∞ 𝜕 , since for any v ∈ 𝒞(0,q−1) (ℂn ), ̃ v) 2 n . (u,̃ 𝜕v)L2 (ℂn ) = (u, 𝜕v)L2 (Ω) = (ϑu, v)L2 (Ω) = (ϑu, L (ℂ ) We assume again without loss of generality that Ω is star-shaped with 0 as a center. We first approximate ũ by ̃ ũ −ϵ (z) = u(

z ). 1−ϵ

Now we have forms ũ −ϵ with compact support in Ω and ϑ ũ −ϵ → ϑ ũ in L2 (ℂn ). Regularizing ũ −ϵ as before, we define u(−ϵ) = ũ −ϵ ∗ χδϵ .

(5.16)

Then the u(−ϵ) are (0, q)-forms with coefficients in 𝒞0∞ (Ω) such that u(−ϵ) → u and ϑu(−ϵ) → ϑu in L2 (Ω). However, compactly supported smooth forms are not dense in dom(𝜕) in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 )1/2 . Nevertheless, we have the following: ∗

k Lemma 5.18. Let Ω be as in Proposition 5.14. Then 𝒞(0,q) (Ω) ∩ dom(𝜕 ) is dense in dom(𝜕)

in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 )1/2 .

∞ Proof. By Lemma 5.16 it suffices to show that for any u ∈ 𝒞(0,q) (Ω), we can find a sequence ∗

k um ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ) such that um → u and 𝜕um → 𝜕u in L2 (Ω).

Let r be a 𝒞 k+1 defining function such that |dr| = 1 on bΩ. We now introduce some special vector fields and (1, 0)-forms associated with bΩ. Near a point p ∈ bΩ, we choose fields L1 , L2 , . . . , Ln−1 of type (1, 0) that are orthonormal and span Tp1,0 (bΩ). This can be done by choosing a basis and then using the Gram–Schmidt process. To this collection add Ln , the complex normal, normalized to have length 1. So Ln is a smooth multiple of n

∑ j=1

𝜕r 𝜕 . 𝜕zj 𝜕zj

Now denote by w1 , w2 , . . . , wn the (1, 0)-forms such that wj (Lk ) = δjk . Note that Ln is defined globally, in contrast to L1 , . . . , Ln−1 . The wj then form an orthonormal basis for 𝜕r the (1, 0)-forms near p. The (1, 0)-form wn is a smooth multiple of ∑nj=1 𝜕z dzj and is again j

globally defined. Taking the wedge products of the wj yields (local) orthonormal bases for the (1, 0)-forms. We will regularize near a boundary point p ∈ bΩ. Let U be a small neighborhood of p. By a partition of unity we may assume that Ω ∩ U is star-shaped and u is supported in U ∩ Ω. Shrinking U if necessary, we can choose a special boundary chart

122 � 5 Density of smooth forms (t1 , t2 , . . . , t2n−1 , r), where (t1 , t2 , . . . , t2n−1 , 0) are coordinates on bΩ near p. Let w1 , . . . , wn be an orthonormal basis for the (0, 1)-forms on U such that 𝜕r = wn . Let Lj = ∑ns=1 ajs 𝜕z𝜕 , wj = ∑ns=1 bjs dzs , 1 ≤ j ≤ n. Then s

n

n

s=1

ℓ=1

δjk = wj (Lk ) = ∑ bjs dzs ( ∑ akℓ

n 𝜕 ) = ∑ bjs aks . 𝜕zℓ s=1

Consequently, if f is a function, then n

𝜕f = ∑

s=1

n n 𝜕f dzs = ∑ ask (Lk f ) bsj wj = ∑ (Lj f ) wj , 𝜕zs j=1 j,k,s=1

(5.17)

where the superscripts denote the entries of the inverses of the corresponding matrices ∗ with subscripts. Since multiplication by functions in 𝒞 1 (Ω) preserves dom(𝜕 ), we may ′ assume that the form u is supported in a special boundary chart. So u = ∑ |J|=q uJ wJ , where wJ = wj1 ∧ ⋅ ⋅ ⋅ ∧ wjq , and each uJ is a function in 𝒞 k (Ω). Then, in view of (5.17), ′

′

|J|=q

|J|=q

𝜕u = 𝜕( ∑ uJ wJ ) = ∑ (𝜕uJ ∧ wJ + uJ 𝜕wJ ) ′

n

′

= ∑ ∑(Lj uJ ) wj ∧ wJ + ∑ uJ 𝜕wJ . |J|=q j=1

|J|=q

(5.18) (5.19)

Using the special boundary chart, from (4.27) we get that ∗

u ∈ dom(𝜕 ) ⇐⇒ uJ = 0

on bΩ when n ∈ J.

(5.20)

Indeed, the only boundary terms that arise when proving (4.27) come from integrating ∫ Ln αK unK dλ Ω

by parts, and they equal ∫ αK unK (Ln r) dσ. bΩ

Since αK can be arbitrary on bΩ and Ln r ≠ 0 on bΩ, we conclude that unK = 0 on bΩ for all K. To see that the condition is sufficient, note that the computation to prove (4.30) ∞ shows that (u, 𝜕α) = (ϑu, α) when (4.27) holds and α ∈ 𝒞(0,q) (Ω). In view of Lemma 5.16, ∞ 𝒞(0,q) (Ω) is dense in dom(𝜕) in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 )1/2 . Hence (u, 𝜕α) = ∗

∗

(ϑu, α) for all α ∈ dom(𝜕), which implies u ∈ dom(𝜕 ) and 𝜕 u = ϑu.

5.2 Density in the graph norm

� 123

∗

These arguments also give a formula for ϑ and 𝜕 in special boundary frames: ′

n

′

ϑu = ϑ( ∑ uJ wJ ) = − ∑ (∑ Lj ujK ) wK + 0th order(u); |J|=q

j=1

|K|=q−1

(5.21)

0th order(u) indicates the terms that contain no derivatives of the uJ . We note that both 𝜕 and ϑ are first-order differential operators with variable coefficients in 𝒞 k (Ω) when computed in the special frame w1 , . . . , wn . We write u = uτ + uν , where uτ =

∑

′

|J|=q,n∉J

uν =

uJ wJ ,

∑

′

|J|=q,n∈J

uJ wJ ;

uτ is the complex tangential part of u, and uν is the complex normal part of u. Our arguments from above imply that ∗

k u ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ) ⇐⇒ uν = 0

on bΩ.

∞ ∞ For u ∈ 𝒞(0,q) (Ω) and α ∈ 𝒞(0,q+1) (Ω), by (4.31) we have

(𝜕u, α) = (u, ϑα) + ∫ ⟨𝜕r ∧ u, α⟩ dσ bΩ

and 𝜕r ∧ u = 𝜕r ∧ uτ on bΩ, which follows from the representation in special boundary charts: 𝜕r ∧ uν = cwn ∧ ∑

′

|J|=q,n∈J

uJ wJ = 0.

(5.22) ∗

∞ k To approximate a form u ∈ 𝒞(0,q) (Ω) by forms in 𝒞(0,q) (Ω) ∩ dom(𝜕 ), we only change ν the complex normal part u and leave the complex tangential part uτ unchanged: for ∗ ∞ k u ∈ 𝒞(0,q) (Ω), it follows that uτ ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ), and we denote by ũ ν the extension of uν to ℂn by setting ũ ν equal to zero outside of Ω. We approximate ũ ν as in Lemma 5.17 by ν u(−ϵ) = (ũ ν )

−ϵ

∗ χδϵ .

ν Then u(−ϵ) is smooth and supported in a compact subset of Ω∩U. By this we approximate ν ν u by u(−ϵ) ∈ 𝒞0∞ (Ω∩U) in the L2 -norm. Furthermore, by extending 𝜕uν to be zero outside ̃ Ω ∩ U and denoting the extension by 𝜕uν we have

124 � 5 Density of smooth forms ̃ 𝜕ũ ν = 𝜕uν in L2 (ℂn ) in the sense of distributions. This follows from (4.31) and (5.22), since uν ∈ k ∞ 𝒞(0,q) (Ω), and for α ∈ 𝒞(0,q+1) (ℂn ), we have ̃ (ũ ν , ϑα)L2 (ℂn ) = (𝜕uν , α)L2 (Ω) − ∫ ⟨𝜕r ∧ uν , α⟩ dσ = (𝜕uν , α)L2 (ℂn ) . bΩ

Since 𝜕 is a first-order differential operator with variable coefficients, by Friedrichs’ Lemma 5.6 we get ν 𝜕u(−ϵ) → 𝜕ũ ν

in L2 (ℂn ).

(5.23)

We set ν u(−ϵ) = uτ + u(−ϵ) . ∗

ν k , and wj is in It follows that u(−ϵ) ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ), since each coefficient of uτ , u(−ϵ) ∗

k 𝒞 k (Ω ∩ U). Therefore we get u(−ϵ) ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ) and u(−ϵ) → u in L2 (Ω).

To see that 𝜕u(−ϵ) → 𝜕u in the L2 (Ω)-norm, using (5.23), we find that ν 𝜕u(−ϵ) = 𝜕uτ + 𝜕u(−ϵ) → 𝜕u

in L2 (Ω) as ϵ → 0. To finish the proof of Proposition 5.14, we consider an arbitrary u ∈ dom(𝜕) ∩ ∗ dom(𝜕 ) and use a partition of unity and the same notation as before to regularize u in each small star-shaped neighborhood near the boundary. We regularize the complex tangential and normal part separately by setting τ ν u((ϵ)) = u(ϵ) + u(−ϵ) ,

which means that we first consider u(ϵ) as it was defined in (5.15), then take the tangential τ components u(ϵ) , then we consider u(−ϵ) as it is defined in (5.16), and finally take the ν ν normal components u(−ϵ) . It follows that for sufficiently small ϵ > 0, u(−ϵ) has coefficients ∞ τ ∞ in 𝒞0 (Ω) and u(ϵ) has coefficients in 𝒞 (Ω). ∗

k Thus we see that u((ϵ)) ∈ 𝒞(0,q) (Ω) ∩ dom(𝜕 ). We get from Lemma 5.16 that u(ϵ) → u ∗

τ in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 , and hence u(ϵ) → uτ in the graph norm ∗

u 󳨃→ (‖u‖2 +‖𝜕u‖2 +‖𝜕 u‖2 )1/2 . By Lemma 5.17 we obtain u(−ϵ) → u in the graph norm u 󳨃→ ∗

∗

ν (‖u‖2 +‖𝜕 u‖2 )1/2 , and hence u(−ϵ) → uν in the graph norm u 󳨃→ (‖u‖2 +‖𝜕 u‖2 )1/2 . Finally, ν we use Lemma 5.18, in particular, formula (5.23), and see that 𝜕u(−ϵ) → 𝜕ũ ν in L2 (ℂn ), ∗

and hence u((ϵ)) → u in the graph norm u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 .

5.3 Notes

∗

� 125

∗

k This shows that 𝒞(0,q) (Ω) ∩ dom(𝜕 ) is dense in dom(𝜕) ∩ dom(𝜕 ) in the graph norm ∗

u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 .

5.3 Notes Friedrichs’ lemma can be found in [16]. It will also be important in the following chapters. In the proof of the density result, we mainly follow the presentation in [16], Straube [91] uses a different approach for the same result, whereas Hörmander [57] proves a corresponding density result in weighted L2 -spaces.

6 The weighted 𝜕-complex In this chapter, we investigate the basic properties of the 𝜕-operator on weighted L2 spaces on the whole of ℂn . It turns out that the behavior of the 𝜕-operator depends very much on properties of the Levi matrix of the weight function. For n = 1, there is an interesting connection between 𝜕 and the theory of Schrödinger operators with magnetic fields, and for n ≥ 2, the corresponding notion is the Witten Laplacian. The corresponding density result from the last chapter is much easier to handle in the weighted case, therefore also the basic estimate and existence of the 𝜕-Neumann operator.

6.1 The 𝜕-Neumann operator on (0, 1)-forms Let φ : ℂn 󳨀→ ℝ+ be a plurisubharmonic 𝒞 2 -weight function and define the space L2 (ℂn , e−φ ) = {f : ℂn 󳨀→ ℂ : ∫ |f |2 e−φ dλ < ∞}, ℂn

where λ denotes the Lebesgue measure, the space L2(0,1) (ℂn , e−φ ) of (0, 1)-forms with coefficients in L2 (ℂn , e−φ ), and the space L2(0,2) (ℂn , e−φ ) of (0, 2)-forms with coefficients in L2 (ℂn , e−φ ). Let (f , g)φ = ∫ f ge−φ dλ ℂn

denote the inner product and ‖f ‖2φ = ∫ |f |2 e−φ dλ ℂn

the norm in L2 (ℂn , e−φ ). We define dom(𝜕) to be the space of all functions f ∈ L2 (ℂn , e−φ ) such that 𝜕f , in the sense of distributions, belongs to L2(0,1) (ℂn , e−φ ), and consider the weighted 𝜕-complex L2 (ℂn , e−φ ) 󳨀→ L2(0,1) (ℂn , e−φ ) 󳨀→ L2(0,2) (ℂn , e−φ ), 𝜕

𝜕

←󳨀

←󳨀

∗ 𝜕φ

(6.1)

∗ 𝜕φ

∗

where 𝜕φ is the adjoint operator to 𝜕 with respect to the weighted inner product. For a smooth (0, 1)-form u = ∑nj=1 uj dzj ∈ dom(𝜕φ ), we have ∗

∗

n

𝜕φ u = − ∑( j=1

https://doi.org/10.1515/9783111182926-006

𝜕φ 𝜕 − )u . 𝜕zj 𝜕zj j

(6.2)

6.1 The 𝜕-Neumann operator on (0, 1)-forms

� 127

The complex Laplacian on (0, 1)-forms is defined as ∗

∗

◻φ := 𝜕 𝜕φ + 𝜕φ 𝜕, and dom(◻φ ) is the space of all f ∈ L2(0,1) (ℂn , e−φ ) such that ∗

f ∈ dom(𝜕) ∩ dom(𝜕φ ), ∗

∗

𝜕f ∈ dom(𝜕φ ), and 𝜕φ f ∈ dom(𝜕). The operator ◻φ is self-adjoint and positive (see Chapter 4), which means that (◻φ f , f )φ ≥ 0

for f ∈ dom(◻φ ).

The associated Dirichlet form is denoted by ∗

∗

Qφ (f , g) = (𝜕f , 𝜕g)φ + (𝜕φ f , 𝜕φ g)φ

(6.3)

∗

for f , g ∈ dom(𝜕) ∩ dom(𝜕φ ). The weighted 𝜕-Neumann operator Nφ , if it exists, is the bounded inverse of ◻φ . In the weighted space L2(0,1) (ℂn , e−φ ), we can give a simple characterization of ∗

dom(𝜕φ ).

Proposition 6.1. Let f = ∑ fj dzj ∈ L2(0,1) (ℂn , e−φ ). Then f ∈ dom(𝜕φ ) if and only if ∗

n

eφ ∑ j=1

𝜕 (f e−φ ) ∈ L2 (ℂn , e−φ ), 𝜕zj j

(6.4)

where the derivative is to be taken in the sense of distributions. Proof. Suppose first that eφ ∑nj=1

𝜕 (f e−φ ) 𝜕zj j

∈ L2 (ℂn , e−φ ). We have to show that there

exists a constant C such that |(𝜕g, f )φ | ≤ C‖g‖φ for all g ∈ dom(𝜕). To this end, let (χR )R∈ℕ be a family of radially symmetric smooth cut-off functions, which are identically one on 𝔹R , the ball with radius R, such that the support of χR is contained in 𝔹R+1 , supp(χR ) ⊂ 𝔹R+1 , and such that all first-order derivatives of all functions in this family are uniformly bounded by a constant M. Then for all g ∈ 𝒞0∞ (ℂn ), n

(𝜕g, χR f )φ = ∑( j=1

n 𝜕g 𝜕 , χR fj ) = − ∫ ∑ g (χR f j e−φ )dλ 𝜕zj 𝜕z φ j j=1 ℂn

by integration by parts, which in particular means that 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝜕 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 −φ 󵄨 (χR f j e ) dλ󵄨󵄨󵄨. 󵄨󵄨(𝜕g, f )φ 󵄨󵄨 = lim 󵄨󵄨(𝜕g, χR f )φ 󵄨󵄨 = lim 󵄨󵄨 ∫ ∑ g 󵄨󵄨 R→∞ R→∞󵄨󵄨 𝜕z j 󵄨ℂn j=1 󵄨

128 � 6 The weighted 𝜕-complex Now we use the triangle inequality and then the Cauchy–Schwarz inequality to get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n n 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 ∫ ∑ g 𝜕 (χR f e−φ ) dλ󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨 ∫ χR g ∑ 𝜕 (f e−φ ) dλ󵄨󵄨󵄨 + 󵄨󵄨󵄨 ∫ ∑ f g 𝜕χR e−φ dλ󵄨󵄨󵄨 j j j 󵄨 󵄨 󵄨 󵄨 󵄨󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 𝜕zj 𝜕zj 󵄨󵄨 󵄨󵄨 n 󵄨󵄨 󵄨󵄨 n j=1 󵄨󵄨 󵄨󵄨 n j=1 𝜕zj j=1 ℂ ℂ ℂ 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝜕 ≤ ‖χR g‖φ 󵄩󵄩󵄩eφ ∑ (fj e−φ )󵄩󵄩󵄩 + M‖g‖φ ‖f ‖φ 󵄩󵄩 󵄩󵄩 󵄩 j=1 𝜕zj 󵄩φ 󵄩󵄩 n 󵄩󵄩 󵄩󵄩 󵄩󵄩 𝜕 ≤ ‖g‖φ 󵄩󵄩󵄩eφ ∑ (fj e−φ )󵄩󵄩󵄩 + M‖g‖φ ‖f ‖φ . 󵄩󵄩 󵄩󵄩 󵄩 j=1 𝜕zj 󵄩φ Hence by assumption, 󵄩󵄩 󵄩󵄩 n 󵄩󵄩 𝜕 󵄨󵄨 󵄨 󵄩󵄩 φ (fj e−φ )󵄩󵄩󵄩 + M‖g‖φ ‖f ‖φ ≤ C‖g‖φ 󵄨󵄨(𝜕g, f )φ 󵄨󵄨󵄨 ≤ ‖g‖φ 󵄩󵄩󵄩e ∑ 󵄩󵄩 󵄩󵄩󵄩 j=1 𝜕zj 󵄩φ for all g ∈ 𝒞0∞ (ℂn ), and by the denseness of 𝒞0∞ (ℂn ) this is true for all g ∈ dom(𝜕). ∗ Conversely, let f ∈ dom(𝜕φ ), which means that there exists a uniquely determined element 𝜕φ f ∈ L2 (ℂn , e−φ ) such that for each g ∈ dom(𝜕), we have ∗

∗

(𝜕g, f )φ = (g, 𝜕φ f )φ . Now take g ∈ 𝒞0∞ (ℂn ). Then g ∈ dom(𝜕), and ∗

(g, 𝜕φ f )φ = (𝜕g, f )φ n

= ∑( j=1

𝜕g ,f ) 𝜕zj j φ n

= −(g, ∑ j=1

𝜕 (f e−φ )) 𝜕zj j

L2

n

= −(g, eφ ∑ j=1

𝜕 (f e−φ )) . 𝜕zj j φ

Since 𝒞0∞ (ℂn ) is dense in L2 (ℂn , e−φ ), we conclude that n

𝜕φ f = −eφ ∑ ∗

j=1

which in particular implies that eφ ∑nj=1

𝜕 (f e−φ ), 𝜕zj j

𝜕 (f e−φ ) 𝜕zj j

∈ L2 (ℂn , e−φ ).

The following lemma will be important for our considerations.

6.1 The 𝜕-Neumann operator on (0, 1)-forms

� 129

Lemma 6.2. The forms with coefficients in 𝒞0∞ (ℂn ) are dense in dom(𝜕) ∩ dom(𝜕φ ) in the ∗

1

graph norm f 󳨃→ (‖f ‖2φ + ‖𝜕f ‖2φ + ‖𝜕φ f ‖2φ ) 2 . ∗

Proof. First, we show that compactly supported L2 -forms are dense in the graph norm. So let {χR }R∈ℕ be a family of smooth radially symmetric cut-offs identically one on 𝔹R and supported in 𝔹R+1 , such that all first-order derivatives of the functions in this family are uniformly bounded in R by a constant M. ∗ ∗ Let f ∈ dom(𝜕) ∩ dom(𝜕φ ). Then, clearly, χR f ∈ dom(𝜕) ∩ dom(𝜕φ ) and χR f → f in L2(0,1) (ℂn , e−φ ) as R → ∞. As observed in Proposition 6.1, we have n

𝜕φ f = −eφ ∑ ∗

j=1

𝜕 (f e−φ ), 𝜕zj j

and hence n

𝜕φ (χR f ) = −eφ ∑ ∗

j=1

𝜕 (χ f e−φ ). 𝜕zj R j

We need to estimate the difference of these expressions, ∗

∗

∗

∗

n

𝜕φ f − 𝜕φ (χR f ) = 𝜕φ f − χR 𝜕φ f + ∑ j=1

𝜕χR f. 𝜕zj j

By the triangle inequality, n

∗ ∗ 󵄩 󵄩 󵄩 ∗ 󵄩󵄩 ∗ 2 −φ 󵄩󵄩𝜕φ f − 𝜕φ (χR f )󵄩󵄩󵄩φ ≤󵄩󵄩󵄩𝜕φ f − χR 𝜕φ f 󵄩󵄩󵄩φ + M ∑ ∫ |fj | e dλ. j=1 ℂn \𝔹

R

Now both terms tend to 0 as R → ∞, and we can similarly see that also 𝜕(χR f ) → 𝜕f as R → ∞. So we have the denseness of compactly supported forms in the graph norm, and the denseness of forms with coefficients in 𝒞0∞ (ℂn ) will follow by applying Friedrichs’ Lemma 5.6. As in the case of bounded domains, the canonical solution operator to 𝜕, which we ∗ denote by Sφ , is given by 𝜕φ Nφ . The existence and compactness of Nφ and Sφ are closely related. For the rest of this chapter, we will suppose that n ≥ 2. The case n = 1 will be discussed separately in Chapter 10. Now we suppose that the lowest eigenvalue μφ of the Levi matrix Mφ = (

𝜕2 φ ) 𝜕zj 𝜕zk jk

130 � 6 The weighted 𝜕-complex of φ satisfies μφ (z) > ϵ

(6.5)

for all z ∈ ℂn , where ϵ > 0. Now we can prove the basic estimate. Proposition 6.3. For a plurisubharmonic weight function φ satisfying (6.5), there is C > 0 such that 󵄩 ∗ 󵄩2 ‖u‖2φ ≤ C(‖𝜕u‖2φ + 󵄩󵄩󵄩𝜕φ u󵄩󵄩󵄩φ )

(6.6)

∗

for each (0, 1)-form u ∈ dom(𝜕) ∩ dom(𝜕φ ). Proof. By Lemma 6.2 and the assumption on φ it suffices to show that n

𝜕2 φ 󵄩 ∗ 󵄩2 uj uk e−φ dλ ≤ ‖𝜕u‖2φ + 󵄩󵄩󵄩𝜕φ u󵄩󵄩󵄩φ 𝜕z 𝜕z j k j,k=1

∫ ∑ ℂn

for each (0, 1)-form u = ∑nk=1 uk dzk with coefficients uk ∈ 𝒞0∞ (ℂn ), k = 1, . . . , n. 𝜕φ For this purpose, we set δk = 𝜕z𝜕 − 𝜕z , and since k

𝜕u = ∑ ( j ϵ

(6.12)

for all z ∈ ℂn , where ϵ > 0. Then there exists a uniquely determined bounded linear operator Nφ,q : L2(0,q) (ℂn , e−φ ) 󳨀→ L2(0,q) (ℂn , e−φ ) such that ◻φ ∘ Nφ,q u = u for all u ∈ L2(0,q) (ℂn , e−φ ).

134 � 6 The weighted 𝜕-complex Proof. Let μφ,1 ≤ μφ,2 ≤ ⋅ ⋅ ⋅ ≤ μφ,n denote the eigenvalues of Mφ and suppose that Mφ is diagonalized. Then, in a suitable basis, ′

∑

|K|=q−1

n ′ 𝜕2 φ ujK ukK = ∑ ∑ μφ,j |ujK |2 𝜕zj 𝜕zk |K|=q−1 j=1 j,k=1 n

∑

=

∑

′

J=(j1 ,...,jq )

(μφ,j1 + ⋅ ⋅ ⋅ + μφ,jq )|uJ |2

≥ sq |u|2 . The last equality results as follows: for J = (j1 , . . . , jq ) fixed, |uJ |2 occurs precisely q times in the second sum, once as |uj1 K1 |2 , once as |uj2 K2 |2 , etc. At each occurrence, it is multiplied by μφ,jℓ . For the rest of the proof, we use (6.11) and proceed as in the proof of Proposition 6.5. ̄ Remark 6.8. For the 𝜕-Neumann operator Nφ,q on (0, q)-forms, we obtain in a similar ∗ way as in Proposition 6.3 that Nφ,q = jφ,q ∘ jφ,q , where jφ,q : dom (𝜕) ∩ dom (𝜕φ ) 󳨀→ L2(0,q) (ℂn , e−φ ), ∗

and dom (𝜕) ∩ dom (𝜕φ ) is endowed with the graph norm (‖𝜕u‖2φ + ‖𝜕φ u‖2φ )1/2 . ∗

∗

∗ It also follows in this case that both 𝜕 and 𝜕φ have closed ranges; see Proposition 4.58.

6.3 Notes Most of the material in this chapter is taken from [35, 36, 46]. It will be fundamental for a closer analysis of the 𝜕-Neumann operator and its connection to Schrödinger operators with magnetic fields and the Witten Laplacian. Using spectral analysis and Persson’s theorem on the bottom of the essential spectrum, we can show that assumption (6.5) in Proposition 6.3 can be replaced by the more general condition lim inf|z|→∞ μφ (z) > 0; see [46]. We indicate that formula (6.11) has an interesting differential geometric interpretation: let (M, ω) be a Kähler manifold with fundamental form ω, and let (E, M, π) be a holomorphic vector bundle over M. Let ∇ : Γ(T ℂ (M)) × Γ(E) 󳨀→ Γ(E) be the uniquely determined connection on E that is both holomorphic and compatible with the metric. The operator Θ := ∇2 is called the curvature of the connection ∇.

6.3 Notes

� 135

We consider the weighted 𝜕-complex on ℂn with fundamental form n

ω = i ∑ dzk ∧ dzk . k=1

The weight factor e−φ can be interpreted as a metric on the trivial line bundle over ℂn and n

𝜕2 φ dz ∧ dzk ; 𝜕zj 𝜕zk j j,k=1

Θ = 𝜕𝜕φ = ∑

see [94] for the details. Let Λ denote the interior multiplication with the fundamental form ω, (Λα, w) = (α, ω ∧ w) for suitable differential forms α and w, and let u = ∑′|J|=q uJ dzJ be a (0, q)-form with coefficients in 𝒞0∞ (ℂn ). We want to interpret the term ∑

′

n

∑ ∫

|K|=q−1 j,k=1 ℂn

𝜕2 φ u u e−φ dλ 𝜕zj 𝜕zk jK kK

of (6.11) by the curvature Θ and operator Λ. For this purpose, we consider (n, q)-forms ′

ξ = ∑ ξI dz ∧ dzI , |I|=q

instead of (0, q)-forms, where dz = dz1 ∧ ⋅ ⋅ ⋅ ∧ dzn . We use the notation ̂ ∧ ⋅ ⋅ ⋅ ∧ dz , d ẑj := dz1 ∧ ⋅ ⋅ ⋅ ∧ dz j n which means that dzj is excluded. It follows that n

′

Λξ = i ∑ ∑ ξjJ d ẑj ∧ dzJ , j=1 |J|=q−1

and since Θξ = 0, we obtain for the commutator [Θ, Λ] that ′

(i[Θ, Λ]ξ, ξ)φ = ∑

n

∑ ∫

|J|=q−1 j,k=1 ℂn

𝜕2 φ ξ ξ e−φ dλ. 𝜕zj 𝜕zk jJ kJ

The commutator [Θ, Λ] appears in the Nakano vanishing theorem; see [94].

7 The twisted 𝜕-complex We consider the twisted 𝜕-complex T

S

L2 (Ω) 󳨀→ L2(0,1) (Ω) 󳨀→ L2(0,2) (Ω)

(7.1)

for operators T = 𝜕 ∘ √τ and S = √τ ∘ 𝜕, where τ ∈ 𝒞 2 (Ω) and τ > 0 on Ω. It turns out that this approach leads to generalized basic estimates, which are seminal for the applications explained in the next chapter. The key method consists of a keen use of integration by parts.

7.1 An exact sequence of unbounded operators First, we prove a general result about operators like T and S. Proposition 7.1. Let H1 , H2 , H3 be Hilbert spaces, let T : H1 󳨀→ H2 and S : H2 󳨀→ H3 be densely defined linear operators such that S(T(f )) = 0 for each f ∈ dom(T), and let P : H2 󳨀→ H2 be a positive invertible operator such that 󵄩 󵄩2 ‖Pu‖22 ≤ 󵄩󵄩󵄩T ∗ u󵄩󵄩󵄩1 + ‖Su‖23

(7.2)

for all u ∈ dom(S) ∩ dom(T ∗ ), where 󵄨 󵄨 dom(T ∗ ) = {u ∈ H2 : 󵄨󵄨󵄨(u, Tf )2 󵄨󵄨󵄨 ≤ C ‖f ‖1 for all f ∈ dom(T)}. Suppose (7.2) holds and let α ∈ H2 be such that Sα = 0. Then there exists σ ∈ H1 such that (i) T(σ) = α and (ii) ‖σ‖21 ≤ ‖P−1 α‖22 . Proof. Since P is positive, it follows that P = P∗ . Now let α ∈ H2 be such that Sα = 0. We consider the linear functional T ∗ u 󳨃→ (u, α)2 for u ∈ dom(T ∗ ). We show that this linear functional is well defined and continuous. If u ∈ kerS, then 󵄨󵄨 󵄨 󵄨 󵄨 󵄩 −1 󵄩 󵄩 ∗ 󵄩2 −1 2 1/2 󵄩 −1 󵄩 󵄨󵄨(u, α)2 󵄨󵄨󵄨 = 󵄨󵄨󵄨(Pu, P α)2 󵄨󵄨󵄨 ≤ ‖Pu‖2 󵄩󵄩󵄩P α󵄩󵄩󵄩2 ≤ (󵄩󵄩󵄩T u󵄩󵄩󵄩1 + ‖Su‖3 ) 󵄩󵄩󵄩P α󵄩󵄩󵄩2 󵄩 󵄩 󵄩 󵄩 = 󵄩󵄩󵄩T ∗ u󵄩󵄩󵄩1 󵄩󵄩󵄩P−1 α󵄩󵄩󵄩2 , and if u⊥2 kerS, then (u, α)2 = 0. In addition, we have that T ∗ w = 0 for all w⊥2 kerS; this follows from the assumption that Tf ∈ kerS, so 0 = (w, Tf )2 ≤ C‖f ‖1 , which means that https://doi.org/10.1515/9783111182926-007

7.2 The twisted basic estimates

� 137

w ∈ dom(T ∗ ) and T ∗ w = 0, since (T ∗ w, f )1 = (w, Tf )2 = 0 for all f ∈ dom(T). If T ∗ u = 0, then from the above estimate it follows that (u, α)2 = 0. We apply the Hahn–Banach theorem, where we keep the constant for the estimate of the functional and the Riesz representation theorem to get σ ∈ H1 such that (T ∗ u, σ)1 = (u, α)2 , which implies that (u, Tσ)2 = (u, α)2 . Hence Tσ = α, and, again by the above estimate, ‖σ‖1 ≤ ‖P−1 α‖2 .

7.2 The twisted basic estimates Let Ω be a smoothly bounded pseudoconvex domain in ℂn with defining function r such that |∇r(z)| = 1 on bΩ. Let τ ∈ 𝒞 2 (Ω) and τ > 0 on Ω. For f ∈ 𝒞 ∞ (Ω), we define n

𝜕 (√τf ) dzk , 𝜕z k k=1

Tf = (𝜕 ∘ √τ)f = ∑

(7.3)

and for u = ∑nj=1 uj dzj with coefficients uj in 𝒞 ∞ (Ω) (we will write u ∈ Λ0,1 (Ω)), we define Su = ∑ √τ( j 0 on Ω, and let u ∈ 𝒟0,1 . Then ∗ 󵄩2 󵄩󵄩√ −φ 2 2 −φ 󵄩󵄩 τ + A 𝜕φ u󵄩󵄩󵄩φ + ‖√τ 𝜕u‖φ ≥ ‖√τu‖φ,z + ∫ Θ(u, u) e dλ + ∫ τi𝜕𝜕r(u, u) e dσ, Ω

bΩ

where Θ(u, u) = τi𝜕𝜕φ(u, u) − i𝜕𝜕τ(u, u) −

|⟨𝜕τ, u⟩|2 . A

Proof. As in the case without twists, we get 󵄩󵄩 𝜕uj 󵄩󵄩󵄩2 𝜕u 󵄩 ‖√τ 𝜕u‖2φ = ∑ 󵄩󵄩󵄩√τ( k − )󵄩󵄩 󵄩 𝜕zj 𝜕zk 󵄩󵄩󵄩φ j k} for k = 0, 1, . . . , n. We will prove inductively that for every k, there is a holomorphic function uk in Ωk such that uk (z0 ) = 1 and −3k 󵄨2 󵄨 ∫ 󵄨󵄨󵄨uk (z)󵄨󵄨󵄨 e−φ(z) (1 + |z|2 ) dλ(z) < ∞.

Ωk

When k = 0, we can take u0 (z) ≡ 1, and un will have the desired properties. Assume that 0 < k ≤ n and that uk−1 has already been constructed. Choose ψ ∈ 𝒞0∞ (ℂ) such that ψ(zk ) = 0 when |zk | > r/2 and ψ(zk ) = 1 when |zk | < r/3, and set uk (z) := ψ(zk )uk−1 (z) − zk v(z). Notice that ψ(zk )uk−1 (z) = 0 in Ωk \ Ωk−1 . To make uk holomorphic, we must choose v as a solution of the equation 𝜕v = z−1 k uk−1 𝜕ψ = f . By the inductive hypothesis we have −3(k−1) 󵄨 󵄨2 dλ(z) < ∞. ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−φ(z) (1 + |z|2 )

Ωk

Hence it follows from Theorem 8.8 that we can find v such that 1−3k 󵄨 󵄨2 dλ(z) < ∞. ∫ 󵄨󵄨󵄨v(z)󵄨󵄨󵄨 e−φ(z) (1 + |z|2 )

Ωk

Together with the inductive hypothesis on uk−1 , this implies that −3k 󵄨 󵄨2 ∫ 󵄨󵄨󵄨uk (z)󵄨󵄨󵄨 e−φ(z) (1 + |z|2 ) dλ(z) < ∞.

Ωk

Since 𝜕v = 0 in a neighborhood of 0, v is a 𝒞 ∞ function there, and we have uk (0) = uk−1 (0) = 1, so uk has the required properties. Lemma 8.10. Let ζ ∈ ℂn and K > 0 and define g(z) = log(1 + K|z − ζ |2 ). Then for each w ∈ ℂn , we have K K |w|2 ≤ i𝜕𝜕g(w, w)(z) ≤ |w|2 , (1 + K|z − ζ |2 )2 1 + K|z − ζ |2 where we use notation (7.5). Proof. An easy computation shows K 2 (zj − ζ j )(zk − ζk ) Kδjk 𝜕2 g (z) = − + 2 2 𝜕zj 𝜕zk (1 + K|z − ζ | ) 1 + K|z − ζ |2

(8.11)

8.2 Weighted spaces of entire functions

=

� 147

K [(1 + K|z − ζ |2 ) δjk − K(zj − ζ j )(zk − ζk )]. (1 + K|z − ζ |2 )2

This implies i𝜕𝜕g(w, w)(z) =

K 󵄨2 󵄨 [(1 + K|z − ζ |2 ) |w|2 − K 󵄨󵄨󵄨(w, z − ζ )󵄨󵄨󵄨 ], 2 2 (1 + K|z − ζ | )

and hence K [(1 + K|z − ζ |2 ) |w|2 − K|w|2 |z − ζ |2 ] ≤ i𝜕𝜕g(w, w)(z) (1 + K|z − ζ |2 )2 K ≤ |w|2 . 1 + K|z − ζ |2 We are now able to show that weighted spaces of entire functions are of infinite dimension if the weight function has an appropriate behavior at infinity. Theorem 8.11. Let W : ℂn 󳨀→ ℝ be a 𝒞 ∞ -function, and let μ(z) denote the lowest eigenvalue of the Levi matrix i𝜕𝜕W (z) = (

n

𝜕2 W (z) ) . 𝜕zj 𝜕zk j,k=1

Suppose that lim |z|2 μ(z) = ∞.

|z|→∞

Then the Hilbert space A2 (ℂn , e−W ) of all entire functions f such that 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 exp(−W (z)) dλ(z) < ∞

ℂn

is of infinite dimension. Proof. Assumption (8.12) implies that there exists a constant K > 0 such that i𝜕𝜕W (w, w)(z) ≥ −K|w|2 for all z, w ∈ ℂn and that i𝜕𝜕W (z) is strictly positive for large |z|. From Lemma 8.10 we have i𝜕𝜕g(w, w)(z) ≥ where g(z) = log(1 + 4K|z − ζ |2 ).

4K |w|2 , (1 + 4K|z − ζ |2 )2

(8.12)

148 � 8 Applications Hence, for |z − ζ | ≤ 1/√4K, we have i𝜕𝜕g(w, w)(z) ≥ K|w|2 . Since i𝜕𝜕W (w, w)(z) is negative at most on a compact set in ℂn , there exist finitely many points ζ1 , . . . , ζM ∈ ℂn such that this compact set is covered by the balls {z : |z − ζl | < 1/√4K}. Hence M

̃ φ(z) := W (z) + ∑ gl (z) l=1

is strictly plurisubharmonic, where gl (z) = log(1 + 4K|z − ζl |2 ), l = 1, . . . , M. ̃ be the least eigenvalue of i𝜕𝜕φ.̃ Then by assumption (8.12) we have Let μ(z) ̃ = ∞. lim |z|2 μ(z)

|z|→∞

For each N ∈ ℕ, there exists R > 0 such that ̃ ≥ μ(z)

N +M +1 |z|2

for |z| > R.

̃ : |z| ≤ R}. Then μ̃ 0 > 0. Set Let μ̃ 0 := inf{μ(z) κ=

μ̃ 0 2(N + M)

and M

φ(z) := W (z) + ∑ gl (z) − (N + M) log(1 + κ|z|2 ). l=1

It follows that e−φ is locally integrable. Next, we claim that φ is strictly plurisubharmonic. Notice that ̃ − i𝜕𝜕φ(w, w)(z) ≥ |w|2 (μ(z)

(N + M)κ ). 1 + κ|z|2

For |z| ≤ R, we have ̃ − μ(z)

(N + M)μ̃ 0 μ̃ 0 (N + M)κ ≥ μ̃ 0 − (N + M)κ = μ̃ 0 − = > 0, 2(N + M) 2 1 + κ|z|2

and for |z| > R, we have ̃ − μ(z)

(N + M)κ N + M + 1 N + M 1 ≥ − = 2, 1 + κ|z|2 |z|2 |z|2 |z|

8.2 Weighted spaces of entire functions

� 149

which implies that φ is strictly plurisubharmonic. Therefore we can apply Theorem 8.9 and get an entire function f such that f (0) = 1 and −3n 󵄨2 󵄨 ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 + |z|2 ) e−φ(z) dλ(z) < ∞.

ℂn

Now we set Ñ = N − 3n and get ̃

N 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 + |z|2 ) e−W (z) dλ(z)

ℂn

=∫ ℂn

2 ∏M 󵄨2 2 Ñ −φ(z) l=1 (1 + 4K|z − ζl | ) 󵄨󵄨 dλ(z) 󵄨󵄨f (z)󵄨󵄨󵄨 (1 + |z| ) e (1 + κ|z|2 )N+M

≤ sup { z∈ℂn

2 (1 + |z|2 )N ∏M −3n 󵄨 󵄨2 l=1 (1 + 4K|z − ζl | ) } ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 (1 + |z|2 ) e−φ(z) dλ(z) < ∞. 2 N+M (1 + κ|z| ) n ℂ

Hence fp ∈ A2 (ℂn , e−W ) for any polynomial p of degree < N,̃ and since N = Ñ + 3n was arbitrary, we are done. The following example in ℂ2 shows that Theorem 8.11 is not sharp. Let φ(z, w) = |z|2 |w|2 + |w|4 . In this case, we have that A2 (ℂ2 , e−φ ) contains all the functions fk (z, w) = wk for k ∈ ℕ, since ∞∞

2 2

4

∞ ∞

2 2

4

∫ ∫ r22k e−(r1 r2 +r2 ) r1 r2 dr1 dr2 = ∫ ( ∫ r1 r22 e−r1 r2 dr1 ) r22k−1 e−r2 dr2 0 0

0 0 ∞ ∞

∞

0

0

4 4 1 1 = ∫ ( ∫ e−s ds) r22k−1 e−r2 dr2 = ∫ r22k−1 e−r2 dr2 < ∞. 2 2

0

The Levi matrix of φ has the form i𝜕𝜕φ = (

|w|2 wz

zw ), |z| + 4|w|2 2

and hence φ is plurisubharmonic, and the lowest eigenvalue has the form 1 1/2 μφ (z, w) = (5|w|2 + |z|2 − (9|w|4 + 10|z|2 |w|2 + |z|4 ) ) 2 16|w|4 = , 2(5|w|2 + |z|2 + (9|w|4 + 10|z|2 |w|2 + |z|4 )1/2 ) and hence

150 � 8 Applications lim |z|2 μφ (z, 0) = 0,

|z|→∞

which implies that condition (8.12) of Theorem 8.11 is not satisfied.

8.3 Notes The original proof of Hörmander’s L2 -estimates uses a density lemma in weighted spaces. The L2 -estimates turned out to be an important method in different problems of analysis such as approximation and interpolation. The method is known as a real method in complex analysis (see [60]). It can also be used to prove the Levi problem that domains of holomorphy coincide with pseudoconvex domains [57]. The basic estimates (4.38) also hold for general bounded pseudoconvex domains (see [91]). We simplified the proof of Shikegawa’s result, Theorem 8.11. In the following, we will show that the infinite dimension of weighted spaces of entire functions is closely related to the compactness of the 𝜕-Neumann operator on the corresponding weighted L2 -spaces, a phenomenon that also appears in applications to Dirac and Pauli operators. Rozenblum and Shirokov [82] showed that A2 (ℂ, e−φ ) is of infinite dimension if φ is a subharmonic 𝒞 2 -function such that ∫ℂ △φ(z) dλ(z) = ∞; see also Chapter 13. Borichev, Van An, and Youssfi [11] extended and completed these results.

9 Spectral analysis In this chapter, we discuss the spectral theorem for self-adjoint unbounded operators on a separable Hilbert space and important consequences of it. For the general approach, we use the basic notions of measure theory such as σ-algebras and Borel sets. Moreover, we prove Ruelle’s lemma and show how it can be used to interpret Hörmander’s L2 -estimates. In the following, we will discuss applications of these results to the 𝜕-Neumann operator, Schrödinger operators with magnetic fields, Dirac operators, and the Witten Laplacian.

9.1 Resolutions of the identity We start with some definitions and basic facts. Definition 9.1. Let Ω be a subset of ℂ, let M be a σ-algebra in Ω, and let H be a Hilbert space. A resolution of the identity is a mapping E : M 󳨀→ ℒ(H) of M to the algebra ℒ(H) of bounded linear operators on H with the following properties: (a) E(0) = 0 (zero operator), E(Ω) = I (identity on H). (b) For each ω ∈ M, the image E(ω) is an orthogonal projection on H. (c) E(ω′ ∩ ω′′ ) = E(ω′ )E(ω′′ ). (d) If ω′ ∩ ω′′ = 0, then E(ω′ ∪ ω′′ ) = E(ω′ ) + E(ω′′ ). (e) For all x ∈ H and y ∈ H, the set function Ex,y defined by Ex,y (ω) = (E(ω)x, y) is a complex measure on M. We collect some immediate consequences of these properties. Since each E(ω) is an orthogonal (i. e., self-adjoint) projection, for x ∈ H, we have 󵄩 󵄩2 Ex,x (ω) = (E(ω)x, x) = (E(ω)2 x, x) = (E(ω)x, E(ω)x) = 󵄩󵄩󵄩E(ω)x 󵄩󵄩󵄩 ,

(9.1)

and hence each Ex,x is a positive measure on M with total variation Ex,x (Ω) = ‖x‖2 .

(9.2)

By (c) any two of the projections E(ω) commute with each other; if ω ∩ ω′ = 0, then (a) and (c) show that im(E(ω)) ⊥ im(E(ω′ )), which follows from (E(ω)x, E(ω′ )y) = (E(ω)2 x, E(ω′ )y) = (E(ω)x, E(ω)E(ω′ )y) = 0. https://doi.org/10.1515/9783111182926-009

152 � 9 Spectral analysis By (d), E is finitely additive. Concerning countable additivity, we have the following result. Proposition 9.2. If E is a resolution of the identity, and if x ∈ H, then ω 󳨃→ E(ω)x is a countably additive H-valued measure on M. If ωn ∈ M and E(ωn ) = 0 for n ∈ ℕ, and if ω = ⋃∞ n=1 ωn , then E(ω) = 0. Proof. By (d), ω 󳨃→ (E(ω)x, y) is a complex measure, and hence ∞

∑ (E(ωn )x, y) = (E(ω)x, y)

n=1

(9.3)

for every y ∈ H. For n ≠ m, we have E(ωn )x ⊥ E(ωm )x. Let N

ΛN (y) = ∑ (y, E(ωn )x). n=1

By (9.3) the sequence (ΛN (y))N converges for every y ∈ H. The uniform boundedness principle (Theorem 4.14) implies that (‖ΛN ‖)N is bounded, where 󵄩 󵄩 󵄩 󵄩2 󵄩 󵄩2 1/2 ‖ΛN ‖ = 󵄩󵄩󵄩E(ω1 )x + ⋅ ⋅ ⋅ + E(ωN )x 󵄩󵄩󵄩 = (󵄩󵄩󵄩E(ω1 )x 󵄩󵄩󵄩 + ⋅ ⋅ ⋅ + 󵄩󵄩󵄩E(ωN )x 󵄩󵄩󵄩 ) , and hence by the orthogonality the partial sums N

∑ E(ωn )x

n=1

form a Cauchy sequence in H, so ∞

∑ E(ωn )x = E(ω)x,

n=1

and ω 󳨃→ E(ω)x is countably additive and therefore a complex measure on M. For the second claim, observe that E(ωn ) = 0 implies Ex,x (ωn ) = 0 for every x ∈ H. Since Ex,x is countably additive, it follows that Ex,x (ω) = 0. Since ‖E(ω)x‖2 = Ex,x (ω), E(ω) = 0. Definition 9.3. Let E be a resolution of the identity on M, and let f be a complex M-measurable function on Ω. There is a countable family (Dk )k of open discs forming a base for the topology of ℂ. Let V be the union of those Dk for which E(f −1 (Dk )) = 0.

9.1 Resolutions of the identity

� 153

By Proposition 9.2, E(f −1 (V )) = 0. Also, V is the largest open subset of ℂ with this property. The essential range of f is, by definition, the complement of V . It is the smallest closed subset of ℂ that contains f (z) for all z ∈ Ω except those that lie in some set ω ∈ M with E(ω) = 0. We say that f is essentially bounded if its essential range is bounded, hence compact. The largest value of |λ| as λ runs through the essential range of f is called the essential supremum ‖f ‖∞ of f . Let B be the algebra of all bounded complex M-measurable functions on Ω with the norm 󵄨 󵄨 ‖f ‖ = sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨, z∈Ω

and let N = {f ∈ B : ‖f ‖∞ = 0}, which by Proposition 9.2 is a closed ideal. Hence B/N is a Banach algebra, which is denoted by L∞ (E). The norm of a coset [f ] = f + N is ‖f ‖∞ , the spectrum σ([f ]) is the essential range of f , and the spectrum of an element g in a Banach algebra is the set of all complex numbers λ such that λe − g is not invertible. In the next step, we describe that a resolution of the identity induces an isometric isomorphism of the Banach algebra L∞ (E) onto a closed normal subalgebra 𝒜 of ℒ(H), the algebra of all bounded linear operators from H to H (a normal subalgebra is a commutative one that contains T ∗ for every T ∈ 𝒜). For this purpose, let {ω1 , . . . , ωn } be a partition of Ω with ωj ∈ M, and let s be a simple function, such that s = αj on ωj . Define Ψ(s) ∈ ℒ(H) by n

Ψ(s) = ∑ αj E(ωj ). j=1

Since each E(ωj ) is self-adjoint, Ψ(s)∗ = Ψ(s). If t is another simple function and α, β ∈ ℂ, then we have Ψ(s)Ψ(t) = Ψ(st)

and

Ψ(αs + βt) = αΨ(s) + βΨ(t).

For x, y ∈ H, we get n

n

j=1

j=1

(Ψ(s)x, y) = ∑ αj (E(ωj )x, y) = ∑ αj Ex,y (ωj ) = ∫ s dEx,y . Ω

In addition, we have Ψ(s)∗ Ψ(s) = Ψ(|s|2 ) and

󵄩󵄩 󵄩2 2 󵄩󵄩Ψ(s)x 󵄩󵄩󵄩 = ∫ |s| dEx,x . Ω

154 � 9 Spectral analysis By (9.1) this implies (9.4)

󵄩 󵄩󵄩 󵄩󵄩Ψ(s)x 󵄩󵄩󵄩 ≤ ‖s‖∞ ‖x‖, and if x ∈ im(E(ωj )), then Ψ(s)x = αj E(ωj )x = αj x,

since the projections E(ωj ) have mutually orthogonal ranges. If j is chosen so that |αj | = ‖s‖∞ , then by (9.4) it follows that 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩Ψ(s)󵄩󵄩󵄩 = sup 󵄩󵄩󵄩Ψ(s)x 󵄩󵄩󵄩 = ‖s‖∞ .

(9.5)

‖x‖≤1

Now suppose that f ∈ L∞ (E). There is a sequence of simple measurable functions sk that converges to f in the norm of L∞ (E). By (9.5) the corresponding operators Ψ(sk ) form a Cauchy sequence in ℒ(H), which is therefore norm-convergent to an operator that we call Ψ(f ). By (9.5) we get 󵄩󵄩 󵄩 󵄩󵄩Ψ(f )󵄩󵄩󵄩 = ‖f ‖∞ .

(9.6)

Thus Ψ is an isometric isomorphism of L∞ (E) into ℒ(H). Since L∞ (E) is complete, 𝒜 = Ψ(L∞ (E)) is closed in ℒ(H). In addition, we have (Ψ(f )x, y) = ∫ f dEx,y Ω

󵄩 󵄩2 and 󵄩󵄩󵄩Ψ(f )x 󵄩󵄩󵄩 = ∫ |f |2 dEx,x , Ω

which justifies the notation Ψ(f ) = ∫ f dE. Ω

9.2 Spectral decomposition of bounded normal operators We will now relate a resolution of the identity to a certain operator in ℒ(H). Definition 9.4. The spectrum σ(T) of an operator T ∈ ℒ(H) is the set of all λ ∈ ℂ such that λI −T has no inverse in ℒ(H). The complement ρ(T) = ℂ\σ(T) is called the resolvent set. If λ ∈ ρ(T), then the operator (λI − T)−1 ∈ ℒ(H) is called the resolvent of T at λ and is denoted by RT (λ). We have an operator-valued function RT : ρ(T) 󳨀→ ℒ(H). An operator T ∈ ℒ(H) is called normal if TT ∗ = T ∗ T.

9.2 Spectral decomposition of bounded normal operators

� 155

Proposition 9.5. Let T ∈ ℒ(H). Then the spectrum σ(T) of T is a compact subset of ℂ, and |λ| ≤ ‖T‖ for every λ ∈ σ(T). Proof. First, we show that ρ(T) is open. Let λ ∈ ρ(T). Then 󵄩−1 󵄩 α = 󵄩󵄩󵄩RT (λ)󵄩󵄩󵄩 > 0. Let μ ∈ ℂ with |μ| < α. We will show that (λ + μ)I − T has a bounded inverse. Then we prove that ρ(T) is open. We have (λ + μ)I − T = λI − T + μI

= (λI − T)[I + μ(λI − T)−1 ]

= (λI − T)(I + μRT (λ)). Formally, −1

(I + μRT (λ))

∞

k

= I + ∑ (−1)k (μRT (λ)) , k=1

but as |μ| < α, we have 󵄩󵄩 ∞ 󵄩󵄩 ∞ ∞ 󵄩󵄩 󵄩 󵄩󵄩 ∑ (−1)k (μRT (λ))k 󵄩󵄩󵄩 ≤ ∑ |μ|k 󵄩󵄩󵄩RT (λ)󵄩󵄩󵄩k = ∑ (|μ|/α)k < ∞, 󵄩󵄩 󵄩󵄩 󵄩 󵄩 󵄩󵄩k=1 󵄩󵄩 k=1 k=1 k k and therefore the partial sums of ∑∞ k=1 (−1) (μRT (λ)) form a Cauchy sequence in ℒ(H). Since ℒ(H) is complete, we obtain that ∞

k

∑ (−1)k (μRT (λ)) ∈ ℒ(H),

k=1

and ρ(T) is open. If η ∈ ℂ with |η| > ‖T‖, then I − T/η has a bounded inverse, since ∞

(I − T/η)−1 = I + ∑ (T/η)k . k=1

This implies that ηI − T has a bounded inverse. Hence, if λ ∈ σ(T), then |λ| ≤ ‖T‖, and σ(T) is a bounded set. The resolvent has the following properties: Lemma 9.6. If λ, μ ∈ ρ(T), then RT (λ) − RT (μ) = (μ − λ)RT (λ)RT (μ).

(9.7)

156 � 9 Spectral analysis If λ ∈ ρ(T) and |λ − μ| < ‖RT (λ)‖−1 , then ∞

k+1

RT (μ) = ∑ (λ − μ)k [RT (λ)] k=0

,

(9.8)

and therefore we say that RT is a holomorphic operator-valued function. Proof. Relation (9.7) follows from RT (λ) = RT (λ)(μI − T)RT (μ) = RT (λ)[(λI − T) + (μ − λ)I]RT (μ) = RT (μ) + (μ − λ)RT (λ)RT (μ).

Equality (9.8) follows immediately from the proof of Proposition 9.5. The spectral theorem indicates that every bounded normal operator T on a Hilbert space induces a resolution E of the identity on the Borel subsets of its spectrum σ(T) and that T can be reconstructed from E by an integral of the type discussed before. Using Banach algebra techniques such as the Gelfand transform (see [85]), we obtain the spectral decomposition for a single normal operator. Proposition 9.7. If T ∈ ℒ(H) and T is normal, then there exists a uniquely determined resolution of the identity E on the Borel subsets of the spectrum σ(T) that satisfies T = ∫ λ dE(λ) and (Tx, y) = ∫ λ dEx,y (λ).

(9.9)

σ(T)

σ(T)

Furthermore, every projection E(ω) commutes with every S ∈ ℒ(H) that commutes with T. We will refer to this E as the spectral decomposition of T. – –

We list a few consequences of the spectral decomposition. If ω ⊆ σ(T) is a nonempty open set, then E(ω) ≠ 0. If f is a bounded Borel function on σ(T), then it is customary to denote the operator Ψ(f ) = ∫ f dE σ(T)

by f (T). The mapping f 󳨃→ f (T) establishes a homomorphism of the algebra of all bounded Borel functions on σ(T) into ℒ(H), which carries the function 1 to I, the identity function on σ(T) to T, and satisfies f (T) = f (T)∗

and

󵄩󵄩 󵄩 󵄨 󵄨 󵄩󵄩f (T)󵄩󵄩󵄩 ≤ sup{󵄨󵄨󵄨f (λ)󵄨󵄨󵄨 : λ ∈ σ(T)}.

The procedure explained above is also called symbolic calculus.

9.2 Spectral decomposition of bounded normal operators

� 157

If f ∈ 𝒞 (σ(T)), then f 󳨃→ f (T) is an isomorphism on 𝒞 (σ(T)) satisfying 󵄩2 󵄩󵄩 2 󵄩󵄩f (T)x 󵄩󵄩󵄩 = ∫ |f | dEx,x . σ(T)

The eigenvalues of a normal operator can be characterized in terms of the spectral decomposition. For this purpose, we mention the following applications of the symbolic calculus. Proposition 9.8. Let T ∈ ℒ(H) be a normal operator, and let E be its spectral decomposition. If f ∈ 𝒞 (σ(T)) and ω0 = f −1 (0), then ker(f (T)) = im(E(ω0 )). Proof. We set h(λ) = 1 on ω0 and h(λ) = 0 on ω̃ = σ(T) \ ω0 . Then fh = 0, and by the symbolic calculus, f (T)h(T) = 0. Since h(T) = E(ω0 ), it follows that im(E(ω0 )) ⊆ ker(f (T)). For the opposite inclusion, we define for n ∈ ℕ the set 󵄨 󵄨 ωn = {λ ∈ σ(T) : 1/n ≤ 󵄨󵄨󵄨f (λ)󵄨󵄨󵄨 < 1/(n − 1)}. Then ω̃ is the union of the disjoint Borel sets ωn . We define fn (λ) = 1/f (λ) on ωn and fn (λ) = 0 on σ(T) \ ωn . Then each fn is a bounded Borel function on σ(T), and fn (T)f (T) = E(ωn ),

n ∈ ℕ.

If f (T)x = 0, then it follows that E(ωn )x = 0. Since the mapping ω 󳨃→ E(ω)x is countably ̃ = 0. We also have that additive (Proposition 9.2), we obtain E(ω)x E(ω)̃ + E(ω0 ) = I. Hence E(ω0 )x = x, and therefore ker(f (T)) ⊆ im(E(ω0 )). Proposition 9.9. Let T ∈ ℒ(H) be a normal operator, and let E be its spectral decomposition. Let λ0 ∈ σ(T) and E0 = E({λ0 }). Then (a) ker(T − λ0 I) = im(E0 ); (b) λ0 is an eigenvalue of T if and only if E0 ≠ 0; (c) Every isolated point of σ(T) is an eigenvalue of T;

158 � 9 Spectral analysis (d) If σ(T) = {λ1 , λ2 , . . . } is a countable set, then every x ∈ H has a unique expansion of the form ∞

x = ∑ xj , j=1

where Txj = λj xj , and xj ⊥ xk for j ≠ k; (e) If σ(T) has no limit point except possibly 0 and if dim ker(T − λI) < ∞ for λ ≠ 0, then T is compact (compare with Section 2.1). Proof. (a) is an immediate consequence of Proposition 9.8 with f (λ) = λ − λ0 . (b) follows from (a). (c) If λ0 is an isolated point of σ(T), then {λ0 } is a nonempty open subset of σ(T), and by the properties of the spectral decomposition listed above we get E0 ≠ 0, and therefore (c) follows from (b). (d) Let Ej = E({λj }) for j ∈ ℕ. The projections Ej have pairwise orthogonal ranges, and the mapping ω 󳨃→ E(ω)x is countably additive (Proposition 9.2). Hence for each x ∈ H, we have ∞

∑ Ej x = E(σ(T))x = x, j=1

and the series converges in the norm of H. Now set xj = Ej x and observe that the uniqueness follows from the orthogonality of the vectors xj and that Txj = λj xj follows from (a). (e) Let {λj } be an enumeration of the nonzero points of σ(T) such that |λ1 | ≥ |λ2 | ≥ ⋅ ⋅ ⋅. Define fn (λ) = λ if λ = λj and j ≤ n, and put fn (λ) = 0 at the other points of σ(T). We set again Ej = E({λj }) and obtain fn (T) = λ1 E1 + ⋅ ⋅ ⋅ + λn En . Since dim im(Ej ) = dim ker(T − λj I) < ∞, each fn (T) is a compact operator. We have |λ − fn (λ)| ≤ |λn | for all λ ∈ σ(T) and, by the symbolic calculus, 󵄩󵄩 󵄩 󵄩󵄩T − fn (T)󵄩󵄩󵄩 ≤ |λn | → 0

as n → ∞.

By Proposition 2.2, T is compact. The symbolic calculus is a powerful tool in operator theory. Finally, we mention important applications to positive operators: Definition 9.10. An operator T ∈ ℒ(H) is called positive if (Tx, x) ≥ 0 for every x ∈ H. We write T ≥ 0.

9.2 Spectral decomposition of bounded normal operators

� 159

Proposition 9.11. (a) T ∈ ℒ(H) is positive if and only if T = T ∗ and σ(T) ⊂ [0, ∞). (b) Every positive T ∈ ℒ(H) has a unique positive square root S ∈ ℒ(H), i. e., S 2 = T. (c) If T ∈ ℒ(H), then T ∗ T is positive, and the positive square root P of T ∗ T is the only positive operator in ℒ(H) satisfying ‖Px‖ = ‖Tx‖ for every x ∈ H. (d) If T ∈ ℒ(H) is normal, then T has a polar decomposition T = UP, where U is unitary, and P is positive. Proof. (a) (Tx, x) and (x, Tx) are complex conjugates of each other. If T is positive, then (Tx, x) is real, so that (x, T ∗ x) = (Tx, x) = (x, Tx) for every x ∈ H. Hence T = T ∗ (by the proof of Theorem 2.7). Let λ = α + iβ ∈ σ(T) and put Tλ = T − λI. Then ‖Tλ x‖2 = ‖Tx − αx‖2 + β2 ‖x‖2 , so that ‖Tλ x‖ ≥ |β| ‖x‖. If β ≠ 0, then it follows that Tλ is invertible, which means that λ ∉ σ(T). So we get that σ(T) lies in the real axis. If λ > 0, then we obtain 󵄩 󵄩 λ‖x‖2 = (λx, x) ≤ ((T + λI)x, x) ≤ 󵄩󵄩󵄩(T + λI)x 󵄩󵄩󵄩 ‖x‖, so that ‖(T + λI)x‖ ≥ λ‖x‖, which implies that T + λI is invertible in ℒ(H) and −λ ∉ σ(T), and hence σ(T) ⊂ [0, ∞). Now assume that T = T ∗ and σ(T) ⊂ [0, ∞). Let E be the spectral decomposition of T. We have (Tx, x) = ∫ λ dEx,x (λ). σ(T)

Since Ex,x is a positive measure and λ ≥ 0 on σ(T), we obtain that T is positive. (b) By (a), Proposition 9.5, and the symbolic calculus, σ(T) is a compact subset of ℝ+ , and there exists a uniquely determined spectral decomposition E such that T = ∫ λ dE(λ). σ(T)

Define S = ∫ λ1/2 dE(λ). σ(T)

160 � 9 Spectral analysis Then S is a positive self-adjoint operator with S 2 = T. In addition, there is a sequence of polynomials pn such that pn (λ) → λ1/2 uniformly on σ(T) (see Corollary 12.5) and lim 󵄩p (T) n→∞󵄩 n 󵄩󵄩

󵄩 − S 󵄩󵄩󵄩 = 0.

Let S̃ be an arbitrary positive self-adjoint operator with S̃ 2 = T. Since T S̃ = S̃ 3 and ̃ = S̃ 3 , the operator S̃ commutes with T, and so with polynomials of T. Hence also with ST ̃ Using that SS ̃ = S S̃ and S 2 = S̃ 2 , we S = limn→∞ pn (T). Let x ∈ H and put y = (S − S)x. obtain

̃ y) = ((S + S)(S ̃ − S)x, ̃ y) = ((S 2 − S̃ 2 )x, y) = 0. (Sy, y) + (Sy, ̃ y) = 0. Hence Sy = Sy ̃ = 0, because (S⋅, ⋅) is a Since S and S̃ are positive, (Sy, y) = (Sy, positive semidefinite sesquilinear form, so we can apply the Cauchy–Schwarz inequality 󵄨󵄨 󵄨2 󵄨󵄨(Sy, z)󵄨󵄨󵄨 ≤ (Sy, y)(Sz, z) for all z ∈ H. Now we get 󵄩󵄩 ̃ 󵄩󵄩󵄩2 = ((S − S)̃ 2 x, x) = ((S − S)y, ̃ x) = 0, 󵄩󵄩(S − S)x 󵄩 ̃ and S = S.̃ which yields Sx = Sx (c) Note first that (T ∗ Tx, x) = (Tx, Tx) = ‖Tx‖2 ≥ 0

for x ∈ H,

so that T ∗ T is positive. If P ∈ ℒ(H) and P∗ = P, then (P2 x, x) = (Px, Px) = ‖Px‖2 . Then by the proof of Theorem 2.7 it follows that ‖Px‖ = ‖Tx‖ for every x ∈ H if and only if P2 = T ∗ T. (d) Put p(λ) = |λ| and u(λ) = λ/|λ| for λ ≠ 0 and u(0) = 1. Then p and u are bounded Borel functions on σ(T). Put P = p(T) and U = u(T). As p ≥ 0, we get from (a) that P ≥ 0. Since uu = 1, we get UU ∗ = U ∗ U = I, and since λ = u(λ)p(λ), the relation T = UP follows from the symbolic calculus. Remark 9.12. We apply the last results to the 𝜕-Neumann operator N to prove the following: Suppose that ◻ : dom(◻) 󳨀→ L2(0,1) (Ω) is bijective and has a bounded inverse N. Then we have the basic estimate

9.3 Spectral decomposition of unbounded self-adjoint operators

󵄩 ∗ 󵄩2 ‖u‖2 ≤ C (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ),

� 161

u ∈ dom(◻). (4.38)

The operator N is self-adjoint and bounded and therefore, by Proposition 9.11, has a bounded self-adjoint root N 1/2 , which is again injective. By Lemma 4.28, N 1/2 has a self-adjoint inverse, which we denote by N −1/2 . Let u ∈ dom(◻). Then there exists w ∈ L2(0,1) (Ω) such that Nw = u. Hence we have N 1/2 v = u, where v = N 1/2 w, and N −1/2 v = w = N −1/2 N −1/2 u is well defined. Now we get 󵄩 󵄩2 ‖u‖2 = 󵄩󵄩󵄩N 1/2 v󵄩󵄩󵄩 ≤ C‖v‖2 = C (N −1/2 u, N −1/2 u)

= C (N −1/2 N −1/2 u, u) = C (N −1/2 N −1/2 Nw, Nw)

= C (w, Nw) = C (◻u, u) 󵄩 ∗ 󵄩2 = C (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ), which is the basic estimate (4.38).

9.3 Spectral decomposition of unbounded self-adjoint operators Let Ω be a subset of ℂ, let M be a σ-algebra in Ω, and let H be a Hilbert space. Let E : M 󳨀→ ℒ(H) be a resolution of the identity. The symbolic calculus associates with every f ∈ L∞ (E) an operator Ψ(f ) ∈ ℒ(H) by the formula (Ψ(f )x, y) = ∫ f dEx,y ,

x, y ∈ H.

Ω

Now we will extend this for unbounded measurable functions f . Lemma 9.13. Let f : Ω 󳨀→ ℂ be a measurable function. Put 2

𝒟f = {x ∈ H : ∫ |f | dEx,x < ∞}. Ω

Then 𝒟f is a dense subspace of H. If x, y ∈ H, then 1/2

∫ |f | d|Ex,y | ≤ ‖y‖ [∫ |f |2 dEx,x ] . Ω

(9.10)

Ω

If f is bounded and u = Ψ(f )v for v ∈ H, then dEx,u = f dEx,v ,

x ∈ H.

(9.11)

162 � 9 Spectral analysis Proof. Let z = x + y and ω ∈ M. Then 󵄩2 󵄩2 󵄩 󵄩 󵄩2 󵄩 󵄩 󵄩 󵄩2 󵄩󵄩 󵄩󵄩E(ω)z󵄩󵄩󵄩 ≤ (󵄩󵄩󵄩E(ω)x 󵄩󵄩󵄩 + 󵄩󵄩󵄩E(ω)y󵄩󵄩󵄩) ≤ 2(󵄩󵄩󵄩E(ω)x 󵄩󵄩󵄩 + 󵄩󵄩󵄩E(ω)y󵄩󵄩󵄩 ). Recall that Ex,x (ω) = (E(ω)x, x) = (E(ω)2 x, x) = ‖E(ω)x‖2 , so we get from the above Ez,z (ω) ≤ 2(Ex,x (ω) + Ey,y (ω)), which implies that 𝒟f is closed under addition. It is clear that 𝒟f is also closed under scalar multiplication. Therefore 𝒟f is a subspace of H. For n ∈ ℕ, let ωn be the subset of Ω where |f | < n. If x ∈ im(E(ωn )), then E(ω)x = E(ω)E(ωn )x = E(ω ∩ ωn )x,

ω ∈ M.

Hence Ex,x (ω) = Ex,x (ω ∩ ωn ), and therefore ∫ |f |2 dEx,x = ∫ |f |2 dEx,x ≤ n2 ‖x‖2 < ∞. Ω

ωn

Thus im(E(ωn )) ⊂ 𝒟f . Since Ω = ⋃∞ n=1 ωn , the countable additivity of ω 󳨃→ E(ω)y implies that y = limn→∞ E(ωn )y for every y ∈ H. Hence y lies in the closure of 𝒟f , and 𝒟f is dense in H. If x, y ∈ H and f is bounded and measurable, the Radon–Nikodym theorem (see, for instance, [83]) implies that there is a measurable function g on Ω such that |g| = 1 on Ω and gf dEx,y = |f | d|Ex,y |. Hence 󵄩 󵄩 ∫ |f | d|Ex,y | = (Ψ(gf )x, y) ≤ 󵄩󵄩󵄩Ψ(gf )x 󵄩󵄩󵄩 ‖y‖.

(9.12)

Ω

As in Section 9.1, we get 󵄩󵄩 󵄩2 2 2 󵄩󵄩Ψ(gf )x 󵄩󵄩󵄩 = ∫ |gf | dEx,x = ∫ |f | dEx,x , Ω

Ω

which implies (9.10) for a bounded function f . The general case is done as in the first statement of this proposition.

9.3 Spectral decomposition of unbounded self-adjoint operators

� 163

To show (9.11), for an arbitrary bounded measurable function h, we have ∫ h dEx,u = (Ψ(h)x, u) = (Ψ(h)x, Ψ(f )v) Ω

= (Ψ(f )Ψ(h)x, v) = (Ψ(f h)x, v) = ∫ hf dEx,v . Ω

In the next step, we carry over the results of Section 9.1 (symbolic calculus) for unbounded measurable functions. Proposition 9.14. Let E be a resolution of identity on Ω. (a) To every measurable f : Ω 󳨀→ ℂ, there corresponds a densely defined closed operator Ψ(f ) on H with domain dom(Ψ(f )) = 𝒟f , which is characterized by (Ψ(f )x, y) = ∫ f dEx,y ,

x ∈ 𝒟f ,

y ∈ H,

(9.13)

Ω

and satisfies 󵄩󵄩 󵄩2 2 󵄩󵄩Ψ(f )x 󵄩󵄩󵄩 = ∫ |f | dEx,x ,

x ∈ 𝒟f .

(9.14)

Ω

(b) If f and g are measurable, then Ψ(f )Ψ(g) ⊂ Ψ(fg), which means that dom(Ψ(f )Ψ(g)) ⊂ dom(Ψ(fg)) and Ψ(f )Ψ(g) = Ψ(fg) on dom(Ψ(f )Ψ(g)), and dom(Ψ(f )Ψ(g)) = 𝒟g ∩ 𝒟fg . Hence Ψ(f )Ψ(g) = Ψ(fg) if and only if 𝒟fg ⊆ 𝒟g . (c) For every measurable f : Ω 󳨀→ ℂ, Ψ(f )∗ = Ψ(f )

and Ψ(f )Ψ(f )∗ = Ψ(|f |2 ) = Ψ(f )∗ Ψ(f ).

Proof. Fix x ∈ 𝒟f . Then the conjugate-linear functional y 󳨃→ ∫Ω f dEx,y is bounded on H (Lemma 9.13). Hence there is a unique element Ψ(f )x ∈ H satisfying (9.13) and 󵄩󵄩 󵄩2 2 󵄩󵄩Ψ(f )x 󵄩󵄩󵄩 ≤ ∫ |f | dEx,x ,

x ∈ 𝒟f .

Ω

The linearity of Ψ(f ) on 𝒟f follows from (9.13) and the fact that Ex,y is linear in x.

(9.15)

164 � 9 Spectral analysis Now we associate with each f its truncations fn = fϕn , where ϕn (p) = 1 if |f (p)| ≤ n and ϕn (p) = 0 if |f (p)| > n. Then 𝒟f −fn = 𝒟f , since each fn is bounded, and therefore (9.15) shows, using the dominated convergence theorem, that 󵄩2 󵄩󵄩 2 󵄩󵄩Ψ(f )x − Ψ(fn )x 󵄩󵄩󵄩 ≤ ∫ |f − fn | dEx,x → 0

as n → ∞

(9.16)

Ω

for every x ∈ 𝒟f . Since fn is bounded, (9.14) holds for fn . Hence (9.16) implies (9.14) for f . This proves (a), except for the statement that Ψ(f ) is closed. This will follow from (c) (to be proved) and Lemma 4.5 with f instead of f . (b) First, assume that f is bounded. Then 𝒟fg ⊂ 𝒟g . If v ∈ H and u = Ψ(f )v, then we get from Section 9.1 and (9.11) that (Ψ(f )Ψ(g)x, v) = (Ψ(g)x, Ψ(f )v) = (Ψ(g)x, u) = ∫ g dEx,u = ∫ fg dEx,v Ω

= (Ψ(fg)x, v).

Ω

So we have shown that Ψ(f )Ψ(g)x = Ψ(fg)x,

x ∈ 𝒟g ,

f ∈ L∞ .

The last line implies that for y = Ψ(g)x, ∫ |f |2 dEy,y = ∫ |fg|2 dEx,x , Ω

x ∈ 𝒟g ,

f ∈ L∞ .

(9.17)

Ω

Using truncation, we see that (9.17) also holds for arbitrary f (possibly unbounded). Since dom(Ψ(f )Ψ(g)) consists of all x ∈ 𝒟g such that y = Ψ(g)x ∈ 𝒟f and since (9.17) shows that y ∈ 𝒟f if and only if x ∈ 𝒟fg , we see that dom(Ψ(f )Ψ(g)) = 𝒟g ∩ 𝒟fg . If x ∈ 𝒟g ∩ 𝒟fg and y = Ψ(g)x and if the truncations fn are defined as above, we obtain fn → f in L2 (Ex,x ) and fn g → fg in L2 (Ey,y ) and finally Ψ(f )Ψ(g)x = Ψ(f )y = lim Ψ(fn )y = lim Ψ(fn g)x = Ψ(fg)x. n→∞

n→∞

This proves (b). (c) Suppose that x ∈ 𝒟f and y ∈ 𝒟f = 𝒟f . It follows from (9.16) and Section 9.1 that (Ψ(f )x, y) = lim (Ψ(fn )x, y) = lim (x, Ψ(f n )y) = (x, Ψ(f )y). n→∞

n→∞

9.3 Spectral decomposition of unbounded self-adjoint operators

� 165

Hence y ∈ dom(Ψ(f )∗ ), and dom(Ψ(f )) ⊆ dom(Ψ(f )∗ ). If we can show that each u ∈ dom(Ψ(f )∗ ) lies in 𝒟f , then we obtain Ψ(f )∗ = Ψ(f ). Fix u for this purpose and put v = Ψ(f )∗ u. Since fn = fϕn , the multiplication theorem yields Ψ(fn ) = Ψ(f )Ψ(ϕn ). Since Ψ(ϕn ) is self-adjoint and bounded, by Lemma 4.3 and Section 9.1 we have that ∗

Ψ(ϕn )Ψ(f )∗ = [Ψ(f )Ψ(ϕn )] = Ψ(fn )∗ = Ψ(f n ). Hence Ψ(ϕn )v = Ψ(f n )u,

n ∈ ℕ.

Since |ϕn | ≤ 1, we now have ∫ |fn |2 dEu,u = ∫ |ϕn |2 dEv,v ≤ Ev,v (Ω), Ω

n ∈ ℕ.

Ω

Hence u ∈ 𝒟f . Finally, since 𝒟f f ⊂ 𝒟f , another application of the multiplication theorem gives the last statement of (c). Definition 9.15. The resolvent set of a linear operator T : dom(T) 󳨀→ H is the set of all λ ∈ ℂ such that λI − T is an injective mapping of dom(T) onto H whose inverse belongs to ℒ(H). The spectrum σ(T) of T is the complement of the resolvent set of T. First, we collect some information about the spectrum of an unbounded operator. Lemma 9.16. If the spectrum σ(T) of an operator T does not coincide with the whole complex plane ℂ, then T must be a closed operator. The spectrum of a linear operator is always closed. Moreover, if ζ ∉ σ(T) and c := ‖RT (ζ )‖ = ‖(ζI − T)−1 ‖, then the spectrum σ(T) does not intersect the ball {w ∈ ℂ : |ζ − w| < c−1 }. The resolvent operator RT is a holomorphic operator-valued function. Proof. For ζ ∉ σ(T), let S = (ζI − T)−1 , which is a bounded operator. Let xn ∈ dom(T) with x = limn→∞ xn = x and limn→∞ Txn = y and set un = (ζI − T)xn . Then lim u n→∞ n

= lim (ζxn − Txn ) = ζx − y, n→∞

and therefore S(ζx − y) = lim Sun = lim xn = x. n→∞

n→∞

This implies x ∈ dom(T) and (ζI − T)x = ζx − y, and so Tx = y. Hence T is closed.

166 � 9 Spectral analysis The remainder of the proof is similar to the case where T is bounded; see Lemma 9.6. In the next proposition, we refer to the notion of the essential range of a function with respect to a given resolution of the identity (Definition 9.3). Proposition 9.17. Let E be a resolution of the identity on Ω, and let f : Ω 󳨀→ ℂ be a measurable function. For α ∈ ℂ, put ωα = {p ∈ Ω : f (p) = α}. (a) If α is in the essential range of f and E(ωα ) ≠ 0, then αI − Ψ(f ) is not injective. (b) If α is in the essential range of f but E(ωα ) = 0, then αI − Ψ(f ) is an injective mapping of 𝒟f onto a proper dense subspace of H, and there exist vectors xn ∈ H with ‖xn ‖ = 1 such that lim [αxn − Ψ(f )xn ] = 0.

n→∞

(c) σ(Ψ(f )) is the essential range of f . We say that α lies in the point spectrum of Ψ(f ) in case (a) and in the continuous spectrum of Ψ(f ) in case (b). Proof. Without loss of generality, we can assume that α = 0. (a) If E(ω0 ) ≠ 0, then there exists x0 ∈ im(E(ω0 )) with ‖x0 ‖ = 1. Let ϕ0 be the characteristic function of ω0 . Then fϕ0 = 0, and Ψ(f )Ψ(ϕ0 ) = 0. Since Ψ(ϕ0 ) = E(ω0 ), it follows that Ψ(f )x0 = Ψ(f )E(ω0 )x0 = Ψ(f )Ψ(ϕ0 )x0 = 0. (b) Now we have E(ω0 ) = 0 but E(ωn ) ≠ 0 for n ∈ ℕ, where 󵄨 󵄨 ωn = {p ∈ Ω : 󵄨󵄨󵄨f (p)󵄨󵄨󵄨 < 1/n}. Let xn ∈ im(E(ωn )) with ‖xn ‖ = 1, and let ϕn be the characteristic functions of ωn . As in (a), we obtain 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩Ψ(f )xn 󵄩󵄩󵄩 = 󵄩󵄩󵄩Ψ(fϕn )xn 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩Ψ(fϕn )󵄩󵄩󵄩 = ‖fϕn ‖∞ ≤ 1/n. Thus Ψ(f )xn → 0 although ‖xn ‖ = 1. If Ψ(f )x = 0 for some x ∈ 𝒟f , then 󵄩 󵄩2 ∫ |f |2 dEx,x = 󵄩󵄩󵄩Ψ(f )x 󵄩󵄩󵄩 = 0.

Ω

9.3 Spectral decomposition of unbounded self-adjoint operators

� 167

Since |f | > 0 almost everywhere (Ex,x ), we must have Ex,x (Ω) = 0. However, Ex,x (Ω) = ‖x‖2 . Hence Ψ(f ) is injective. Similarly, Ψ(f )∗ = Ψ(f ) is injective. If y ⊥ im(Ψ(f )), then x 󳨃→ (Ψ(f )x, y) = 0 is continuous in 𝒟f ; hence y ∈ dom(Ψ(f )∗ ), and (x, Ψ(f )y) = (Ψ(f )x, y) = 0,

x ∈ 𝒟f .

Hence Ψ(f )y = 0 and y = 0. Therefore im(Ψ(f )) is dense in H. Since Ψ(f ) is closed, so is Ψ(f )−1 . If im(Ψ(f )) = H, then the closed graph theorem would imply that Ψ(f )−1 ∈ ℒ(H). This is impossible in view of the sequence (xn )n constructed above. Hence (b) is proved. (c) It follows from (a) and (b) that the essential range of f is a subset of σ(Ψ(f )). Now assume that 0 is not in the essential range of f . Then g = 1/f ∈ L∞ (E), and fg = 1, hence Ψ(f )Ψ(g) = Ψ(1) = I, which proves that im(Ψ(f )) = H. Since |f | > 0, we have that Ψ(f ) is injective, as in the proof of (b). By the closed graph theorem, Ψ(f )−1 ∈ ℒ(H). Therefore 0 ∉ σ(Ψ(f )), and (c) is proved. In the following proposition, we describe the change of measure principle. Proposition 9.18. Let M and M′ be σ-algebras in the sets Ω, Ω′ ⊆ ℂ, let E : M 󳨀→ ℒ(H) be a resolution of identity, and suppose that Φ : Ω 󳨀→ Ω′ has the property that Φ−1 (ω′ ) ∈ M for every ω′ ∈ M′ . If E ′ (ω′ ) = E(Φ−1 (ω′ )), then E ′ : M′ 󳨀→ ℒ(H) is a resolution of the identity, and ′ = ∫(f ∘ Φ) dEx,y ∫ f dEx,y Ω′

(9.18)

Ω

for every M′ -measurable f : Ω′ 󳨀→ ℂ for which either of these integrals exists. Proof. A straightforward verification gives that E ′ is again a resolution of the identity. For characteristic functions, (9.18) is just the definition of E ′ . So (9.18) follows for simple functions and also in the general case. To derive the general spectral theorem for unbounded self-adjoint operators, we will use the Cayley transform. The mapping t 󳨃→

t−i t+i

sets up a bijection between the real line and the unit circle minus the point 1. The symbolic calculus developed in Section 9.2 therefore shows that every self-adjoint operator T ∈ ℒ(H) gives rise to a unitary operator U = (T − iI)(T + iI)−1

168 � 9 Spectral analysis and that every unitary U whose spectrum does not contain the point 1 is obtained in this way. This relation will now be extended to unbounded symmetric operators. If T is a symmetric operator, then we have ‖Tx + ix‖2 = (Tx + ix, Tx + ix) = ‖x‖2 + ‖Tx‖2 = ‖Tx − ix‖2 for x ∈ dom(T). This implies that (T + iI) is injective and that there is an isometry U with dom(U) = im(T + iI) and im(U) = im(T − iI) defined by U(Tx + ix) = T − ix,

x ∈ dom(T).

Since (T + iI)−1 maps dom(U) onto dom(T), we can write U = (T − iI)(T + iI)−1 . This operator U is called the Cayley transform of T. Lemma 9.19. Let U be an isometry, i. e., ‖Ux‖ = ‖x‖ for all x ∈ dom(U). (a) For x, y ∈ dom(U), we have (Ux, Uy) = (x, y). (b) If im(I − U) is dense in H, then I − U is injective. (c) If any of the three spaces dom(U), im(U), and 𝒢 (U) is closed, so are the other two. Proof. (a) follows from the polarization identity: (x, y) =

1 (‖x + y‖2 − ‖x − y‖2 + i‖x + iy‖2 − i‖x − iy‖2 ). 4

(9.19)

(b) Let x ∈ dom(U) and (I − U)x = 0. Then x = Ux, and (x, (I − U)y) = (x, y) − (x, Uy) = (Ux, Uy) − (x, Uy) = 0 for every y ∈ dom(U). This implies x ⊥ im(I − U), so that x = 0 if im(I − U) is dense in H. (c) follows from ‖Ux − Uy‖ = ‖x − y‖ =

1 󵄩󵄩 󵄩 󵄩(x, Ux) − (y, Uy)󵄩󵄩󵄩, √2 󵄩

x, y ∈ dom(U),

where in the last term, (x, Ux), (y, Uy) ∈ 𝒢 (U) are elements of the graph of U, and 󵄩󵄩 󵄩 2 2 1/2 󵄩󵄩(x, Ux) − (y, Uy)󵄩󵄩󵄩 = (‖x − y‖ + ‖Ux − Uy‖ ) . Proposition 9.20. Let T be a symmetric operator on H (not necessarily densely defined), and let U be its Cayley transform. The following statements are true:

9.3 Spectral decomposition of unbounded self-adjoint operators

� 169

(a) U is closed if and only if T is closed. (b) im(I − U) = dom(T), I − U is injective, and T can be reconstructed from U by T = i(I + U)(I − U)−1 . The Cayley transforms of distinct symmetric operators are distinct. (c) U is unitary if and only if T is self-adjoint. Conversely, if V is an operator in H that is an isometry and if I − V is injective, then V is the Cayley transform of a symmetric operator in H. Proof. (a) The identity ‖Tx + ix‖2 = ‖x‖2 + ‖Tx‖2 implies that (T + iI)x ↔ (x, Tx) is an isometric one-to-one correspondence between im(T + iI) and the graph 𝒢 (T) of T. Hence T is closed if and only if im(T + iI) is closed. By Lemma 9.19, U is closed if and only if dom(U) is closed. However, by the definition of the Cayley transform, dom(U) = im(T + iI), which proves (a). (b) The one-to-one correspondence x ↔ z between dom(T) and dom(U) = im(T +iI) given by z = Tx + ix,

Uz = Tx − ix

can be written in the form (I − U)z = 2ix,

(I + U)z = 2Tx.

Hence (I − U) is injective, and im(I − U) = dom(T), and therefore (I − U)−1 maps dom(T) onto dom(U), and 2Tx = (I + U)z = (I + U)(I − U)−1 (2ix),

x ∈ dom(T).

This proves (b). (c) Assume that T is self-adjoint. Then by Lemma 4.29 im(I + T ∗ T) = im(I + T 2 ) = H.

(9.20)

We have (T + iI)(T − iI) = T 2 + I = (T − iI)(T + iI), where all operators have domain dom(T 2 ). Hence (9.20) implies that dom(U) = im(T + iI) = H and

(9.21)

170 � 9 Spectral analysis im(U) = im(T − iI) = H.

(9.22)

Now (U ∗ Ux, x) = (Ux, Ux) = (x, x) for every x ∈ H, which implies U ∗ U = I, and U is unitary. Now assume that U is unitary. Then ⊥

(im(I − U)) = ker(I − U)∗ = {0}, and dom(T) = im(I − U) is dense in H. Thus T ∗ is defined, and T ⊂ T ∗ . Fix y ∈ dom(T ∗ ). Since im(T + iI) = dom(U) = H, there exists y0 ∈ dom(T) such that (T ∗ + iI)y = (T + iI)y0 = (T ∗ + iI)y0 . Set y1 = y − y0 . Then y1 ∈ dom(T ∗ ), and for every x ∈ dom(T), we have ((T − iI)x, y1 ) = (x, (T ∗ + iI)y1 ) = (x, 0) = 0. Thus y1 ⊥ im(T − iI) = im(U) = H, so y1 = 0 and y = y0 ∈ dom(T). Hence dom(T) = dom(T ∗ ), and (c) is proved. Finally, let V be as in the statement of the converse. Then there is a one-to-one correspondence z ↔ x between dom(V ) and im(I − V ) given by x = z − Vz. Define S on dom(S) = im(I − V ) by Sx = i(z + Vz) if x = z − Vz.

(9.23)

If x, y ∈ dom(S), then x = z − Vz and y = u − Vu for some z, u ∈ dom(V ). Since V is an isometry, it follows from Lemma 9.19 that (Sx, y) = i(z + Vz, u − Vu) = i(Vz, u) − i(z, Vu) = (z − Vz, iu + iVu) = (x, Sy).

Hence S is symmetric. For z ∈ dom(V ), (9.23) can be written in the form 2iVz = Sx − ix,

2iz = Sx + ix,

and hence if x ∈ dom(S), then V (Sx + ix) = Sx − ix, and dom(V ) = im(S + iI). Therefore V is the Cayley transform of S.

9.3 Spectral decomposition of unbounded self-adjoint operators �

171

At this point, we use the methods developed above to prove a key result about the spectrum of an unbounded self-adjoint operator. It transfers properties of unbounded self-adjoint operators to the bounded resolvent operators. Let T ∈ ℒ(H) be a self-adjoint operator. If Iλ ≠ 0, then 󵄩 󵄨 󵄩 󵄨 |Iλ| ‖u‖2 = 󵄨󵄨󵄨I((T − λI)u, u)󵄨󵄨󵄨 ≤ 󵄩󵄩󵄩(T − λI)u󵄩󵄩󵄩 ‖u‖ for all u ∈ H, where we used that (Tu, u) = (u, Tu) is real. This implies that T − λI is injective and has a closed range. As ⊥

̄ (im(T − λI)) = ker(T − λI) and this kernel reduces to {0}, we obtain that T − λI is bijective. This follows also from Proposition 4.35, once we have observed that 󵄨󵄨 󵄨 2 󵄨󵄨((T − λI)u, u)󵄨󵄨󵄨 ≥ |Iλ| ‖u‖ . Proposition 9.21. Let T be a bounded self-adjoint operator. The spectrum of T is contained in the interval [m, M], where m = inf

u=0 ̸

(Tu, u) (u, u)

and

M = sup u=0 ̸

(Tu, u) , (u, u)

and m and M belong to the spectrum of T. Proof. We have already shown that the spectrum is real (Proposition 9.23). Let λ be a real number with λ > M. We consider the scalar product ⟨u, v⟩ := M(u, v) − (Tu, v). Notice that ⟨u, u⟩ ≥ 0. By the Cauchy–Schwarz inequality we have 󵄨󵄨 󵄨 1/2 1/2 󵄨󵄨(Mu − Tu, v)󵄨󵄨󵄨 ≤ (Mu − Tu, u) (Mv − Tv, v) ; in particular, setting v = (M − T)u, we have 1/2 󵄩󵄩 󵄩2 1/2 2 󵄩󵄩(M − T)u󵄩󵄩󵄩 ≤ (Mu − Tu, u) ((M − T) u, (M − T)u) 󵄩 󵄩2 1/2 ≤ (Mu − Tu, u)1/2 (‖M − T‖󵄩󵄩󵄩(M − T)u󵄩󵄩󵄩 ) ,

which implies that 󵄩󵄩 󵄩 1/2 1/2 󵄩󵄩(M − T)u󵄩󵄩󵄩 ≤ ‖M − T‖ (Mu − Tu, u) .

(9.24)

Now let (un )n be a sequence such that ‖un ‖ = 1 for all n and (Tun , un ) → M as n → ∞. By (9.24) we obtain that (M − T)un → 0. Hence M ∈ σ(T); otherwise, we would obtain that

172 � 9 Spectral analysis un = (MI − T)−1 (MI − T)un → 0, in contradiction with ‖un ‖ = 1. Corollary 9.22. Let T ∈ ℒ(H) be a self-adjoint operator such that σ(T) = {0}. Then T = 0. Proof. By Proposition 9.21 we have m = M = 0, and hence (Tu, u) = 0 for each u ∈ H. As (Tu, v) can be written as a linear combination of terms of the form (Tw, w), we obtain T = 0. Proposition 9.23. The spectrum σ(T) of any self-adjoint operator T is real and nonempty. If ζ ∉ σ(T), then 󵄩󵄩 −1 󵄩 −1 󵄩󵄩(ζI − T) 󵄩󵄩󵄩 ≤ |Iζ | .

(9.25)

Moreover, ∗

(ζ I − T)−1 = ((ζI − T)−1 ) .

(9.26)

Proof. Let ζ = ξ + iη and η ≠ 0 and set K = η1 (T − ξI). Using Lemma 4.4, it follows that K ∗ = K. Let f ∈ dom(K) be such that Kf = K ∗ f = if . Then i(f , f ) = (Kf , f ) = (f , Kf ) = −i(f , f ), which implies that f = 0 and K − iI is injective. In a similar way, we show that K + iI is injective. The proof of Proposition 9.20(a) implies that im(K ± iI) is closed. Now we obtain from Lemma 4.8 that im(K ± iI)⊥ = ker(K ± iI) = {0}. Therefore (K ± iI)−1 is defined on the whole of H. Since ‖Kx ± ix‖2 = ‖Kx‖2 + ‖x‖2 ,

x ∈ dom(K),

we get 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 −1 󵄩 −1 󵄩󵄩(K ± iI) y󵄩󵄩󵄩 = 󵄩󵄩󵄩(K ± iI) (K ± iI)x 󵄩󵄩󵄩 = ‖x‖ ≤ 󵄩󵄩󵄩(K ± iI)x 󵄩󵄩󵄩 = ‖y‖ for each y ∈ H, which implies that 󵄩󵄩 −1 󵄩 󵄩󵄩(K ± iI) 󵄩󵄩󵄩 ≤ 1.

(9.27)

Thus ±i ∉ σ(K), and hence ζ ∉ σ(T). In addition, (9.27) implies (9.25). Now let x1 , x2 ∈ dom(T). Then ((T − ζI)x1 , x2 ) = (x1 , (T − ζ I)x2 ). Putting y1 = (T − ζI)x1 and y2 = (T − ζ I)x2 and rewriting the last equation in terms of y1 and y2 yield (9.26). Finally, suppose T has the empty spectrum. Then T −1 is a bounded self-adjoint operator. We claim that σ(T −1 ) = {0}. For λ ≠ 0, we can write the inverse of T −1 − λI in the form

9.3 Spectral decomposition of unbounded self-adjoint operators

−1

(T −1 − λI)

= ((λ−1 I − T)λT −1 )

−1

� 173

−1

= λ−1 T(λ−1 I − T) ,

and −1

λ−1 T(λ−1 I − T)

− λ−2 (λ−1 I − T)

−1

−1

= −λ−1 (λ−1 I − T)(λ−1 I − T)

= −λ−1 I;

hence −1

(T −1 − λI)

−1

= −λ−1 I + λ−2 (λ−1 I − T) ,

which is a bounded operator. So σ(T −1 ) = {0}. The spectrum of the bounded self-adjoint operator T is contained in the interval [m, M], where m = inf

u=0 ̸

(Tu, u) (u, u)

and

M = sup u=0 ̸

(Tu, u) , (u, u)

and m and M belong to the spectrum of T; see Proposition 9.21. So we get T −1 = 0, which contradicts to TT −1 = I. The Cayley transform is now used to reduce the construction of the spectral decomposition of an unbounded self-adjoint operator to the spectral decomposition of a unitary operator. Proposition 9.24. Let T be an unbounded self-adjoint operator (T = T ∗ and dom(T) = dom(T ∗ )). Then there exists a uniquely determined resolution of the identity E on the Borel subsets of ℝ such that ∞

(Tx, y) = ∫ t dEx,y (t),

x ∈ dom(T),

y ∈ H.

(9.28)

−∞

Moreover, E is concentrated on the spectrum σ(T) ⊂ ℝ of T in the sense that E(σ(T)) = I. Proof. Let U be the Cayley transform of T, and let Ω be the unit circle with the point 1 removed. Let E ′ be the spectral decomposition of U (Proposition 9.7). By Proposition 9.20, I − U is injective, and E ′ ({1}) = 0 by Proposition 9.9. Hence ′ (Ux, y) = ∫ λ dEx,y (λ),

x, y ∈ H.

Ω

Define f (λ) =

i(1 + λ) , (1 − λ)

λ ∈ Ω.

We define Ψ(f ) as in Proposition 9.14 with E ′ in place of E.

(9.29)

174 � 9 Spectral analysis ′ (Ψ(f )x, y) = ∫ f dEx,y ,

x ∈ 𝒟f ,

y ∈ H.

(9.30)

Ω

Since f is real-valued, Ψ(f ) is self-adjoint (Proposition 9.14). Since f (λ)(1 − λ) = i(1 + λ), the symbolic calculus gives Ψ(f )(I − U) = i(I + U).

(9.31)

im(I − U) ⊂ dom(Ψ(f )).

(9.32)

T(I − U) = i(I + U),

(9.33)

This in particular implies

By Proposition 9.20

and dom(T) = im(I − U) ⊂ dom(Ψ(f )). Relations (9.32) and (9.33) imply that Ψ(f ) is a self-adjoint extension of the self-adjoint operator T. Thus we have T ⊂ Ψ(f ) and Ψ(f ) = Ψ(f )∗ ⊂ T ∗ = T, and hence Ψ(f ) = T. In addition, we get ′ (Tx, y) = ∫ f dEx,y ,

x ∈ dom(T),

y ∈ H.

(9.34)

Ω

By Proposition 9.17(c), σ(T) is the essential range of f . Thus σ(T) ⊂ ℝ. Since f is injective in Ω, we can define E(f (ω)) = E ′ (ω) for every Borel set ω ⊂ Ω to obtain the desired resolution E, which converts (9.34) into (9.28). The uniqueness of E follows from the uniqueness of representation (9.29). The symbolic calculus is now used to prove the following statements. Proposition 9.25. Let T be a self-adjoint operator on H. (a) (Tx, x) ≥ 0 for every x ∈ dom(T) (briefly T ≥ 0) if and only if σ(T) ⊂ [0, ∞). (b) If T ≥ 0, then there exists a unique self-adjoint operator S ≥ 0 such that dom(S) ⊇ dom(T) and S 2 = T on dom(T). The symbolic calculus implies that x ∈ dom(T) if and only if x ∈ dom(S) and also Sx ∈ dom(S). In addition, dom(T) is a core of S. Proof. (a) see Proposition 9.11(a). (b) Assume that T ≥ 0, so that σ(T) ⊂ [0, ∞), and ∞

(Tx, y) = ∫ t dEx,y (t), 0

x ∈ dom(T),

y ∈ H,

(9.35)

9.3 Spectral decomposition of unbounded self-adjoint operators

� 175

∞

where dom(T) = {x ∈ H : ∫0 t 2 dEx,x (t) < ∞}. Let s(t) be the nonnegative square root of t ≥ 0 and put Ψ(s) = S; explicitly, ∞

(Sx, y) = ∫ s(t) dEx,y (t),

x ∈ 𝒟s ,

y ∈ H.

(9.36)

0

By Proposition 9.14 we obtain that S 2 = T on dom(T) and S ≥ 0. To prove the uniqueness, suppose R is self-adjoint, R ≥ 0, and R2 = T, and let E R be its spectral decomposition: ∞

R (Rx, y) = ∫ t dEx,y (t),

x ∈ dom(R),

y ∈ H.

(9.37)

0

We apply Proposition 9.18 with Ω = [0, ∞), ϕ(t) = t 2 , f (t) = t, and E ′ (ϕ(ω)) = E R (ω) for ω ⊂ [0, ∞)

(9.38)

to obtain ∞

∞

R ′ (Tx, y) = (R2 x, y) = ∫ t 2 dEx,y(t) = ∫ t dEx,y (t). 0

(9.39)

0

Relations (9.35) and (9.39) and the uniqueness statement in Proposition 9.24 show that E ′ = E. By (9.38), E determines E R and hence R. The statement about the domains of the operators S and T follows from the symbolic calculus and the definition of the domains there (see Proposition 9.14). As S ∗ = S, Lemma 4.32 immediately implies that dom(T) is a core of S. As applications of these results, we prove a useful characterization of self-adjoint and essentially self-adjoint operators. Proposition 9.26. Let T be a closed symmetric operator. Then the following statements are equivalent: (i) T is self-adjoint; (ii) ker(T ∗ + iI) = {0} and ker(T ∗ − iI) = {0}; (iii) im(T + iI) = H and im(T − iI) = H. Proof. (i) implies (ii): By Proposition 9.23, ±i ∉ σ(T). (ii) implies (iii): Notice that ker(T ∗ ± iI) = {0} if and only if im(T ∓ iI) is dense in H. This follows easily from (Tu ± iu, v) = (u, T ∗ ∓ iv)

176 � 9 Spectral analysis for u, v ∈ dom(T). So it remains to show that im(T ∓ iI) is closed. The symmetry of T implies that 󵄩2 󵄩󵄩 2 2 󵄩󵄩(T ∓ iI)u󵄩󵄩󵄩 = ‖Tu‖ + ‖u‖

(9.40)

for u ∈ dom(T). Now since T is closed, we easily obtain that im(T ∓ iI) is closed. (iii) implies (i): Let u ∈ dom(T ∗ ). By (iii) there exists v ∈ dom(T) such that (T − iI)v = (T ∗ − iI)u. Since T is symmetric, we have also (T ∗ − iI)(v − u) = 0. However, if (T + iI) is surjective, then (T ∗ − iI) is injective (Lemma 4.8), and we obtain u = v. This proves that u ∈ dom(T) and that T is self-adjoint. We proved during the statement that (ii) implies (iii) the following: Lemma 9.27. If T is closed and symmetric, then im(T ± iI) is closed. In a similar way, we obtain a characterization for essentially self-adjoint operators. Proposition 9.28. Let A be a symmetric operator. Then the following statements are equivalent: (i) A is essentially self-adjoint; (ii) ker(A∗ + iI) = {0} and ker(A∗ − iI) = {0}; (iii) im(A + iI) and im(A − iI) are dense in H. Proof. We apply Proposition 9.26 to A and notice that A is symmetric and that Lemma 4.5 implies that A∗ = (A)∗ . In addition, we use Lemma 9.27. If A is also a positive operator, then we get the following: Proposition 9.29. Let A be a positive symmetric operator. Then the following statements are equivalent: (i) A is essentially self-adjoint; (ii) ker(A∗ + bI) = {0} for some b > 0; (iii) im(A + bI) is dense in H. Proof. We proceed in a similar way as before and notice that for a positive symmetric operator A, we have ((A + bI)u, u) ≥ b‖u‖2

(9.41)

for u ∈ dom(A), which is a good substitute for (9.40). By Lemma 4.8, (ii) and (iii) are equivalent. Since the closure of a positive symmetric operator is again positive and symmetric, it remains to show that a closed positive symmetric operator T is self-adjoint if and only if ker(T ∗ + bI) = {0} for some b > 0.

9.3 Spectral decomposition of unbounded self-adjoint operators �

177

We can suppose that b = 1. If T is self-adjoint, then the spectrum σ(T) ⊆ ℝ+ , and hence ker(T + I) = ker(T ∗ + I) = {0}. For the converse, we first show that im(T + I) is closed: let (yk )k ⊂ im(T + I) be a convergent sequence. There exists a sequence (xk )k ⊂ dom(T) such that yk = (T + I)xk . Then (xk , yk ) = (xk , Txk ) + ‖xk ‖2 ≥ ‖xk ‖2 , and, by the Cauchy–Schwarz inequality, ‖xk ‖ ≤ ‖yk ‖.

(9.42)

Since (yk )k is convergent, supk ‖yk ‖ < ∞, and, by (9.42), supk ‖xk ‖ < ∞. Now the positivity implies ‖xk − xℓ ‖2 ≤ (xk − xℓ , (T + I)(xk − xℓ )) ≤ (‖xk ‖ + ‖xℓ ‖)‖yk − yℓ ‖

≤ C‖yk − yℓ ‖.

Hence (xk )k is a Cauchy sequence. Since we supposed that T is closed, there exists x ∈ dom(T) such that x = limk→∞ xk and (T +I)x = y = limk→∞ yk . Hence im(T +I) is closed. The assumption ker(T ∗ + I) = {0} now gives im(T + I) = H. To show that T is self-adjoint, it suffices to show that dom(T ∗ ) ⊆ dom(T). Let x ∈ dom(T ∗ ). There exists y ∈ dom(T) such that (T + I)y = (T ∗ + I)y = (T ∗ + I)x, since dom(T) ⊆ dom(T ∗ ). This implies (T ∗ +I)(x −y) = 0, and hence x = y ∈ dom(T). Finally, we mention that every self-adjoint operator is unitarily equivalent to a multiplication operator, which is important for applications to the solution of other spectral problems. We omit the proof, as we will not use this version of the spectral theorem in the sequel. Using the Riesz representation theorem in measure theory (see, for instance, [83]), this version follows easily from Proposition 9.24. Theorem 9.30. Let T be a self-adjoint operator on H with spectrum σ(T). Then there exist a finite measure μ on σ(T) × ℕ and a unitary operator U : H 󳨀→ L2 (σ(T) × ℕ, dμ) with the following properties: if g : σ(T) × ℕ 󳨀→ ℝ

178 � 9 Spectral analysis is the function g(t, n) = t, then x ∈ H lies in dom(T) if and only if g ⋅ Ux ∈ L2 (σ(T) × ℕ, dμ). In addition, we have UTU −1 h = gh for all h ∈ U(dom(T)), and Uf (T)U −1 h = f (g)h for all bounded Borel functions f on σ(T). In particular, f (T) is a bounded operator, and 󵄩󵄩 󵄩 󵄩󵄩f (T)󵄩󵄩󵄩 = ‖f ‖∞ .

(9.43)

In the proof of this version of the spectral theorem, we start with a function f ∈ 𝒞0 (σ(T)), a continuous function vanishing at infinity. Then we consider the linear functional Λ(f ) := (f (T)x, x), where x ∈ H is an appropriate vector (a cyclic vector). The Riesz representation theorem (see [83]) implies that there exists a finite countably additive measure μ on ℝ such that (f (T)x, x) = ∫ f (t) dμ(t). σ(T)

This is the main step. The rest of the proof consists of an application of the symbolic calculus (see [24] for all details).

9.4 Determination of the spectrum In this part, we prove some results of the spectral theory of unbounded self-adjoint operators, which are used later on for the applications to the ◻-operator, to Schrödinger operators with magnetic field, and to Pauli and Dirac operators. These results enable us to determine the spectrum of the ◻-operator in some particular cases, and they yield methods to decide whether the corresponding differential operators are with compact resolvent. First, we prove some general results about the spectrum of an unbounded selfadjoint operator. Lemma 9.31. Let T be an unbounded self-adjoint operator. Then λ ∈ σ(T) if and only if there exists a sequence (xk )k in dom(T) such that ‖xk ‖ = 1 for each k ∈ ℕ and

9.4 Determination of the spectrum

� 179

󵄩 󵄩 lim 󵄩󵄩󵄩(T − λI)xk 󵄩󵄩󵄩 = 0.

k→∞

If T and S are self-adjoint operators such that T = U −1 SU for some unitary operator U, where dom(T) = U −1 (dom(S)), then σ(T) = σ(S). Proof. If λ ∈ σ(T), then T − λI is not injective, and we can find x ∈ dom(T) such that ‖x‖ = 1 and (T − λI)x = 0. So the constant sequence xk = x has the desired property. Conversely, suppose that λ ∉ σ(T). Then (T − λI)−1 is a bounded operator. If there exists a sequence (xk )k in dom(T) such that ‖xk ‖ = 1 for each k ∈ ℕ and limk→∞ ‖(T − λI)xk ‖ = 0, then lim (T − λI)−1 (T − λI)xk = lim xk = 0

k→∞

k→∞

yields a contradiction to ‖xk ‖ = 1 for each k ∈ ℕ. To prove the second statement, take λ ∈ σ(S) and a sequence (yk )k in dom(S) such that ‖yk ‖ = 1 for each k ∈ ℕ and 󵄩 󵄩 lim 󵄩󵄩󵄩(S − λI)yk 󵄩󵄩󵄩 = 0.

k→∞

Then for U −1 yk = zk ∈ dom(T), we have ‖zk ‖ = 1, since U is unitary and 󵄩󵄩 󵄩 󵄩 −1 󵄩 󵄩 󵄩 󵄩󵄩(T − λI)zk 󵄩󵄩󵄩 = 󵄩󵄩󵄩U (Syk − λyk )󵄩󵄩󵄩 = 󵄩󵄩󵄩(S − λI)yk 󵄩󵄩󵄩. This shows that λ ∈ σ(T). Lemma 9.32. Let T be an unbounded self-adjoint operator, and let E be the uniquely determined resolution of the identity E on the Borel subsets of ℝ such that ∞

(Tx, y) = ∫ t dEx,y (t),

x ∈ dom(T),

y∈H

−∞

(see Proposition 9.24). Then σ(T) = {λ ∈ ℝ : E((λ − ϵ, λ + ϵ)) ≠ 0, ∀ϵ > 0}.

(9.44)

Proof. Let λ ∈ σ(T). Suppose that there exists ϵ0 > 0 such that E((λ − ϵ0 , λ + ϵ0 )) = 0. Then there exists a continuous function f on ℝ such that f (t) = (t − λ)−1 on the support of the measure dE(t). By the symbolic calculus we obtain a bounded operator (T − λI)−1 and hence a contradiction. Conversely, let λ belong to the right-hand side of (9.44). For each k ∈ ℕ, we can now find xk ∈ dom(T) such that ‖xk ‖ = 1 and E((λ − 1/k, λ + 1/k))xk = xk .

180 � 9 Spectral analysis Consider the bounded Borel functions fk (t) = (t − λ) χ(λ−1/k,λ+1/k) (t),

t ∈ ℝ,

k ∈ ℕ,

where χ(λ−1/k,λ+1/k) is the characteristic function of the interval (λ − 1/k, λ + 1/k). Then by (9.43) 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(T − λI)xk 󵄩󵄩󵄩 = 󵄩󵄩󵄩(T − λI) E((λ − 1/k, λ + 1/k))xk 󵄩󵄩󵄩 ≤ 1/k ‖xk ‖ = 1/k. Using Lemma 9.31, we get λ ∈ σ(T). Lemma 9.33. Let T be a symmetric operator on H with domain dom(T), and suppose that {xk }k is a complete orthonormal system in H. If each xk lies in dom(T), and there exist λk ∈ ℝ such that Txk = λk xk for every k ∈ ℕ, then T is essentially self-adjoint. Moreover, the spectrum of T is the closure in ℝ of the set of all λk . Proof. If x = ∑∞ k=1 αk xk belongs to dom(T) and ∞

y = Tx = ∑ βk xk , k=1

then βk = (y, xk ) = (Tx, xk ) = (x, Txk ) = λk (x, xk ) = λk αk . Since x, y ∈ H, we have ∞

∑ |αk |2 < ∞,

k=1

∞

∑ |βk |2 < ∞,

k=1

and hence ∞

∑ (1 + λ2k )|αk |2 < ∞.

k=1

We define an operator T̃ as follows: let ∞

∞

k=1

k=1

dom(T)̃ = {x ∈ H : x = ∑ αk xk with ∑ (1 + λ2k )|αk |2 < ∞} and define

9.4 Determination of the spectrum

� 181

∞

̃ = ∑ αk λk xk Tx k=1

̃ It follows that T̃ is an extension of T. for x ∈ dom(T). Let Σ be the closure of the set {λk : k ∈ ℕ}. Each λk is an eigenvalue of T,̃ and, by ̃ For z ∉ Σ and x = ∑∞ αk xk , we define the Lemma 9.16, σ(T)̃ is closed, so Σ ⊆ σ(T). k=1 operator S on H by ∞

∞

k=1

k=1

Sx = S( ∑ αk xk ) := ∑ αk (z − λk )−1 xk . It follows that S is injective and that 1/2 󵄩󵄩 ∞ 󵄩󵄩 ∞ 󵄩󵄩 󵄩󵄩 2 −1 󵄩󵄩S( ∑ αk xk )󵄩󵄩 ≤ sup |z − λk | ( ∑ |αk | ) = C ‖x‖, 󵄩󵄩 󵄩󵄩 󵄩󵄩 k=1 󵄩󵄩 k k=1

̃ and (zI − which implies that the operator S is bounded. Its range is precisely dom(T), −1 ̃ ̃ ̃ ̃ T)Sx = x for all x ∈ H. Thus z ∉ σ(T) and S = (zI − T) . This implies Σ = σ(T). Now we claim that T̃ is the closure of T. Since σ(T)̃ is not equal to ℂ, Lemma 9.16 implies that T̃ is a closed operator. Let ∞

u = ∑ αk xk ∈ dom(T)̃ k=1

and put um = ∑m k=1 αk xk . Then limm→∞ um = u, and m

∞

k=1

k=1

̃ lim Tum = lim ∑ αk λk xk = ∑ αk λk xk = Tu.

m→∞

m→∞

Hence T̃ = T. Finally, we prove that T̃ is self-adjoint. For x ∈ dom(T̃ ∗ ) and T̃ ∗ x = y, we have ̃ k ) = λk (x, xk ). (y, xk ) = (T̃ ∗ x, xk ) = (x, Tx ∞ ̃ If x = ∑∞ k=1 αk xk , then the above implies that y = ∑k=1 αk λk xk . Hence x ∈ dom(T), thus ∗ ∗ ̃ ̃ ̃ ̃ dom(T ) = dom(T), and T = T .

Definition 9.34. Let T be a self-adjoint operator on H. The discrete spectrum σd (T) of T is the set of all eigenvalues λ of finite multiplicity that are isolated in the sense that the intervals (λ − ϵ, λ) and (λ, λ + ϵ) are disjoint from the spectrum for some ϵ > 0. The nondiscrete part of the spectrum of T is called the essential spectrum of T and is denoted by σe (T).

182 � 9 Spectral analysis A closed linear subspace L of H is called invariant if (ζI − T)−1 (L) ⊆ L for all ζ ∉ ℝ. Lemma 9.35. Let T be a self-adjoint operator. Then σd (T) = {λ ∈ σ(T) : ∃ϵ > 0, dim im(E((λ − ϵ, λ + ϵ))) < ∞}.

(9.45)

Proof. Let λ belong to the right-hand side of (9.45). Then there exists ϵ0 > 0 such that for each ϵ ∈ (0, ϵ0 ), the projection E((λ − ϵ, λ + ϵ)) becomes a projection with finite range independent of ϵ. This is in fact the projection E({λ}), and we observe that E((λ − ϵ0 , λ)) = 0 and E((λ, λ + ϵ0 )) = 0. This shows that λ ∈ σd (T). If λ ∈ σd (T), then λ is an eigenvalue of finite multiplicity, and there exists ϵ > 0 such that the intervals (λ − ϵ, λ) and (λ, λ + ϵ) are disjoint from the spectrum, which means that dim im(E((λ − ϵ, λ + ϵ))) < ∞. As the whole spectrum of a self-adjoint operator is closed, it now follows that the essential spectrum is closed. The following characterization of the essential spectrum of a self-adjoint operator is similar to the characterization of the whole spectrum; compare with Lemma 9.31. Lemma 9.36. Let T be a self-adjoint operator. Then λ belongs to the essential spectrum σe (T) if and only if there exists a sequence (xk )k in dom(T) such that ‖xk ‖ = 1 for each k ∈ ℕ, xk converges weakly to 0, and 󵄩 󵄩 lim 󵄩󵄩󵄩(T − λI)xk 󵄩󵄩󵄩 = 0.

k→∞

Proof. The sequence appearing in the lemma is called a Weyl sequence. We say that λ belongs to the Weyl spectrum W (T) if there exists an associated Weyl sequence. By Lemma 9.31 we already know that W (T) ⊆ σ(T). Let λ ∈ W (T) and suppose that λ ∈ σd (T). Then the spectral projection E({λ}) has a finitedimensional range and hence is compact. So, by Proposition 4.17, lim E({λ})xk = 0

k→∞

in H. Now let yk := (I − E({λ}))xk . Then, as E({λ})T = TE({λ}) (Proposition 9.24), we get limk→∞ ‖yk ‖ = 1 and lim (T − λI)yk = lim (I − E({λ}))(T − λI)xk = 0.

k→∞

k→∞

9.4 Determination of the spectrum

�

183

Since (T − λI) is invertible on im(I − E({λ})), we obtain limk→∞ yk = 0, which is a contradiction. Hence λ ∈ σe (T). Conversely, let λ ∈ σe (T). Then, by Lemma 9.35, dim im(E((λ − ϵ, λ + ϵ))) = ∞ for any ϵ > 0. Let ϵk be a decreasing sequence of positive numbers such that limk→∞ ϵk = 0. We can now choose an orthonormal system (xk )k such that xk ∈ im(E((λ − ϵk , λ + ϵk ))). Then, by Bessel’s inequality (Proposition 1.9), xk converges weakly to 0, and the same reasoning as in Lemma 9.32 yields 󵄩 󵄩 lim 󵄩󵄩󵄩(T − λI)xk 󵄩󵄩󵄩 = 0.

k→∞

Next, we will characterize the situation where σe (T) = 0. For this purpose, we need some preparations. Lemma 9.37. Let T be a self-adjoint operator on H, and let λ ∈ ℝ be an eigenvalue of T. Then Lλ = {x ∈ dom(T) : Tx = λx} is a closed invariant subspace of H. If L is an invariant subspace of T, then L⊥ is also invariant. Proof. Since Lλ = ker(T − λI) and T − λI is a closed operator, it follows by Lemma 4.4 that Lλ is a closed subspace. Now let x ∈ Lλ and ζ ∉ ℝ. Then (ζI − T)−1 x = (λ − ζ )−1 (ζI − T)−1 (λ − ζ )x

= (λ − ζ )−1 (ζI − T)−1 (T − ζI)x = −(λ − ζ )−1 x,

and hence (ζI − T)−1 (Lλ ) ⊆ Lλ . Let L be an invariant subspace and y ∈ L⊥ . Then by (9.26) ((ζI − T)−1 y, x) = (y, (ζ I − T)−1 x) = 0 for all x ∈ L and ζ ∉ ℝ. Therefore (ζI − T)−1 y ∈ L⊥ , and L⊥ is invariant. For the following results, we always suppose that the underlying Hilbert space H is separable and infinite-dimensional, i. e., each complete orthonormal system is countably infinite.

184 � 9 Spectral analysis Proposition 9.38. Let T be a self-adjoint operator on H. The essential spectrum σe (T) of T is empty if and only if there exists a complete orthonormal system of eigenvectors {xn }n of T such that the corresponding eigenvalues λn converge in absolute value to ∞ as n → ∞. Proof. If σe (T) = 0, then the spectrum σ(T) consists of a set {rn }n of isolated eigenvalues of finite multiplicity, which can only converge to ±∞. We enumerate the eigenvalues in order of increasing absolute values and repeat each eigenvalue according to its multiplicity. In this way, we get an associated orthonormal system of eigenvectors {xn }n of T. Suppose that this system is not complete. Then by Lemma 9.37 the subspace L = {x ∈ H : (x, xn ) = 0, n ∈ ℕ} is invariant with respect to T in the sense of Definition 9.33. If x ∈ dom(T) ∩ L, then (Tx, xn ) = (x, Txn ) = rn (x, xn ) = 0, and hence T(dom(T) ∩ L) ⊆ L. The essential spectrum of the restriction of T to L is also empty. Since the spectrum of this restriction is nonempty (Proposition 9.23), T has further eigenvalues and eigenvectors not accounted for in the above list. Conversely, suppose that a sequence of eigenvalues and eigenvectors with the stated properties exists, and let {sn }n be the set of distinct eigenvalues. By the assumption that λn converge in absolute value to ∞ as n → ∞ we deduce that sn are isolated eigenvalues of finite multiplicity. It follows from Lemma 9.33 that ∞

σ(T) = ⋃ {sn }. n=1

Thus σe (T) = 0. Proposition 9.39. Let T be an unbounded self-adjoint operator on H that is nonnegative in the sense that σ(T) ⊆ [0, ∞). Then the following conditions are equivalent: (i) The resolvent operator (I + T)−1 is compact. (ii) σe (T) = 0. (iii) There exists a complete orthonormal system of eigenvectors {xn }n of T with corresponding eigenvalues μn ≥ 0 that converge to +∞ as n → ∞. Proof. (i) ⇒ (iii): The operator (I + T)−1 : H 󳨀→ dom(T) is compact and self-adjoint and has the dense image. By Proposition 2.7 there exists a complete orthonormal system of eigenvectors xn of (I + T)−1 and eigenvalues λn (of finite multiplicity) tending to 0 such that ∞

(I + T)−1 x = ∑ λn (x, xn )xn . n=1

9.4 Determination of the spectrum

�

185

Since all xn ∈ dom(T), we have T(I + T)−1 xn = λn Txn . Adding (I + T)−1 xn to this equality, we get λn Txn + (I + T)−1 xn = T(I + T)−1 xn + (I + T)−1 xn = (I + T)(I + T)−1 xn = xn , which implies that λn Txn + λn xn = xn , and therefore Txn =

1 − λn x . λn n

1−λ

Setting μn = λ n , we get (iii). n Now suppose that (iii) holds. We rearrange the eigenvectors xn so that the sequence {μn }n is nondecreasing. Each x ∈ H can be written in the form x = ∑∞ n=1 (x, xn )xn , and we obtain ∞

∞

1 (x, xn )xn . 1 + μn n=1

(I + T)−1 x = ∑ (x, xn )(I + T)−1 xn = ∑ n=1

Let N

1 (x, xn )xn . 1 + μn n=1

AN = ∑

Then the operators AN are of finite rank, and by Bessel’s inequality (1.6) we obtain 󵄩󵄩 󵄩󵄩 ∞ 󵄩󵄩 1 1 󵄩󵄩 󵄩 󵄩󵄩 −1 (x, xn )xn 󵄩󵄩󵄩 ≤ ‖x‖, 󵄩󵄩((I + T) − AN )x 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ 󵄩󵄩 1 + μN 󵄩󵄩󵄩n=N+1 1 + μn 󵄩 from which we see that AN converges in operator norm to (I + T)−1 as N → ∞. Now, by Proposition 2.6, (I + T)−1 is a compact operator. The equivalence of (ii) and (iii) is a consequence of Proposition 9.38. The following general result explains the approach to the 𝜕-Neumann operator (4.42) by means of the embedding ∗

j : dom(𝜕) ∩ dom(𝜕 ) 󳨅→ L2(0,q) (Ω), ∗

where dom(𝜕) ∩ dom(𝜕 ) is endowed with the graph norm 󵄩 ∗ 󵄩2 1/2 u 󳨃→ (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ) .

186 � 9 Spectral analysis It will also be crucial for the question of compactness of the 𝜕-Neumann operator, which will be discussed in the following chapters. Proposition 9.40. Let A be a nonnegative self-adjoint operator (i. e., σ(A) is contained in [0, ∞)). There exists a unique self-adjoint square root A1/2 of A, and dom(A1/2 ) ⊇ dom(A). In addition, dom(A) endowed with the norm 1/2 󵄩2 󵄩 ‖f ‖𝒟 := (󵄩󵄩󵄩A1/2 f 󵄩󵄩󵄩 + ‖f ‖2 )

becomes a Hilbert space; the norm ‖ ⋅ ‖𝒟 stems from the inner product (f , g)𝒟 = (A1/2 f , A1/2 g) + (f , g). Let dom(A) be endowed with the norm ‖ ⋅ ‖𝒟 . Then A has a compact resolvent if and only if the canonical embedding j : dom(A) 󳨅→ H is a compact linear operator. Furthermore, A has a compact resolvent if and only if A1/2 has a compact resolvent. Proof. In Proposition 9.25, we proved the existence and uniqueness of the square root of A. For n ∈ ℕ, we define the functions qn (t) =

nt , n+t

t ∈ [0, ∞).

These are continuous functions such that qn (t) ≤ qn+1 (t) and limn→∞ qn (t) = t. Moreover, qn (t) ≤ n for each t ∈ [0, ∞). By Theorem 9.30 the operator nA(nI + A)−1 is bounded on H, and the function Qn (x) := (nA(nI + A)−1 x, x),

x ∈ H,

is bounded on the unit ball of H and continuous. The functional calculus implies that Qn (x) increases monotonically to Q(x), where (A1/2 x, A1/2 x)

Q(x) = {

+∞

for x ∈ dom(A), otherwise.

A function Θ : H 󳨀→ (−∞, +∞] is said to be lower semicontinuous if for every convergent sequence xn → x in H, we have Θ(x) ≤ lim inf Θ(xn ). n→∞

It is easily seen that a function Θ is lower semicontinuous if and only if

9.4 Determination of the spectrum

� 187

{x : Θ(x) > α} is open for every real α and that the pointwise limit of an increasing sequence of continuous functions is a lower semicontinuous function. Therefore the function Q is lower semicontinuous. Now let {xn }n be a Cauchy sequence with respect to ‖⋅‖𝒟 . Then {xn }n is also a Cauchy sequence with respect to ‖ ⋅ ‖ and therefore converges to x ∈ H. Given ϵ > 0, there exists N ∈ ℕ such that Q(xm − xn ) + ‖xm − xn ‖2 < ϵ2 for all m, n ≥ N. Letting m → ∞ and using the lower semicontinuity of Q, we deduce that x ∈ dom(A) and Q(x − xn ) + ‖x − xn ‖2 ≤ ϵ2 for all n > N. Hence ‖x − xn ‖𝒟 ≤ ϵ and dom(A) endowed with the norm ‖ ⋅ ‖𝒟 is complete. Since −1 ∉ σ(A), we know that (I + A)−1 is a bounded operator on H. From (9.7) we get that RA (−1) = (I + A)−1 is compact if and only if RA (z) is compact for all z ∉ σ(A). Let u ∈ H and v ∈ dom(A). Then (j∗ u, v)𝒟 = (u, jv) = (u, v) = ((I + A)(I + A)−1 u, v) = ((I + A)−1 u, (I + A)v)

= ((I + A)−1 u, Av) + ((I + A)−1 u, v)

= (A1/2 (I + A)−1 u, A1/2 v) + ((I + A)−1 u, v)

= ((I + A)−1 u, v)𝒟 .

This implies that j∗ = (I + A)−1 as an operator on dom(A) and j ∘ j∗ = (I + A)−1 as an operator on H. So we deduce the desired conclusion by the fact that j is compact if and only if j ∘ j∗ is compact (Theorem 2.5). The last statement follows from (iI + A1/2 )∗ = −iI + A1/2 and (I + A) = (iI + A1/2 )(−iI + A1/2 ). Our next aim is to compare two self-adjoint strictly positive operators and to prove Ruelle’s lemma. Remark 9.41. Let S be a self-adjoint operator. Suppose that (Sx, x) ≥ 0 for each x ∈ dom(S). We say that S is strictly positive if (Sx, x) > 0 for each x ∈ dom(S) \ {0}. Using the fact that dom(S) is a core of S 1/2 (Proposition 9.25), we can easily show that S is strictly positive if and only if ‖S 1/2 x‖ > 0 for each x ∈ dom(S 1/2 ) \ {0}. We also indicate that a strictly positive self-adjoint operator S is injective and im(S) is dense in H and S −1 is self-adjoint (Lemma 4.28).

188 � 9 Spectral analysis Lemma 9.42. Let S and T be strictly positive self-adjoint operators. Then the following statements are equivalent. (a) dom(T 1/2 ) ⊂ dom(S 1/2 ) and ‖S 1/2 x‖ ≤ ‖T 1/2 x‖ for x ∈ dom(T 1/2 ); (b) dom(T 1/2 ) ⊂ dom(S 1/2 ) and ‖S 1/2 T −1/2 x‖ ≤ ‖x‖ for x ∈ dom(T −1/2 ); (c) dom(S −1/2 ) ⊂ dom(T −1/2 ) and ‖T −1/2 S 1/2 x‖ ≤ ‖x‖ for x ∈ dom(S 1/2 ); (d) dom(S −1/2 ) ⊂ dom(T −1/2 ) and ‖T −1/2 x‖ ≤ ‖S −1/2 x‖ for x ∈ dom(S −1/2 ). Proof. We show that (a) and (b) are equivalent and that (b) implies (c). The remaining implications follow by symmetry and by replacing S and T by S −1 and T −1 . Suppose that (a) holds. If x ∈ dom(T −1/2 ), then T −1/2 x ∈ dom(T 1/2 ), so we get 1/2 −1/2 ‖S T x‖ ≤ ‖x‖. If (b) holds and x ∈ dom(T 1/2 ), then we have T 1/2 x ∈ dom(T −1/2 ), and hence 󵄩󵄩 1/2 󵄩󵄩 󵄩󵄩 1/2 −1/2 1/2 󵄩󵄩 󵄩󵄩 1/2 󵄩󵄩 T x 󵄩󵄩 ≤ 󵄩󵄩T x 󵄩󵄩. 󵄩󵄩S x 󵄩󵄩 = 󵄩󵄩S T Suppose that (b) holds. We observe that dom(S −1/2 ) = im(S 1/2 ) and that we have to show im(S 1/2 ) ⊂ dom(T −1/2 ) and

󵄩󵄩 −1/2 1/2 󵄩󵄩 S x 󵄩󵄩 ≤ ‖x‖ for x ∈ dom(S 1/2 ). 󵄩󵄩T

If x ∈ dom(S 1/2 ), then S 1/2 x ∈ im(S 1/2 ). We consider the linear functional ψx (y) := (S 1/2 x, T −1/2 y)

for y ∈ dom(T −1/2 ).

Then 󵄨󵄨 󵄨 󵄨 1/2 󵄨 󵄨 −1/2 󵄨󵄨 y)󵄨󵄨 = 󵄨󵄨󵄨(x, S 1/2 T −1/2 y)󵄨󵄨󵄨 ≤ ‖x‖ ‖y‖. 󵄨󵄨ψx (y)󵄨󵄨󵄨 = 󵄨󵄨󵄨(S x, T This implies that S 1/2 x ∈ dom((T −1/2 )∗ ) = dom(T −1/2 ). Finally, we get 󵄨󵄨 −1/2 1/2 󵄨 S x, y)󵄨󵄨󵄨 ≤ ‖x‖ ‖y‖, 󵄨󵄨(T

for y ∈ dom(T −1/2 ).

As dom(T −1/2 ) is dense in H, we obtain ‖T −1/2 S 1/2 x‖ ≤ ‖x‖. For Ruelle’s lemma, we consider strictly positive self-adjoint operators S and T, and we write S ≤ T if and only if dom(T) ⊆ dom(S) and (Sx, x) ≤ (Tx, x) for each x ∈ dom(T). By Proposition 9.25 the square roots of S and T exist and are themselves positive selfadjoint operators. Lemma 9.43 (Ruelle’s lemma). Let S and T be strictly positive self-adjoint operators. Suppose that S ≤ T and 0 ∈ ρ(S). Then T −1 ≤ S −1 . Proof. We have dom(T) ⊆ dom(S) and 󵄩󵄩 1/2 󵄩󵄩2 󵄩 1/2 󵄩2 󵄩󵄩S x 󵄩󵄩 = (Sx, x) ≤ (Tx, x) = 󵄩󵄩󵄩T x 󵄩󵄩󵄩

(9.46)

9.4 Determination of the spectrum

�

189

for all x ∈ dom(T). Next, we show that dom(T 1/2 ) ⊂ dom(S 1/2 ). Let x ∈ dom(T 1/2 ). By Proposition 9.25, dom(T) is a core of dom(T 1/2 ). Hence there exists a sequence (xk )k in dom(T) such that xk → x and T 1/2 xk → T 1/2 x. By (9.46) we have 󵄩 󵄩 󵄩 1/2 󵄩󵄩 1/2 󵄩󵄩S (xm − xk )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩T (xm − xk )󵄩󵄩󵄩, which implies that (S 1/2 xk )k is a Cauchy sequence. Since S 1/2 is also a closed operator, we have that x ∈ dom(S 1/2 ) and S 1/2 xk → S 1/2 x. In addition, by (9.46) we have 󵄩󵄩 1/2 󵄩󵄩 󵄩 1/2 󵄩 󵄩 1/2 󵄩 󵄩 1/2 󵄩 󵄩󵄩S x 󵄩󵄩 = lim 󵄩󵄩󵄩S xk 󵄩󵄩󵄩 ≤ lim 󵄩󵄩󵄩T xk 󵄩󵄩󵄩 = 󵄩󵄩󵄩T x 󵄩󵄩󵄩 k→∞ k→∞ for each x ∈ dom(T 1/2 ). Now we can apply Lemma 9.42 (d) and get dom(S −1/2 ) ⊂ dom(T −1/2 ) and ‖T −1/2 x‖ ≤ −1/2 ‖S x‖ for x ∈ dom(S −1/2 ). Our assumption 0 ∈ ρ(S) implies that H = dom(S −1 ) = dom(S −1/2 ), which proves the lemma. Example 9.44. We return to the 𝜕-Neumann operator on L2(0,1) (ℂn , e−φ ) and remark that the Kohn–Morrey formula from Proposition 6.3 can be written in the form (Mφ u, u)φ ≤ (◻φ u, u)φ ∗

for a (0, 1)-form u ∈ dom(𝜕) ∩ dom(𝜕φ ). So, under the assumptions of Theorem 8.4, we obtain using Ruelle’s Lemma 9.43 that (Nφ u, u)φ ≤ (Mφ−1 u, u)φ , and setting 𝜕v = u, we get ∗

‖v‖2φ = (v, v)φ = (v, 𝜕φ Nφ u)φ = (𝜕v, Nφ u)φ = (u, Nφ u)φ ≤ (Mφ−1 𝜕v, 𝜕v)φ for each v ∈ dom(𝜕) orthogonal to ker(𝜕). This gives a different proof of Hörmander’s L2 -estimates similar to the Brascamp–Lieb inequality (see [46] and [61]): 󵄨 󵄨2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨v(z)󵄨󵄨󵄨 e−φ(z) dλ(z) ≤ ∫ 󵄨󵄨󵄨𝜕v(z)󵄨󵄨󵄨i𝜕𝜕φ e−φ(z) dλ(z)

ℂn

(9.47)

ℂn

for each v ∈ dom(𝜕) orthogonal to ker(𝜕), where we used the notation in formula (8.7). ∗ Let 1 ≤ q ≤ n. If u is a (0, q)-form in dom(𝜕) ∩ dom(𝜕φ ), then by (6.11) we get ∑

′

n

∑ ∫

|K|=q−1 j,k=1 ℂn

𝜕2 φ u u e−φ dλ ≤ (◻φ u, u)φ . 𝜕zj 𝜕zk jK kK

190 � 9 Spectral analysis The left-hand side can be written in the form (M̃ φ u, u)φ . We suppose that M̃ φ is invertible and get as above ‖v‖2φ ≤ (M̃ φ−1 𝜕v, 𝜕v)φ

(9.48)

for each (0, q − 1)-form v ∈ dom(𝜕) orthogonal to ker(𝜕).

9.5 Variational characterization of the discrete spectrum Here we explain the max-min principle to describe the lowest part of the spectrum of a self-adjoint operator when it is discrete. This is done in the setting of semibounded operators. Definition 9.45. Let T be a symmetric unbounded operator with dom(T). We say that T is semibounded (from below) if there exists a constant C > 0 such that (Tu, u) ≥ −C‖u‖2 ∀u ∈ dom(T). See the next chapter for examples of semibounded operators. Using Proposition 4.36, we can show that a symmetric semibounded operator T admits a self-adjoint extension. For this purpose, set a(u, v) = (Tu, v) for the sesquilinear form from there. The resulting self-adjoint extension is called the Friedrichs extension. Proposition 9.46. Let A be a self-adjoint semibounded operator. Let Σ = inf σe (A). The set σ(A) ∩ (−∞, Σ) can be described as a sequence (finite or infinite) of eigenvalues λj ordered increasingly. Then λ1 = inf{(Aϕ, ϕ) ‖ϕ‖−2 : ϕ ∈ dom(A), ϕ ≠ 0},

(9.49)

⊥ λk = inf{(Aϕ, ϕ) ‖ϕ‖−2 : ϕ ∈ dom(A) ∩ 𝒦k−1 , ϕ ≠ 0},

(9.50)

and for k ≥ 2,

where 𝒦j = ⨁ ker(A − λm I). m≤j

Proof. Let μ1 denote the right-hand side of (9.49). If ϕ1 is an eigenfunction for the eigenvalue λ1 , then we get μ1 ≤ λ1 . On the other side, we have σ(A − λ1 I) ⊆ [0, ∞), and hence, by Proposition 9.25,

9.5 Variational characterization of the discrete spectrum

� 191

(Aϕ − λ1 ϕ, ϕ) ≥ 0 for each ϕ ∈ dom(A), which implies λ1 ≤ μ1 . In fact, we have shown that if μ1 < Σ, then the spectrum below Σ is not empty. In particular, the bottom of the spectrum is an eigenvalue equal to μ1 . For k = 2, 3, . . . , we apply the first step of the proof to A |dom(A)∩𝒦⊥ and use the k−1 spectral theorem. Our next aim is to generalize the following min-max lemma for positive compact operators. Lemma 9.47. Let A : H 󳨀→ H be a positive compact operator (i. e., (Ax, x) ≥ 0 for all x ∈ H) with spectral decomposition ∞

Ax = ∑ λn (x, xn )xn , n=0

where λ0 ≥ λ1 ≥ ⋅ ⋅ ⋅. Then (i) λ0 = max x∈H

(ii)

(Ax, x) , (x, x)

where the maximum is attained by an eigenvector x0 with eigenvalue λ0 .

λj = min max ⊥ L∈Nj x∈L

(Ax, x) , (x, x)

j ≥ 1,

where Nj denotes the set of all j-dimensional subspaces of H. The minimum is attained by the subspace L = Lj = ⟨x0 , . . . , xj−1 ⟩, i. e., λj = max ⊥ x∈Lj

(Ax, x) . (x, x)

Proof. (i) follows directly from the proof of the spectral Theorem 2.7. For j ≥ 1, we have λj = (Axj , xj )/(xj , xj ). The statement follows if we can show that for j

each j-dimensional subspace L, there exists z0 ⊥ L with z0 ≠ 0 and z0 = ∑k=0 (z0 , xk )xk . In that case, we then have j

2 (Az0 , z0 ) ∑k=0 λk |(z0 , xk )| = ≥ λj , j (z0 , z0 ) ∑k=0 |(z0 , xk )|2

as λj ≤ λi for 0 ≤ i ≤ j and

192 � 9 Spectral analysis max ⊥ x∈L

(Ax, x) ≥ λj . (x, x)

The existence of z0 follows from the fact that for a basis {yk : k = 0, . . . , j − 1} of L, the system of linear equations j

∑ ai (xi , yk ) = 0,

i=0

k = 0, . . . , j − 1,

j

has a nontrivial solution. Set z0 = ∑i=0 ai xi ; then z0 ⊥ L. So if we have positive compact operators A and B such that A ≤ B, which means (Ax, x) ≤ (Bx, x), then the eigenvalues of A and B satisfy λj (A) ≤ λj (B),

j = 0, 1, 2, . . . .

The corresponding result for unbounded operators yields information about the bottom of the spectrum and of the essential spectrum. Proposition 9.48. Let H be a Hilbert space of infinite dimension. Let A be a self-adjoint semibounded operator with domain dom(A). Let μ1 (A) = inf{(Aϕ, ϕ) : ϕ ∈ dom(A), ‖ϕ‖ = 1} and, for n ≥ 2, μn (A) = sup inf{(Aϕ, ϕ) : ϕ ∈ L⊥ ∩ dom(A), ‖ϕ‖ = 1}, L∈Nn−1

(9.51)

where Nn−1 denotes the set of all subspaces of H of dimension ≤ n − 1. Then either (a) μn (A) is the nth eigenvalue when the eigenvalues are increasingly ordered (counting the multiplicities) and A has a discrete spectrum in (−∞, μn (A)], or (b) μn (A) corresponds to the bottom of the essential spectrum. In this case, we have μj (A) = μn (A) for all j ≥ n. Proof. Let E be the uniquely determined resolution of identity on the Borel subsets of ℝ. First, we show that dim im E((−∞, a)) < n

dim im E((−∞, a)) ≥ n

if a < μn (A), if a > μn (A).

To prove (9.52), let a < μn (A) and suppose that dim im E((−∞, a)) ≥ n. If y ∈ im E((−∞, a)), then we get y = E((−∞, a))x for some x ∈ H, and therefore

(9.52) (9.53)

9.5 Variational characterization of the discrete spectrum

Ey,y (ω) = (E(ω)y, y) = (E(ω)E((−∞, a))x, E((−∞, a))x) = 0

� 193

(9.54)

for each Borel subset ω of ℝ such that (−∞, a) ∩ ω = 0. Since A is bounded from below, we have (Ax, x) ≥ −C‖x‖2 for all x ∈ dom(A), and Proposition 9.25 and Lemma 9.13 imply that ∞

dom(A) = {u ∈ H : ∫ t 2 dEu,u (t) < ∞}, −C

so we obtain for y ∈ im E((−∞, a)) by (9.54) that ∞

∫ t 2 dEy,y (t) ≤ C ′ max(C 2 , a2 ) < ∞, −C

which implies that im E((−∞, a)) ⊆ dom(A). So we can find an n-dimensional subspace L ⊂ dom(A) such that a

(Au, u) = ∫ t dEu,u ≤ a(u, u)

∀u ∈ L.

(9.55)

−C

Then, given any ψ1 , . . . , ψn−1 ∈ H, we can find ϕ ∈ L ∩ ⟨ψ1 , . . . , ψn−1 ⟩⊥ such that ‖ϕ‖ = 1 and (Aϕ, ϕ) ≤ a. Returning to the definition of μn (A), we would have μn (A) ≤ a, which is a contradiction. Hence we have shown (9.52). To prove (9.53), let a > μn (A) and suppose that dim im E((−∞, a)) ≤ n − 1. Then we can find (n − 1) generators ψ1 , . . . , ψn−1 of this space, and any ϕ ∈ dom(A) ∩ ⟨ψ1 , . . . , ψn−1 ⟩⊥ is in im E([a, +∞)), so (Aϕ, ϕ) ≥ a‖ϕ‖2 , which is again a contradiction, and we get (9.53). In the next step, we will show that μn (A) < +∞ for each n ∈ ℕ. Since A is semibounded from below, μn (A) has a uniform lower bound. Suppose that μn (A) = +∞. By (9.52) this means that

194 � 9 Spectral analysis dim im E((−∞, a)) < n for all a ∈ ℝ. Hence H must be of finite dimension, and we arrive at a contradiction. (If H is finite-dimensional, then we have μn (A) ≥ ‖A‖.) For the rest of the proof, we distinguish between the following two cases: dim im E((−∞, μn (A) + ϵ)) = ∞

∀ϵ > 0

(9.56)

for some ϵ0 > 0.

(9.57)

and dim im E((−∞, μn (A) + ϵ0 )) < ∞

Assuming (9.56), we claim that statement (b) of the proposition holds: using (9.52) in this case, we obtain dim im E((μn (A) − ϵ, μn (A) + ϵ)) = ∞

∀ϵ > 0.

By Lemma 9.35, this shows that μn (A) ∈ σe (A). Using (9.52) once more, we see that the interval (−∞, μn (A)) does not contain any point of the essential spectrum. Hence μn (A) = inf{λ : λ ∈ σe (A)}. From (9.51) we obtain that μn+1 (A) ≥ μn (A). If we had μn+1 (A) > μn (A), then (9.52) would also be satisfied for μn+1 (A). This is a contradiction to (9.56). Hence statement (b) is proved. Finally, suppose that (9.57) holds. By Lemma 9.35 it is clear that the spectrum of A is discrete in (−∞, μn (A) + ϵ0 ). Then for ϵ1 > 0 small enough, im E((−∞, μn (A)]) = im E((−∞, μn (A) + ϵ1 )), and by (9.53) dim im E((−∞, μn (A)]) ≥ n. So there are at least n eigenvalues λ1 ≤ λ2 ≤ ⋅ ⋅ ⋅ ≤ λn ≤ μn (A) for A. If λn were strictly less than μn (A), then we would have dim im E((−∞, λn ]) = n, which yields a contradiction to (9.52). So λn = μn (A), and μn (A) is an eigenvalue. This proves statement (a).

9.6 Notes

� 195

We note that the proof of (9.52) gives the following: Proposition 9.49. Suppose that there exist a and an n-dimensional subspace L ⊂ dom(A) such that (9.55) is satisfied. Then μn (A) ≤ a. Using Proposition 9.48, we now get the following: Corollary 9.50. Under the same assumptions as in Proposition 9.49, if a is below the bottom of the essential spectrum of A, then A has at least n eigenvalues (counted with multiplicities). The last results allow us to compare the spectra of two operators. If (Au, u) ≤ (Bu, u) for all u ∈ dom(B) ⊆ dom(A), then λn (A) ≤ λn (B).

9.6 Notes Most of the material on spectral theory is taken from [85] and [24]. It would be beyond the scope of this book to provide a complete proof of the spectral theorem. The notion of the spectral decomposition is explained in detail for both the bounded and the unbounded self-adjoint operators. For the complete proof, some aspects of Banach algebras and the Gelfand transform are needed, which appear to be not important enough for a thorough understanding of the spectral properties of the 𝜕-Neumann operator. See [85] for a complete proof of the spectral theorem and [24] for a different approach using the Helffer–Sjöstrand formula. The use of the square root of the 𝜕-Neumann operator and of the ◻-operator, as done in Remark 9.12 and Example 9.44, gives a better understanding of the basic estimate (4.38) and other fundamental results such as Theorem 8.6. Lemma 9.33 is taken from [24]. It is very useful for operators whose eigenvectors can be determined explicitly; it will be of importance in Chapter 14. In the next chapters on operators of mathematical physics, certain aspects of spectral theory developed in this chapter will be crucial. Variational methods in spectral analysis (see [51, 82]) are important to explain special properties of Schrödinger operators like diamagnetism and paramagnetism.

10 Schrödinger operators and Witten Laplacians In this chapter, we collect basic information about Schrödinger operators with magnetic field in real dimension 2 and the Witten Laplacian for complex dimension ≥ 2. It turns out that there is a close relationship between these operators and the weighted ◻-operator of the 𝜕-complex. In addition, we include the definition of Dirac and Pauli operators, for which, later on, spectral properties are investigated as applications of corresponding results on the weighted ◻-operator of the 𝜕-complex. First, we will study the regularity of the solutions for elliptic differential equations of order two.

10.1 Difference quotients To apply Sobolev space theory, we are forced to study difference quotient approximations to weak derivatives. Definition 10.1. Let Ω ⊂ ℝn be a bounded domain, let u ∈ L2loc (Ω), and suppose that V ⋐ Ω. The jth difference quotient of size h is Dhj u(x) =

u(x + hej ) − u(x) h

for j = 1, . . . , n, where x ∈ V and h ∈ ℝ, 0 < |h| < dist(V , bΩ). Further, we denote Dh u := (Dh1 u, . . . , Dhn u). Proposition 10.2. (i) Let u ∈ W 1 (Ω). Then for each V ⋐ Ω, we have 󵄩󵄩 h 󵄩󵄩 󵄩󵄩D u󵄩󵄩L2 (V ) ≤ C‖∇u‖L2 (Ω)

(10.1)

for some constant C > 0 and all h with 0 < |h| < 21 dist(V , bΩ). (ii) Assume that u ∈ L2 (V ) and that there exists a constant C > 0 such that 󵄩󵄩 h 󵄩󵄩 󵄩󵄩D u󵄩󵄩L2 (V ) ≤ C for all h with 0 < |h|
0 such that n

∑ ajk (x)tj t k ≥ C|t|2

j,k=1

(10.6)

for almost every x ∈ Ω and all t ∈ ℂn . Ellipticity means that the symmetric (n × n) matrix A(x) = (ajk (x))nj,k=1 is positive definite with the smallest eigenvalue greater than or equal to C. If ajk = δjk and bj = c = 0, then L = −△. Definition 10.4. Let f ∈ L2 (Ω). We say that a function u ∈ H 1 (Ω) is a weak solution to the elliptic partial differential equation Lu = f

in Ω

if for the bilinear form n

n

j,k=1

j=1

a(u, v) = ∫( ∑ ajk uxj vxk + ∑ bj uxj v + cuv) dλ, Ω

we have a(u, v) = (f , v)

10.2 Interior regularity

� 199

for all v ∈ H01 (Ω), where (⋅, ⋅) denotes the inner product in L2 (Ω). In the next proposition, we show what is called the interior H 2 -regularity of the operator L. Proposition 10.5. Let L be as in the above definition, and let f ∈ L2 (Ω). Suppose that 2 u ∈ H 1 (Ω) is a weak solution of Lu = f . Then u ∈ Hloc (Ω), and for each open V ⋐ Ω, we have ̃ ‖ 2 + ‖u‖ 1 ), ‖u‖H 2 (V ) ≤ C(‖f H (Ω) L (Ω)

(10.7)

where C̃ > 0 is a constant only depending on V , Ω, and the coefficients of L. Proof. Choose an open set W such that V ⋐ W ⋐ Ω. Next, select a smooth cut-off function ψ with 0 ≤ ψ ≤ 1, ψ = 1 on V , and ψ = 0 on ℝn \ W . Since u is a weak solution of Lu = f , we have a(u, v) = (f , v) for all v ∈ H01 (Ω), and hence n

∑ ∫ ajk uxj vxk dλ = ∫ f ̃v dλ,

j,k=1 Ω

(10.8)

Ω

where n

f ̃ := f − ∑ bj uxj − cu.

(10.9)

j=1

Let ℓ ∈ {1, . . . , n}, and let h ∈ ℝ be such that |h| > 0 is small. We substitute 2 h v = −D−h ℓ (ψ Dℓ u)

(10.10)

into (10.8), where Dhℓ u denotes the difference quotient (Definition 10.1). For the left-hand side of (10.8), we get n

n

2 h h 2 h − ∑ ∫ ajk uxj [D−h ℓ (ψ Dℓ u)]x dλ = ∑ ∫ Dℓ (ajk uxj )(ψ Dℓ u)x dλ j,k=1 Ω

−

k

−

k

j,k=1 Ω n

h = ∑ ∫[ajk (Dhℓ uxj )(ψ2 Dhℓ u)x + (Dhℓ ajk )uxj (ψ2 Dhℓ u)x ] dλ, −

k

j,k=1 Ω

where we used the formulas h ∫ vD−h ℓ w dλ = − ∫ wDℓ v dλ Ω

and

Ω

−

k

200 � 10 Schrödinger operators and Witten Laplacians Dhℓ (vw) = vh Dhℓ w + wDhℓ v for vh (x) := v(x + heℓ ). We continue the computation of the left-hand side of (10.8) and obtain n

n

h h (Dhℓ uxj )(Dhℓ uxk ) ψ2 dλ + ∑ ∫[ajk (Dhℓ uxj )(Dhℓ u) 2ψψxk ∑ ∫ ajk −

j,k=1 Ω

−

j,k=1 Ω

− 2

+ (Dhℓ ajk )uxj (Dhℓ uxk ) ψ + (Dhℓ ajk )uxj (Dhℓ u) 2ψψxk ] dλ = T1 + T2 . −

The first term can be estimated from below using ellipticity (10.6): 󵄨 󵄨2 |T1 | ≥ C ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ.

(10.11)

Ω

For the second term T2 , we have by the assumptions on ajk , bj , and c that there exists a constant C ′ > 0 such that 󵄨 󵄨󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 |T2 | ≤ C ′ ∫[ψ󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 󵄨󵄨󵄨Dhℓ u󵄨󵄨󵄨 + ψ󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 |∇u| + ψ󵄨󵄨󵄨Dhℓ u󵄨󵄨󵄨 |∇u|] dλ. Ω

Taking into account that ψ = 0 on ℝn \ W and using the small–large constant trick, we get C ′ 󵄨󵄨 h 󵄨󵄨2 󵄨 󵄨2 |T2 | ≤ ϵ ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ + ∫ [󵄨󵄨Dℓ u󵄨󵄨 + |∇u|2 ] dλ. ϵ W

Ω

Now choose ϵ = C/2 and estimate (10.1) to obtain |T2 | ≤

C 󵄨 󵄨2 ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ + C ′′ ∫ |∇u|2 dλ. 2 Ω

Ω

Hence by (10.10) we see that the left-hand side of (10.8) can be estimated from below by C 󵄨 󵄨2 ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ − C ′′ ∫ |∇u|2 dλ. 2 Ω

(10.12)

Ω

The absolute value of the right-hand side of (10.8) is certainly less than C ′′ ∫(|f | + |∇u| + |u|)|v| dλ. Ω

Using Proposition 10.2(i), we derive 󵄨 󵄨2 ∫ |v|2 dλ ≤ C ′′ ∫󵄨󵄨󵄨∇(ψ2 Dhℓ u)󵄨󵄨󵄨 dλ

Ω

Ω

(10.13)

10.2 Interior regularity

� 201

󵄨 󵄨2 󵄨 󵄨2 ≤ C ′′ ∫ (󵄨󵄨󵄨Dhℓ u󵄨󵄨󵄨 + ψ4 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 ) dλ W

󵄨 󵄨2 ≤ C ∫(|∇u|2 + ψ4 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 ) dλ. ′′

Ω

Again by the small–large constant trick and by (10.13) we obtain now that the absolute value of the right-hand side of (10.8) can be estimated from above by C′ 󵄨 󵄨2 ϵ ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ + ∫(|f |2 + |u|2 + |∇u|2 ) dλ. ϵ Ω

Ω

Let ϵ = C/4. Then the absolute value of the right-hand side of (10.8) can be estimated from above by C 󵄨 󵄨2 ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ + C ′′′ ∫(|f |2 + |u|2 + |∇u|2 ) dλ. 4 Ω

(10.14)

Ω

Finally, combining (10.8), (10.12), and (10.14), we see that 󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ ≤ ∫ ψ2 󵄨󵄨󵄨Dhℓ ∇u󵄨󵄨󵄨 dλ ≤ C̃ ∫(|f |2 + |u|2 + |∇u|2 ) dλ

V

Ω

Ω

for some constant C̃ > 0, ℓ = 1, . . . , n, and all sufficiently small |h| ≠ 0. 1 2 Using Proposition 10.2(ii), we derive that ∇u ∈ Hloc (Ω) and hence that u ∈ Hloc (Ω) with the estimate ̃ ‖ 2 + ‖u‖ 1 ). ‖u‖H 2 (V ) ≤ C(‖f H (Ω) L (Ω) Remark 10.6. (a) It is not difficult to show that even ̃ ‖ 2 + ‖u‖ 2 ) ‖u‖H 2 (V ) ≤ C(‖f L (Ω) L (Ω) in Proposition 10.5. 2 (b) The result that u ∈ Hloc (Ω) implies that Lu = f almost everywhere in Ω. To see this, note that for each v ∈ 𝒞0∞ (Ω), we have a(u, v) = (f , v), 2 and since u ∈ Hloc (Ω), we can integrate by parts and obtain

a(u, v) = (Lu, v). Thus (Lu − f , v) = 0 for all v ∈ 𝒞0∞ (Ω), and so Lu = f almost everywhere in Ω.

202 � 10 Schrödinger operators and Witten Laplacians

10.3 Schrödinger operators with magnetic field We consider differential operators H(A, V ) of the form H(A, V ) = − △A +V ,

(10.15)

where V : ℝn 󳨀→ ℝ is the electric potential, and n

A = ∑ Aj dxj , j=1

Aj : ℝn 󳨀→ ℝ,

j = 1, . . . , n,

is a 1-form, and n

△A = ∑( j=1

2

𝜕 − iAj ) . 𝜕xj

The 2-form B = dA = ∑ ( j 0 is a positive constant. Let dom(H(A, V )) = 𝒞0∞ (ℝn ). Then H(A, V ) is a symmetric semibounded operator on L2 (ℝn ). Proof. For u ∈ 𝒞0∞ (ℝn ), we have (H(A, V )u, u) = ∫ (− △A u + Vu) u dλ ℝn

n

= ∫ ∑ |Xj u|2 dλ + ∫ V |u|2 dλ ℝn j=1

ℝn

2

≥ −C ‖u‖ . Using the Friedrichs extension (Proposition 4.36), we obtain the following: Proposition 10.8. Let H(A, V ) be as in Proposition 10.7. Then H(A, V ) admits a selfadjoint extension. Proof. Define a(u, v) := (H(A, V )u, v) + (C + 1)(u, v), and define 𝒱 to be the completion of 𝒞0∞ (ℝn ) with respect to the inner product a(u, v). Then we can apply Proposition 4.36 to get the desired result. Let f , g ∈ L1loc (ℝn ). We say that f ≥ g in the distributional sense if (f , ϕ) ≥ (g, ϕ) for all positive ϕ ∈ 𝒞0∞ (ℝn ). A useful tool for spectral analysis of Schrödinger operators is Kato’s inequality, sometimes also called the diamagnetic inequality. Proposition 10.9. Let A ∈ 𝒞 2 (ℝn , ℝn ). Then for all f ∈ L1loc (ℝn ) such that (−i∇ − A)2 f ∈ L2loc (ℝn ), we have △ |f | ≥ −R(sgn(f )(−i∇ − A)2 f ) = R(sgn(f ) △A f ) in the distributional sense, where sgn is defined as in Chapter 5. Proof. Let A1 , . . . , An be the components of A. Notice that n

− △A f = (−i∇ − A)2 f = ∑(−i j=1

2

𝜕 − Aj ) f . 𝜕xj

(10.18)

204 � 10 Schrödinger operators and Witten Laplacians The assumption (−i∇−A)2 f ∈ L2loc (ℝn ) and the regularity property of second-order ellip2 tic operators (see Proposition 10.5) imply that f ∈ Wloc (ℝn ) and, in particular, △f , ∇f ∈ 1 n Lloc (ℝ ). First, suppose that u is smooth. Then, with |u|ϵ = √|u|2 + ϵ2 − ϵ, we get ∇|u|ϵ =

R(u∇u) R(u(∇ − iA)u) = . 2 2 √|u| + ϵ √|u|2 + ϵ2

(10.19)

A straightforward calculation shows that for a smooth function g, we have g △ g = div(g∇g) − |∇g|2 . Hence we obtain √|u|2 + ϵ2 △ |u|ϵ = div(√|u|2 + ϵ2 ∇|u|ϵ ) − 󵄨󵄨󵄨󵄨∇|u|ϵ 󵄨󵄨󵄨󵄨2

󵄨 󵄨2 = R[∇u ⋅ (∇ − iA)u + u div((∇ − iA)u)] − 󵄨󵄨󵄨∇|u|ϵ 󵄨󵄨󵄨

= R[(∇u − iAu) ⋅ (∇ − iA)u

󵄨 󵄨2 + (−iAu) ⋅ (∇ − iA)u + u div((∇ − iA)u)] − 󵄨󵄨󵄨∇|u|ϵ 󵄨󵄨󵄨 󵄨 󵄨2 󵄨 󵄨2 = 󵄨󵄨󵄨(∇ − iA)u󵄨󵄨󵄨 − 󵄨󵄨󵄨∇|u|ϵ 󵄨󵄨󵄨 + R[(−iAu) ⋅ (∇ − iA)u + u div((∇ − iA)u)].

An easy calculation shows that (−iAu) ⋅ (∇ − iA)u + u div((∇ − iA)u) = u (∇ − iA)2 u. From (10.19) we get |∇|u|ϵ |2 ≤

|u(∇ − iA)u|2 |u|2 |(∇ − iA)u|2 󵄨󵄨 󵄨2 = ≤ 󵄨󵄨(∇ − iA)u󵄨󵄨󵄨 . 2 2 2 2 |u| + ϵ |u| + ϵ

So we finally see that △ |u|ϵ ≥ R

u (∇ − iA)2 u . √|u|2 + ϵ2

(10.20)

The rest of the proof uses approximative units and follows the same lines as the proof of Proposition 5.7. Using Kato’s inequality and a criterion for essential self-adjointness, we obtain the following: Proposition 10.10. Let A ∈ 𝒞 2 (ℝn , ℝn ), V ∈ L2loc (ℝn ), and V ≥ 0. Then the Schrödinger operator H(A, V ) = − △A +V is essentially self-adjoint on 𝒞0∞ (ℝn ). In this case the Friedrichs

10.3 Schrödinger operators with magnetic field

� 205

extension is the uniquely determined self-adjoint extension (see Remark 4.27(b) and Proposition 10.8). Proof. By Proposition 9.29 it is sufficient to show that ker(H(A, V )∗ + I) = {0}. Since dom(H(A, V )∗ ) ⊆ L2 (ℝn ), the triviality of the kernel follows from the following statement: if − △A u + Vu + u = 0

(10.21)

for u ∈ L2 (ℝn ), then u = 0. If u ∈ L2 (ℝn ) and V ∈ L2loc (ℝn ), we have uV ∈ L1loc (ℝn ). In addition, we have the inclusion L2 (ℝn ) ⊂ L2loc (ℝn ) ⊂ L1loc (ℝn ), which follows from the estimate 1/2

∫ |u| dλ ≤ |K|1/2 (∫ |u|2 dλ) . K

K

Hence we have that u ∈ L1loc (ℝn ) and, by (10.21), that △u ∈ L1loc (ℝn ), where the derivative is taken in the sense of distributions. From (10.18) and (10.21) we obtain △|u| ≥ R(sgn(u) △A u)

= R(sgn(u) (V + 1)u) = |u| (V + 1) ≥ 0.

If (χϵ )ϵ is an approximate unit, then we get △ (χϵ ∗ |u|) = χϵ ∗ △|u| ≥ 0.

(10.22)

Since χϵ ∗ |u| ∈ dom(△), we have 󵄩 󵄩2 (△(χϵ ∗ |u|), χϵ ∗ |u|) = −󵄩󵄩󵄩∇(χϵ ∗ |u|)󵄩󵄩󵄩 ≤ 0.

(10.23)

By (10.22) the left side of (10.23) is nonnegative, so ∇(χϵ ∗|u|) = 0, and hence χϵ ∗|u| = c ≥ 0. Since |u| ∈ L2 (ℝn ) and χϵ ∗ |u| → |u| in L2 (ℝn ), it follows that c = 0. Hence χϵ ∗ |u| = 0, so |u| = 0, and thus u = 0. Before we return to applications of methods from complex analysis, we prove two results on the spectrum of Schrödinger operators:

206 � 10 Schrödinger operators and Witten Laplacians Proposition 10.11. Let A ∈ 𝒞 2 (ℝn , ℝn ), and let V ∈ L2loc (ℝn ) with V ≥ 0. Then inf σ(H(A, V )) ≥ inf σ(H(0, V )).

(10.24)

Proof. By Kato’s inequality (10.18) we have − △ |f | ≤ R(sgn(f )(− △A f )), so we get (|f |, H(0, V )|f |) ≤ ∫ |f | [R(sgn(f )(− △A f )) + V ] dλ ℝn

≤ R ∫ f H(A, V )f dλ ℝn

= (H(A, V )f , f ). Now we can apply Proposition 9.48 to obtain the desired result. Finally, we still mention the gauge invariance of the spectrum of H(A, V ). Proposition 10.12. Let A, A′ ∈ 𝒞 2 (ℝn , ℝn ) be such that dA = dA′ . Suppose that V ∈ L2loc (ℝn ) and V ≥ 0. Then σ(H(A, V )) = σ(H(A′ , V )). Proof. By the Poincaré lemma we have A′ = A + dg, where g ∈ 𝒞 1 (ℝn ). Let Xj = (−i 𝜕x𝜕 − Aj ) and Xj′ = (−i 𝜕x𝜕 − A′j ) for j = 1, . . . , n. Then j

j

Xj′ = eig Xj e−ig . Hence H(A′ , V ) = eig H(A, V ) e−ig . Therefore the operators H(A′ , V ) and H(A, V ) are unitarily equivalent and hence by Lemma 9.31 have the same spectrum. In our first application of methods from complex analysis, we will concentrate on Schrödinger operators with magnetic field on ℝ2 of the form − △A +B, where the 2-form dA is given by dA = Bdx ∧ dy. So in this case the Schrödinger operator − △A +B = H(A, B) is completely determined by the vector field A.

10.3 Schrödinger operators with magnetic field

� 207

Let φ be a subharmonic 𝒞 2 -function. We consider the inhomogeneous 𝜕-equation 𝜕u = f for f ∈ L2 (ℂ, e−φ ). The canonical solution operator to 𝜕 gives a solution with minimal L2 (ℂ, e−φ )-norm. We substitute v = u e−φ/2 and g = f e−φ/2 , and the equation becomes Dv = g , where D = e−φ/2

𝜕 φ/2 e . 𝜕z

(10.25)

Then u is the minimal solution to the 𝜕-equation in L2 (ℂ, e−φ ) if and only if v is the solution to Dv = g that is minimal in L2 (ℂ) . ∗ 𝜕 −φ/2 The formal adjoint of D is D = −eφ/2 𝜕z e . We define dom(D) = {f ∈ L2 (ℂ) : Df ∈ L2 (ℂ)} ∗

∗

and likewise for D . Then D and D are closed unbounded linear operators from L2 (ℂ) to itself. Further, we define ∗

∗

∗

dom(D D ) = {u ∈ dom(D ) : D u ∈ dom(D)}, ∗

∗

and we define D D as D ∘ D on this domain. Any function of the form eφ/2 g with g ∈ ∗ ∗ 𝒞02 (ℂ) belongs to dom(D D ), and hence dom(D D ) is dense in L2 (ℂ). Since D=

1 𝜕φ 𝜕 + 𝜕z 2 𝜕z

∗

and D = −

1 𝜕φ 𝜕 + , 𝜕z 2 𝜕z

we see that ∗

𝒮 = DD = −

2 𝜕2 1 𝜕φ 𝜕 1 𝜕φ 𝜕 1 󵄨󵄨󵄨 𝜕φ 󵄨󵄨󵄨 1 𝜕2 φ − + + 󵄨󵄨󵄨 󵄨󵄨󵄨 + . 𝜕z𝜕z 2 𝜕z 𝜕z 2 𝜕z 𝜕z 4 󵄨󵄨 𝜕z 󵄨󵄨 2 𝜕z𝜕z

(10.26)

It follows that 4𝒮 = −(

2

2

𝜕φ 𝜕φ 𝜕2 𝜕2 i 𝜕φ 𝜕 i 𝜕φ 𝜕 1 1 + )+ − + (( ) + ( ) ) + △ φ, 2 2 2 𝜕x 𝜕y 2 𝜕y 𝜕x 4 𝜕x 𝜕y 2 𝜕x 𝜕y

which implies that 𝒮 is a Schrödinger operator with magnetic field, namely 𝒮=

1 (− △A +B), 4

(10.27)

where the 1-form A = A1 dx + A2 dy is related to the weight φ by A1 = −𝜕y φ/2,

A2 = 𝜕x φ/2,

(10.28)

208 � 10 Schrödinger operators and Witten Laplacians 2

△A = (

2

𝜕 𝜕 − iA1 ) + ( − iA2 ) , 𝜕x 𝜕y

(10.29)

and the magnetic field Bdx ∧ dy satisfies B(x, y) = ∗

1 △ φ(x, y). 2

∗

Both operators D D and D D are nonnegative self-adjoint operators; see Lemmas 4.28 and 4.29. By Proposition 10.10 we now know that the operator 𝒮 with its domain described above is the uniquely determined self-adjoint extension of the Schrödinger operator with magnetic field 41 (− △A +B). ∗

Since 4D D = − △A + 21 △ φ, it follows that ((− △A + 21 △ φ)f , f ) ≥ 0 for f ∈ 𝒞02 (ℂ). ∗

Similarly, we show that 4D D = − △A − 21 △ φ, which implies 1 −2△A ≥ − △A + △ φ ≥ − △A . 2 It follows that ∗

∗

D D = e−φ/2 𝜕φ 𝜕 eφ/2

(10.30)

and that ∗

∗

D D = e−φ/2 𝜕 𝜕φ eφ/2 , ∗

𝜕 where 𝜕φ = − 𝜕z +

𝜕φ . 𝜕z

(10.31)

For n = 1, we have ∗

◻φ = 𝜕 𝜕φ , which means that ∗

D D = e−φ/2 ◻φ eφ/2 .

(10.32)

Theorem 10.13. Let φ be a subharmonic 𝒞 2 -function on ℂ such that △φ(z) > ϵ > 0. Then the Schrödinger operator with magnetic field ∗

𝒮 = DD =

1 1 (− △A + △ φ) 4 2

(10.33)

= e−φ/2 Nφ eφ/2 ,

(10.34)

has a bounded inverse on L2 (ℂ), ∗ −1

(D D )

10.4 Witten Laplacians

�

209

where Nφ = ◻−1 φ . ∗

Proof. We know that 4D D = − △A − 21 △ φ, which implies that −△A ≥

1 △ φ ≥ 0. 2

Hence, for u ∈ 𝒞0∞ (ℂ), we have ((− △A +B)u, u) = (− △A u, u) + (Bu, u) ≥ (Bu, u),

(10.35)

and by a similar density argument as in Proposition 6.2 we obtain from (10.35) together with the assumption on φ that 󵄩󵄩 ∗ 󵄩󵄩 󵄩󵄩D u󵄩󵄩 ≥ C‖u‖

(10.36)

∗

for all u ∈ dom(D ). ∗ Now we take 𝒱 = dom(D ) endowed with the topology given by the inner prod∗ ∗ ∗ ∗ uct (D u, D v), the sesquilinear form a(u, v) = (D u, D v), and H = L2 (ℂ) and apply Proposition 4.36 to get the desired result.

10.4 Witten Laplacians For several complex variables, the situation is more complicated. Let φ : ℂn 󳨀→ ℝ be a 𝒞 2 -weight function. We consider the 𝜕-complex 𝜕

𝜕

L2 (ℂn , e−φ ) 󳨀→ L2(0,1) (ℂn , e−φ ) 󳨀→ L2(0,2) (ℂn , e−φ ) . For v ∈ L2 (ℂn ), let n

D1 v = ∑ ( k=1

𝜕v 1 𝜕φ + v) dzk , 𝜕zk 2 𝜕zk

and for g = ∑nj=1 gj dzj ∈ L2(0,1) (ℂn ), let ∗

n

D1 g = ∑( j=1

𝜕gj 1 𝜕φ g − ), 2 𝜕zj j 𝜕zj

where the derivatives are taken in the sense of distributions. It is easy to see that 𝜕u = f for u ∈ L2 (ℂn , e−φ ) and f ∈ L2(0,1) (ℂn , e−φ ) if and only if D1 v = g, where v = u e−φ/2 and g = f e−φ/2 . It is also clear that the necessary condition 𝜕f = 0 for solvability holds if and only if D2 g = 0. Here

210 � 10 Schrödinger operators and Witten Laplacians n

D2 g = ∑ ( j,k=1

𝜕gj 𝜕zk

1 𝜕φ g ) dzk ∧ dzj , 2 𝜕zk j

+

and n

∗

D2 h = ∑ ( j,k=1

′

for a suitable (0, 2)-form h = ∑

𝜕hkj 1 𝜕φ hkj − ) dzj 2 𝜕zk 𝜕zk

hJ dzJ .

|J|=2

We consider the corresponding D-complex D1

D2

←󳨀

←󳨀

L2 (ℂn ) 󳨀→ L2(0,1) (ℂn ) 󳨀→ L2(0,2) (ℂn ). ∗ D1

∗ D2

The so-called Witten Laplacians (see [50]) Δ(0,0) and Δ(0,1) are defined by φ φ ∗

Δ(0,0) = D1 D1 , φ ∗

∗

Δ(0,1) = D1 D1 + D2 D2 . φ

(10.37)

A computation shows that n

∗

D1 D1 v = ∑( j=1

1 𝜕φ 𝜕v 1 𝜕2 φ 𝜕2 v 1 𝜕φ 𝜕v 1 𝜕φ 𝜕φ + v− − v− ) 2 𝜕zj 𝜕zj 4 𝜕zj 𝜕zj 2 𝜕zj 𝜕zj 2 𝜕zj 𝜕zj 𝜕zj 𝜕zj

and that ∗

∗

n

n

(D1 D1 + D2 D2 )g = ∑ [∑( k=1 j=1

−

𝜕2 gk 𝜕2 φ 1 𝜕φ 𝜕gk 1 𝜕2 φ − gk − + g )] dzk . 2 𝜕zj 𝜕zj 2 𝜕zj 𝜕zj 𝜕zj 𝜕zj 𝜕zj 𝜕zk j

More generally, we set Zk = ∑′|J|=q

1 𝜕φ 𝜕gk 1 𝜕φ 𝜕φ + g 2 𝜕zj 𝜕zj 4 𝜕zj 𝜕zj k

𝜕 𝜕zk

+ ′

1 𝜕φ 2 𝜕zk

and Zk∗ = − 𝜕z𝜕 + k

1 𝜕φ , 2 𝜕zk

(10.38)

and we consider (0, q)-

forms h = hJ dzJ , where ∑ means that we sum up only increasing multi-indices J = (j1 , . . . , jq ), and where dzJ = dzj1 ∧ ⋅ ⋅ ⋅ ∧ dzjq . We define n

′

Dq+1 h = ∑ ∑ Zk (hJ ) dzk ∧ dzJ k=1 |J|=q

and ∗

n

′

Dq h = ∑ ∑ Zk∗ (hJ ) dzk ⌋dzJ , k=1 |J|=q

10.4 Witten Laplacians

� 211

where dzk ⌋dzJ denotes the contraction or interior multiplication by dzk , i. e., we have ⟨α, dzk ⌋dzJ ⟩ = ⟨dzk ∧ α, dzJ ⟩ for each (0, q − 1)-form α. The complex Witten Laplacian on (0, q)-forms is then given by ∗

∗

Δ(0,q) = Dq Dq + Dq+1 Dq+1 φ

(10.39)

for q = 1, . . . , n − 1. The general D-complex has the form Dq

Dq+1

L2(0,q−1) (ℂn ) 󳨀→ L2(0,q) (ℂn ) 󳨀→ L2(0,q+1) (ℂn ). ←󳨀

←󳨀

∗ Dq

∗ Dq+1

It follows that Dq+1 Δ(0,q) = Δ(0,q+1) Dq+1 φ φ

∗

∗

Dq+1 Δ(0,q+1) = Δ(0,q) Dq+1 . φ φ

and

We remark that n

∗

′

′

Dq h = ∑ ∑ Zk∗ (hJ ) dzk ⌋dzJ = ∑ k=1 |J|=q

n

∑ Zk∗ (hkK ) dzK .

|K|=q−1 k=1

In particular, for a function v ∈ L2 (ℂn ), we get n

∗

∗ Δ(0,0) φ v = D1 D1 v = ∑ Zj Zj (v), j=1

and for a (0, 1)-form g = ∑nℓ=1 gℓ dzℓ ∈ L2(0,1) (ℂn ), we obtain ∗

∗

Δ(0,1) φ g = (D1 D1 + D2 D2 )g n

= ∑ {Zj (Zk∗ (gℓ )) dzj ∧ (dzk ⌋dzℓ ) + Zk∗ (Zj (gℓ )) dzk ⌋(dzj ∧ dzℓ )} j,k,ℓ=1 n

= ∑ {Zk∗ (Zj (gℓ )) (dzj ∧ (dzk ⌋dzℓ ) + dzk ⌋(dzj ∧ dzℓ )) j,k,ℓ=1

+ [Zj , Zk∗ ](gℓ ) dzj ∧ (dzk ⌋dzℓ )} n

n

𝜕2 φ g δ dz 𝜕zj 𝜕zk ℓ kℓ j j,k,ℓ=1

= ∑ Zj∗ Zj (gℓ ) dzℓ + ∑ j,ℓ=1

= (Δ(0,0) ⊗ I)g + Mφ g, φ

212 � 10 Schrödinger operators and Witten Laplacians where we used that for (0, 1)-forms α, a, b, we have α⌋(a ∧ b) = (α⌋a) ∧ b − a ∧ (α⌋b), which implies that dzj ∧ (dzk ⌋dzℓ ) + dzk ⌋(dzj ∧ dzℓ )

= dzj ∧ (dzk ⌋dzℓ ) + (dzk ⌋dzj ) ∧ dzℓ − dzj ∧ (dzk ⌋dzℓ )

= (dzk ⌋dzj ) ∧ dzℓ = δkℓ dzℓ , and where we set n

n

𝜕2 φ g ) dzj 𝜕zk 𝜕zj k k=1

Mφ g = ∑( ∑ j=1

and n

gk dzk . (Δ(0,0) ⊗ I) g = ∑ Δ(0,0) φ φ k=1

By Proposition 6.5 we now obtain the following: Theorem 10.14. Let φ : ℂn 󳨀→ ℝ be a 𝒞 2 -plurisubharmonic function and suppose that the lowest eigenvalue μφ of the Levi matrix Mφ of φ satisfies (6.5). Then the operator Δ(0,1) φ has a bounded inverse on L2(0,1) (ℂn ), ∗

∗

−1

(D1 D1 + D2 D2 )

= (Δ(0,1) φ )

−1

= e−φ/2 Nφ eφ/2 ,

(10.40)

where Nφ = ◻−1 φ .

10.5 Dirac and Pauli operators There is an interesting connection to Dirac and Pauli operators: recall (10.29) and define the Dirac operator 𝒟 by 𝒟 = (−i

𝜕 𝜕 − A1 ) σ1 + (−i − A2 ) σ2 = 𝒜1 σ1 + 𝒜2 σ2 , 𝜕x 𝜕y

where σ1 = ( Hence we can write

0 1

1 ), 0

σ2 = (

0 i

−i ). 0

(10.41)

10.6 Notes

𝒟=(

0

𝒜1 + i 𝒜2

𝒜1 − i 𝒜2

� 213

).

0

We remark that i(𝒜2 𝒜1 − 𝒜1 𝒜2 ) = B, and hence it turns out that the square of 𝒟 is diagonal with the Pauli operators P± on the diagonal: 2

𝒟 =(

=(

𝒜21 − i(𝒜2 𝒜1 − 𝒜1 𝒜2 ) + 𝒜22

0

P− 0

0

𝒜21 + i(𝒜2 𝒜1 − 𝒜1 𝒜2 ) + 𝒜22

)

0 ), P+

where P± = (−i

2

2

𝜕 𝜕 − A1 ) + (−i − A2 ) ± B = − △A ±B. 𝜕x 𝜕y

By Lemmas 4.28 and 4.29 the Pauli operators P± are nonnegative self-adjoint operators. It follows that 4𝒮 = P+ is the Schrödinger operator with magnetic field and that (10.42)

4 Δ(0,0) = P− . φ

In addition, we obtain that 𝒟2 is self-adjoint and likewise 𝒟 by the spectral theorem. Finally, we consider decoupled weights φ(z1 , . . . , zn ) = ∑nj=1 φj (zj ). In this case the

operator Δ(0,1) acts diagonally on (0, 1)-forms, each component Ek of the diagonal being φ Ek =

1 (k) 1 (j) P + ∑ P− , 4 + 4 j=k̸

(10.43)

where P±(ℓ) = (−i 1 with zℓ = xℓ + iyℓ , A(ℓ) 1 = −2

2

2

𝜕 𝜕 (ℓ) (ℓ) − A(ℓ) 1 ) + (−i 𝜕y − A2 ) ± B 𝜕xℓ ℓ

𝜕φℓ , A(ℓ) 2 𝜕yℓ

=

1 𝜕φℓ , and B(ℓ) 2 𝜕xℓ

from (10.38) for a decoupled weight and from (10.42).

=

1 2

(10.44)

△ φℓ , ℓ = 1, . . . , n. This follows

10.6 Notes The regularity result on second-order elliptic partial differential operators is taken from [27]. Basic facts about Schrödinger operators with magnetic fields can be found in [21,

214 � 10 Schrödinger operators and Witten Laplacians 51, 52, 54]. Kato’s inequality, sometimes also called the diamagnetic inequality, is an important tool for basic questions about Schrödinger operators; see, for instance, [51]. The relationship to the 𝜕-equation appears in [8, 40, 46]. In the following, we will describe the spectral analysis of these operators in more detail. The weighted 𝜕-complex in several variables leads to the Witten Laplacian. The Witten Laplacians were introduced by E. Witten on a compact manifold in connection with a Morse function, which is a function with nondegenerate critical points. The main idea of E. Witten was that the dimension of the kernels of these Laplacians, which are related by the Hodge theory to the Betti numbers, can be estimated from above by the dimension of the eigenspace associated with the small eigenvalues of these operators (see [50, 55, 56, 61, 95, 96]). The Dirac and Pauli operators are discussed in [21, 54, 92].

11 Compactness In this chapter, we study the compactness of the 𝜕-Neumann operator. We give a general characterization using a description of precompact subsets in L2 -spaces. Therefore we begin this part with a general characterization of precompact subsets in L2 -spaces, followed by basic facts about Sobolev spaces, such as Gårding’s inequality and the Rellich– Kondrachov lemma. We point out that our approach includes the case of L2 (ℂn , e−φ ), where the classical Rellich–Kondrachov lemma cannot be used. In our discussion of Schrödinger operators with magnetic field, we follow the approach of Iwatsuka [59] and give a general characterization of compact resolvents and a necessary condition for compact resolvents in terms of the magnetic field. A sufficient condition for the compactness of the 𝜕-Neumann operator is given in terms of the behavior at infinity of the eigenvalues of the Levi matrix of the weight function. We also show that properties P and P̃ imply the compactness of the 𝜕-Neumann operator for smoothly bounded pseudoconvex domains. In addition, we characterize the compactness in the Segal–Bargmann space.

11.1 Precompact sets in L2 -spaces A set A is precompact (i. e., A is compact) in a Banach space X if and only if for every positive number ϵ, there is a finite subset Nϵ of points of X such that A ⊂ ⋃y∈Nϵ Bϵ (y). Such a set Nϵ is called a finite ϵ-net for A. We recall the Arzelà–Ascoli theorem: Let Ω be a bounded domain in ℝn . A subset K of 𝒞 (Ω) is precompact in 𝒞 (Ω) if the following two conditions hold: (i) There exists a constant M such that |ϕ(x)| ≤ M for all ϕ ∈ K and x ∈ Ω (boundedness); (ii) For every ϵ > 0, there exists δ > 0 such that if ϕ ∈ K, x, y ∈ Ω, and |x − y| < δ, then |ϕ(x) − ϕ(y)| < ϵ (equicontinuity). Let (χϵ )ϵ be an approximation to the identity (see Section 5.1). Recall that u∗χϵ ∈ 𝒞 ∞ (ℝn ) if u ∈ L1loc (ℝn ) (Lemma 5.3). In a similar way, we prove the following result: If Ω is a domain in ℝn and u ∈ L2 (Ω), then u ∗ χϵ ∈ L2 (Ω), and ‖u ∗ χϵ ‖2 ≤ ‖u‖2 ,

lim ‖u ∗ χϵ − u‖2 = 0.

ϵ→0+

Let Ω ⊆ ℝn be a domain, and let u bea complex-valued function on Ω. Let u(x), x ∈ Ω,

̃ u(x) ={

0,

https://doi.org/10.1515/9783111182926-011

x ∈ ℝn \ Ω.

216 � 11 Compactness Theorem 11.1. A bounded subset 𝒜 of L2 (Ω) is precompact in L2 (Ω) if and only if for every ϵ > 0, there exist δ > 0 and a subset ω ⋐ Ω such that for all u ∈ 𝒜 and h ∈ ℝn with |h| < δ, we have the following inequalities: 2 󵄨 ̃ ̃ 󵄨󵄨󵄨󵄨 dλ(x) < ϵ2 , + h) − u(x) ∫󵄨󵄨󵄨u(x

Ω

󵄨2 󵄨 ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x) < ϵ2 .

(11.1)

Ω\ω

Proof. Let τh u(x) = u(x + h) denote the translate of u by h. First, assume that 𝒜 is precompact. Since 𝒜 has a finite ϵ/6-net, and since 𝒞0 (Ω), the space of continuous functions with compact support in Ω, is dense in L2 (Ω), there exists a finite set S ⊂ 𝒞0 (Ω) such that for each u ∈ 𝒜, there exists ϕ ∈ S satisfying ‖u − ϕ‖2 < ϵ/3. Let ω be the union of the supports of finitely many functions in S. Then ω ⋐ Ω, and the second inequality follows immediately. To prove the first inequality, choose a closed ball Br of radius r centered at the origin and containing ω. Note that (τh ϕ − ϕ)(x) = ϕ(x + h) − ϕ(x) is uniformly continuous and vanishes outside Br+1 , provided that |h| < 1. Hence 󵄨 󵄨2 lim ∫ 󵄨󵄨󵄨τh ϕ(x) − ϕ(x)󵄨󵄨󵄨 dλ(x) = 0, |h|→0 ℝn

the convergence being uniform for ϕ ∈ S. For |h| sufficiently small, we have ‖τh ϕ − ϕ‖2 < ϵ/3. If ϕ ∈ S satisfies ‖u − ϕ‖2 < ϵ/3, then also ‖τh ũ − τh ϕ‖2 < ϵ/3. Hence for |h| sufficiently small (independent of u ∈ 𝒜), we have ‖τh ũ − u‖̃ 2 ≤ ‖τh ũ − τh ϕ‖2 + ‖τh ϕ − ϕ‖2 + ‖ϕ − u‖̃ 2 < ϵ, and the first inequality follows. It is sufficient to prove the converse for the particular case Ω = ℝn , as it follows for general Ω from its application in this case to the set 𝒜̃ = {ũ : u ∈ 𝒜}. Let ϵ > 0 be given and choose ω ⊂⊂ ℝn such that for all u ∈ 𝒜, ϵ 󵄨 󵄨2 ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x) < . 3

ℝn \ω

For any η > 0, the function u ∗ χη ∈ 𝒞 ∞ (ℝn ), and in particular it belongs to 𝒞 (ω). If ϕ ∈ 𝒞0 (ℝn ), then by Hölder’s inequality 󵄨 󵄨󵄨󵄨2 󵄨󵄨 󵄨2 󵄨󵄨 󵄨󵄨χη ∗ ϕ(x) − ϕ(x)󵄨󵄨󵄨 = 󵄨󵄨󵄨 ∫ χη (y)(ϕ(x − y) − ϕ(x)) dλ(y)󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 n ℝ

󵄨 󵄨2 ≤ ∫ χη (y)󵄨󵄨󵄨τ−y ϕ(x) − ϕ(x)󵄨󵄨󵄨 dλ(y). Bη

Hence

11.1 Precompact sets in L2 -spaces �

217

‖χη ∗ ϕ − ϕ‖2 ≤ sup ‖τh ϕ − ϕ‖2 . h∈Bη

If u ∈ L2 (ℝn ), then let (ϕj )j be a sequence in 𝒞0 (ℝn ) converging to u in L2 norm. Then (χη ∗ ϕj )j is a sequence converging to χη ∗ u in L2 (ℝn ). Since also τh ϕj → τh u in L2 (ℝn ), we have ‖χη ∗ u − u‖2 ≤ sup ‖τh u − u‖2 . h∈Bη

From the first inequality in our assumption we derive that lim|h|→0 ‖τh u − u‖2 = 0 uniformly for u ∈ 𝒜. Hence limη→0 ‖χη ∗ u − u‖2 = 0 uniformly for u ∈ 𝒜. Fix η > 0 such that ϵ 󵄨 󵄨2 ∫󵄨󵄨󵄨χη ∗ u(x) − u(x)󵄨󵄨󵄨 dλ(x) < 6

ω

for all u ∈ 𝒜. We show that {χη ∗ u : u ∈ 𝒜} satisfies the conditions of the Arzelà–Ascoli theorem on ω and hence is precompact in 𝒞 (ω). We have 1/2 󵄨󵄨 󵄨 2 󵄨󵄨χη ∗ u(x)󵄨󵄨󵄨 ≤ (sup χη (y)) ‖u‖2 , n y∈ℝ

which is bounded uniformly for x ∈ ℝn and u ∈ 𝒜, since 𝒜 is bounded in L2 (ℝn ) and η is fixed. Similarly, 1/2 󵄨󵄨 󵄨 2 󵄨󵄨χη ∗ u(x + h) − χη ∗ u(x)󵄨󵄨󵄨 ≤ (sup χη (y)) ‖τh u − u‖2 , n y∈ℝ

and so lim|h|→0 χη ∗ u(x + h) = χη ∗ u(x) uniformly for x ∈ ℝn and u ∈ 𝒜. Thus {χη ∗ u : u ∈ 𝒜} is precompact in 𝒞 (ω), and there exists a finite set {ψ1 , . . . , ψm } of functions in 𝒞 (ω) such that if u ∈ 𝒜, then for some j, 1 ≤ j ≤ m, and all x ∈ ω, we have ϵ 󵄨󵄨 󵄨 . 󵄨󵄨ψj (x) − χη ∗ u(x)󵄨󵄨󵄨 < √ 6|ω| This, together with the inequality (|a| + |b|)2 ≤ 2(|a|2 + |b|2 ), implies that 󵄨 󵄨 󵄨 󵄨2 󵄨2 󵄨2 ∫ 󵄨󵄨󵄨u(x) − ψ̃ j (x)󵄨󵄨󵄨 dλ(x) = ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x) + ∫󵄨󵄨󵄨u(x) − ψj (x)󵄨󵄨󵄨 dx

ℝn

ℝn \ω

ω

ϵ 󵄨 󵄨2 󵄨 󵄨2 < + 2 ∫(󵄨󵄨󵄨u(x) − χη ∗ u(x)󵄨󵄨󵄨 + 󵄨󵄨󵄨χη ∗ u(x) − ψj (x)󵄨󵄨󵄨 ) dλ(x) 3 ω


0 and for each ω ⋐ Ω, there exists δ > 0 such that for all u ∈ 𝒜 and h ∈ ℝn with |h| < δ, we have 2 󵄨 ̃ ̃ 󵄨󵄨󵄨󵄨 dλ(x) < ϵ2 ; + h) − u(x) ∫󵄨󵄨󵄨u(x

(11.2)

ω

(ii) for every ϵ > 0, there exists ω ⋐ Ω such that for every u ∈ 𝒜, 󵄨 󵄨2 ∫ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x) < ϵ2 .

(11.3)

Ω\ω

(b) An analogous result holds in weighted spaces L2 (ℂn , e−φ ).

11.2 Sobolev spaces and Gårding’s inequality In the following, we will prove a simple version of Gårding’s inequality (coercive estimate), which will be used to investigate the compactness of the 𝜕-Neumann operator. For a comprehensive treatment of Gårding’s inequality, see, for instance, [31] or [16]. Definition 11.3. If Ω is a bounded open set in ℝn , then we define the Sobolev space H k (Ω) for a nonnegative integer k to be the completion of 𝒞 ∞ (Ω) with respect to the norm 1/2

󵄨 󵄨2 [ ∑ ∫󵄨󵄨󵄨𝜕α f 󵄨󵄨󵄨 dλ] , |α|≤k Ω

where α = (α1 , . . . , αn ) is a multi-index, |α| = ∑nj=1 αj , and 𝜕α f =

𝜕|α| f α . α 𝜕x1 1 . . . 𝜕xn n

By H0k (Ω) we denote the closure of 𝒞0∞ (Ω) with respect to the norm (11.4). If Ω is a domain with a 𝒞 1 -boundary, then H k (Ω) coincides with W k (Ω) = {f ∈ L2 (Ω) : 𝜕α f ∈ L2 (Ω), |α| ≤ k}, where the derivatives are taken in the sense of distributions (Proposition 5.10).

(11.4)

11.2 Sobolev spaces and Gårding’s inequality

� 219

Theorem 11.4. Let Ω be a bounded open set in ℝn with 𝒞 1 -boundary. Let D be a Dirichlet form of order 1 given by n

n

n

j,k=1

k=1

k=1

D(u, v) = ∑ (𝜕j u, bjk 𝜕k v) + ∑ (𝜕k u, bk v) + ∑ (u, b′k 𝜕k v) + (u, bv) for u, v ∈ H 1 (Ω), where 𝜕j =

𝜕 , 𝜕xj

(11.5)

bjk , bk , b′k , b are 𝒞 ∞ -coefficients, and the bjk are real-

valued. Suppose that there exists a constant C0 > 0 such that n

R ∑ bjk (x)ξj ξk ≥ C0 |ξ|2 , j,k=1

ξ ∈ ℝn ,

x ∈ Ω;

(11.6)

u ∈ H 1 (Ω);

(11.7)

then we say that D is strongly elliptic on Ω. Then there exist constants C > 0 and M ≥ 0 such that RD(u, u) ≥ C‖u‖2W 1 (Ω) − M‖u‖2L2 (Ω) ,

then we say that D is coercive over H 1 (Ω) (Gårding’s inequality). Proof. We first set ajk = 21 (bjk + bkj ). Since the bjk are real, strong ellipticity means that for some constant C0 > 0, n

n

j,k=1

j,k=1

∑ ajk ξj ξk = ∑ bjk ξj ξk ≥ C0 |ξ|2

for all ξ ∈ ℝn . Thus (ajk ) is symmetric (ajk = akj ), so if ξ ∈ ℂn is any complex vector, then n

n

j,k=1

j,k=1

R ∑ bjk ξj ξ k = ∑ ajk ξj ξ k ≥ C0 |ξ|2 . Setting ξ = ∇u for u ∈ H 1 (Ω), we obtain n

n

j,k=1

k=1

R ∑ bjk (𝜕j u)(𝜕k u) ≥ C0 ∑ |𝜕k u|2 , so integration over Ω yields n

n

j,k=1

k=1

R ∑ (𝜕j u, bjk 𝜕k v) ≥ C0 ∑ ‖𝜕k u‖2L2 (Ω) = C0 (‖u‖2W 1 (Ω) − ‖u‖2L2 (Ω) ). Also, for some C1 > 0 (independent of u), we have 󵄨󵄨 󵄨 󵄨󵄨(𝜕k u, bk u)󵄨󵄨󵄨 ≤ ‖u‖W 1 (Ω) ‖bk u‖L2 (Ω) ≤ C1 ‖u‖W 1 (Ω) ‖u‖L2 (Ω) ,

220 � 11 Compactness 󵄩 ′ 󵄩 󵄨󵄨 󵄨 ′ 󵄨󵄨(u, bk 𝜕k u)󵄨󵄨󵄨 ≤ ‖u‖L2 (Ω) 󵄩󵄩󵄩bk 𝜕k u󵄩󵄩󵄩L2 (Ω) ≤ C1 ‖u‖W 1 (Ω) ‖u‖L2 (Ω) , 󵄨 󵄨󵄨 2 󵄨󵄨(u, bu)󵄨󵄨󵄨 ≤ C1 ‖u‖L2 (Ω) ≤ C1 ‖u‖W 1 (Ω) ‖u‖L2 (Ω) . Setting C2 = (2n + 1)C1 , we have RD(u, u) ≥ C0 (‖u‖2W 1 (Ω) − ‖u‖2L2 (Ω) ) − C2 ‖u‖W 1 (Ω) ‖u‖L2 (Ω) . By the inequality cd ≤ 21 (c2 + d 2 ) for c, d > 0 we have C2 ‖u‖W 1 (Ω) ‖u‖L2 (Ω) ≤

C2 C0 ‖u‖2W 1 (Ω) + 2 ‖u‖2L2 (Ω) , 2 2C0

so RD(u, u) ≥

2C 2 + C22 C0 ‖u‖2L2 (Ω) , ‖u‖2W 1 (Ω) − 0 2 2C0

(11.8)

which proves Gårding’s inequality. Our next aim is to prove the classical Rellich–Kondrachov lemma, which states that the embedding of W 1 (Ω) into L2 (Ω) is compact, provided that Ω is a bounded domain with 𝒞 1 -boundary. Lemma 11.5 (Rellich–Kondrachov). Let Ω be a bounded domain with 𝒞 1 -boundary. Then the embedding j : W 1 (Ω) 󳨀→ L2 (Ω) is compact. Proof. We have to show that the unit ball in W 1 (Ω) is precompact in L2 (Ω). For this purpose, we apply Proposition 5.11 and consider the extension of elements of the unit ball in W 1 (Ω) to ℝn . Let ℱ denote the set of all these extensions. Then, by Proposition 5.11(iii), ℱ is a bounded set in L2 (ℝn ). By Lemma 5.3 we know that for each ϵ > 0, there exists N > 0 such that ‖χ1/k ∗ f − f ‖L2 (ℝn ) ≤ ϵ

(11.9)

for all f ∈ ℱ and k > N. By Hölder’s inequality we have ‖χ1/k ∗ f ‖L∞ (ℝn ) ≤ Ck ‖f ‖L2 (ℝn )

(11.10)

for all f ∈ ℱ , where Ck = ‖χ1/k ‖L2 (ℝn ) . Hence we can now verify the second condition in Theorem 11.1: Given ϵ > 0, there exists ω ⋐ Ω such that ‖f ‖L2 (Ω\ω) < ϵ for each f in the unit ball of W 1 (Ω): indeed, we consider the extensions to ℝn and write

11.2 Sobolev spaces and Gårding’s inequality

� 221

‖f ‖L2 (Ω\ω) ≤ ‖f − χ1/k ∗ f ‖L2 (ℝn ) + ‖χ1/k ∗ f ‖L2 (Ω\ω) using Proposition 5.11(iii) and (11.9), and in view of (11.10), we have to choose ω such that |Ω \ ω| is small enough. We are left to verify the first condition of Remark 11.2: Let ω ⋐ Ω and ϵ > 0, and consider first a function u ∈ 𝒞 ∞ (Ω). Let h ∈ ℝn be such that |h| < dist(ω, bΩ), and set v(t) := u(x + th),

t ∈ [0, 1].

Then v′ (t) = h ⋅ ∇u(x + th), and 1

1

u(x + h) − u(x) = v(1) − v(0) = ∫ v (t) dt = ∫ h ⋅ ∇u(x + th) dt. ′

0

(11.11)

0

Hence we obtain 1

󵄨󵄨 󵄨2 󵄨2 2 󵄨 󵄨󵄨u(x + h) − u(x)󵄨󵄨󵄨 ≤ |h| ∫󵄨󵄨󵄨∇u(x + th)󵄨󵄨󵄨 dt

(11.12)

0

and 1

󵄨 󵄨2 󵄨 󵄨2 ∫󵄨󵄨󵄨u(x + h) − u(x)󵄨󵄨󵄨 dλ(x) ≤ |h|2 ∫ ∫󵄨󵄨󵄨∇u(x + th)󵄨󵄨󵄨 dt dλ(x)

ω

ω 0 1

󵄨 󵄨2 = |h| ∫ ∫󵄨󵄨󵄨∇u(x + th)󵄨󵄨󵄨 dλ(x) dt 2

0 ω 1

󵄨 󵄨2 = |h|2 ∫ ∫ 󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dλ(x) dt. 0 ω+th

If |h| < dist(ω, bΩ), then there exists ω′ ⋐ Ω such that ω + th ⊂ ω′ for each t ∈ [0, 1]. Therefore we get the estimate 󵄨 󵄨2 ‖τh u − u‖2L2 (ω) ≤ |h|2 ∫ 󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dλ(x).

(11.13)

ω′

If u belongs to the unit ball in W 1 (Ω), then we approximate u by functions in 𝒞 ∞ (Ω) (Proposition 5.10), apply (11.13), and pass to the limit, getting 󵄨 󵄨2 ‖τh u − u‖2L2 (ω) ≤ |h|2 ∫ 󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dλ(x) ≤ |h|2 , ω′

which shows that the first condition of Remark 11.2 holds.

222 � 11 Compactness Remark 11.6. (a) If Ω ⊂ ℝn , n ≥ 2, is a bounded domain with 𝒞 1 -boundary, it even follows that W 1 (Ω) ⊂ Lq (Ω),

q ∈ [1, 2n/(n − 2))

and that the embedding is also compact (see, for instance, [12]). (b) It follows immediately from Lemma 11.5 that the embedding of ι : H01 (Ω) 󳨀→ L2 (Ω) is compact. By Theorem 2.5 the adjoint ι∗ : L2 (Ω) 󳨀→ (H01 (Ω)) =: H0−1 (Ω) ′

is also compact. In the following, we will still describe H0−1 (Ω) as a certain space of distributions, which will be helpful later on. Recall that the dual-norm is given by 󵄨 󵄨 ‖f ‖H −1 (Ω) = sup{󵄨󵄨󵄨f (u)󵄨󵄨󵄨 : u ∈ H01 (Ω), ‖u‖H 1 (Ω) ≤ 1}. 0

0

Proposition 11.7. Let f ∈ H0−1 (Ω). Then there exist functions f0 , f1 , . . . , fn in L2 (Ω) such that n

f (v) = ∫(f0 v + ∑ fj vxj ) dλ , Ω

1/2

n

‖f ‖H −1 (Ω) 0

(11.14)

j=1

󵄨 󵄨 = inf{(∫ ∑󵄨󵄨󵄨fj2 󵄨󵄨󵄨 dλ)

: f satisfies (11.14)}.

Ω j=0

(11.15)

We write n

f = f0 − ∑ j=1

𝜕fj 𝜕xj

whenever (11.14) holds. Proof. For u, v ∈ H01 (Ω), the inner product is given by (u, v) = ∫ ∇u ⋅ ∇v dλ + ∫ uv dλ. Ω

Ω

If f ∈ H0−1 (Ω), then the Riesz representation (Theorem 1.6) implies that there exists a unique function u ∈ H01 (Ω) such that

11.3 Magnetic Schrödinger operators with compact resolvent

� 223

f (v) = (u, v) ∀v ∈ H01 (Ω), and hence f (v) = ∫ ∇u ⋅ ∇v dλ + ∫ uv dλ, Ω

(11.16)

Ω

which gives (11.14), where f0 = u and fj = uxj , j = 1, . . . , n. By the Cauchy–Schwarz inequality we obtain 1/2

n

‖f ‖H −1 (Ω) ≤ (∫ ∑ |fj |2 dλ) , 0

Ω j=0

and setting v = u/‖u‖H 1 (Ω) in (11.16), we deduce 0

n

2

1/2

‖f ‖H −1 (Ω) = (∫ ∑ |fj | dλ) , 0

Ω j=0

which gives (11.15).

11.3 Magnetic Schrödinger operators with compact resolvent In this section, we consider the Schrödinger operators with magnetic field H(A, V ) as in Proposition 10.8 and suppose that the electric potential V satisfies V (x) ≥ 0 for all x ∈ ℝn . Recall that H(A, V ) has a compact resolvent if and only if the essential spectrum σe (H(A, V )) is empty (Proposition 9.39). We follow Iwatsuka [59] for a general characterization of magnetic Schrödinger operators with compact resolvent and a necessary condition in terms of A and V ; compare also with Propositions 9.46 and 9.48 about the bottom of the spectrum and of the essential spectrum of a self-adjoint semibounded operator. Proposition 11.8. Let s ∈ ℝ. The following conditions are equivalent: (a) inf σe (H(A, V )) ≥ s. (b) limR→∞ eA,V (ΩR ) ≥ s, where ΩR = {x ∈ ℝn : |x| > R}, and eA,V (ΩR ) = inf{

(H(A, V )ϕ, ϕ) : ϕ ∈ 𝒞0∞ (ΩR ), ϕ ≢ 0}. (ϕ, ϕ)

(c) There exists a continuous function Λ : ℝn 󳨀→ ℝ such that lim inf Λ(x) ≥ s |x|→∞

and

224 � 11 Compactness 󵄨2 󵄨 (H(A, V )ϕ, ϕ) ≥ ∫ Λ(x)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x) ℝn

for all ϕ ∈ 𝒞0∞ (ℝn ). Proof. (a)⇒(b): Since eA,V (ΩR ) is increasing in R, the limit limR→∞ eA,V (ΩR ) exists. Suppose that (a) holds but (b) does not. Then s > 0, and there is s′ < s such that eA,V (ΩR ) < s′ for all R > 0. We can find a sequence (ϕk )k ⊂ 𝒞0∞ (ℝn ) such that ‖ϕk ‖ = 1 and (H(A, V )ϕk , ϕk ) < s′ , where supp ϕk ⊂ {x ∈ ℝn : ak < |x| < ak+1 } for some increasing sequence (ak )k tending to infinity. Let E be the uniquely determined resolution of the identity on the Borel subsets of ℝ such that ∞

󵄩 󵄩2 (H(A, V )ϕk , ϕk ) = ∫ t d 󵄩󵄩󵄩E(t)ϕk 󵄩󵄩󵄩 . −∞

By (a) we have that E(t) = E((−∞, t]) is compact for t < s. Hence E(t)ϕk converges strongly to 0 as k → ∞, since ϕk converges weakly to 0 in L2 (ℝn ) (see Proposition 4.17). Now we get for 0 ≤ t < s that ∞

󵄩 󵄩2 (H(A, V )ϕk , ϕk ) = ∫ μ d 󵄩󵄩󵄩E(μ)ϕk 󵄩󵄩󵄩 0 ∞

󵄩 󵄩2 ≥ ∫ t d 󵄩󵄩󵄩E(μ)ϕk 󵄩󵄩󵄩 t

󵄩 󵄩2 = t(‖ϕk ‖2 − 󵄩󵄩󵄩E(t)ϕk 󵄩󵄩󵄩) → t

as k → ∞.

Hence lim infk→∞ (H(A, V )ϕk , ϕk ) ≥ t for all t ∈ [0, s), contradicting lim sup(H(A, V )ϕk , ϕk ) ≤ s′ k→∞

for our choice of the sequence (ϕk )k . (b)⇒(c): Let ϕ0 , ϕ1 be real-valued smooth functions on ℝ such that supp ϕ0 ⊂ (−∞, 2), supp ϕ1 ⊂ (1, ∞), and ϕ0 (r)2 + ϕ1 (r)2 = 1 for all r ∈ ℝ. We define the sequence (ζk )k≥0 of real-valued functions in 𝒞0∞ (ℝn ) by ζ0 (x) = ϕ0 (|x|) for x ∈ ℝn , ϕ1 (|x|) ζ1 (x) = { ϕ0 (|x|/2)

if |x| ≤ 2,

if |x| ≥ 2,

(11.17)

and ζk (x) = ζ1 (x/2k−1 ) for x ∈ ℝn and k ≥ 2. Then we have supp ζ0 ⊂ {x : |x| < 2}, 2 supp ζk ⊂ {x : 2k−1 < |x| < 2k+1 } for k ≥ 1, and ∑∞ k=0 ζk = 1. By Proposition 10.8 we know that

11.3 Magnetic Schrödinger operators with compact resolvent

� 225

n

(H(A, V )ϕ, ψ) = ∑(Xj ϕ, Xj ψ) + (V 1/2 ϕ, V 1/2 ψ) j=1

for ϕ, ψ ∈ 𝒞0∞ (ℝn ), where Xj = (−i 𝜕x𝜕 − Aj ), and as (H(A, V )ϕ, ϕ) is always real, we can j

write

∞

∞

k=0

k=0

(H(A, V )ϕ, ϕ) = (H(A, V )ϕ, ∑ ζk2 ϕ) = R( ∑ (H(A, V )ϕ, ζk2 ϕ). Now we compute 𝜕ζ − 󵄨 󵄨2 R((Xj (ϕ)(Xj (ζk2 ϕ)) ) = ζk2 󵄨󵄨󵄨Xj (ϕ)󵄨󵄨󵄨 + 2ζk k R(Xj (i|ϕ|2 )) 𝜕xj 󵄨2 󵄨 󵄨 󵄨2 󵄨󵄨 𝜕ζ 󵄨󵄨 = 󵄨󵄨󵄨Xj (ζk2 ϕ)󵄨󵄨󵄨 − 󵄨󵄨󵄨 k ϕ󵄨󵄨󵄨 . 󵄨󵄨 𝜕xj 󵄨󵄨

Hence we get that ∞

∞

k=0

k=0

󵄩 󵄩2 (H(A, V )ϕ, ϕ) = ∑ (H(A, V )(ζk ϕ), ζk ϕ) − ∑ 󵄩󵄩󵄩(∇ζk )ϕ󵄩󵄩󵄩 . Let e0 = eA,V ({x : |x| < 2}) and ek = eA,V (Ω2k−1 ) for k ≥ 1. Define ∞

∞

k=0

k=0

󵄨 󵄨2 Λ(x) = ∑ ek ζk (x)2 − ∑ 󵄨󵄨󵄨∇ζk (x)󵄨󵄨󵄨 . Then ∞

∞

k=0

k=0

󵄩 󵄩2 󵄨 󵄨2 (H(A, V )ϕ, ϕ) ≥ ∑ ek ‖ζk ϕ‖2 − ∑ 󵄩󵄩󵄩(∇ζk )ϕ󵄩󵄩󵄩 = ∫ Λ(x)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x). ℝn

Now let ϵ > 0. Using (b), we can find some k0 ≥ 2 such that ek ≥ s − ϵ for all k ≥ k0 . If 2k ≤ |x| < 2k+1 for k ≥ k0 , then by the definitions of (ζk )k and Λ we get that 󵄨 󵄨2 󵄨 󵄨2 Λ(x) = ek ζk (x)2 + ek+1 ζk+1 (x)2 − (󵄨󵄨󵄨∇ζk (x)󵄨󵄨󵄨 + 󵄨󵄨󵄨∇ζk+1 (x)󵄨󵄨󵄨 ) 5 󵄨 󵄨2 ≥ (s − ϵ) − 2k sup 󵄨󵄨󵄨∇ζ1 (x)󵄨󵄨󵄨 . 2 x∈ℝn Letting |x| → ∞, we get (c). (c)⇒(a): Let τ ∈ σe (H(A, V )). By Lemma 9.36 there exists a sequence (uk )k in dom(H(A, V )) such that ‖uk ‖ = 1 for each k ∈ ℕ, uk converges weakly to 0, and 󵄩 󵄩 lim 󵄩󵄩󵄩(H(A, V ) − τI)uk 󵄩󵄩󵄩 = 0.

k→∞

226 � 11 Compactness ̃ ̃ We denote Λ(x) = min(Λ(x), s) and have by (c) that Λ(x) → s as |x| → ∞ and ̃ 󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨2 dλ(x). (H(A, V )ϕ, ϕ) ≥ ∫ Λ(x) 󵄨 󵄨 ℝn

̃ Then H(A, V ) ≥ s + K, where K denotes the multiplication operator by (Λ(x) − s)1/2 . 2 2 1/2 We endow dom(H(A, V )) with the norm (‖u‖ + ‖H(A, V )u‖ ) and consider K as an operator defined on this normed space. We claim that the image of the unit ball of this space under K is a precompact set in L2 (ℝn ). For this, we can apply Theorem 11.1 and show that the first condition in Theorem 11.1 is satisfied by using the method in the proof of Lemma 11.5. The second condition in Theorem 11.1 is satisfied since for each ϵ > 0, there exists ω ⋐ ℝn such that 󵄨 󵄨2 ̃ − s)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x) < ϵ, ∫ (Λ(x)

ℝn \ω

̃ − s) → 0 as |x| → ∞. We have that (uk )k is a bounded sequence in L2 (ℝn ) because (Λ(x) and 󵄩󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩󵄩H(A, V )uk 󵄩󵄩󵄩 = 󵄩󵄩󵄩(H(A, V ) − τI)uk + τuk 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(H(A, V ) − τI)uk 󵄩󵄩󵄩 + ‖τuk ‖, and hence also (H(A, V )uk )k is bounded. On the other side, we now know that Kuk → 0 strongly as k → ∞, and hence (H(A, V )uk , uk ) ≥ s + (Kuk , uk ), where the left-hand side converges to τ, and the right-hand side converges to s as k → ∞. So τ ≥ s for any τ ∈ σe (H(A, V )), and we get (a). Now we are able to prove a general characterization of magnetic Schrödinger operators with compact resolvent. Theorem 11.9. The following conditions are equivalent: (a) H(A, V ) has a compact resolvent. (b) eA,V (ΩR ) → ∞ as R → ∞, where ΩR = {x ∈ ℝn : |x| > R}. (c) eA,V (Qx ) → ∞ as |x| → ∞, where Qx = {y ∈ ℝn : |x − y| < 1}. (d) There exists a continuous function Λ : ℝn 󳨀→ ℝ such that Λ(x) → ∞

as |x| → ∞ and

󵄨 󵄨2 (H(A, V )ϕ, ϕ) ≥ ∫ Λ(x)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x) ℝn

for all ϕ ∈ 𝒞0∞ (ℝn ).

11.3 Magnetic Schrödinger operators with compact resolvent � 227

Proof. Since H(A, V ) has compact resolvent if and only if σe (H(A, V )) = 0, the equivalence of (a) and (b) follows from the equivalence of (a) and (b) of Proposition 11.8. Since eA,V (Ω) ≥ eA,V (Ω′ ) if Ω ⊂ Ω′ , (b) implies (c). It is also easy to see that (d) implies (a): since Λ(x) → ∞ as |x| → ∞, we have lim inf|x|→∞ Λ(x) ≥ s for any s ∈ ℝ. Hence we can apply (c) of Proposition 11.8 and get (a) of Proposition 11.8 for any s ∈ ℝ. This implies σe (H(A, V )) = 0, and we get (a). It remains to prove that (c) implies (d): for this purpose, let ϕ0 be a real-valued 𝒞 ∞ (ℝn ) such that ϕ0 (x) = 1 if |x| ≤ 1/2 and supp ϕ0 ⊂ Q(0,...,0) . Now let L = {(l1 , . . . , ln ) : lj = kj /√n; kj ∈ ℤ, j = 1, . . . , n} and define Φ(x) = ∑l∈L ϕ0 (x − l)2 . Then we have Φ(x) ≥ 1 for all x ∈ ℝn , since for all x ∈ ℝn , there exists a point l ∈ L such that |x − l| ≤ 1/2. Set ζl (x) = ϕ0 (x − l)/√Φ(x). Then we have ζl ∈ 𝒞0∞ (ℝn ) and ζl (x) = ζ(0,...,0) (x − l) for l ∈ L, ∑l∈L ζl (x)2 = 1, and supp ζl ⊂ Ql . In a similar way as in the proof that (b) implies (c) of Proposition 11.8, we get now 󵄨 󵄨2 (H(A, V )ϕ, ϕ) ≥ ∫ Λ(x)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x) ℝn

for all ϕ ∈ 𝒞0∞ (ℝn ), where 󵄨 󵄨2 Λ(x) = ∑ el ζl (x)2 − ∑󵄨󵄨󵄨∇ζl (x)󵄨󵄨󵄨 l∈L

l∈L

with el = eA,V (Ql ). Since, by (c), el → ∞ as |l| → ∞, we obtain ∑l∈L el ζl (x)2 → ∞ as |x| → ∞. Since M(x + l) = M(x) for l ∈ L, M(x) ≡ ∑l∈L |∇ζl (x)|2 is bounded on ℝn , so we get Λ(x) → ∞ as |x| → ∞ and thus arrive at (d). The next aim is to find a necessary condition for magnetic Schrödinger operators with compact resolvent in terms of the magnetic field and electric potential. For this purpose, the gauge invariance of the spectrum of H(A, V ) (Proposition 10.12) will be of importance. A vector potential A is considered as a 1-form A = ∑j Aj dxj . The 2-form B = dA = ∑ ( j 0, there exists R > 0 such that Λ ≥ 1/ϵ on ℂn \ 𝔹R . This implies 1 󵄩 ∗ 󵄩2 ‖𝜕u‖2φ + 󵄩󵄩󵄩𝜕φ u󵄩󵄩󵄩φ ≥ ∫ Λ |u|2 e−φ dλ ≥ ϵ ℂn

∫ |u|2 e−φ dλ, ℂn \𝔹R

which means that (11.30) holds. We indicate that the condition of Theorem 11.17 can be written in the following form: for each ϵ > 0, there exists R(ϵ) > 0 such that

11.5 Compactness in the Segal–Bargmann space

‖u‖L2

(0,1)

(ℂn \𝔹R(ϵ) ,φ)

� 247

≤ ϵ‖u‖Qφ .

Hence for all u ∈ 𝒲 Qφ and j ∈ ℕ, we have 2j

|u|2 e−φ dλ ≤

∫ ℂn \𝔹R(1/2j )

1 ‖u‖2Qφ , 2j

and hence ∫ |u|2 e−φ dλ ≤ ℂn

∫ 1 ⋅ |u|2 e−φ dλ + 𝔹R(1/2)

2 ⋅ |u|2 e−φ dλ

∫ 𝔹R(1/4) \𝔹R(1/2)

+

4 ⋅ |u|2 e−φ dλ + ⋅ ⋅ ⋅

∫ 𝔹R(1/8) \𝔹R(1/4)

≤ (C + 1)‖u‖2Qφ . Now it is easy to define a smooth function Λ tending to ∞ as |z| tends to ∞ such that (11.41) holds.

11.5 Compactness in the Segal–Bargmann space In this chapter, we use Montel’s theorem (see, for instance, [43]) to characterize compact 2 subsets of the Segal–Bargmann space A2 (ℂn , e−|z| ). Later (Chapter 16), we will use this characterization to describe properties of certain unbounded operators on the Segal– Bargmann space, which are connected to the uncertainty principle. 2

Theorem 11.32. A bounded closed subset 𝒜 of A2 (ℂn , e−|z| ) is compact if and only if for every ϵ > 0, there exists a number R > 0 such that for every f ∈ 𝒜, 2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z) < ϵ2 .

(11.42)

ℂn \𝔹R 2

Proof. For f ∈ A2 (ℂn , e−|z| ) and R > 0, let 1/2

2 󵄨 󵄨2 ‖f ‖R = ( ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z)) ,

1/2

2 󵄨 󵄨2 ‖f ‖(R) = ( ∫ 󵄨󵄨󵄨f (z)󵄨󵄨󵄨 e−|z| dλ(z)) .

(11.43)

󵄨󵄨 2 2 󵄨 󵄨󵄨 ‖f ‖ − ‖f ‖R 󵄨󵄨󵄨 → 0

(11.44)

ℂn \𝔹R

𝔹R

Then ‖f ‖2(R) = ‖f ‖2 − ‖f ‖2R

and

as R → ∞. Assume that 𝒜 is compact and that (11.42) fails. Then there is ϵ > 0 such that there is no R > 0 with

248 � 11 Compactness ‖f ‖2 − ‖f ‖2R < ϵ2 for all f ∈ 𝒜. Hence, for each R1 > 0, there is f1 ∈ 𝒜 with ‖f1 ‖2 − ‖f1 ‖2R1 > ϵ2 . We construct a sequence in 𝒜 that has no Cauchy subsequence. We have 1/2

>ϵ

1/2

ϵ < . 2

‖f1 ‖(R1 ) = (‖f1 ‖2 − ‖f1 ‖2R1 ) and can find R2 > R1 such that ‖f1 ‖(R2 ) = (‖f1 ‖2 − ‖f1 ‖2R2 ) For R2 , we can find f2 ∈ 𝒜 with ‖f2 ‖(R2 ) > ϵ. Hence

󵄨 󵄨 ϵ ‖f2 − f1 ‖ ≥ ‖f2 − f1 ‖(R2 ) ≥ 󵄨󵄨󵄨 ‖f2 ‖(R2 ) − ‖f1 ‖(R2 ) 󵄨󵄨󵄨 > . 2 We continue in the same way to find f3 and notice that ‖f3 − f2 ‖ > ϵ/2, and since ‖f1 ‖(R3 ) ≤ ‖f1 ‖(R2 ) , we have also ‖f3 − f1 ‖ > ϵ/2. Now it is clear that there is no subsequence of (fk )k that is a Cauchy sequence, and we proved that (11.42) is necessary for compactness. To show the sufficiency, we recall (1.23). For the bounded set 𝒜, we have ‖f ‖ ≤ CA for each f ∈ 𝒜, where CA > 0 is a constant. By (1.23), for each compact subset K ⊂ ℂn , there is a constant MK > 0 such that 󵄨 󵄨 sup󵄨󵄨󵄨f (z)󵄨󵄨󵄨 ≤ MK z∈K

for all f ∈ 𝒜. This means that 𝒜 is uniformly bounded in the compact open topology, and Montel’s theorem implies that each sequence (fk )k in 𝒜 has a convergent subsequence, which is again denoted by (fk )k ; it is convergent in the compact open topology, and it is also a Cauchy sequence in this topology, which means that for each compact ball 𝔹R and for each ϵ > 0, there exists N > 0 such that 󵄨 󵄨 sup 󵄨󵄨󵄨fk (z) − fm (z)󵄨󵄨󵄨 < ϵ

z∈𝔹R

(11.45) 2

for all k, m > N. It remains to show that (fk )k is a Cauchy sequence in A2 (ℂn , e−|z| ). For this purpose, we use assumption (11.42) and estimate 2 2 󵄨 󵄨2 󵄨 󵄨2 ‖fk − fm ‖2 ≤ ∫ 󵄨󵄨󵄨fk (z) − fm (z)󵄨󵄨󵄨 e−|z| dλ(z) + ∫ 󵄨󵄨󵄨fk (z) − fm (z)󵄨󵄨󵄨 e−|z| dλ(z)

ℂn \𝔹R

𝔹R

󵄨 󵄨2 ≤ sup 󵄨󵄨󵄨fk (z) − fm (z)󵄨󵄨󵄨 ∫ e−|z| dλ(z) + ϵ 2

z∈𝔹R 2

2

≤ Cϵ + ϵ .

ℂn

11.6 Bounded pseudoconvex domains

� 249

11.6 Bounded pseudoconvex domains Finally, we investigate the compactness of the 𝜕-Neumann operator on a bounded pseudoconvex domain. For this purpose we consider the following potential theoretic concepts. Definition 11.33. Let Ω ⋐ ℂn be a smoothly bounded pseudoconvex domain. It satisfies property (P) if for each M > 0, there exist a neighborhood U of 𝜕Ω and a plurisubharmonic function φM ∈ 𝒞 2 (U) with 0 ≤ φM ≤ 1 on U such that n

𝜕2 φM (p)tj t k ≥ M‖t‖2 𝜕zj 𝜕zk j,k=1 ∑

for all p ∈ U and t ∈ ℂn . Definition 11.34. Ω satisfies property (P)̃ if the following holds: there is a constant C such that for all M > 0, there exists a 𝒞 2 -function φM in a neighborhood U (depending on M) of bΩ such that (i) 󵄨󵄨2 󵄨󵄨 n n 󵄨󵄨 󵄨󵄨 𝜕φM 𝜕2 φM 󵄨 󵄨󵄨 ∑ (z)t (z)tj t k ≤ C ∑ 󵄨 j 󵄨󵄨 󵄨󵄨 𝜕zj 𝜕zk 󵄨󵄨 󵄨󵄨 j=1 𝜕zj j,k=1 (ii)

and n

∑ j=1

𝜕2 φM (z)tj t k ≥ M‖t‖2 𝜕zj 𝜕zk

for all z ∈ U and t ∈ ℂn . Remark 11.35. Catlin [13] showed that property (P) implies the compactness of the 𝜕-operator N on L2(0,1) (Ω), and McNeal [75] showed that property (P)̃ also implies the compactness of the 𝜕-operator N on L2(0,1) (Ω). It is not difficult to show that property (P) implies property (P)̃ : If (φM ) is the family of functions from the definition of property (P), then (eφM ) will work for (P)̃ ; see also [91]. We can now use a similar approach as for weighted spaces to prove Catlin’s result. We use again Theorem 11.1. ∗ To show that the unit ball in dom(𝜕) ∩ dom(𝜕 ) in the graph norm f 󳨃→ (‖𝜕f ‖2 + 1 ∗ ‖𝜕 f ‖2 ) 2 satisfies condition (i) of Theorem 11.1, we first remark that the compactly sup∗ ∗ ported forms are not dense in dom(𝜕) ∩ dom(𝜕 ), but the forms in dom(𝜕 ) with coefficients in 𝒞 ∞ (Ω) are dense (Proposition 5.14). So if ω ⋐ Ω, the we choose ω ⋐ ω1 ⋐ ω2 ⋐ Ω and a cut-off function ψ such that ψ(z) = 1 for z ∈ ω1 and ψ(z) = 0 for z ∈ Ω \ ω2 .

250 � 11 Compactness ∗

∗

For u ∈ dom(𝜕) ∩ dom(𝜕 ), we define ũ = ψu and remark that the domain of 𝜕 is preserved under multiplication by a function in 𝒞 1 (Ω) (Remark 5.15), and therefore ũ has ∗ compactly supported coefficients and belongs to dom(𝜕) ∩ dom(𝜕 ). The graph norm of ũ is bounded by a constant C depending only on ω, ω1 , ω2 , Ω if u belongs to the unit ball in the graph norm. By construction we have ‖τh u − u‖L2 (ω) = ‖τh ũ − u‖̃ L2 (ω) if |h| is small enough, and hence we can use Gårding’s inequality for ω ⋐ Ω to show that condition (i) holds. To verify condition (ii), we use property (P) and the following version of the Kohn– Morrey formula: n

𝜕2 φM 󵄩 ∗ 󵄩2 uj uk e−φM dλ ≤ ‖𝜕u‖2φM + 󵄩󵄩󵄩𝜕φM u󵄩󵄩󵄩φ , M 𝜕z 𝜕z j k j,k=1

(11.46)

∫ ∑ Ω

here we used that Ω is pseudoconvex, which means that the boundary terms in the Kohn–Morrey formula can be neglected; this follows easily from Theorem 7.2. Now we point out that the weighted 𝜕-complex is equivalent to the unweighted one and that ∗ M the expression ∑nj=1 𝜕φ u appearing in 𝜕φM u can be controlled by the complex Hessian 𝜕z j 𝜕2 φM ∑nj,k=1 𝜕z j 𝜕zk

j

uj uk , which follows from the fact that property (P) implies property (P)̃ . Of

course, we also use the boundedness of the weight φM on Ω ⋐ ℂn . In this way, the same reasoning as in the weighted case shows that property (P) implies condition (11.3). ∗ Therefore condition (P) gives that the unit ball of dom(𝜕) ∩ dom(𝜕 ) in the graph norm 1 ∗ f 󳨃→ (‖𝜕f ‖2 + ‖𝜕 f ‖2 ) 2 is relatively compact in L2(0,1) (Ω) and hence that the 𝜕-Neumann operator is compact. Now let j : dom(𝜕) ∩ dom(𝜕 ) 󳨅→ L2(0,1) (Ω) ∗

denote the embedding. By (4.42) we know that N = j ∘ j∗ . ∗

Hence N is compact if and only if j is compact, where dom(𝜕) ∩ dom(𝜕 ) is endowed with 1 ∗ the graph norm f 󳨃→ (‖𝜕f ‖2 + ‖𝜕 f ‖2 ) 2 . Theorem 11.36. Let Ω ⋐ ℂn be a smoothly bounded pseudoconvex domain. The 𝜕-Neumann operator N is compact if and only if for each ϵ > 0, there exists ω ⋐ Ω such that 󵄨 󵄨2 󵄩 ∗ 󵄩2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 dλ(z) ≤ ϵ(‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 )

Ω\ω

11.6 Bounded pseudoconvex domains

� 251

∗

for each u ∈ dom(𝜕) ∩ dom(𝜕 ). This follows from the above remarks about the embedding j and the fact that the two conditions (11.1) are also necessary for a bounded set in L2 to be relatively compact. Remark 11.37. (a) Let 1 ≤ q ≤ n. We say that Ω satisfies property (Pq ) if the following holds: for every positive number M, there exist a neighborhood U of bΩ and a 𝒞 2 -function φM on U such that 0 ≤ φM (z) ≤ 1, z ∈ U, and such that for any z ∈ U, the sum of any q 2

𝜕 φM (equivalently, the smallest q) eigenvalues of the Hermitian form ( 𝜕z (z))j,k is at 𝜕z j

k

least M. Using the methods from above, we can show that property (Pq ) implies the compactness of the operator Nq ; compare with Proposition 11.16 and [91]. For this purpose, we consider the following version of the Kohn–Morrey formula: ∫ ∑

′

Ω |K|=q−1

n

𝜕2 φM 󵄩 ∗ 󵄩2 ujK ukK e−φM dλ ≤ ‖𝜕u‖2φM + 󵄩󵄩󵄩𝜕φM u󵄩󵄩󵄩φ , M 𝜕z 𝜕z j k j,k=1 ∑

(11.47)

∗

where u ∈ dom(𝜕)∩dom(𝜕 ) is a (0, q)-form, and follow the lines of Proposition 11.16 (see also [91, Theorem 4.8]). (b) In view of Remark 11.6(a), it is interesting to compare the Sobolev embedding W 1 (Ω) ⊂ Lq (Ω),

q ∈ [1, 2n/(n − 2)),

where the derivatives are taken with respect to the real variables xj = Rzj and yj = Izj for j = 1, . . . , n, with the embedding of the space 2

∗

𝒲 (Ω) := {u ∈ L(0,1) (Ω) : u ∈ dom(𝜕) ∩ dom(𝜕 )},

endowed with graph norm, into L2(0,1) (Ω). We have the following result: If Ω ⊂⊂ ℂn is a smoothly bounded pseudoconvex domain and 󵄩 ∗ 󵄩2 1/2 ‖u‖q ≤ C(‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 )

(11.48)

∗

for some q > 2 and all u ∈ dom(𝜕) ∩ dom(𝜕 ), then the 𝜕-Neumann operator N : L2(0,1) (Ω) 󳨀→ L2(0,1) (Ω) is compact. To show this, we have to check that the unit ball in 𝒲 (Ω) is precompact in L2(0,1) (Ω). By Theorem 11.36 we have to show that for each ϵ > 0, there exists ω ⋐ Ω such that 󵄨 󵄨2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 dλ(z) < ϵ2

Ω\ω

252 � 11 Compactness for all u in the unit ball of 𝒲 (Ω). By (11.48) and Hölder’s inequality we have 1

1

2 q 1 1 − 󵄨q 󵄨2 󵄨 󵄨 ( ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 dλ(z)) ≤ ( ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 dλ(z)) ⋅ |Ω \ ω| 2 q

Ω\ω

Ω\ω

1

≤ C |Ω \ ω| 2

− q1

.

Now we can choose ω ⊂⊂ Ω such that the last term is < ϵ. In a similar way as for Proposition 11.27, we obtain compactness estimates for the 𝜕-Neumann operator on a smoothly bounded domain. Here we use the standard Sobolev spaces W 1 (Ω) and the classical Rellich–Kondrachov lemma without weights. Proposition 11.38. Let Ω be a smoothly bounded pseudoconvex domain. Then the following statements are equivalent. (1) The 𝜕-Neumann operator N1 is a compact operator from L2(0,1) (Ω) into itself. ∗

(2) The embedding of the space dom(𝜕) ∩ dom(𝜕 ), provided with the graph norm u 󳨃→ ∗ (‖u‖2 + ‖𝜕u‖2 + ‖𝜕 u‖2 )1/2 , into L2(0,1) (Ω) is compact. (3) For every positive ϵ, there exists a constant Cϵ such that 󵄩 ∗ 󵄩2 ‖u‖2 ≤ ϵ(‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ) + Cϵ ‖u‖2−1 ∗

for all u ∈ dom(𝜕) ∩ dom(𝜕 ). (4) For every positive ϵ, there exists ω ⋐ Ω such that 󵄨 󵄨2 󵄩 ∗ 󵄩2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 dλ(z) ≤ ϵ(‖𝜕u‖2 + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 )

Ω\ω

∗

for all u ∈ dom(𝜕) ∩ dom(𝜕 ). (5) The operators 𝜕 N1 : L2(0,1) (Ω) ∩ ker(𝜕) 󳨀→ L2 (Ω) ∗

and

𝜕 N2 : L2(0,2) (Ω) ∩ ker(𝜕) 󳨀→ L2(0,1) (Ω) ∗

are both compact.

11.7 Notes In this chapter, we tried to point out some parallel aspects in the classical theory of Sobolev spaces and in the properties of the 𝜕-Neumann operator. The treatment of compact subsets in L2 -spaces stems from [1], and basic properties of Sobolev spaces and

11.7 Notes

� 253

the Rellich–Kondrachov lemma are taken from [12] and [27]. We refer the reader to these books for more general results on higher derivatives and Lp -spaces. The sufficient condition for the compactness of the 𝜕-Neumann operator in weighted L2 -spaces was originally discovered from the theory of Schrödinger operators with magnetic field and from the Witten Laplacian (see [46]); here we presented the complex analytic approach from [42] and [36]. We also discuss necessary conditions for compactness, where we use a corresponding result on Schrödinger operators with magnetic fields [59] and methods; from complex analysis; see [6], [7], and [22], where there are many other results concerning the question of compactness. A different proof of the compactness criterium in the Segal–Bargmann space can be found in [25]. A thorough treatment of the compactness of the 𝜕-Neumann operator for bounded pseudoconvex domains can be found in the book of Straube [91], with all its important implications concerning regularity of the 𝜕-Neumann operator in Sobolev spaces and far-reaching connections to potential theory. A characterization of the compactness of the 𝜕-Neumann operator in terms of geometric properties of pseudoconvex domains or in terms of the weight functions is still open. Fu and Straube [33] solved the problem for bounded convex domains. It turned out that in this case, the compactness of the 𝜕-Neumann operator on the level of holomorphic (0, q)-forms already implies the compactness on the whole L2(0,q) (Ω). In counterexamples to compactness, the obstruction often already occurs on the level of holomorphic (0, q)-forms. We will draw attention to this phenomenon in the next chapters.

12 The 𝜕-Neumann operator and the Bergman projection In this chapter, we investigate the connection between the 𝜕-Neumann operator and commutators of the Bergman projection with multiplication operators. In [14], it is shown that the compactness of the 𝜕-Neumann operator N on L2(0,1) (Ω) implies the compactness of the commutator [P, M], where P is the Bergman projection, and M is a pseudodifferential operator of order 0. Here we show that the compactness of the 𝜕-Neumann operator N restricted to (0, 1)-forms with holomorphic coefficients is equivalent to the compactness of the commutator [P, M] defined on the whole L2 (Ω). In addition, we derive a formula for the 𝜕-Neumann operator restricted to (0, 1)-forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplication operators by z and z.̄ To show the equivalences to the compactness mentioned above, we will need a complex version of the Stone–Weierstraß theorem.

12.1 The Stone–Weierstraß theorem In the sequel, we will need a generalization of the well-known theorem of Weierstraß that any real-valued continuous function on a compact interval [a, b] is the uniform limit of a sequence of polynomials on [a, b]. We will consider a compact subset K in ℂn and will show that any complex-valued continuous function on K can be uniformly approximated on K by polynomials in the variables z = (z1 , . . . , zn ) and z = (z1 , . . . , zn ). We prove a more general version and will need some appropriate notions and several lemmas. A subset 𝒜 of the space 𝒞 (K) of continuous real-valued or complex-valued functions (endowed with the sup-norm) is said to separate points if for every z, w ∈ K such that z ≠ w, there exists f ∈ 𝒜 such that f (z) ≠ f (w). We call 𝒜 an algebra if it is a real (complex) vector subspace of 𝒞 (K) such that fg ∈ 𝒜 (pointwise multiplication) for all f , g ∈ 𝒜. If 𝒜 is a subset of the space 𝒞 (K) of real-valued continuous functions, then 𝒜 is called a lattice if max(f , g) and min(f , g) are in 𝒜 for all f , g ∈ 𝒜. Since the algebra and lattice operations are continuous in sup-norm of 𝒞 (K), we easily sees that if 𝒜 is an algebra or a lattice, then is its the closure 𝒜 in the sup-norm. Lemma 12.1. For each ϵ > 0, there is a polynomial p on ℝ such that p(0) = 0 and 󵄨󵄨 󵄨 󵄨󵄨 |x| − p(x)󵄨󵄨󵄨 < ϵ

for x ∈ [−1, 1].

Proof. For |t| < 1, the function (1 − t)1/2 has the Taylor series expansion (−1) 1 2k − 3 t k ... 2 2 2 k! k=1 ∞

(1 − t)1/2 = 1 + ∑ https://doi.org/10.1515/9783111182926-012

12.1 The Stone–Weierstraß theorem �

255

∞

= 1 − ∑ ck t k k=1

with ck > 0. It is also absolutely convergent for t = −1. The monotone convergence theorem applied to the counting measure on ℕ implies that ∞

∞

∑ ck = lim ∑ ck t k = 1 − lim(1 − t)1/2 = 1,

k=1

t→1

t→1

k=1

k from which we get that the series 1 − ∑∞ k=1 ck t converges absolutely and uniformly on 1/2 [−1, 1] to (1 − t) . Given ϵ > 0, we can find a suitable partial sum q(t) such that

󵄨󵄨 󵄨 1/2 󵄨󵄨(1 − t) − q(t)󵄨󵄨󵄨 < ϵ/2,

t ∈ [−1, 1].

We set t = 1 − x 2 and r(x) = q(1 − x 2 ) and obtain a polynomial r such that 󵄨󵄨 󵄨 󵄨󵄨 |x| − r(x)󵄨󵄨󵄨 < ϵ/2,

x ∈ [−1, 1].

In particular, |r(0)| < ϵ/2, so setting p(x) = r(x)−r(0), we get the desired polynomial. First, we consider the space of real-valued continuous functions. Lemma 12.2. Let 𝒜 be a closed subalgebra of the space 𝒞 (K) of real-valued continuous functions. Then |f | ∈ 𝒜 whenever f ∈ 𝒜, and 𝒜 is a lattice. Proof. We write ‖f ‖ = supz∈K |f (z)| for f ∈ 𝒞 (K). If f ∈ 𝒜 and f ≠ 0, then h := f /‖f ‖ maps K into [−1, 1]. If p is as in Lemma 12.1, then we have 󵄩󵄩 󵄩 󵄩󵄩 |h| − p ∘ h󵄩󵄩󵄩 < ϵ. Since p(0) = 0, p has no constant term, and therefore p ∘ h ∈ 𝒜, as 𝒜 is an algebra. Since 𝒜 is closed, it follows that |h| ∈ 𝒜 and |f | = ‖f ‖ |h| ∈ 𝒜. This proves the first statement. The second follows from 1 max(f , g) = (f + g + |f − g|), 2

1 min(f , g) = (f + g − |f − g|). 2

Lemma 12.3. Let 𝒜 be a closed lattice in the space 𝒞 (K) of real-valued continuous functions. Let f ∈ 𝒞 (K) and suppose that for all z, w ∈ K, there exists gzw ∈ 𝒜 such that f (z) = gzw (z) and f (w) = gzw (w). Then f ∈ 𝒜. Proof. For ϵ > 0 and z, w ∈ K, let Uzw = {ζ ∈ K : f (ζ ) < gzw (ζ ) + ϵ},

Vzw = {ζ ∈ K : f (ζ ) > gzw (ζ ) − ϵ}.

256 � 12 The 𝜕-Neumann operator and the Bergman projection

These sets are open and contain z and w. Fix w ∈ K. Then {Uzw }z∈K covers K, so there is a finite subcover {Uzj w }nj=1 . Let gw = max(gz1 w , . . . , gzn w ). Then f < gw + ϵ on K, and f > gw − ϵ on Vw = ⋂nj=1 Vzj w , which is open and contains w. So {Vw }w∈K is an open covering of K, which has a finite subcover {Vwk }m k=1 . Let g = min(gw1 , . . . , gwm ). Then ‖f − g‖ < ϵ, and since 𝒜 is a closed lattice, g ∈ 𝒜 and f ∈ 𝒜. Before we prove the first version of the Stone–Weierstraß theorem, we remark that ℝ2 as an algebra under coordinatewise addition and multiplication has only the following subalgebras: ℝ2 , {(0, 0)}, and the linear spans of (1, 0), (0, 1), and (1, 1), because a nonzero subalgebra 𝒜 ⊂ ℝ2 has the property that with (0, 0) ≠ (a, b) ∈ 𝒜, the pair (a2 , b2 ) ∈ 𝒜, so if a ≠ 0, b ≠ 0, and a ≠ b, then (a, b) and (a2 , b2 ) are linearly independent, so 𝒜 = ℝ2 . The other cases – a ≠ 0 = b, a = 0 ≠ b, and a = b ≠ 0 – give the other three subalgebras. With this in mind, we can now prove the first version of the Stone–Weierstraß theorem. Theorem 12.4. If 𝒜 is a closed subalgebra of the space 𝒞 (K) of real-valued continuous functions that separates points, then either 𝒜 = 𝒞 (K), or 𝒜 = {f ∈ 𝒞 (K) : f (z0 ) = 0} for some z0 ∈ K. The first alternative holds if and only if 𝒜 contains the constant functions. Proof. For z, w ∈ K such that z ≠ w, let 𝒜zw = {(f (z), f (w)) : f ∈ 𝒜}. Then since f 󳨃→ (f (z), f (w)) is an algebra homomorphism, 𝒜zw is a subalgebra of ℝ2 in the sense of the remark from above. If 𝒜zw = ℝ2 for all z, w ∈ K, then Lemmas 12.1 and 12.2 imply that 𝒜 = 𝒞 (K). Otherwise, there exist z, w ∈ K for which 𝒜zw is a proper subalgebra of ℝ2 . It cannot be {(0, 0)} or the linear span of (1, 1) because 𝒜 separates points. So 𝒜zw is the linear span of (1, 0) or (0, 1). In either case, there exists z0 ∈ K such that f (z0 ) = 0 for all f ∈ 𝒜. There is only one such z0 since 𝒜 separates points, so if neither z nor w is z0 , then 𝒜zw = ℝ2 . Then Lemmas 12.1 and 12.2 imply that 𝒜 = {f ∈ 𝒞 (K) : f (z0 ) = 0}. If 𝒜 contains constant functions, then this cannot be the case, so then 𝒜 equals 𝒞 (K). The set of polynomials is an algebra that separates points and contains constants. Hence we get the following: Corollary 12.5. The restrictions of the real polynomial functions to K are dense in the space 𝒞 (K) of real-valued continuous functions. Theorem 12.4 is false for the space 𝒞 (K) of complex-valued continuous functions. Nevertheless, there is a complex version of the Stone–Weierstraß theorem. Theorem 12.6. If 𝒜 is a closed complex subalgebra of the space 𝒞 (K) of complex-valued continuous functions that separates points and is closed under complex conjugation, then either 𝒜 = 𝒞 (K), or 𝒜 = {f ∈ 𝒞 (K) : f (z0 ) = 0} for some z0 ∈ K. Proof. Since Rf = (f + f )/2 and If = (f − f )/2i, the set 𝒜ℝ of real and imaginary parts of functions in 𝒜 is a subalgebra of the space of real-valued continuous functions to which Theorem 12.4 applies. Since 𝒜 = {f + ig : f , g ∈ 𝒜ℝ }, the desired result follows.

12.2 Commutators of the Bergman projection � 257

Corollary 12.7. The set of all polynomials p of the form p(z, z) = ∑ λα zα1 zα2 , |α|≤N

where α = (α1 , α2 ) is a multi-index in ℕ2n , is dense in the space 𝒞 (K) of complex-valued continuous functions. Proof. It is clear that the set of polynomials p of the above form separates points and is invariant under complex conjugation. Therefore we can apply Theorem 12.6.

12.2 Commutators of the Bergman projection Let Ω be a bounded pseudoconvex domain in ℂn , and let A2(0,1) (Ω) denote the space of all (0, 1)-forms with holomorphic coefficients belonging to L2 (Ω). In Section 3.2, we showed that the canonical solution operator S : A2(0,1) (Ω) 󳨀→ L2 (Ω) has the form S(g)(z) = ∫ K(z, w)⟨g(w), z − w⟩ dλ(w),

(12.1)

Ω

where K denotes the Bergman kernel of Ω, g = ∑nj=1 gj dzj is a (0, 1)-form belonging to A2(0,1) (Ω), and

n

⟨g(w), z − w⟩ = ∑ gj (w)(zj − wj ) j=1

for z = (z1 , . . . , zn ) and w = (w1 , . . . , wn ). The restriction of the canonical solution operator to forms with holomorphic coefficients has many interesting aspects, which in most cases correspond to certain growth properties of the Bergman kernel. It is also of great interest to clarify to what extent the compactness of the restriction already implies the compactness of the original solution operator to 𝜕. This is the case for convex domains; see [33]. There are many other examples of noncompactness where the obstruction already occurs for forms with holomorphic coefficients (see Chapter 14 and [68, 70]). We define the operator T : L2(0,1) (Ω) 󳨀→ L2 (Ω) by Tf (z) = ∫ K(z, w)⟨f (w), z − w⟩ dλ(w), Ω

(12.2)

258 � 12 The 𝜕-Neumann operator and the Bergman projection where f = ∑nk=1 fk dzk and ⟨f (w), z − w⟩ = ∑nk=1 fk (w)(zk − wk ). The operator T can be written as a sum of commutators n

n

Tf = ∑ [Mk , P]fk ,

f = ∑ fk dzk ,

k=1

k=1

(12.3)

where Mk v(z) = zk v(z), v ∈ L2 (Ω), k = 1, . . . , n, and P is the Bergman projection; see (1.5). Let 𝒫 : L2(0,1) (Ω) 󳨀→ A2(0,1) (Ω) be the orthogonal projection on the space of (0, 1)forms with holomorphic coefficients. We claim that Tf = T 𝒫 f ,

f ∈ L2(0,1) (Ω).

It suffices to show that Tg = 0 for g⊥A2(0,1) (Ω): n

n

Tg(z) = − ∑ PMk gk (z) = − ∑ ∫ K(z, w)wk gk (w) dλ(w) k=1 Ω

k=1 n

−

= − ∑ ∫ gk (w) [K(w, z)wk ] dλ(w) = 0, k=1 Ω

because w 󳨃→ K(w, z)wk is holomorphic and gk ⊥A2 (Ω) for k = 1, . . . , n. Now let S denote the canonical solution operator to 𝜕 restricted to A2(0,1) (Ω). From (12.1) we have, for f ∈ L2(0,1) (Ω), S(𝒫 f ) = T(𝒫 f ) = Tf .

(12.4)

Hence we have proved the following: Theorem 12.8. If f ∈ L2(0,1) (Ω), then T(𝒫 f ) = Tf . The operator S is compact as an operator from A2(0,1) (Ω) to L2 (Ω) if and only if the operator T is compact as an operator from L2(0,1) (Ω) to L2 (Ω). Remark 12.9. The adjoint operator T ∗ : L2 (Ω) 󳨀→ L2(0,1) (Ω) is given by n

T ∗ (g) = ∑ [P, Mk ] g dzk , g ∈ L2 (Ω), k=1

where Mk v(z) = zk v(z). Here we have T ∗ (I − P)(g) = T ∗ (g), since [P, Mk ] Pg = PMk Pg − Mk Pg = 0.

(12.5)

12.2 Commutators of the Bergman projection

� 259

In a similar way, the following results can be proved. Lemma 12.10. (1) PMj P = Mj P,

(2) PMj P = PMj . Let 2 B(0,1) (Ω) = {f ∈ L2(0,1) (Ω) : f ∈ ker 𝜕}.

Now suppose that Ω is a bounded pseudoconvex domain in ℂn . The 𝜕-Neumann 2 2 operator N can be viewed as an operator from B(0,1) (Ω) to B(0,1) (Ω). The operator ∗

2 𝜕 N : B(0,1) (Ω) 󳨀→ A2 (Ω)⊥

is the canonical solution operator to 𝜕 (see Proposition 4.64). 2 Theorem 12.11. If f = ∑nk=1 fk dzk ∈ B(0,1) (Ω), then n

n

k=1

j=1

𝒫 N 𝒫 f = ∑ (∑(PMk M j Pfj − Mk PM j fj ))dzk .

(12.6)

If f = ∑nk=1 fk dzk ∈ A2(0,1) (Ω), then n

n

k=1

j=1

𝒫 Nf = ∑ [P, Mk ](∑ M j fj )dzk .

(12.7)

2 Proof. First, we observe that for f ∈ B(0,1) (Ω), we have ∗

∗

N𝜕𝜕 Nf = N(I − 𝜕 𝜕N)f = Nf , where we used the fact that 2 2 N : B(0,1) (Ω) 󳨀→ B(0,1) (Ω).

If f ∈ A2(0,1) (Ω), then by Theorem 12.8 it follows that ∗

𝜕 Nf = Tf . 2 Let f ∈ A2(0,1) (Ω), and let g ∈ B(0,1) (Ω) with orthogonal decomposition g = h + h,̃ where 2 ̃ h ∈ A(0,1) (Ω) and h = (I − 𝒫 )g. Then ∗ ∗ ̃ Tf ) = (𝜕∗ Nh, Tf ) + (𝜕∗ N h,̃ Tf ) (g, N𝜕𝜕 Nf ) = (𝜕 N(h + h),

260 � 12 The 𝜕-Neumann operator and the Bergman projection ∗ ∗ = (Th, Tf ) + (𝜕 N h,̃ Tf ) = (Tg, Tf ) + (𝜕 N h,̃ Tf ) ∗

= (g, T ∗ Tf ) + (𝜕 N h,̃ Tf ). Since ∗ (𝜕 N h,̃ Tf ) = (N h,̃ 𝜕Tf ) = (N h,̃ f ) = (h,̃ Nf ),

we obtain ∗ (g, Nf ) = (g, N𝜕𝜕 Nf ) = (g, T ∗ Tf ) + (h,̃ Nf )

= (g, T ∗ Tf ) + ((I − 𝒫 )g, Nf ) = (g, T ∗ Tf ) + (g, (I − 𝒫 )Nf ).

2 Now since g ∈ B(0,1) (Ω) was arbitrary, we get

Nf = T ∗ Tf + Nf − 𝒫 Nf , and therefore ∗

𝒫 Nf = T Tf . 2 If we take into account that for f ∈ B(0,1) (Ω), we have Tf = T 𝒫 f , we can now apply the last formula for 𝒫 f and get ∗

𝒫 N 𝒫 f = T Tf . 2 It remains to compute T ∗ T. If f ∈ B(0,1) (Ω), then n

n

T ∗ Tf = ∑ [P, Mk ](∑[M j , P]fj )dzk j=1

k=1 n

n

k=1

j=1

n

n

k=1

j=1

= ∑ (∑(PMk M j P − Mk PM j P − PMk PM j + Mk PM j )fj )dzk = ∑ (∑(PMk M j Pfj − Mk PM j fj ))dzk , where we used Lemma 12.10. If f ∈ A2(0,1) (Ω), then Pfj = fj , and we obtain the second formula in Theorem 12.11. Using the last results, we get a criterion for the compactness of commutators [P, Mk ].

12.2 Commutators of the Bergman projection � 261

Theorem 12.12. Let Ω be a bounded pseudoconvex domain in ℂn . Then the following conditions are equivalent: (1) N|A2 (Ω) is compact; ∗

(0,1)

(2) 𝜕 N|A2

(0,1)

(Ω)

is compact;

(3) [P, Mk ] is compact on L2 (Ω) for k = 1, . . . , n; (4) (I − P)M k P is compact on L2 (Ω) for k = 1, . . . , n; (5) [Mφ , P] is compact on L2 (Ω) for each continuous function φ on Ω. ∗

2 Proof. Let S1 = 𝜕 N1 : B(0,1) (Ω) 󳨀→ A2 (Ω)⊥ be the canonical solution operator to 𝜕, and ∗

2 2 (Ω) 󳨀→ B(0,1) (Ω)⊥ . Then similarly S2 = 𝜕 N2 : B(0,2)

N1 = S1∗ S1 + S2 S2∗ (see Proposition 4.64). Since S2∗ |A2

(0,1)

(Ω)

N1 |A2

(0,1)

= 0, we have (Ω)

= S1∗ S1 |A2

(0,1)

(Ω) ,

and (1) is equivalent to (2). ∗ Now suppose that (2) holds. Then since the restriction of 𝜕 N to A2(0,1) (Ω) is of the form ∗

n

𝜕 Nf = ∑ [Mk , P]fk , k=1

where f = ∑nk=1 fk dzk ∈ A2(0,1) (Ω), by Theorem 12.8 it follows that the operators [Mk , P] are compact on L2 (Ω). Since [Mk , P]∗ = [P, Mk ], we obtain property (3). It is also clear by Theorem 12.8 that (3) implies (2). Now suppose that (3) holds. It follows that [Mk , P]P is also compact, and since [Mk , P]P = Mk P − PMk P = (I − P)Mk P, the Hankel operators (I − P)Mk P are compact. So we have shown that (3) implies (4). Suppose that (4) holds. The Hankel operators Hzj zk with symbol zj zk can be written in the form Hzj zk = (I − P)Mj (P + (I − P))Mk P = (I − P)Mj (I − P)Mk P, and hence it follows that Hzj zk is compact. Similarly, we can show that for any polynomial p(z, z) = ∑ λα zα1 zα2 , |α|≤N

262 � 12 The 𝜕-Neumann operator and the Bergman projection where α = (α1 , α2 ) is a multi-index in ℕ2n , the corresponding Hankel operator Hp = (I −P)Mp P is compact. Now let φ ∈ 𝒞 (Ω). Then by Corollary 12.7 there exists a polynomial p of the above form such that ‖φ − p‖∞ < ϵ. Hence 󵄩 󵄩 ‖Hφ − Hp ‖ = 󵄩󵄩󵄩(I − P)Mφ−p P󵄩󵄩󵄩 ≤ ‖φ − p‖∞ , where ‖ ⋅ ‖ denotes the operator norm, and ‖ ⋅ ‖∞ denote the sup-norm on Ω. Since the compact operators form a closed two-sided ideal in the operator norm (Corollary 2.4) and since for g = g1 + g2 with g1 ∈ A2 (Ω) and g2 ∈ A2 (Ω)⊥ , we have [Mφ , P]g = −Hφ∗ g2 + Hφ g1 , it follows that [Mφ , P] is compact. Remark 12.13. (a) If Ω is a bounded convex domain, then the compactness of ∗

𝜕 N|A2

(0,1)

(Ω)

∗

implies already the compactness of 𝜕 N on all of L2(0,1) (Ω) (see [33]), and hence in this case, property (1) of Theorem 12.12 can be replaced by the compactness of N on L2(0,1) (Ω), and property (2) of Theorem 12.12 can be replaced by the compactness of ∗

𝜕 N on L2(0,1) (Ω). (b) For n ≥ 2 and 1 ≤ q ≤ n, let

2 B(0,q) (Ω) = {f ∈ L2(0,q) (Ω) : f ∈ ker 𝜕}.

From Proposition 4.65 we know that the orthogonal projection 2 Pq : L2(0,q) (Ω) 󳨀→ B(0,q) (Ω)

can be written in the form ∗

Pq = I − 𝜕 Nq+1 𝜕. Recently, Celik and Sahutoglu [15] showed that for 2 ≤ q ≤ n, the following statements are equivalent: (1) Nq is compact on L2(0,q) (Ω), ∗

2 (2) 𝜕 Nq is compact on B(0,q) (Ω),

2 (3) [Mφ , Pq ] is compact on B(0,q) (Ω) for each φ ∈ 𝒞 (Ω).

12.3 Notes

� 263

2 This is mainly due to the fact that B(0,q) (Ω) is a much larger space than the space of

(0, q)-forms with holomorphic coefficients belonging to L2 (Ω). Whether the same is true for q = 1 is still not known; compare with Theorem 12.12, where in the first statement, we have the compactness only on A2(0,1) (Ω).

12.3 Notes The proof of the Stone–Weierstraß theorem stems from [29]. Salinas [86] showed that the ∗ compactness of the canonical solution operator 𝜕 Nq+1 restricted to forms with holomorphic coefficients is a consequence of the compactness of the commutators [M j , Pq ], 1 ≤ j ≤ n. The compactness of Nq for 1 ≤ q ≤ n can also be characterized in terms of C ∗ algebras (see [86, 87] and [15]). Catlin and D’Angelo [14] used the compactness of the commutators [Mφ , P] in conjunction with a complex-variable analogue of Hilbert’s 17th problem. They showed that the compactness of N1 implies that the commutators [M, P] are compact for all tangential pseudodifferential operators M of order 0. Theorems 12.11 and 12.12 are from [41].

13 Compact resolvents In this chapter, we return to the differential operators introduced in Chapter 10 and discuss the question whether these operators are with compact resolvents. For this purpose, we use the general characterization of the compactness of the 𝜕-Neumann operator.

13.1 Schrödinger operators Here we characterize the compactness of the 𝜕-Neumann operator Nφ on L2 (ℂ, e−φ ), which was originally done in [46] using methods from Schrödinger operators and later in [74] using estimates of the Bergman kernel in A2 (ℂ, e−φ ). We follow the method from [46], showing that the necessary condition for compactness (11.22) is also sufficient for n = 1 if we suppose that the Laplacian of the weight function belongs to the reverse Hölder class B2 (ℝ2 ) consisting of positive and almost nonzero everywhere L2 functions V for which there exists a constant C > 0 such that (

1 ∫ V 2 dλ) |B|

1/2

B

≤ C(

1 ∫ V dλ) |B|

(13.1)

B

for any ball B in ℝ2 . Note that any positive (nonzero) polynomial is in B2 (ℝ2 ). Before we proceed characterizing the compactness of Nφ on L2 (ℂ, e−φ ), we collect important properties of functions belonging to the reverse Hölder class. A function w(x) ≥ 0 is called an A1 weight if [w]A1 =

sup

ℝ2

Q cubes in

(

1 󵄩 󵄩 ∫ w(x) dλ(x)) 󵄩󵄩󵄩w−1 󵄩󵄩󵄩L∞ (Q) < ∞; |Q| Q

for 1 < p < ∞, a weight w is said to be of class Ap if [w]Ap =

sup

p−1

Q cubes in ℝ2

(

1 1 − 1 ∫ w(x) dλ(x)) ( ∫ w(x) p−1 dλ(x)) |Q| |Q| Q

< ∞;

Q

a weight w is called an A∞ weight if [w]A∞ =

sup

Q cubes in ℝ2

(

1 1 ∫ w(x) dλ(x)) exp( ∫ log w(x)−1 dλ(x)) < ∞; |Q| |Q| Q

Q

the quantity [w]Ap is called the Ap Muckenhoupt characteristic constant of w. It follows that for all 1 ≤ p < ∞, we have https://doi.org/10.1515/9783111182926-013

13.1 Schrödinger operators

[w]A∞ ≤ [w]Ap

and

� 265

⋃ Ap ⊆ A∞ .

1≤p 0,

and where w(Q) = ∫Q w dλ; (4) For all C > 1 and all cubes Q, we have

2

2

w(CQ) ≤ 2C [w]CA∞ w(Q), and we say that the measure wdλ is doubling. Proof. (1) immediately follows from the definition. To prove (2), we have to recall the consequence of Jensen’s inequality 1/q

󵄨 󵄨 (∫󵄨󵄨󵄨h(t)󵄨󵄨󵄨 dμ(t)) X

󵄨 󵄨 ≥ exp(∫ log󵄨󵄨󵄨h(t)󵄨󵄨󵄨 dμ(t)), X

which holds for all measurable functions h on a probability space (X, μ) and all 0 < q < ∞. To prove (3), observe that by taking f = w−1 the expression on the right of (3) is at least as large as [w]A∞ . From Jensen’s inequality we get exp(

1 1 󵄨󵄨 󵄨 󵄨 󵄨 ∫ log(󵄨󵄨󵄨f (x)󵄨󵄨󵄨w(x)) dλ(x)) ≤ ∫󵄨f (x)󵄨󵄨󵄨w(x) dλ(x), |Q| |Q| 󵄨 Q

which can be written as

Q

266 � 13 Compact resolvents w(Q) w(Q) 1 1 exp(− exp( ∫ log |f |dλ) ≤ ∫ log |w|dλ) Q |Q| |Q| ∫Q |f |wdλ Q

Q

when f does not vanish almost everywhere on Q, and we get (3). To show (4), we apply (3) to the cube CQ instead of Q and to the function f (x) = c on n Q and f (x) = 1 on ℝ2 \ Q, where c is chosen such that c1/C = 2[w]A∞ . We obtain log c w(CQ) exp( n ) ≤ [w]A∞ , w(CQ \ Q) + cw(Q) C which implies (4). Lemma 13.2. (a) Let w ∈ A∞ . Then there exists ϵ > 0, independent of Q, such that E = {x ∈ Q : w(x) > ϵ avQ w} satisfies |E| ≥ 21 |Q|, where avQ w = |Q| ∫Q wdλ. (b) If w ∈ Aα for 1 ≤ α < ∞, then there exists a constant C > 0 such that α−1

avE w |E| ≥C( ) avQ w |Q| for each measurable subset E of Q.

Proof. (a) We show that there exists ϵ > 0, independent of Q, such that 󵄨󵄨 󵄨 1 󵄨󵄨{x ∈ Q : w(x) ≤ ϵ avQ w}󵄨󵄨󵄨 ≤ |Q|, 2 which implies (a). By Lemma 13.1(1) and the log-term in the definition of the characteristic [w]A∞ we can assume that ∫Q log w dλ = 0. Hence we have avQ w ≤ [w]A∞ , and we get 󵄨󵄨 󵄨 󵄨 󵄨 󵄨󵄨{x ∈ Q : w(x) ≤ ϵ avQ w}󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨{x ∈ Q : w(x) ≤ ϵ [w]A∞ }󵄨󵄨󵄨 −1 󵄨 󵄨 = 󵄨󵄨󵄨{x ∈ Q : log(1 + w(x)−1 ) ≥ log(1 + (ϵ[w]A∞ ) )}󵄨󵄨󵄨 1 1 + w(x) dλ(x) ≤ ∫ log w(x) log(1 + (ϵ[w]A∞ )−1 ) Q

1 = ∫ log(1 + w(x)) dλ(x) log(1 + (ϵ[w]A∞ )−1 ) Q

1 ≤ ∫ w(x) dλ(x) log(1 + (ϵ[w]A∞ )−1 ) Q

≤

[w]A∞ |Q|

log(1 + (ϵ[w]A∞ )−1 ) 1 = |Q|, 2 2[w]A∞ where ϵ = [w]−1 − 1)−1 . A∞ (e

� 267

13.1 Schrödinger operators

1 p

To prove (b), we apply Hölder’s inequality with exponents p and p′ such that + p1′ = 1, and for a suitable measurable function f , we get that p

(

1 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨 ∫󵄨󵄨f (x)󵄨󵄨󵄨 dλ(x)) = ( ∫󵄨f (x)󵄨󵄨󵄨w(x)1/p w(x)−1/p dλ(x)) |Q| |Q| 󵄨 Q

p

Q

′ 1 󵄨p 󵄨 ≤ (∫󵄨󵄨f (x)󵄨󵄨󵄨 w(x) dλ(x))(∫ w(x)−p /p dλ(x)) |Q|p 󵄨

Q

Q

=(

p/p′

1 1 󵄨 󵄨p ∫󵄨󵄨f (x)󵄨󵄨󵄨 w(x) dλ(x))( ∫ w(x) dλ(x)) w(Q) 󵄨 |Q| Q

×(

Q

1 − 1 ∫ w(x) p−1 dλ(x)) |Q|

p−1

Q

≤ [w]Ap

1 󵄨 󵄨p ∫󵄨󵄨󵄨f (x)󵄨󵄨󵄨 w(x) dλ(x). w(Q) Q

Now take for f the characteristic function of E. Then p

(

w(E) |E| ) ≤ [w]Ap , |Q| w(Q)

which proves (b). Theorem 13.3. Let φ be a subharmonic 𝒞 2 -function such that △φ(z) > ϵ > 0 and suppose that △φ ∈ B2 (ℝ2 ). The 𝜕-Neumann operator Nφ is compact on L2 (ℂ, e−φ ) if and only if lim ∫ △φ dλ = +∞,

|z|→∞

(13.2)

Qz

where Qz = {w ∈ ℂ : |w − z| < 1}. Proof. Suppose that Nφ is compact. In Chapter 10, we showed that ◻φ = eφ/2 D D e−φ/2 ∗

and that D D = − 41 △A + 81 △ φ. Now Theorem 11.20 implies that ∗

lim ∫ (△φ)2 dλ = +∞.

|z|→∞

(13.3)

Qz

Since △φ ∈ B2 (ℝ2 ), we get (13.2). For the other direction, we first use a version of the diamagnetic property for Schrödinger operators (see, for example, [67]) saying that H(A, △φ) = − △A + △ φ has a compact resolvent if H(0, △φ) = − △ + △ φ has a compact resolvent. To see this, let

268 � 13 Compact resolvents − △ + △ φ be with compact resolvent. Then by Theorem 11.9 there exists a continuous function Λ : ℝn 󳨀→ ℝ such that Λ(x) → ∞ as |x| → ∞ and 󵄨2 󵄨 (H(0, △φ)ϕ, ϕ) ≥ ∫ Λ(x)󵄨󵄨󵄨ϕ(x)󵄨󵄨󵄨 dλ(x)

(13.4)

ℝn

for all ϕ ∈ 𝒞0∞ (ℝn ). Kato’s inequality implies that (H(A, △φ)ϕ, ϕ) ≥ (H(0, △φ)|ϕ|, |ϕ|); see the proof of Proposition 10.11. So we get (H(A, △φ)ϕ, ϕ) ≥ (H(0, △φ)|ϕ|, |ϕ|) ≥ (Λ|ϕ|, |ϕ|) = (Λϕ, ϕ), and Theorem 11.9 gives that H(A, △φ) has a compact resolvent. Now, again by Theorem 11.9(c), it suffices to show that e0,△φ (Qz ) → ∞

(13.5)

as |z| → ∞. For this purpose, we have to estimate ∫Q (H(0, △φ)ϕ)ϕ dλ z

∫Q |ϕ|2 dλ

=

∫Q (|∇ϕ|2 + △φ|ϕ|2 ) dλ z

z

∫Q |ϕ|2 dλ

.

z

Without loss of generality, we can consider cubes instead of balls when using the following improved version of the Fefferman–Phong lemma as given in [2]. Lemma 13.4. Let V ∈ A∞ . Then there exist constants CV > 0 and βV ∈ (0, 1) such that for all cubes Q with side length R and for all u ∈ 𝒞0∞ (Q), CV

mβ (R2 avQ V ) R2

∫ |u|2 dλ ≤ ∫(|∇u|2 + V |u|2 ) dλ,

where mβ (t) = t for t ≤ 1 and mβ (t) = t βV for t ≥ 1. We apply this lemma for R = 1 and V = △φ. Then by (13.6) we obtain β

e0,△φ (Qz ) ≥ CV avVQV z

if avVQz = ∫Q △φ dλ ≥ 1. Hence (13.2) implies (13.5), and we are done. z

It remains to prove Lemma 13.4. From (11.12) we have 1

󵄨󵄨 󵄨2 󵄨2 2 󵄨 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 ≤ |x − y| ∫󵄨󵄨󵄨∇u(x + t(y − x))󵄨󵄨󵄨 dt, 0

(13.6)

13.1 Schrödinger operators

� 269

and hence 󵄨2 󵄨2 󵄨 󵄨 ∫ ∫󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 dλ(x)dλ(y) ≤ R4 ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dλ(x). Q

Q Q

In addition, we also have 󵄨2 󵄨 󵄨2 󵄨 R2 ∫ V (x)󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x) = ∫ ∫ V (x)󵄨󵄨󵄨u(x)󵄨󵄨󵄨 dλ(x)dλ(y), Q Q

Q

so we get 󵄨 󵄨2 ∫(|∇u|2 + V |u|2 ) dλ ≥ R−4 ∫ ∫󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 dλ(x)dλ(y)

Q

Q Q

󵄨 󵄨2 + R−2 ∫ ∫ V (y)󵄨󵄨󵄨u(y)󵄨󵄨󵄨 dλ(x)dλ(y) Q Q

󵄨 󵄨2 󵄨 󵄨2 ≥ R−2 ∫ ∫ min(V (y), R−2 )(󵄨󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 ) dλ(x)dλ(y) Q Q

1 ≥ avQ [min(V , R−2 )] ∫ |u|2 dλ, 2 Q

where we used the inequality 󵄨2 󵄨󵄨 󵄨2 󵄨 󵄨2 1 󵄨 󵄨󵄨u(x) − u(y)󵄨󵄨󵄨 + 󵄨󵄨󵄨u(y)󵄨󵄨󵄨 ≥ 󵄨󵄨󵄨u(x)󵄨󵄨󵄨 . 2 By Lemma 13.2(a) there exists ϵ > 0, independent of Q, such that E = {x ∈ Q : V (x) > ϵavQ V } satisfies |E| ≥ 21 |Q|. Hence avQ [min(V , R−2 )] ≥

1 min(ϵavQ V , R−2 ). 2

So we get the desired inequality if R2 avQ V ≤ 1. It remains to analyze the case R2 avQ V ≥ 1. For this purpose, we subdivide Q in a dyadic manner and stop for the first time when R(Q′ )2 avQ′ V < 1. In this way, we obtain a collection {Qi } of strict dyadic subcubes of Q that are maximal for the property R2i avQi V < 1. Since, by Lemma 13.1 (4), w dλ is a doubling measure and the ancestor Q̂ i of Qi satisfies (2Ri )2 avQ̂ V ≥ 1, there exists a constant A > 0 such that i

R2i avQi V ≥ A.

(13.7)

This implies that the Qi form a disjoint covering of Q up to a set of null measure. Now we get

270 � 13 Compact resolvents ∫(|∇u|2 + V |u|2 ) dλ = ∑ ∫(|∇u|2 + V |u|2 ) dλ i Q i

Q

2 ≥ C ′ ∑ min(avQi V , R−2 i ) ∫ |u| dλ i

≥ AC

′

Qi

2 ∑ R−2 i ∫ |u| dλ i Qi

≥ AC ′ min( i

2

R ) R−2 ∫ |u|2 dλ, Ri Qi

where we used (13.7). By Lemma 13.2(b) there exists α ≥ 1 such that for any cube Q and any measurable subset E of Q, we have α−1

(

avE V |E| ) ≥ C( ) avQ V |Q|

.

We apply this to E = Qi and get (

2 R2 avQ V avQi V R ) = 2 Ri Ri avQi V avQ V

≥ R2 avQ V

avQi V avQ V

≥ CR2 avQ V (

2(α−1)

Ri ) R

.

Finally, this implies min( i

2

R β ) ≥ C(R2 avQ V ) Ri

with β = 1/α, which proves the lemma. Corollary 13.5. Let φ be a subharmonic 𝒞 2 -function such that △φ(z) > ϵ > 0 and △ φ ∈ B2 (ℝ2 ). The canonical solution operator Sφ to 𝜕 is compact on L2 (ℂ, e−φ ) if and only if lim|z|→∞ ∫Q △φ dλ = +∞. z

Proof. Since Nφ = Sφ∗ Sφ , the result follows from Theorems 13.3 and 2.5.

13.2 Dirac and Pauli operators

� 271

Remark 13.6. Using the notations of Chapter 8 and the assumptions on φ, we can express the last theorem in the following way: Let φ be a subharmonic 𝒞 2 -function such that △φ(z) > ϵ > 0

and

△ φ ∈ B2 (ℝ2 ).

The Schrödinger operator with magnetic field 𝒮=

1 (− △A +B), 4

where ΔA = (

2

2

𝜕 i 𝜕φ 𝜕 i 𝜕φ + ) +( − ) 𝜕x 2 𝜕y 𝜕y 2 𝜕x

and

B=

1 △ φ, 2

has a compact resolvent if and only if lim|z|→∞ ∫Q △φ dλ = +∞. z

13.2 Dirac and Pauli operators We return to the Dirac and Pauli operators related to the weight function φ: 𝜕 𝜕 − A1 ) σ1 + (−i − A2 ) σ2 , 𝜕x 𝜕y

𝒟 = (−i

where A1 = − 21 𝜕φ , A2 = 𝜕y

1 𝜕φ 2 𝜕x

and

σ1 = (

0 1

1 ), 0

σ2 = (

0 i

−i ) . 0

The square of 𝒟 is diagonal with the Pauli operators P± on the diagonal: 2

𝒟 =(

P− 0

0 ), P+

where P± = (−i with B =

1 2

2

2

𝜕 𝜕 − A1 ) + (−i − A2 ) ± B = −ΔA ± B 𝜕x 𝜕y

△ φ.

Theorem 13.7. Suppose that |z|2 △φ(z) → +∞ as |z| → ∞. Then the corresponding Dirac operator 𝒟 has a noncompact resolvent.

272 � 13 Compact resolvents Proof. By Proposition 9.40, 𝒟2 has a compact resolvent if and only if 𝒟 has a compact resolvent. Suppose that 𝒟 has a compact resolvent. Since 2

𝒟 =(

P− 0

0 ), P+

this would imply that both P± have compact resolvents. We know from (10.30) that P− = 4D D = 4e−φ/2 𝜕φ 𝜕 eφ/2 ∗

∗

and that P− is a nonnegative self-adjoint operator. It follows from Theorem 8.11 that the space of entire functions A2 (ℂ, e−φ ) is of infinite dimension. This means that 0 belongs to the essential spectrum of P− . Hence, by Proposition 9.39, P− fails to have a compact resolvent, and we arrive at a contradiction. A similar conclusion can be drawn in several variables for the Witten Laplacian Δ(0,0) = D1 D1 = e−φ/2 𝜕φ 𝜕 eφ/2 ; φ ∗

∗

if lim|z|→∞ |z|2 μφ (z) = +∞, then Δ(0,0) fails to have a compact resolvent (μφ is the lowest φ eigenvalue of the Levi matrix Mφ ). Remark 13.8. The notion of a doubling measure appeared already in Lemma 13.1(4). Recall that a nonnegative Borel measure μ on ℂ is doubling if there exists a constant C > 0 such that for all z ∈ ℂ and r > 0, μ(D(z, r)) ≤ Cμ(D(z, r/2)).

(13.8)

μ(D(z, 2r)) ≥ (1 + C −3 )μ(D(z, r))

(13.9)

It can be shown that

for all z ∈ ℂ and r > 0; in particular, μ(ℂ) = ∞, unless μ(ℂ) = 0 (see [90]). Examples: If p(z, z) is a polynomial on ℂ of degree d, then 󵄨 󵄨a dμ(z) = 󵄨󵄨󵄨p(z, z)󵄨󵄨󵄨 dλ(z),

1 a>− , d

is a doubling measure on ℂ; see [90]. If V belongs to the reverse Hölder class B2 (ℝ2 ) or to A∞ , then V (x)dλ(x) is a doubling measure. Let φ : ℂ 󳨀→ ℝ+ be a subharmonic 𝒞 2 -function. Suppose that dμ = △φ dλ is a nontrivial doubling measure.

13.2 Dirac and Pauli operators

� 273

We show that the weighted space of entire functions A2 (ℂ, e−φ ) = {f entire : ‖f ‖2φ = ∫ |f |2 e−φ dλ < ∞} ℂ

is of infinite dimension. We know that μ(ℂ) = ∫ △φ dλ = ∞.

(13.10)

ℂ

Let Kφ (z, w) denote the Bergman kernel of A2 (ℂ, e−φ ). So if f ∈ A2 (ℂ, e−φ ), then f (z) = ∫ Kφ (z, w)f (w)e−φ(w) dλ(w).

(13.11)

ℂ

We claim that ∞

∫ Kφ (z, z)e−φ(z) dλ(z) = ∑ ‖fk ‖2φ k=1

ℂ

(13.12)

for any complete orthonormal basis (fk )k of A2 (ℂ, e−φ ), finite or infinite. The partial sums N

󵄨 󵄨2 FN (z) = ∑ 󵄨󵄨󵄨fk (z)󵄨󵄨󵄨 k=1

form a pointwise nondecreasing sequence: FN (z) ≤ FN+1 (z) for all z ∈ ℂ and N ∈ ℕ. In addition, we have ∞

lim FN (z) = ∑ fk (z)fk (z) = Kφ (z, z);

N→∞

k=1

see (1.9). The monotone convergence theorem implies that ∞

∞

k=1

ℂ k=1

󵄩 󵄨 󵄩2 󵄨2 ∑ 󵄩󵄩󵄩fk (z)󵄩󵄩󵄩φ = ∫ ∑ 󵄨󵄨󵄨fk (z)󵄨󵄨󵄨 e−φ(z) dλ(z) = ∫ Kφ (z, z)e−φ(z) dλ(z).

(13.13)

ℂ

Now we use the following pointwise inequality for the Bergman kernel, which is due to Marzo and Ortega-Cerdà ([74]): there exists a constant C > 0 such that C −1 △ φ(z) ≤ Kφ (z, z)e−φ(z) ≤ C △ φ(z) for each z ∈ ℂ. From this we conclude that A2 (ℂ, e−φ ) is of infinite dimension.

(13.14)

274 � 13 Compact resolvents Relations (13.10) and (13.13) imply that the corresponding Dirac operator 𝒟 fails to be with compact resolvent if φ is a subharmonic function such that dμ = △φ dλ is a nontrivial doubling measure.

13.3 Notes Iwatsuka [59] and Avron, Herbst, and Simon [3] were the first to obtain results on the problem of compact resolvence of Schrödinger operators with magnetic field. Theorem 13.3 was inspired by the work of Iwatsuka. Christ [17] initiated the study of weighted spaces of entire functions, where the weight φ has the property that dμ = △φ dλ is a nontrivial doubling measure. There is another approach to characterize the compactness of Nφ by means of estimates of the Bergman kernel of A2 (ℂ, e−φ ) [74]. The results there read as follows: Suppose that △φ defines a doubling measure. Then Nφ is compact if and only if ∫𝔹(z,1) △φ dλ → ∞ as |z| → ∞, where 𝔹(z, 1) = {w : |w − z| < 1}. Witten Laplacians, as they appear in the context of semiclassical analysis, are studied in [50, 53] and [61]. A thorough treatment of Dirac and Pauli operators can be found in [21, 54] and [92].

14 Spectrum of ◻ on the Fock space In this chapter, we investigate the spectral properties of the ◻-operator for the Fock space using solutions of the corresponding partial differential equations without boundary conditions. It is important that in this case the ◻-operator acts diagonally, which is related to the fact that the weight function is decoupled, a property that turns out to be an obstruction for compactness (see Chapter 15). We show that the spectrum of the ◻-operator for the Fock space consists of positive integers of infinite multiplicity and indicates consequences of this result related to the essential spectrum.

14.1 The general setting We consider the weighted 𝜕-complex 𝜕

𝜕

←󳨀

←󳨀

L2(0,q−1) (ℂn , e−φ ) 󳨀→ L2(0,q) (ℂn , e−φ ) 󳨀→ L2(0,q+1) (ℂn , e−φ ), ∗ 𝜕φ

∗ 𝜕φ

where for (0, q)-forms u = ∑′|J|=q uJ dzJ with coefficients in 𝒞0∞ (ℂn ), we have ′

n

𝜕u = ∑ ∑ |J|=q j=1

𝜕uJ 𝜕zj

dzj ∧ dzJ

and ∗

′

𝜕φ u = − ∑

n

∑ δk ukK dzK ,

|K|=q−1 k=1

𝜕φ where δk = 𝜕z𝜕 − 𝜕z . k k The complex Laplacian on (0, q)-forms is defined as ∗

∗

◻φ,q := 𝜕 𝜕φ + 𝜕φ 𝜕. The operator ◻φ,q is self-adjoint and positive, which means that (◻φ,q f , f )φ ≥ 0

for f ∈ dom(◻φ ).

The associated Dirichlet form is defined as ∗

∗

Qφ (f , g) = (𝜕f , 𝜕g)φ + (𝜕φ f , 𝜕φ g)φ ∗

(14.1)

for f , g ∈ dom(𝜕) ∩ dom(𝜕φ ). The weighted 𝜕-Neumann operator Nφ,q is – if it exists – the bounded inverse of ◻φ,q . https://doi.org/10.1515/9783111182926-014

276 � 14 Spectrum of ◻ on the Fock space ∗

We indicate that a square-integrable (0, 1)-form f = ∑nj=1 fj dzj belongs to dom(𝜕φ ) if and only if n

eφ ∑ j=1

𝜕 (f e−φ ) ∈ L2 (ℂn , e−φ ), 𝜕zj j

where the derivative is to be taken in the sense of distributions, and that the forms with ∗ coefficients in 𝒞0∞ (ℂn ) are dense in dom(𝜕) ∩ dom(𝜕φ ) in the graph norm f 󳨃→ (‖𝜕f ‖2φ + ∗

1

‖𝜕φ f ‖2φ ) 2 (see Chapter 6 and [36]). We consider the Levi matrix

Mφ = (

𝜕2 φ ) 𝜕zj 𝜕zk jk

of φ and suppose that the sum sq of any q (equivalently, the smallest q) eigenvalues of Mφ satisfies lim inf sq (z) > 0.

(14.2)

|z|→∞

In Proposition 6.7, we showed that (14.2) implies that there exists a continuous linear operator Nφ,q : L2(0,q) (ℂn , e−φ ) 󳨀→ L2(0,q) (ℂn , e−φ ) such that ◻φ,q ∘ Nφ,q u = u for all u ∈ L2(0,q) (ℂn , e−φ ). If the sum sq of any q (equivalently, the smallest q) eigenvalues of Mφ satisfies lim sq (z) = ∞,

(14.3)

|z|→∞

then the 𝜕-Neumann operator Nφ,q : L2(0,q) (ℂn , e−φ ) 󳨀→ L2(0,q) (ℂn , e−φ ) is compact (see Proposition 11.16). To find the canonical solution to 𝜕f = u, where u ∈ L2(0,1) (ℂn , e−φ ) is a given (0, 1)∗

form such that 𝜕u = 0, we can take f = 𝜕φ Nφ,1 u, which will also satisfy f ⊥ ker𝜕.

If the weight function is φ(z) = |z|2 , then the corresponding Levi matrix Mφ is the 2

2

identity. The space A2 (ℂn , e−|z| ) of entire functions belonging to L2 (ℂn , e−|z| ) is the Fock space, which plays an important role in quantum mechanics. In this case, ∗

◻φ,0 u = 𝜕φ 𝜕u = − 2

for u ∈ dom(◻φ,0 ) ⊆ L2 (ℂn , e−|z| ), and

n 1 △ u + ∑ zj uzj 4 j=1

(14.4)

14.2 Determination of the spectrum

∗

◻φ,n u = 𝜕 𝜕φ u = −

n 1 △ u + ∑ zj uzj + n u 4 j=1

�

277

(14.5)

2

for u ∈ dom(◻φ,n ) ⊆ L2(0,n) (ℂn , e−|z| ). For n > 1 and 1 ≤ q ≤ n − 1, the 𝜕-Neumann Laplacian ◻φ,q acts diagonally (see Chapter 10): for 2

′

u = ∑ uJ dzJ ∈ dom(◻φ,q ) ⊆ L2(0,q) (ℂn , e−|z| ), |J|=q

we have ∗

∗

′

◻φ,q u = (𝜕 𝜕φ + 𝜕φ 𝜕) u = ∑ (− |J|=q

n 1 △ uJ + ∑ zj uJzj + q uJ ) dzJ . 4 j=1

(14.6)

14.2 Determination of the spectrum To determine the spectrum of ◻φ,q for φ(z) = |z|2 , we use Lemma 9.33. For simplicity, to explain the general method, we start with the complex onedimensional case. Looking for the eigenvalues μ of ◻φ,0 , we have to consider ◻φ,0 u = −uz z + zuz = μu.

(14.7)

2

It is clear that the space A2 (ℂn , e−|z| ) is contained in the eigenspace of the eigenvalue μ = 0. For any positive integer k, the antiholomorphic monomial zk is an eigenfunction for the eigenvalue μ = k. In the following, we denote ℕ0 = ℕ ∪ {0}. Lemma 14.1. Let n = 1. For k ∈ ℕ0 and m ∈ ℕ, the functions m

uk,m (z, z) = zk+m zm + ∑ j=1

(−1)j (k + m)! m! zk+m−j zm−j j! (k + m − j)! (m − j)!

(14.8)

are eigenfunctions for the eigenvalue k + m of the operator ◻φ,0 u = −uz z + zuz . For k ∈ ℕ and m ∈ ℕ0 , the functions k

vk,m (z, z) = zk zk+m + ∑ j=1

(−1)j (k + m)! k! zk−j zk+m−j j! (k + m − j)! (k − j)!

are eigenfunctions for the eigenvalue k of the operator ◻φ,0 u = −uz z + zuz .

(14.9)

278 � 14 Spectrum of ◻ on the Fock space Proof. To prove (14.8), we set uk,m (z, z) = zk+m zm + a1 zk+m−1 zm−1 + a2 zk+m−2 zm−2 + ⋅ ⋅ ⋅ + am−1 zk+1 z + am zk and compute 𝜕2 u (z, z) 𝜕z𝜕z k,m = (k + m)mzk+m−1 zm−1 + a1 (k + m − 1)(m − 1)zk+m−2 zm−2 + ⋅ ⋅ ⋅ + am−1 (k + 1)zk and z

𝜕 u (z, z) 𝜕z k,m = (k + m)zk+m zm + a1 (k + m − 1)zk+m−1 zm−1 + ⋅ ⋅ ⋅ + am−1 (k + 1)zk+1 z + am kzk ,

which implies that the function uk,m is an eigenfunction for the eigenvalue μ = k + m, and from (14.7) we obtain, comparing the highest exponents of z and z, (k + m)m − a1 (k + m − 1) = −(k + m)a1 , and hence a1 = −(k + m)m. Comparing the next lower exponents, we get a1 (k + m − 1)(m − 1) − a2 (k + m − 2) = −a2 (k + m) and a2 =

1 2

(k + m)(k + m − 1)m(m − 1). In general, we find that for j = 1, 2, . . . , m, aj =

(−1)j (k + m)! m! , j! (k + m − j)! (m − j)!

which proves (14.8). To show (14.9), we set vk,m (z, z) = zk zk+m + b1 zk−1 zk+m−1 + b2 zk−2 zk+m−2 + ⋅ ⋅ ⋅ + bk−1 z zm+1 + bk zm and compute 𝜕2 v (z, z) 𝜕z𝜕z k,m = k(k + m)zk−1 zk+m−1 + b1 (k − 1)(k + m − 1)zk−2 zk+m−2 + ⋅ ⋅ ⋅ + bk−1 z zm+1 and z

𝜕 v (z, z) = kzk zk+m + b1 (k − 1)zk−1 zk+m−1 + ⋅ ⋅ ⋅ + bk−1 z zm+1 , 𝜕z k,m

14.2 Determination of the spectrum

� 279

which implies that the function vk,m is an eigenfunction for the eigenvalue μ = k, for each m ∈ ℕ, and from (14.7) we obtain, comparing the highest exponents of z and z, k(k + m) − b1 (k − 1) = −kb1 , and hence b1 = −(k + m)k. Comparing the next lower exponents, we get b1 (k − 1)(k + m − 1) − b2 (k − 2) = −b2 k and b2 =

1 2

(k + m)(k + m − 1)k(k − 1). In general, we find that for j = 1, 2, . . . , k, bj =

(−1)j (k + m)! k! , j! (k + m − j)! (k − j)!

which proves (14.9). Now we are able to prove the following: Theorem 14.2. Let n = 1 and φ(z) = |z|2 . The spectrum of ◻φ,0 consists of all nonnegative integers {0, 1, 2, . . . } each of which is of infinite multiplicity, so 0 is the bottom of the essential spectrum. The spectrum of ◻φ,1 consists of all positive integers {1, 2, 3, . . . } each of which is of infinite multiplicity. 2

Proof. We already know that the whole Bergman space A2 (ℂ, e−|z| ) is contained in the eigenspace of the eigenvalue 0 of the operator ◻φ,0 and for any positive integer k, the

antiholomorphic monomial zk is an eigenfunction for the eigenvalue μ = k. In addition, all functions of the form zν zκ with ν, κ ∈ ℕ0 can be expressed as linear combinations of functions of the form (14.8) and (14.9). For a fixed k ∈ ℕ, the functions of the form (14.9) have infinite multiplicity as the parameter m ∈ ℕ0 is free. So these eigenvalues are of infinite multiplicity. All the eigenfunctions considered so far yield a complete orthogonal 2 basis of L2 (ℂ, e−|z| ), since the Hermite polynomials {H0 (x)Hk (y), H1 (x)Hk−1 (y), . . . , Hk (x)H0 (y)} 2

2

for k = 0, 1, 2, . . . form a complete orthogonal system in L2 (ℝ2 , e−x −y ) (see, for instance, [29]), and since x = 1/2(z + z) , y = i/2(z − z), we can apply Lemma 9.33 and obtain that the spectrum of ◻φ,0 is ℕ0 . The statement for the spectrum of ◻φ,1 follows from (14.5). For several variables, we can adopt the method from above to obtain the following result. Theorem 14.3. Let n > 1, φ(z) = |z1 |2 + ⋅ ⋅ ⋅ + |zn |2 , and 0 ≤ q ≤ n. The spectrum of ◻φ,q consists of all integers {q, q + 1, q + 2, . . . } each of which is of infinite multiplicity.

280 � 14 Spectrum of ◻ on the Fock space Proof. Recall that the 𝜕-Neumann Laplacian ◻φ,q acts diagonally and that ′

◻φ,q u = ∑ (− |J|=q

n 1 △ uJ + ∑ zj uJzj + q uJ ) dzJ . 4 j=1

The factor q in the last formula is responsible for the fact that the eigenvalues start with q, which can be seen, in each component separately, by −

n 1 △ uJ + ∑ zj uJzj = (μ − q) uJ . 4 j=1

Now let k1 , k2 , . . . , kn ∈ ℕ0 and m1 , m2 , . . . , mn ∈ ℕ. Then the function uk1 ,m1 (z1 , z1 ) uk2 ,m2 (z2 , z2 ) . . . ukn ,mn (zn , zn ) is the component of an eigenfunction for the eigenvalue ∑nj=1 (kj + mj ) of the operator ◻φ,q , which follows from (14.6) and (14.8). Similarly, it follows from (14.6) and (14.9) that for k1 , k2 , . . . , kn ∈ ℕ and m1 , m2 , . . . , mn ∈ ℕ0 , the function vk1 ,m1 (z1 , z1 ) vk2 ,m2 (z2 , z2 ) . . . vkn ,mn (zn , zn ) is an eigenfunction for the eigenvalue ∑nj=1 kj . All other possible n-fold products with factors ukj ,mj or vkj ,mj (also mixed) appear as eigenfunctions of ◻φ,q . α

β

α

From this we obtain that all expressions of the form z1 1 z1 1 . . . znn zβnn for arbitrary αj , βj ∈ ℕ0 , j = 1, . . . , n, can be written as linear combinations of eigenfunctions of ◻φ,q , 2

which proves that all these eigenfunctions constitute a complete basis in L2(0,q) (ℂn , e−|z| ); see the proof of Theorem 14.2. So we can again apply Lemma 9.33.

Remark 14.4. (i) Since in all cases the essential spectrum is nonempty, the corresponding ◻φ,q operator fails to be with compact resolvent (see Proposition 9.39). (ii) For the weight function α

φ(z) = (|z1 |2 + |z2 |2 + ⋅ ⋅ ⋅ + |zn |2 ) + |z|2

for α > 1,

the situation is completely different: The operators ◻φ,q , 1 ≤ q ≤ n, are with compact resolvent (see Proposition 11.16), so the essential spectrum must be empty. We can use the results from above to settle the corresponding questions for the socalled Witten Laplacian, which is defined on L2 (ℂn ).

14.2 Determination of the spectrum

For this purpose, we set Zk =

′

∑′|J|=q

𝜕 𝜕zk

+

1 𝜕φ 2 𝜕zk

and Zk∗ = − 𝜕z𝜕 + k

1 𝜕φ 2 𝜕zk

� 281

and consider (0, q)-

forms h = hJ dzJ , where ∑ means that we sum up only increasing multi-indices J = (j1 , . . . , jq ), and where dzJ = dzj1 ∧ ⋅ ⋅ ⋅ ∧ dzjq . We define n

′

Dq+1 h = ∑ ∑ Zk (hJ ) dzk ∧ dzJ k=1 |J|=q

and n

∗

′

Dq h = ∑ ∑ Zk∗ (hJ ) dzk ⌋dzJ , k=1 |J|=q

where dzk ⌋dzJ denotes the contraction or interior multiplication by dzk , i. e., we have ⟨α, dzk ⌋dzJ ⟩ = ⟨dzk ∧ α, dzJ ⟩ for each (0, q − 1)-form α. The complex Witten Laplacian on (0, q)-forms is then given by ∗

∗

Δ(0,q) = Dq Dq + Dq+1 Dq+1 φ for q = 1, . . . , n − 1. The general D-complex has the form Dq

Dq+1

L2(0,q−1) (ℂn ) 󳨀→ L2(0,q) (ℂn ) 󳨀→ L2(0,q+1) (ℂn ) . ←󳨀

←󳨀

∗ Dq

∗ Dq+1

It follows that Dq+1 Δ(0,q) = Δ(0,q+1) Dq+1 φ φ

and

∗

∗

Dq+1 Δ(0,q+1) = Δ(0,q) Dq+1 . φ φ

We remark that ∗

n

′

′

Dq h = ∑ ∑ Zk∗ (hJ ) dzk ⌋dzJ = ∑ k=1 |J|=q

n

∑ Zk∗ (hkK ) dzK .

|K|=q−1 k=1

In particular, for a function v ∈ L2 (ℂn ), we get ∗

n

∗ Δ(0,0) φ v = D1 D1 v = ∑ Zj Zj (v), j=1

(14.10)

and for a (0, 1)-form g = ∑nℓ=1 gℓ dzℓ ∈ L2(0,1) (ℂn ), we obtain ∗

∗

(0,0) Δ(0,1) ⊗ I)g + Mφ g, φ g = (D1 D1 + D2 D2 )g = (Δφ

(14.11)

282 � 14 Spectrum of ◻ on the Fock space where we set n

n

𝜕2 φ g ) dzj 𝜕zk 𝜕zj k k=1

Mφ g = ∑( ∑ j=1

and n

(Δ(0,0) ⊗ I) g = ∑ Δ(0,0) gk dzk . φ φ k=1

In general, we have that Δ(0,q) = e−φ/2 ◻φ,q eφ/2 φ

(14.12)

for q = 0, 1, . . . , n. In our case φ(z) = |z1 |2 + ⋅ ⋅ ⋅ + |zn |2 , we get ′

Δ(0,q) φ h = ∑ (− |J|=q

1 n n 1 1 △ hJ + ∑(zj hJzj − zj hJzj ) + |z|2 hJ + (q − ) hJ ) dzJ 4 2 j=1 4 2

(14.13)

for ′

2 n h = ∑ hJ dzJ ∈ dom(Δ(0,q) φ ) ⊆ L(0,q) (ℂ ). |J|=q

(0,q)

Using (14.12) and Lemma 9.31, we get that Δφ Hence by Theorem 14.3 we obtain the following:

and ◻φ,q have the same spectrum.

Theorem 14.5. Let φ(z) = |z1 |2 + ⋅ ⋅ ⋅ + |zn |2 and 0 ≤ q ≤ n. The spectrum of the Wit(0,q) ten Laplacian Δφ consists of all integers {q, q + 1, q + 2, . . . } each of which is of infinite multiplicity.

14.3 Notes Only in a few cases it is possible to determine the exact form of the spectrum of a ◻-operator. Folland [28] described it for the unit ball in ℂn using spherical harmonics; in this case the spectrum is discrete consisting of eigenvalues of finite multiplicity. With the help of Bessel functions, Fu [32] showed that the spectrum of the ◻-operator for a polycylinder contains eigenvalues of finite and infinite multiplicity. The spectrum of Δ(0,0) φ , even in a more general form, was calculated by Ma and Marinescu [72, 73]. They directly use Hermite polynomials, the fact that the Hermite polyno2 mials form a complete orthogonal system of the weighted space L2 (ℝ, e−x ), and general properties of the Kodaira Laplacian.

15 Obstructions to compactness In this chapter, we give some examples of domains or weights for which the corresponding 𝜕-Neumann operator or the canonical solution operator to 𝜕 fails to be compact. For this purpose, we will mainly use the characterization of compactness from Chapter 11. The standard examples for noncompactness are polydiscs and so-called decoupled weights.

15.1 The bidisc First, we consider the canonical solution operator to 𝜕 for the bidisc 𝔻 × 𝔻 (see [68]). We know from Chapter 1 that the monomials φn (z) = √

n+1 n z , π

n = 0, 1, 2, . . . ,

constitute a complete orthonormal system in A2 (𝔻). Consider the following (0, 1)forms αn in L2(0,1) (𝔻 × 𝔻) with holomorphic coefficients: αn (z1 , z2 ) = φn (z1 ) dz2 . They are 𝜕-closed, and their norms in L2(0,1) (𝔻 × 𝔻) are ‖αn ‖ = √π,

n = 0, 1, 2, . . . .

The canonical solution to 𝜕u = αn is given by un (z1 , z2 ) = φn (z1 ) z2 ; this means that un ∈ A2 (𝔻 × 𝔻)⊥ , which follows by the fact that for each h ∈ A2 (𝔻 × 𝔻), we have −

∫ un (z1 , z2 ) h(z1 , z2 ) dλ(z1 , z2 ) = ∫ φn (z1 ) (∫ z2 h(z1 , z2 ) dλ(z2 )) dλ(z1 ), 𝔻

𝔻×𝔻

𝔻

where the inner integral vanishes by Cauchy’s theorem applied to the holomorphic function z2 󳨃→ z2 h(z1 , z2 ). Finally, ‖un ‖ = √

π 2

and

un ⊥um

if n ≠ m,

which follows from (un , um ) = ∫ |z2 |2 dλ(z2 ) ∫ φn (z1 )φm (z1 ) dλ(z1 ) 𝔻 https://doi.org/10.1515/9783111182926-015

𝔻

284 � 15 Obstructions to compactness and (φn , φm ) = δn,m . Thus {un } has no convergent subsequence in L2 (𝔻 × 𝔻). This shows that the canon∗ ical solution operator 𝜕 N to 𝜕 fails to be compact. Further obstructions to compactness on weakly pseudoconvex domains can be found in [33, 34] and [91].

15.2 Weighted spaces We continue to calculate the integrals in (11.30) for the weight φ(z) = |z|α in ℂ. We set β = α/2 and uk (z) = zk for k ∈ ℕ. The left-hand side of (11.30) is ∞

α α 󵄨 󵄨2 ∫ 󵄨󵄨󵄨uk (z)󵄨󵄨󵄨 e−|z| dλ(z) = 2π ∫ r 2k+1 e−r dr;

R

ℂ\BR

we indicate that ∞

α

∫ r 2k+1 e−r dr = 0

1 k 1 Γ( + ). 2β β β

The right-hand side of (11.30) reads α α 󵄨 ∗ 󵄨 󵄨2 󵄨2 ∫󵄨󵄨󵄨𝜕φ uk (z)󵄨󵄨󵄨 e−|z| dλ(z) = ∫󵄨󵄨󵄨−kzk−1 + βzβ+k−1 zβ 󵄨󵄨󵄨 e−|z| dλ(z)

ℂ

ℂ ∞

2β

= 2π ∫ (k 2 r 2k−1 − 2kβr 2β+2k−1 + β2 r 4β+2k−1 ) e−r dr 0

= 2π[

β k k k k2 Γ( ) − k Γ( + 1) + Γ( + 2)] 2β β β 2 β

= πβ Γ(

k + 1). β

If α = 2, then it follows that condition (11.30) is not satisfied. For this purpose, we consider the integral ∞

2

∫ r 2k+1 e−r dr, R

and substituting r 2 = s, we obtain ∞

2

∞

∫ r 2k+1 e−r dr = ∫ sk e−s ds. R

R2

15.2 Weighted spaces

� 285

Now k-times integrating by parts, we get ∞

∞

2

2

k

R2j . j! j=0

∫ sk e−s ds = e−R R2k + k ∫ sk−1 e−s ds = e−R k! ∑ R2

R2

Observe that for β = 1, we have Γ(

k k 1 + 1) = Γ( + ) = k!. β β β

There is ϵ0 > 0 such that for each R > 0, there exists k ∈ ℕ such that 2

k

R2j > ϵ0 , j! j=0

e−R ∑

so that condition (11.30) is not satisfied for α = 2. This means that the 𝜕-Neumann oper2 ator Nφ on L2 (ℂ, e−|z| ) fails to be compact, and as Nφ = S ∗ S, where S is the canonical solution operator to 𝜕, the canonical solution operator S also fails to be compact (compare Theorem 2.22). Another proof of this fact uses spectral theory: from (10.3) we know that ◻φ u = 𝜕 𝜕φ u = − ∗

𝜕2 u 𝜕u +z + u, 𝜕z𝜕z 𝜕z 2

and hence it follows immediately that the whole space A2 (ℂ, e−|z| ) is a subspace of the eigenspace to the eigenvalue 1 of the operator ◻φ , which means that the essential spectrum of ◻φ is nonempty and Nφ fails to be compact by Proposition 9.39. In the next examples, we consider the decoupled 𝒞 2 -weights φ(z1 , z2 , . . . , zn ) = φ1 (z1 ) + φ2 (z2 ) + ⋅ ⋅ ⋅ + φn (zn ) and follow an idea of Schneider [88] (see also the results in Chapter 3). Proposition 15.1. Suppose that n ≥ 2 and that there exists ℓ such that A2 (ℂ, e−φℓ ) is infinite dimensional. Suppose also that 1 ∈ L2 (ℂ, e−φj ) for all j. Suppose finally that zk ∈ L2 (ℂ, e−φk ) for some k ≠ ℓ. Then the canonical solution operator to 𝜕 fails to be compact even on the space A2(0,1) (ℂn , e−φ ). Proof. Let Pk denote the Bergman projection Pk : L2 (ℂ, e−φk ) 󳨀→ A2 (ℂ, e−φk ). It is clear that the function (zk − Pk zk ) is not zero. Let (fν )ν be an infinite orthonormal system in A2 (ℂ, e−φℓ ) and define

286 � 15 Obstructions to compactness hν (z) := fν (zℓ )(zk − Pk zk ). Then (hν )ν is an orthogonal family in A2 (ℂn , e−φ )⊥ . To see this, let g ∈ A2 (ℂn , e−φ ) and note that (v, Pk zk )φk = (v, zk )φk for v ∈ A2 (ℂ, e−φk ), which implies that (g, hν )φ = 0. In addition, we have 𝜕hν = fν (zℓ )dzk . Hence (𝜕hν )ν constitutes a bounded sequence in A2(0,1) (ℂn , e−φ ), and for the canonical solution operator S, we have S(fν (zℓ )dzk ) = hν , and since (hν )ν is an orthogonal family, it has no convergent subsequence, which implies the result. Remark 15.2. If the conditions of Proposition 15.1 are satisfied, then the corresponding 𝜕-Neumann operator Nφ,1 also fails to be compact, which follows from Proposition 11.27. In the following example, we consider the 𝜕-Neumann operator Nφ,1 for a decoupled weight φ. Example. Let φ(z1 , z2 ) = |z1 |2 +|z2 |2 and consider the corresponding 𝜕-Neumann operator Nφ,1 . We will investigate the following sequence of (0, 1)-forms: uk (z1 , z2 ) = ψk (z1 ) dz2 , where ψk (z1 ) =

zk1 √πk!

for k ∈ ℕ. It follows that 𝜕uk = 0 for each k ∈ ℕ and 𝜕φ uk (z1 , z2 ) = z2 ψk (z1 ). ∗

This implies ◻φ,1 uk = uk

and

Nφ,1 uk = uk

for each k ∈ ℕ. The set {uk : k ∈ ℕ} is a bounded set of mutually orthogonal (0, 1)-forms in L2(0,1) (ℂn , e−φ ). As Nφ,1 uk = uk , it follows that Nφ,1 fails to be compact. The following computation shows that condition (11.30) is not satisfied for the (0, 1)forms uk , where we consider ∫ℂ2 \Q instead of ∫ℂ2 \𝔹 , where R

R

QR = {(z1 , z2 ) : |z1 | < R , |z2 | < R}. We have R

∞

2 2 2 2 4π 󵄨 󵄨2 ∫ 󵄨󵄨󵄨uk (z1 , z2 )󵄨󵄨󵄨 e−|z1 | −|z2 | dλ(z1 , z2 ) = ∫( ∫ r12k+1 e−r1 dr1 )r2 e−r2 dr2 k!

0

ℂ2 \QR

R

∞ ∞

+

2 2 4π ∫ ( ∫ r12k+1 e−r1 dr1 )r2 e−r2 dr2 . k!

R

0

15.2 Weighted spaces

�

287

After the substitution r12 = s, the first integral is equal to R

∞

0

R2

2 2π ∫( ∫ sk e−s ds)r2 e−r2 dr2 . k!

As in the above example, we get ∞

2

k

R2j , j! j=0

∫ sk e−s ds = e−R k! ∑ R2

and finally substituting r22 = t, R

∞

0

R2

R2

k 2 2 2π R2j ∫( ∫ sk e−s ds)r2 e−r2 dr2 = πe−R ∑ ∫ e−t dt k! j! j=0 0

2

k

2j

2 R (1 − e−R ). j! j=0

= πe−R ∑ On the right-hand side of (11.30), we only have the term

2 2 1 ∫ |z1 |2k |z2 |2 e−|z1 | −|z2 | dλ(z1 , z2 ) πk!

ℂ2

∞

∞

0

0

2 2 4π = ∫ r12k+1 e−r1 dr1 ∫ r23 e−r2 dr2 k!

= π. This implies

k 2 2 2 2 R2j 󵄨 󵄨2 (1 − e−R ). ∫ 󵄨󵄨󵄨uk (z1 , z2 )󵄨󵄨󵄨 e−|z1 | −|z2 | dλ(z1 , z2 ) ≥ πe−R ∑ j! j=0

ℂ2 \QR

There is ϵ0 > 0 such that for each R > 0, there exists k ∈ ℕ such that 2

k

2 R2j (1 − e−R ) > ϵ0 , j! j=0

e−R ∑

so that condition (11.30) is not satisfied. Finally, we discuss the compactness of Nφ,1 and Nφ,2 in ℂ2 in a more general setting: let φ(z1 , z2 ) = φ1 (z1 ) + φ2 (z2 ). The eigenvalues of the Levi matrix are

288 � 15 Obstructions to compactness 𝜕2 φ1 𝜕z1 𝜕z1

and

𝜕2 φ2 . 𝜕z2 𝜕z2

If the (0, 1)-form u = u1 dz1 + u2 dz2 belongs to dom(◻φ,1 ), then ◻φ,1 u = (−

𝜕2 u1 𝜕2 u1 𝜕φ 𝜕u 𝜕φ 𝜕u 𝜕2 φ1 − + 1 1 + 2 1 + u ) dz1 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z1 𝜕z1 1

+ (−

𝜕2 u2 𝜕2 u2 𝜕φ 𝜕u 𝜕φ 𝜕u 𝜕2 φ2 − + 1 2 + 2 2 + u ) dz2 , 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z2 𝜕z2 2

and for V = v dz1 ∧ dz2 ∈ dom(◻φ,2 ), we have that ◻φ,2 V equals (−

𝜕φ 𝜕v 𝜕φ2 𝜕v 𝜕2 φ1 𝜕2 φ2 𝜕2 v 𝜕2 v − + 1 + + v+ v) dz1 ∧ dz2 . 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z1 𝜕z1 𝜕z2 𝜕z2 𝜕z1 𝜕z1 𝜕z2 𝜕z2

Now suppose that A2 (ℂ, e−φ1 ) is infinite dimensional, that 1 ∈ L2 (ℂ, e−φj ) for j = 1, 2, that z2 ∈ L2 (ℂ, e−φ2 ), and finally that 𝜕2 φ1 (z1 ) 𝜕2 φ2 (z2 ) + →∞ 𝜕z1 𝜕z1 𝜕z2 𝜕z2

as |z1 |2 + |z2 |2 → ∞.

Then Nφ,2 is compact, but Nφ,1 fails to be compact. Our assumptions imply that Nφ,2 is compact by Proposition 11.16. In addition, we have that Nφ,2 = S2∗ S2 , where S2 is the canonical solution operator for 𝜕 for (0, 2)-forms. Hence S2 is also compact. Now suppose that Nφ,1 is compact. Since Nφ,1 = S1∗ S1 + S2 S2∗ , this would imply that S1 is compact, contradicting Proposition 15.1. We get the same conclusion if we apply Proposition 11.27. The above assumptions are all satisfied, for instance, for the weight functions φ(z1 , z2 ) = |z1 |2k + |z2 |2k ,

k = 2, 3, . . . .

15.3 Notes A general characterization of compactness of the 𝜕-Neumann operator for smoothly bounded domains in terms of geometric properties of the boundary of the domain is not known. Christ and Fu [18] showed: If Ω ⊂ ℂ2 is a smoothly bounded pseudoconvex Hartogs domain (i. e., with (z, w) ∈ Ω, we have (z, eiθ w) ∈ Ω for every θ ∈ ℝ), then

15.3 Notes

� 289

the 𝜕-Neumann operator N is compact if and only if property (P) holds. Also, analytic discs in the boundary are an obstruction to compactness. However, there are examples of Hartogs domains that fail property (P) although their boundaries contain no analytic discs. In the case of a weighted space L2 (ℂ, e−φ ), the compactness of the 𝜕-Neumann operator can be characterized in terms of the weight function (see Theorem 13.3); a different approach to this problem, using estimates of the Bergman kernel, is contained in [74]. An analogous characterization for the compactness of the 𝜕-Neumann operator on L2 (ℂn , e−φ ), n ≥ 2, has also not been found up to now; see [7] for various partial results.

16 The 𝜕-complex on the Segal–Bargmann space We use the powerful classical methods of the 𝜕-complex based on the theory of unbounded densely defined operators on Hilbert spaces (see Chapter 4 or [91]) to study the 𝜕-complex on an appropriate Hilbert space. The main difference to the classical theory is that the underlying Hilbert space is now not an L2 -space but a closed subspace of an L2 2 space, the Segal–Bargmann space of entire functions A2 (ℂn , e−|z| ). It is well known that 2 the differentiation with respect to zj defines an unbounded operator on A2 (ℂn , e−|z| ). We consider the operator n

𝜕f = ∑ j=1

𝜕f dz , 𝜕zj j

2

2

which is densely defined on A2 (ℂn , e−|z| ) and maps to A21,0 (ℂn , e−|z| ), the space of (1, 0)2

forms with coefficients in A2 (ℂn , e−|z| ). In general, we get the 𝜕-complex 2

2

2

A2(p−1,0) (ℂn , e−|z| ) 󳨀→ A2(p,0) (ℂn , e−|z| ) 󳨀→ A2(p+1,0) (ℂn , e−|z| ), 𝜕

𝜕

←󳨀

←󳨀

𝜕∗

𝜕∗

where 1 ≤ p ≤ n − 1, and 𝜕∗ denotes the adjoint operator of 𝜕. We will choose the domain dom(𝜕) in such a way that 𝜕 becomes a closed operator 2 on A2 (ℂn , e−|z| ). In addition, we get that the corresponding complex Laplacian ◻̃ p = 𝜕∗ 𝜕 + 𝜕𝜕∗ with dom(◻̃ p ) = {f ∈ dom(𝜕) ∩ dom(𝜕∗ ) : 𝜕f ∈ dom(𝜕∗ ) and 𝜕f ∗ ∈ dom(𝜕)} acts as an 2

unbounded self-adjoint operator on A2(p,0) (ℂn , e−|z| ). We point out that in this case the complex Laplacian is a differential operator of order one. Nevertheless, we can use the general features of a Laplacian for these differential operators of order one. Using an estimate analogous to the basic estimate for the 𝜕-complex, we obtain that 2 2 ◻̃ p has a bounded inverse Ñ p : A2 (ℂn , e−|z| ) 󳨀→ A2 (ℂn , e−|z| ), and we show that Ñ p (p,0)

(p,0)

is even compact. In addition, we compute the spectrum of ◻̃ p . The inspiration for this comes from quantum mechanics, where the annihilation 2 operator aj can be represented by the differentiation with respect to zj on A2 (ℂn , e−|z| ) and its adjoint, the creation operator aj∗ , the multiplication by zj , both operators being 2

unbounded densely defined, see Chapter 1. We can show that A2 (ℂn , e−|z| ) with this action of aj and aj∗ is an irreducible representation M of the Heisenberg group; by the Stone–von Neumann theorem it is the only one up to unitary equivalence. Physically, M can be thought of as the Hilbert space of a harmonic oscillator with n degrees of freedom and Hamiltonian operator

https://doi.org/10.1515/9783111182926-016

16.1 Unbounded operators and the complex Laplacian

� 291

n n 1 1 H = ∑ (Pj2 + Qj2 ) = ∑ (aj∗ aj + aj aj∗ ). 2 2 j=1 j=1

Finally, we investigate operators of the form Du = ∑nj=1 pj (u) dzj , where u ∈ 2

A2 (ℂn , e−|z| ) and pj ( 𝜕z𝜕 , . . . , 𝜕z𝜕 ) are polynomial differential operators with constant co1 n efficients. Differential operators of polynomial type on the Segal–Bargmann space were also investigated by Newman and Shapiro [78, 79] and Render [81] in the real analytic setting. Replacing 𝜕 by D, we get the corresponding complex Laplacian ◻̃ D = DD∗ + D∗ D, for which we can use duality and the machinery of the 𝜕-Neumann operator [64, 65] to prove the existence and boundedness of the inverse to ◻̃ D and to find the canonical solutions to the inhomogeneous equations Du = α and D∗ v = β.

16.1 Unbounded operators and the complex Laplacian From Sections 1.4 and 1.5 we know that the operators a(f ) =

𝜕f 𝜕z

2

densely defined unbounded operators on A2 (ℂ, e−|z| ). 2

and a∗ (f ) = zf are

2

Lemma 16.1. Let dom(a) = {f ∈ A2 (ℂ, e−|z| ) : a(f ) ∈ A2 (ℂ, e−|z| )}. The operators a and 2 a∗ are densely defined operators on A2 (ℂ, e−|z| ) with closed graphs. Proof. By general properties of unbounded operators it suffices to prove the statement for a (see Section 4.1). Let (fj )j be a sequence in dom(a) such that limj→∞ fj = f and limj→∞ a(fj ) = g. We have to show that f ∈ dom(a) and a(f ) = g. By (1.23) it follows that (fj )j converges uniformly on each compact subset of ℂ to f , and the same is true for the derivatives: limj→∞ fj′ = f ′ . This implies that f ′ = a(f ) = g. We supposed that g ∈ 2

2

A2 (ℂ, e−|z| ), and we know that taking primitives does not leave A2 (ℂ, e−|z| ). Therefore we get that f ∈ dom(a), which proves the statement.

For the rest of this section, we consider the Segal–Bargmann space in several variables with weight φ(z) = |z1 |2 + ⋅ ⋅ ⋅ + |zn |2 . We will denote the derivative with respect to z by 𝜕, and in the following, we will consider the 𝜕-complex for the Segal–Bargmann space in several variables 2

2

2

A2(p−1,0) (ℂn , e−|z| ) 󳨀→ A2(p,0) (ℂn , e−|z| ) 󳨀→ A2(p+1,0) (ℂn , e−|z| ), 𝜕

←󳨀

𝜕

←󳨀

𝜕∗

𝜕∗

2

where A2(p,0) (ℂn , e−|z| ) denotes the Hilbert space of (p, 0)-forms with coefficients in 2

A2 (ℂn , e−|z| ), and

𝜕f = ∑ |J|=p

for a (p, 0)-form

′

n

∑ j=1

𝜕fJ 𝜕zj

dzj ∧ dzJ

292 � 16 The 𝜕-complex on the Segal–Bargmann space f = ∑

′

fJ dzJ

|J|=p

with summation over increasing multiindices J = (j1 , . . . , jp ), 1 ≤ p ≤ n − 1; and we take 2

2

dom(𝜕) = {f ∈ A2(p,0) (ℂn , e−|z| ) : 𝜕f ∈ A2(p+1,0) (ℂn , e−|z| )}. Now let ◻̃ p = 𝜕∗ 𝜕 + 𝜕𝜕∗ with dom(◻̃ p ) = {f ∈ dom(𝜕) ∩ dom(𝜕∗ ) : 𝜕f ∈ dom(𝜕∗ ) and 𝜕∗ f ∈ dom(𝜕)}. 2

Then ◻̃ p acts as an unbounded self-adjoint operator on A2(p,0) (ℂn , e−|z| ) (see Proposition 4.54).

Remark 16.2. (a) We remark that a (1, 0)-form g = ∑nj=1 gj dzj with holomorphic coefficients is invariant under the pull-back by a holomorphic map F = (F1 , . . . , Fn ) : ℂn 󳨀→ ℂn . We have n

n

n

l=1

j=1

l=1

F ∗ g = ∑ gl dFl = ∑(∑ gl

𝜕Fl ) dzj , 𝜕zj

where we used the fact that n

n n 𝜕Fl 𝜕F 𝜕F dzj + ∑ l dzj = ∑ l dzj . 𝜕zj 𝜕zj 𝜕zj j=1 j=1

dFl = 𝜕Fl + 𝜕 Fl = ∑ j=1

The expressions

𝜕Fl 𝜕zj

are holomorphic.

(b) For p = 0 and a function f ∈ dom(◻̃ 0 ), we have n

◻̃ 0 f = 𝜕∗ 𝜕f = ∑ zj j=1

𝜕f . 𝜕zj

If 1 ≤ p ≤ n − 1 and f = ∑|J|=p ′ fJ dzJ ∈ dom(◻̃ p ) is a (p, 0)-form, then we have ◻̃ p f = ∑ |J|=p

′

n

( ∑ zk k=1

𝜕fJ 𝜕zk

+ pfJ ) dzJ .

(16.1)

For p = n and a (n, 0)-form F ∈ dom(◻̃ n ) (here we identify the (n, 0)-form with a function), we have n

◻̃ n F = 𝜕𝜕∗ F = ∑ zj j=1

𝜕F + nF. 𝜕zj

16.2 Basic estimates and compactness

� 293

16.2 Basic estimates and compactness 2

The (p, 0)-forms with polynomial components are dense in A2(p,0) (ℂn , e−|z| ). In addition, we have Lemma 16.3. The (p, 0)-forms with polynomial components are also dense in dom(𝜕) ∩ dom(𝜕∗ ) endowed with the graph norm 󵄩 󵄩2 1/2 u 󳨃→ (‖u‖2 + ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ) . Proof. Let u = ∑|J|=p ′ uJ dzJ ∈ dom(𝜕) ∩ dom(𝜕∗ ) and consider the partial sums of the Fourier series expansions of uJ = ∑ uJ,α φα , α

where φα (z) =

zα √π n α!

and

∑ |uJ,α |2 < ∞

(16.2)

α

with α! = α1 ! . . . αn !. 2 2 We have that 𝜕u ∈ A2(p+1,0) (ℂn , e−|z| ) and 𝜕∗ u ∈ A2(p−1,0) (ℂn , e−|z| ). Hence the partial sums of the Fourier series of the components of 𝜕u converge to the components of 𝜕u in 2 A2(p+1,0) (ℂn , e−|z| ), and the partial sums of the Fourier series of the components of 𝜕∗ u 2

converge to the components of 𝜕∗ u in A2(p−1,0) (ℂn , e−|z| ).

As an immediate consequence of (16.17), (6.11), and Lemma 16.3, we get the basic estimate. 2

Lemma 16.4. Let 1 ≤ p ≤ n − 1, and let u = ∑′|J|=p uJ dzJ ∈ A2(p,0) (ℂn , e−|z| ). Suppose that u ∈ dom(𝜕) ∩ dom(𝜕∗ ). Then ‖u‖2 ≤

1 󵄩 󵄩2 (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ). p

(16.3)

Proof. Taking (p, 0)-forms with polynomial components and replacing 𝜕 by 𝜕 and taking into account that φ(z) = |z|2 , from (6.11) we derive that 󵄩 󵄩2 ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 = ∑

|J|=p

′

n 󵄩 󵄩󵄩 𝜕uJ 󵄩󵄩󵄩2 󵄩󵄩 + p ∑ ∑ 󵄩󵄩󵄩 󵄩 𝜕zj 󵄩󵄩󵄩 |J|=p j=1 󵄩

′

2

∫ |uJ |2 e−|z| dλ.

(16.4)

ℂn

Since (p, 0)-forms with polynomial components are dense in dom(𝜕) ∩ dom(𝜕∗ ) with the graph-norm (Lemma 16.3), we immediately get (16.3).

294 � 16 The 𝜕-complex on the Segal–Bargmann space At this point, we recall that (16.3) can be seen as a generalization of the uncertainty principle (see Section 1.5). Now we can use the machinery of the classical 𝜕-Neumann operator to show the following results. Lemma 16.5. Both operators 𝜕 and 𝜕∗ have closed range. If we endow dom(𝜕)∩dom(𝜕∗ ) with the graph-norm (‖𝜕f ‖2 +‖𝜕∗ f ‖2 )1/2 , then the dense 2 subspace dom(𝜕) ∩ dom(𝜕∗ ) of A2(p,0) (ℂn , e−|z| ) becomes a Hilbert space. Proof. We notice that ker 𝜕 = (im 𝜕∗ )⊥ , which implies that (ker 𝜕)⊥ = im 𝜕∗ ⊆ ker 𝜕∗ . If u ∈ ker 𝜕 ∩ ker 𝜕∗ , then by (16.3) we have that u = 0. Hence (ker 𝜕)⊥ = ker 𝜕∗ .

(16.5)

If u ∈ dom(𝜕) ∩ (ker 𝜕)⊥ , then u ∈ ker 𝜕∗ , and (16.3) implies ‖u‖ ≤

1 ‖𝜕u‖. p

Now we can use general results on unbounded operators on Hilbert spaces (see Chapter 4) to show that im 𝜕 and im 𝜕∗ are closed. The last statement follows again by (16.3); see also Chapter 4. The next result describes the implication of the basic estimates (16.3) for the ̃ ◻-operator. 2

Theorem 16.6. The operator ◻̃ : dom(◻)̃ 󳨀→ A2(p,0) (ℂn , e−|z| ) is bijective and has a bounded inverse 2

̃ Ñ : A2(p,0) (ℂn , e−|z| ) 󳨀→ dom(◻). In addition, ̃ ≤ 1 ‖u‖ ‖Nu‖ p

(16.6)

2

for each u ∈ A2(p,0) (ℂn , e−|z| ). Proof. The proof follows very closely the proof of Proposition 4.59 (see also [16, Section 4.4]). Using the classical 𝜕-Neumann calculus, we obtain the following:

16.2 Basic estimates and compactness

� 295

2

Theorem 16.7. Let Ñ p denote the inverse of ◻̃ on A2(p,0) (ℂn , e−|z| ). Then Ñ p+1 𝜕 = 𝜕Ñ p

(16.7)

Ñ p−1 𝜕∗ = 𝜕∗ Ñ p

(16.8)

on dom(𝜕), and

on dom(𝜕∗ ). In addition, we have that 𝜕∗ Ñ p is zero on (ker 𝜕)⊥ . Proof. See the proof of Proposition 4.62 (see also [16, Section 4.4]). Now we can also prove a solution formula for the equation 𝜕u = α, where α is a 2 given (p, 0)-form in A2(p,0) (ℂn , e−|z| ) with 𝜕α = 0. 2

Theorem 16.8. Let α ∈ A2(p,0) (ℂn , e−|z| ) with 𝜕α = 0. Then u0 = 𝜕∗ Ñ p α is the canonical solution of 𝜕u = α, which means that 𝜕u0 = α and u0 ∈ (ker 𝜕)⊥ = im 𝜕∗ , and 󵄩󵄩 ∗ ̃ 󵄩󵄩 −1/2 ‖α‖. 󵄩󵄩𝜕 Np α󵄩󵄩 ≤ p

(16.9)

Proof. See the proof of Proposition 4.63. Now we discuss a different approach to the operator N,̃ which is related to the quadratic form Q(u, v) = (𝜕u, 𝜕v) + (𝜕∗ u, 𝜕∗ v). For this purpose, we consider the embedding 2

ι : dom(𝜕) ∩ dom(𝜕∗ ) 󳨀→ A2(p,0) (ℂn , e−|z| ), where dom(𝜕) ∩ dom(𝜕∗ ) is endowed with the graph norm 󵄩 󵄩2 1/2 u 󳨃→ (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ) . The graph norm stems from the inner product ̃ v) = (𝜕u, 𝜕v) + (𝜕∗ u, 𝜕∗ v). Q(u, v) = (u, v)Q = (◻u, The basic estimates (16.3) imply that ι is a bounded operator with operator norm ‖ι‖ ≤

1 . √p

By (16.3) it follows in addition that dom(𝜕) ∩ dom(𝜕∗ ) endowed with the graph norm u 󳨃→ (‖𝜕u‖2 + ‖𝜕∗ u‖2 )1/2 is a Hilbert space (see Lemma 16.5).

296 � 16 The 𝜕-complex on the Segal–Bargmann space Since (u, v) = (u, ιv), we have that (u, v) = (ι∗ u, v)Q . 2

For u ∈ A2(p,0) (ℂn , e−|z| ) and v ∈ dom(𝜕) ∩ dom(𝜕∗ ), we get ̃ v) = ((𝜕𝜕∗ + 𝜕∗ 𝜕)Nu, ̃ v) = (𝜕∗ Nu, ̃ 𝜕∗ v) + (𝜕Nu, ̃ 𝜕v). (u, v) = (◻̃ Nu,

(16.10)

Equation (16.10) suggests that as an operator to dom(𝜕) ∩ dom(𝜕∗ ), Ñ coincides with 2 ι and as an operator to A2(p,0) (ℂn , e−|z| ), Ñ is equal to ι ∘ ι∗ (see (4.42) or [91, p. 30] for the details). Hence Ñ is compact if and only if the embedding ∗

2

ι : dom(𝜕) ∩ dom(𝜕∗ ) 󳨀→ A2(p,0) (ℂn , e−|z| ) is compact. We will use this to prove the following theorem. 2

2

Theorem 16.9. The operator Ñ : A2(p,0) (ℂn , e−|z| ) 󳨀→ A2(p,0) (ℂn , e−|z| ), 1 ≤ p ≤ n, is compact. Proof. First, we consider the case p = 1. We use (16.17) for the graph norm on dom(𝜕) ∩ dom(𝜕∗ ) and indicate that it suffices to consider one component uj of the (1, 0)-form u. For this purpose, we will denote uj by f . We have to handle n 󵄨󵄨 𝜕f 󵄨󵄨2 2 󵄨 󵄨󵄨 −|z|2 dλ + ∫ |f |2 e−|z| dλ ∑ ∫ 󵄨󵄨󵄨 󵄨 e 󵄨󵄨 𝜕zk 󵄨󵄨󵄨 k=1 n n ℂ

ℂ

for the graph norm. We use the complete orthonormal system (16.2) (φα )α of A2 (ℂn , 2 e−|z| ). Let f = ∑α fα φα be an element of dom(𝜕) ∩ dom(𝜕∗ ). We have ι(f ) = f , and hence ι(f ) = ∑(f , φα )φα α

2

2

in A2 (ℂn , e−|z| ). The basis elements φα are normalized in A2 (ℂn , e−|z| ). First, we have to compute the graph norm of the basis elements φα . Notice that α

α −1

α

α k+1 α zk+1 z k−1 αk zkk z1 1 𝜕φα zn 1 = . . . k−1 . . . n = √αk φ(αk−1) , √π n √α1 ! 𝜕zk √αk−1 ! √αk ! √αk+1 ! √αn !

where (αk − 1) = (α1 , . . . αk−1 , αk − 1, αk+1 , . . . , αn ). Hence the graph norm of the basis elements φα equals n 󵄩 󵄩󵄩 𝜕φ 󵄩󵄩󵄩2 ‖φα ‖Q = (‖φα ‖2 + ∑ 󵄩󵄩󵄩 α 󵄩󵄩󵄩 ) 󵄩 𝜕zk 󵄩󵄩 k=1󵄩

where |α| = α1 + ⋅ ⋅ ⋅ + αn .

1/2

= √1 + |α|,

16.2 Basic estimates and compactness

� 297

Now let φα . √1 + |α|

ψα =

Then (ψα )α constitutes a complete orthonormal system in the Hilbert space dom(𝜕) ∩ dom(𝜕∗ ) endowed with the graph norm. Notice that n

(f , ψα )Q = ∑ ( k=1

n αk 𝜕f 𝜕ψα 1 , ) + (f , ψα ) = ∑ fα + fα 𝜕zk 𝜕zk √ √ 1 + |α| k=1 1 + |α|

= √1 + |α| fα , and we have ι(f ) = f = ∑(f , ψα )Q ψα . α

2

For the norm of A2 (ℂn , e−|z| ), we have 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩󵄩2 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩ι(f ) − ∑ (f , ψα )Q ψα 󵄩󵄩󵄩 = 󵄩󵄩󵄩 ∑ (f , ψα )Q ψα 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 󵄩󵄩 |α|≤N |α|≥N+1

󵄩󵄩2 󵄩󵄩 1 󵄩 󵄩 = 󵄩󵄩󵄩 ∑ (f , ψα )Q φα 󵄩󵄩󵄩 󵄩󵄩 󵄩󵄩 √ 1 + |α| |α|≥N+1 =

≤

󵄨󵄨2 󵄨󵄨 1 󵄨 󵄨 (f , ψα )Q 󵄨󵄨󵄨 ∑ 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 √1 + |α| |α|≥N+1 ‖f ‖2Q

N +2

,

where we finally used Bessel’s inequality for the Hilbert space dom(𝜕) ∩ dom(𝜕∗ ) endowed with the graph norm. This proves that 2

ι : dom(𝜕) ∩ dom(𝜕∗ ) 󳨀→ A2(1,0) (ℂn , e−|z| ) is a compact operator and the same is true for 2

2

Ñ : A2(1,0) (ℂn , e−|z| ) 󳨀→ A2(1,0) (ℂn , e−|z| ). For arbitrary p between 1 and n, we can use (16.4) and the same reasoning as before to get the desired conclusion. 2

Compare with the 𝜕-Neumann operator N on L2 (ℂn , e−|z| ): in this case, N fails to be compact (see Chapter 14). This is related to the fact that the kernel of 𝜕 is large (it is the

298 � 16 The 𝜕-complex on the Segal–Bargmann space Segal–Bargmann space) in the case of the 𝜕-complex, but the kernel of 𝜕 consists just of the constant functions in the case of the 𝜕-complex on the Segal–Bargmann space. To compute the spectrum of the operator ◻̃ p , we will use Lemma 9.33. Theorem 16.10. The spectrum of ◻̃ p , where 0 ≤ p ≤ n, consists of all numbers m + p for )(np). m = 0, 1, 2, . . . , where m + p has the multiplicity (n+m−1 n−1 Proof. Recall that the monomials (φα )α , where α = (α1 , . . . , αn ) ∈ ℕn0 is a multi-index, 2

constitute a complete orthonormal system in A2 (ℂn , e−|z| ). We use (16.1) and compute n

( ∑ zk k=1

𝜕φα ) + pφα = (|α| + p)φα . 𝜕zk

We use Lemma 9.33 and indicate that there are (n+|α|−1 ) monomials of degree |α|. Hence n−1 2

we get the statement about the multiplicity from the fact that A2(p,0) (ℂn , e−|z| ) is the direct 2

sum of (np) copies of A2 (ℂn , e−|z| ).

The last result implies also that the inverse Ñ p of ◻̃ p is a compact operator with the eigenvalues 1/(m + p). 2 Note that the complex Laplacian ◻q of the 𝜕-complex on L2 (ℂn , e−|z| ) has also the eigenvalues q + m for m = 0, 1, 2, . . . , but each of them has infinite multiplicity (see [72] and Chapter 14).

16.3 Properties of the graph norm and compactness If we endow dom(𝜕) ∩ dom(𝜕∗ ) with the graph norm, then we obtain Hilbert spaces of entire functions with reproducing kernels. If 1 ≤ p < n, then we write for the multiindex J = (j1 (J), . . . jp (J)) and denote the missing n − p indices by jp+1 (J), . . . , jn (J). Then we can write (16.4) in the form p n 󵄩 󵄩󵄩 𝜕uJ 󵄩󵄩2 󵄩󵄩 𝜕uJ 󵄩󵄩󵄩2 󵄩󵄩 󵄩 󵄩2 󵄩 󵄩󵄩 ) ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 = ∑ ′ ( ∑ (‖uJ ‖2 + 󵄩󵄩󵄩 󵄩󵄩 ) + ∑ 󵄩󵄩󵄩 󵄩󵄩 󵄩 𝜕z 󵄩󵄩 𝜕zj (J) 󵄩󵄩 |J|=p k k=p+1󵄩 jk (J) 󵄩 k=1 p n 󵄩 󵄩󵄩 𝜕uJ 󵄩󵄩󵄩2 󵄩󵄩 ), = ∑ ′ ( ∑ ‖zjk (J) uJ ‖2 + ∑ 󵄩󵄩󵄩 󵄩󵄩 󵄩 𝜕z |J|=p k=1 k=p+1󵄩 jk (J) 󵄩 2

where we used (1.31). For (p, 0)-forms u = ∑′|J|=p uJ dzJ ∈ A2(p,0) (ℂn , e−|z| ) belonging to dom(𝜕) ∩ dom(𝜕∗ ), by (16.4), we have ‖u‖2 = ∑ ′ ‖uJ ‖2 ≤ |J|=p

which implies

1 󵄩 󵄩2 (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ), p

� 299

16.3 Properties of the graph norm and compactness

‖uJ ‖2 ≤

p n 󵄩 󵄩󵄩 𝜕uJ 󵄩󵄩󵄩2 1 󵄩󵄩 ) ( ∑ ‖zjk (J) uJ ‖2 + ∑ 󵄩󵄩󵄩 󵄩󵄩 𝜕zj (J) 󵄩󵄩󵄩 p k=1 k k=p+1

(16.11)

for each component uJ . For p = n, we get ‖uJ ‖2 ≤

p

1 ∑ ‖z u ‖2 . p k=1 jk (J) J

(16.12)

The components uJ dzJ ∈ dom(𝜕) ∩ dom(𝜕∗ ) with the right-hand side of (16.11) or (16.12) as a norm form Hilbert spaces with reproducing kernel, since by (16.11) or (16.12) the point evaluations are continuous linear functionals. We denote these Hilbert spaces by 𝒢p . 2

The Bergman kernel of A2 (ℂn , e−|z| ) is of the form K(z, w) =

1 exp(⟨z, w⟩), π2

where ⟨z, w⟩ = ∑nj=1 zj w̄ j . Theorem 16.11. Let 1 ≤ p ≤ n. The Bergman kernel of 𝒢p is of the form p−1 ζ k k!

ζ p d p−1 e − ∑k=0 Bp (z, w) = n p−1 ( π dζ ζ

󵄨󵄨 󵄨 )󵄨󵄨󵄨 . 󵄨󵄨ζ =⟨z,w⟩

(16.13) 2

𝜕u Proof. We write pk (u)(w) = 𝜕w and p∗k (u)(w) = wk u(w) for u ∈ A2 (ℂn , e−|z| ) and k = k 1, . . . , n. Without loss of generality, we can assume that the graph norm is p

n 1 󵄩 󵄩2 󵄩 󵄩2 ( ∑ 󵄩󵄩󵄩p∗k (u)󵄩󵄩󵄩 + ∑ 󵄩󵄩󵄩pk (u)󵄩󵄩󵄩 ). p k=1 k=p+1

Since ℬp has the reproducing property, we have u(z) =

p

2 1 − ∫ ∑ p∗k (u)(w)[p∗k (vz )(w)] e−|w| dλ(w) p k=1

ℂn

+

n 2 1 − ∫ ∑ pk (u)(w)[pk (vz )(w)] e−|w| dλ(w) p k=p+1 ℂn

p

−

n 2 1 = ∫ u(w)[( ∑ pk p∗k + ∑ p∗k pk )(vz )(w)] e−|w| dλ(w) p k=1 k=p+1 ℂn

for a uniquely determined function vz ∈ 𝒢p . Observe that

300 � 16 The 𝜕-complex on the Segal–Bargmann space p

n

k=1

k=p+1

−

1 ∞ ⟨z, w⟩k . ∑ π n k=0 k!

p[( ∑ pk p∗k + ∑ p∗k pk )(vz )(w)] = K(z, w) = Now we claim that Bp (z, w) =

p ∞ ⟨z, w⟩k . ∑ π n k=0 (k + p)k!

(16.14)

For this aim, we compute p

n

k=1

k=p+1

k

p

k

k−1

( ∑ pk p∗k + ∑ p∗k pk )(⟨w, z⟩) = p(⟨w, z⟩) + k ∑ wj z̄j (⟨w, z⟩) j=1

n

k−1

+ k ∑ wj z̄j (⟨w, z⟩) k=p+1

k

= (k + p)(⟨w, z⟩) , which proves (16.14). Comparison of coefficients in the Taylor series yields p−1 ζ k k!

ζ p d p−1 e − ∑k=0 p ∞ ⟨z, w⟩k = n p−1 ( Bp (z, w) = n ∑ π k=0 (k + p)k! π dζ ζ

󵄨󵄨 󵄨 )󵄨󵄨󵄨 . 󵄨󵄨ζ =⟨z,w⟩

For example, B1 (z, w) =

1 e⟨z,w⟩ − 1 π n ⟨z, w⟩

and B2 (z, w) =

2 ⟨z, w⟩e⟨z,w⟩ − e⟨z,w⟩ + 1 . πn (⟨z, w⟩)2

Now we use the unbounded operators 𝜕∗ and 𝜕 to exhibit compact subsets of 2 A (ℂn , e−|z| ). 2

Theorem 16.12. Let n = 1 and define 󵄩2

𝒰 = {u ∈ dom(𝜕 ) : 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ≤ 1}.

󵄩

∗

∗

2

Then 𝒰 is a relatively compact set in A2 (ℂ, e−|z| ). 2

Proof. The basic estimate implies that 𝒰 is a bounded subset of A2 (ℂn , e−|z| ). By Theorem 11.32 we have to show that for each ϵ > 0, there exists a constant R > 0 such that 2 󵄨 󵄨2 󵄩 󵄩2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 e−|z| dλ(z) ≤ ϵ󵄩󵄩󵄩𝜕∗ (u)󵄩󵄩󵄩 ,

(16.15)

ℂ\𝔹R 2

for all u ∈ dom(𝜕∗ ). For this purpose, we take the orthonormal basis (φk )k of A2 (ℂ, e−|z| ),

where φk (z) =

k

z , √πk!

2

k = 0, 1, 2, . . . . Then, for u ∈ A2 (ℂ, e−|z| ), we have u(z) =

2 ∑∞ k=0 uk φk (z), where (uk )k ∈ l .

16.3 Properties of the graph norm and compactness

� 301

For u ∈ dom(𝜕∗ ), we get ∞ zk+1 zk+1 = ∑ uk √k + 1 . √πk! k=0 √π(k + 1)!

∞

𝜕∗ u(z) = zu(z) = ∑ uk k=0

Then (16.15) reads as ∞ ∞ 2 |u |2 |z|2k −|z|2 dλ(z) = 2 ∑ k ∫ r 2k+1 e−r dr ≤ ϵ ∑ |uk |2 (k + 1), e ∫ ∑ |uk | πk! k! k=0 k=0 k=0 ∞

∞

2

R

ℂ\DR

where DR = {z ∈ ℂ : |z| < R}. Comparing the terms in the sum from above, it remains to show that ∞

2 2 ∫ r 2k+1 e−r dr ≤ ϵ (k + 1)!

R

for all k = 0, 1, 2, . . . . It follows that ∞

2∫r

2k+1 −r 2

e

∞

dr = ∫ sk e−s ds.

R

R2

Now integrating k times by parts, we get ∞

2

k

R2j . j! j=0

∫ sk e−s ds = e−R k! ∑ R2

So we still have to prove that for each ϵ > 0, there exists t > 0 such that 1 −t k t j e ∑ ≤ϵ k+1 j! j=0 for all k = 0, 1, 2, . . . . If k ≥ ϵ1 , then we obtain 1 −t k t j 1 −t t 1 e ∑ ≤ e e = ≤ ϵ, k+1 j! k + 1 k+1 j=0 and if k ≤ ϵ1 , then we can choose t > 0 large enough to get 1 −t k t j e ∑ ≤ ϵ. k+1 j! j=0

(16.16)

302 � 16 The 𝜕-complex on the Segal–Bargmann space Remark 16.13. (a) The operator ◻̃ 1 = 𝜕𝜕∗ has a bounded inverse 2

Ñ 1 : A2(1,0) (ℂ, e−|z| ) 󳨀→ dom(◻̃ 1 ). 2 Let j ̃ : dom(𝜕∗ ) 󳨀→ A2 (ℂ, e−|z| ) be the embedding, where dom(𝜕∗ ) is endowed with the graph norm ‖𝜕∗ u‖. It follows that Ñ 1 = j ̃ ∘ j∗̃ . Since 𝒰 = {u ∈ dom(𝜕∗ ) : ‖𝜕∗ u‖2 ≤ 2 1} is a precompact set in A2 (ℂ, e−|z| ), the operator j ̃ is compact. This implies Ñ 1 is compact; see Theorem 16.9, where we showed directly that the operator j ̃ can be approximated by finite-dimensional operators. It turns out that the method in the proof of Theorem 16.12 can be used in more general cases. (b) In [58], we can find a different proof of compactness using coverings in the corresponding sequence spaces of the Taylor coefficients of the entire functions.

For several variables, for a (1, 0)-form u = ∑nj=1 uj dzj ∈ dom(𝜕) ∩ dom(𝜕∗ ), we get n n 󵄨󵄨 𝜕uj 󵄨󵄨2 2 󵄨󵄨 −|z|2 󵄩 󵄩2 󵄨 dλ + ∑ ∫ |uj |2 e−|z| dλ. ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 = ∑ ∫ 󵄨󵄨󵄨 󵄨󵄨 e 󵄨󵄨 𝜕zk 󵄨󵄨 j=1 n j,k=1 n

(16.17)

ℂ

ℂ

For simplicity, we only handle the case n = 2; now we use (1.31), and (16.17) reads as 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩 󵄩2 󵄩󵄩 𝜕u 󵄩󵄩 󵄩󵄩 𝜕u 󵄩󵄩 󵄩󵄩 𝜕u 󵄩󵄩 󵄩󵄩 𝜕u 󵄩󵄩 ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 = 󵄩󵄩󵄩 1 󵄩󵄩󵄩 + 󵄩󵄩󵄩 1 󵄩󵄩󵄩 + 󵄩󵄩󵄩 2 󵄩󵄩󵄩 + 󵄩󵄩󵄩 2 󵄩󵄩󵄩 + ‖u1 ‖2 + ‖u2 ‖2 󵄩󵄩 𝜕z1 󵄩󵄩 󵄩󵄩 𝜕z2 󵄩󵄩 󵄩󵄩 𝜕z1 󵄩󵄩 󵄩󵄩 𝜕z2 󵄩󵄩 1 = (‖z1 u1 ‖2 + ‖z2 u1 ‖2 + ‖z1 u2 ‖2 + ‖z2 u2 ‖2 ) 2 2 2 2 2 1 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜕u 󵄩󵄩󵄩 + (󵄩󵄩󵄩 1 󵄩󵄩󵄩 + 󵄩󵄩󵄩 1 󵄩󵄩󵄩 + 󵄩󵄩󵄩 2 󵄩󵄩󵄩 + 󵄩󵄩󵄩 2 󵄩󵄩󵄩 ). 2 󵄩󵄩 𝜕z1 󵄩󵄩 󵄩󵄩 𝜕z2 󵄩󵄩 󵄩󵄩 𝜕z1 󵄩󵄩 󵄩󵄩 𝜕z2 󵄩󵄩 Theorem 16.14. Let 2

󵄩2

𝒰 = {u ∈ dom(𝜕) ∩ dom(𝜕 ) : ‖𝜕u‖ + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ≤ 1}. ∗

󵄩

∗

2

Then 𝒰 is a relatively compact set in A2(1,0) (ℂn , e−|z| ). Proof. We have to show that for each ϵ > 0, there exists R > 0 such that ∫ ℂ2 \DR ×DR

2 󵄨 󵄨2 󵄨 󵄨2 󵄩 󵄩2 (󵄨󵄨󵄨u1 (z)󵄨󵄨󵄨 + 󵄨󵄨󵄨u2 (z)󵄨󵄨󵄨 )e−|z| dλ(z) ≤ ϵ(‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 )

for all u ∈ dom(𝜕) ∩ dom(𝜕∗ ), where DR = {z ∈ ℂ : |z| < R}.

16.3 Properties of the graph norm and compactness

� 303

It suffices to show that for uj = v, j = 1, 2, we have ∫ ℂ2 \D

󵄨2 −|z|2 󵄨󵄨 dλ(z) ≤ ϵ(‖z1 v‖2 + ‖z2 v‖2 ). 󵄨󵄨v(z)󵄨󵄨󵄨 e

(16.18)

R ×DR

We use the orthonormal basis φα,β (z) =

β

z2 zα1 √πα! √πβ!

and write

v(z) = ∑ vα,β φα,β (z), α,β

where ∑ |vα,β |2 < ∞. α,β

Then (16.18) reads as ∑ |vα,β |2 ( ∫ α,β

ℂ\DR

= 4∑ α,β

|z1 |2α −|z1 |2 |z2 |2β −|z2 |2 e dλ(z1 ) ∫ e dλ(z2 )) πα! πβ!

|vα,β |2 α!β!

ℂ\DR

∞

∞

2

2

( ∫ r 2α+1 e−r dr ∫ r 2β+1 e−r dr) ≤ ϵ ∑ |vα,β |2 (α + β + 2). α,β

R

R

Since ∞

2 1 ∫ r 2α+1 e−r dr ≤ 1 α!

R

for all α = 0, 1, 2, . . . , we can proceed as in the proof of Theorem 16.12 to show (16.18). In general, for 1 ≤ p ≤ n − 1 and for a (p, 0)-form u = ∑|J|=p ′ uJ dzJ ∈ dom(𝜕) ∩ dom(𝜕∗ ), we have that 󵄩 󵄩2 ‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 = ∑

′

|J|=p

n 󵄩 󵄩󵄩 𝜕uJ 󵄩󵄩󵄩2 󵄩󵄩 + p ∑ ∑ 󵄩󵄩󵄩 󵄩 𝜕zj 󵄩󵄩󵄩 |J|=p j=1 󵄩

′

2

∫ |uJ |2 e−|z| dλ.

(16.19)

ℂn

As an immediate consequence of (16.19), we get the basic estimate ‖u‖2 ≤

1 󵄩 󵄩2 (‖𝜕u‖2 + 󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ). p

(16.20)

It is now clear how to adopt the methods we developed in the case n = 2 for general n and (p, 0)-forms in dom(𝜕) ∩ dom(𝜕∗ ), and we get that the set 2

󵄩2

𝒰 = {u ∈ dom(𝜕) ∩ dom(𝜕 ) : ‖𝜕u‖ + 󵄩󵄩󵄩𝜕 u󵄩󵄩󵄩 ≤ 1} ∗

󵄩

∗

304 � 16 The 𝜕-complex on the Segal–Bargmann space 2

is precompact in A2(p,0) (ℂn , e−|z| ) and that the corresponding complex Laplacian ◻̃ p = 𝜕∗ 𝜕 + 𝜕𝜕∗ has a compact inverse 2

2

Ñ p : A2(p,0) (ℂn , e−|z| ) 󳨀→ A2(p,0) (ℂn , e−|z| ).

16.4 The generalized 𝜕-complex 2

We consider the classical Segal–Bargmann space A2 (ℂn , e−|z| ) and replace a single derivative with respect to zj by a differential operator of the form pj ( 𝜕z𝜕 , . . . , 𝜕z𝜕 ), where 1 n pj is a complex polynomial on ℂn (see [78, 79]). 2

We write pj (u) for pj ( 𝜕z𝜕 , . . . , 𝜕z𝜕 )u, where u ∈ A2 (ℂn , e−|z| ), and consider the 1 n densely defined operators n

Du = ∑ pj (u) dzj ,

(16.21)

j=1

2

where u ∈ A2 (ℂn , e−|z| ) and pj ( 𝜕z𝜕 , . . . , 𝜕z𝜕 ) are polynomial differential operators with 1 n constant coefficients [44]. More generally, we define Du = ∑ |J|=p

′

n

∑ pk (uJ ) dzk ∧ dzJ ,

(16.22)

k=1

2

where u = ∑|J|=p ′ uJ dzJ is a (p, 0)-form with coefficients in A2 (ℂn , e−|z| ), J = (j1 , . . . , jp ) is a multi-index, dzJ = dzj1 ∧ ⋅ ⋅ ⋅ ∧ dzjp , and the summation is taken only over increasing multiindices. It is clear that D2 = 0 and (Du, v) = (u, D∗ v), 2

(16.23) 2

where u ∈ dom(D) = {u ∈ A2(p,0) (ℂn , e−|z| ) : Du ∈ A2(p+1,0) (ℂn , e−|z| )}, D∗ v =

∑ |K|=p−1

′

n

∑ p∗j vjK dzK j=1

for v = ∑|J|=p ′ vJ dzJ , and p∗j (z1 , . . . , zn ) is the polynomial pj with complex conjugate coefficients, taken as the multiplication operator.

16.4 The generalized 𝜕-complex �

305

Now the corresponding D-complex has the form D

2

2

D

2

A2(p−1,0) (ℂn , e−|z| ) 󳨀→ A2(p,0) (ℂn , e−|z| ) 󳨀→ A2(p+1,0) (ℂn , e−|z| ). ←󳨀

←󳨀

D∗

D∗

Similarly to the classical 𝜕-complex (see [44]), we consider the generalized box operator 2 ◻̃ D,p := D∗ D + DD∗ as a densely defined self-adjoint operator on A2(p,0) (ℂn , e−|z| ) with dom(◻̃ D,p ) = {f ∈ dom(D) ∩ dom(D∗ ) : Df ∈ dom(D∗ ) and D∗ f ∈ dom(D)}; see [45] for more detail. 2 The (p, 0)-forms with polynomial components are dense in A2(p,0) (ℂn , e−|z| ). In addition, we have the following: Lemma 16.15. The (p, 0)-forms with polynomial components are also dense in dom(D) ∩ dom(D∗ ) endowed with the graph norm 󵄩 󵄩2 1/2 u 󳨃→ (‖u‖2 + ‖Du‖2 + 󵄩󵄩󵄩D∗ u󵄩󵄩󵄩 ) . Proof. Let u = ∑|J|=p ′ uJ dzJ ∈ dom(D) ∩ dom(D∗ ) and consider the partial sums of the Fourier series expansions of uJ = ∑ uJ,α φα , α

where φα (z) =

zα √π n α!

and

∑ |uJ,α |2 < ∞ α

(16.24)

with α! = α1 ! . . . αn !. 2 2 We have that Du ∈ A2(p+1,0) (ℂn , e−|z| ) and D∗ u ∈ A2(p−1,0) (ℂn , e−|z| ). Hence the partial sums of the Fourier series of the components of Du converge to the components of Du in 2 A2(p+1,0) (ℂn , e−|z| ), and the partial sums of the Fourier series of the components of D∗ u 2

converge to the components of D∗ u in A2(p−1,0) (ℂn , e−|z| ).

We want to find conditions under which ◻̃ D,1 has a bounded inverse. For this purpose, we have to consider the graph norm (‖u‖2 +‖Du‖2 +‖D∗ u‖2 )1/2 on dom(D)∩dom(D∗ ). Theorem 16.16. Suppose that there exists a constant C > 0 such that n

‖u‖2 ≤ C ∑ ([pk , p∗j ]uj , uk ) j,k=1

for all (1, 0)-forms u = ∑nj=1 uj dzj with polynomial components. Then

(16.25)

306 � 16 The 𝜕-complex on the Segal–Bargmann space 󵄩2 󵄩 ‖u‖2 ≤ C(‖Du‖2 + 󵄩󵄩󵄩D∗ u󵄩󵄩󵄩 )

(16.26)

for all u ∈ dom(D) ∩ dom(D∗ ). Proof. First, we have Du = ∑ (pj (uk ) − pk (uj )) dzj ∧ dzk

and

j 0 independent of α. In the following, we discuss some examples. First, we consider the one-dimensional case. Let p denote the polynomial differential operator p = a0 + a1

𝜕m 𝜕 + ⋅ ⋅ ⋅ + am m 𝜕z 𝜕z

with constant coefficients a0 , a1 , . . . , am ∈ ℂ, and let p∗ denote the polynomial p∗ (z) = a0 + a1 z + ⋅ ⋅ ⋅ + am zm with the complex conjugate coefficients a0 , a1 , . . . , am ∈ ℂ. We consider the densely defined operator Du = p(u) dz,

(16.29)

2

where u ∈ A2 (ℂ, e−|z| ), and p(u) dz is considered as a (1, 0)-form. It is clear that D2 = 0, as all (2, 0)-forms are zero if n = 1, and that (Du, v) = (u, D∗ v),

(16.30)

308 � 16 The 𝜕-complex on the Segal–Bargmann space 2

2

where u ∈ dom(D) = {u ∈ A2 (ℂ, e−|z| ) : Du ∈ A2(1,0) (ℂ, e−|z| )} and D∗ v dz = p∗ v. In the sequel, we consider the generalized box operator ◻̃ D,1 := DD∗ 2

as a densely defined self-adjoint operator on A2(1,0) (ℂ, e−|z| ) with dom(◻̃ D,1 ) = {f ∈ dom(D∗ ) : D∗ f ∈ dom(D)}. Lemma 16.19. Let u be an arbitrary polynomial. Then 2

∫[p, p∗ ]u(z) u(z) e−|z| dλ(z) ℂ

󵄨󵄨 m 󵄨󵄨2 2 󵄨󵄨 󵄨󵄨 k (k−ℓ) 󵄨 = ∑ ℓ ! ∫󵄨󵄨 ∑ ( )ak u (z)󵄨󵄨󵄨 e−|z| dλ(z). 󵄨 󵄨 ℓ 󵄨 󵄨󵄨 ℓ=1 ℂ 󵄨k=ℓ m

(16.31)

Proof. On the right-hand side of (16.31), we have the integrand m m k j ∑ ℓ ! ∑ ( )( )ak aj u(k−ℓ) u(j−ℓ) . ℓ ℓ ℓ=1 k,j=ℓ

(16.32)

For the left-hand side of (16.31), we first compute m

[p, p∗ ]u = ∑ ak k=0

m m 𝜕k m (∑ aj zj u) − (∑ aj zj )(∑ aj u(j) ). k 𝜕z j=0 j=0 j=0

(16.33)

We use the Leibniz rule to get j

m m 𝜕k m 𝜕k j k j j ( (z u) = a z u) = a aj ∑ ℓ !( )( )u(k−ℓ) zj−ℓ , ∑ ∑ ∑ j j k ℓ ℓ 𝜕zk j=0 𝜕z ℓ=0 j=0 j=0

(16.34)

noticing that (kℓ ) = 0 in the case k < ℓ. Hence we obtain j

m k j [p, p∗ ]u = ∑ ak aj ∑ ℓ !( )( )u(k−ℓ) zj−ℓ . ℓ ℓ ℓ=1 j,k=1

After integration, we obtain 2

∫[p, p∗ ]u(z) u(z) e−|z| dλ(z) ℂ

(16.35)

16.4 The generalized 𝜕-complex � j

m 2 k j = ∑ ak aj ∑ ℓ !( )( ) ∫ u(k−ℓ) u(j−ℓ) e−|z| dλ(z). ℓ ℓ ℓ=1 j,k=1

309

(16.36)

ℂ

Now it is easy to show that integration of (16.32) coincides with (16.36), and we are done. Lemma 16.20. There exists a constant C > 0 such that 󵄩 󵄩 ‖u‖ ≤ C 󵄩󵄩󵄩D∗ u󵄩󵄩󵄩

(16.37)

for each u ∈ dom(D∗ ). Proof. First, let u be a polynomial and note that the last term in (16.31) equals 2 󵄨 󵄨2 m! ∫󵄨󵄨󵄨am u(z)󵄨󵄨󵄨 e−|z| dλ(z) = m! |am |2 ‖u‖2

ℂ

and all the other terms are nonnegative; see Lemma 16.19. Now we get that 1 󵄩󵄩 ∗ 󵄩󵄩2 2 ∗ ∗ ∗ 󵄩󵄩D u󵄩󵄩 = (p u, p u) = ([p, p ]u, u) + (p(u), p(u)) ≥ ‖u‖ , C where C=

1 m! |am |2

if we suppose that am ≠ 0, and we are done. Finally, apply Lemma 16.15 to obtain the desired result. Theorem 16.21. Let D be as in (16.29). Then ◻̃ D,1 = DD∗ has a bounded inverse 2

Ñ D,1 : A2(1,0) (ℂ, e−|z| ) 󳨀→ dom(◻̃ D,1 ). 2

If α ∈ A2(1,0) (ℂ, e−|z| ), then u0 = D∗ Ñ D,1 α is the canonical solution of Du = α, which means Du0 = α and u0 ∈ (ker D)⊥ = im D∗ , and ‖D∗ Ñ D,1 α‖ ≤ C‖α‖ for some constant C > 0 independent of α. Proof. Using Lemma 16.20, we obtain that 2

◻̃ D,1 : dom(◻̃ D,1 ) 󳨀→ A2(1,0) (ℂ, e−|z| ) is bijective and has the bounded inverse Ñ D,1 , and the rest follows from Theorem 16.18. To show that Ñ D,1 is even compact, we first consider the case m = 1, taking p∗ (z) = ā 0 + ā 1 z. We suppose that a0 , a1 ≠ 0 and choose a complex number c such that

310 � 16 The 𝜕-complex on the Segal–Bargmann space |a1 |2

1+

|a1 |2 |a0 |2

< |c|2 < |a1 |2 .

(16.38)

Now let d :=

a0 ā 1 . c̄

(16.39)

Then using (1.31), for an arbitrary polynomial u, we get that 󵄩󵄩 ∗ 󵄩󵄩2 2 󵄩󵄩D u󵄩󵄩 = ‖ā 0 u + ā 1 zu‖ = |a0 |2 ‖u‖2 + ā 0 a1 (u, zu) + ā 1 a0 (zu, u) + |a1 |2 ‖zu‖2

󵄩 󵄩2 = |a0 |2 ‖u‖2 + ā 0 a1 (u′ , u) + ā 1 a0 (u, u′ ) + |a1 |2 (‖u‖2 + 󵄩󵄩󵄩u′ 󵄩󵄩󵄩 ) 󵄩 󵄩2 󵄩 󵄩2 ≥ (|a0 |2 + |a1 |2 )‖u‖2 + ā 0 a1 (u′ , u) + ā 1 a0 (u, u′ ) + (|a1 |2 )󵄩󵄩󵄩u′ 󵄩󵄩󵄩 − 󵄩󵄩󵄩du + cu′ 󵄩󵄩󵄩 󵄩 󵄩2 = (|a0 |2 + |a1 |2 − |d|2 )‖u‖2 + (|a1 |2 − |c|2 )󵄩󵄩󵄩u′ 󵄩󵄩󵄩 󵄩 󵄩2 ≥ α(‖u‖2 + 󵄩󵄩󵄩u′ 󵄩󵄩󵄩 ) = α‖zu‖2 ,

where α = min(|a0 |2 + |a1 |2 − |d|2 , |a1 |2 − |c|2 ).

(16.40)

Note that (16.38) and (16.39) imply α > 0. For a general polynomial p, from [45] we obtain that there exists a constant C > 0 such that ‖u‖ ≤ C‖D∗ u‖ for each u ∈ dom(D∗ ) and 2

∫[p, p∗ ]u(z) u(z) e−|z| dλ(z) ℂ

󵄨󵄨 m 󵄨󵄨2 2 󵄨󵄨 󵄨󵄨 k (k−ℓ) 󵄨 = ∑ ℓ ! ∫󵄨󵄨 ∑ ( )ak u (z)󵄨󵄨󵄨 e−|z| dλ(z). 󵄨 󵄨 ℓ 󵄨 󵄨󵄨 ℓ=1 ℂ 󵄨k=ℓ m

(16.41)

Now from (16.41), taking ℓ = m and ℓ = m − 1, we get 󵄩󵄩 ∗ 󵄩󵄩2 ∗ ∗ ∗ 󵄩󵄩D u󵄩󵄩 = (p u, p u) = ([p, p ]u, u) + (p(u), p(u)) 󵄩 󵄩2 ≥ m!|am |2 ‖u‖2 + (m − 1)!󵄩󵄩󵄩am−1 u + mam u′ 󵄩󵄩󵄩 . If am−1 = 0, then we get 󵄩󵄩 ∗ 󵄩󵄩2 2 2 󵄩 ′ 󵄩2 2 2 󵄩󵄩D u󵄩󵄩 ≥ m!|am | (‖u‖ + 󵄩󵄩󵄩u 󵄩󵄩󵄩 ) = m!|am | ‖zu‖ .

(16.42)

16.4 The generalized 𝜕-complex

� 311

If am−1 ≠ 0, then we set a = √m! am and b = √(m − 1)! am−1 and get 󵄩 󵄩󵄩 ∗ 󵄩󵄩2 ′ 󵄩2 2 2 󵄩󵄩D u󵄩󵄩 ≥ m!|am | ‖u‖ + (m − 1)!󵄩󵄩󵄩am−1 u + mam u 󵄩󵄩󵄩 ̄ ′ , u) + |a|2 󵄩󵄩󵄩u′ 󵄩󵄩󵄩2 , ̄ u′ ) + ab(u = (|a|2 + |b|2 ) ‖u‖2 + ba(u, 󵄩 󵄩 and we can proceed as before to get 󵄩󵄩 ∗ 󵄩󵄩2 2 󵄩󵄩D u󵄩󵄩 ≥ α‖zu‖

(16.43)

for α > 0. Using the orthonormal basis (φk )k , we write ∞

u = ∑ uk φk k=0

and get ∞

‖zu‖2 = ∑ |uk |2 (k + 1). k=0

Now we want to show that for each ϵ > 0, there exists a constant R > 0 such that 2 󵄨 󵄨2 󵄩 󵄩2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 e−|z| dλ(z) ≤ ϵ󵄩󵄩󵄩D∗ u󵄩󵄩󵄩

(16.44)

ℂ\𝔹R

for all u ∈ dom(D∗ ). As for the operator 𝜕∗ , we get 2 󵄨 󵄨2 󵄩 󵄩2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 e−|z| dλ(z) ≤ αϵ󵄩󵄩󵄩𝜕∗ u󵄩󵄩󵄩 ;

ℂ\𝔹R

by (16.42) and (16.43) 2 󵄩 󵄩2 󵄨 󵄨2 ∫ 󵄨󵄨󵄨u(z)󵄨󵄨󵄨 e−|z| dλ(z) ≤ αϵ󵄩󵄩󵄩D∗ u󵄩󵄩󵄩 ,

ℂ\𝔹R 2

and so {u ∈ dom(D∗ ) : ‖D∗ u‖2 ≤ 1} is a relatively compact set in A2 (ℂ, e−|z| ). We can now summarize in the following: Theorem 16.22. Let Du = pm (u) dz, where p = a0 + a1

𝜕 𝜕m + ⋅ ⋅ ⋅ + am m , 𝜕z 𝜕z

Then the operator ◻̃ D,1 = DD∗ has a compact inverse

am ≠ 0, m ≥ 1.

312 � 16 The 𝜕-complex on the Segal–Bargmann space 2

Ñ D,1 : A2(1,0) (ℂ, e−|z| ) 󳨀→ dom(◻̃ D,1 ). Remark 16.23. We give a simple example in ℂ2 , where the basic estimate does not hold. Let p be a polynomial differential operator without constant summand and set D(u) = p(u)dz1 − p(u)dz2 for a polynomial u. If u = u1 dz1 + u2 dz2 is a (1, 0)-form with polynomial components, then we have D(u) = (p(u2 ) − p(u1 ))dz1 ∧ dz2 and D∗ u = p∗ u1 + p∗ u2 . Now take u = dz1 + dz2 . Then D(u) = 0 and D∗ u = p∗ 1 − p∗ 1 = 0. As ‖u‖ ≠ 0, the basic estimate does not hold. Remark 16.24. The basic estimates can be viewed as generalizations of the uncertainty 𝜕m principle; see Section 1.5. Let n = 1 and p = 𝜕z m . Then we have m−1󵄩 󵄩󵄩 𝜕 j 󵄩󵄩󵄩2 󵄩󵄩 m 󵄩󵄩2 2 󵄩󵄩z u󵄩󵄩 = ‖u‖ + ∑ 󵄩󵄩󵄩 (z u)󵄩󵄩󵄩 . 󵄩 𝜕z 󵄩󵄩 j=0 󵄩

So we get 󵄩 m 󵄩2 󵄩 󵄩2 󵄩󵄩 𝜕 u 󵄩󵄩 ‖u‖2 ≤ 󵄩󵄩󵄩zm u󵄩󵄩󵄩 + 󵄩󵄩󵄩 m 󵄩󵄩󵄩 . 󵄩󵄩 𝜕z 󵄩󵄩 For a general polynomial differential operator p of degree m with leading coefficient am , we obtain by Lemma 16.20 that there exists a constant C > 0 such that ‖u‖ ≤ C‖D∗ u‖ = C‖p∗ u‖ for each u ∈ dom(D∗ ), where C = √ m!|a1 |2 . Hence we have again m

󵄩 󵄩2 󵄩 󵄩2 ‖u‖2 ≤ C(󵄩󵄩󵄩p∗ u󵄩󵄩󵄩 + 󵄩󵄩󵄩p(u)󵄩󵄩󵄩 ). We remark that we can interchange the roles of D and D∗ and obtain the following: Theorem 16.25. Suppose that n > 1 and 1 ≤ p ≤ n − 1. Let D be as in (16.22) and suppose that ‖u‖2 ≤ C ∑ |K|=p−1

′

n

∑ ([pk , p∗j ]ujK , ukK )

j,k=1 2

for all u ∈ dom(◻̃ D,p ). If β ∈ A2(p,0) (ℂn , e−|z| ) satisfies D∗ β = 0, then v0 = DÑ D,p β ∈ 2

A2(p+1,0) (ℂn , e−|z| ) is the canonical solution of D∗ v = β, which means that D∗ v0 = β and v0 ∈ (ker D∗ )⊥ = im D, and ‖DÑ D,p β‖ ≤ C‖β‖ for some constant C > 0 independent of β. Proof. As in Theorem 16.18, we get that ◻̃ D,p has a bounded inverse 2

Ñ D,p : A2(p,0) (ℂn , e−|z| ) 󳨀→ dom(◻̃ D,p ). Now using the 𝜕-Neumann calculus, we obtain that

16.4 The generalized 𝜕-complex �

313

0 = D∗ β = D∗ (D∗ D + DD∗ )Ñ D,p β = D∗ DD∗ Ñ D,p β, and hence 0 = (D∗ DD∗ Ñ D,p β, D∗ Ñ D,p β) = (DD∗ Ñ D,p β, DD∗ Ñ D,p β). This implies that DD∗ Ñ D,p β = 0, and we get D∗ v0 = D∗ DÑ D,p β = (DD∗ + D∗ D)Ñ D,p β = β and (v0 , f ) = (DÑ D,p β, f ) = (Ñ D,p β, D∗ f ) = 0 for all f ∈ ker D∗ . Example 16.26. We take Example 16.28(b) and consider f = f1 dz1 + f2 dz2 ∈ A2(1,0) (ℂ2 , 2

e−|z| ) such that D∗ f = p∗1 f1 + p∗2 f2 = 0. By Theorem 16.25 we get

2

g = g dz1 ∧ dz2 = DÑ D,1 f ∈ A2(2,0) (ℂ2 , e−|z| ) such that D∗ g = −p∗2 g dz1 + p∗1 g dz2 = f and ‖DÑ D,1 f ‖ ≤ C‖f ‖ for some constant C > 0. Remark 16.27. Finally, we point out that the 𝜕-Neumann operator Ñ D,p exists and is 2

bounded on A2(p,0) (ℂn , e−|z| ) if and only if the basic estimate (16.26) holds (see Remark 9.12 for the details). In the following, we exhibit some examples in ℂ2 , where the basic estimates hold. Example 16.28. 𝜕2 ∗ 2 (a) Let pk = 𝜕z 2 . Then pj (z) = zj , and we have k

n

n

n

j,k=1

j,k=1

j,k=1

∑ ([pk , p∗j ]uj , uk ) = ∑ (2δj,k uj , uk ) + ∑ (4δjk zj

𝜕uj 𝜕zk

, uk )

n 󵄩 󵄩󵄩 𝜕uj 󵄩󵄩󵄩2 󵄩󵄩 = 2‖u‖2 + 4 ∑󵄩󵄩󵄩 󵄩 𝜕zj 󵄩󵄩󵄩 j=1 󵄩

for u = ∑nj=1 uj dzj ∈ dom(◻̃ D,1 ). Hence (16.7) is satisfied.

(b) Let n = 2 and take p1 = and we have

𝜕2 𝜕z1 𝜕z2

and p2 =

𝜕2 𝜕z21

([p1 , p∗1 ]u1 , u1 ) = (u1 , u1 ) + (

2

𝜕 ∗ ∗ 2 2 + 𝜕z 2 . Then p1 (z) = z1 z2 and p2 (z) = z1 +z2 , 2

𝜕u1 𝜕u1 𝜕u 𝜕u , ) + ( 1 , 1 ), 𝜕z1 𝜕z1 𝜕z2 𝜕z2

([p1 , p∗2 ]u2 , u1 ) = 2(

𝜕u2 𝜕u1 𝜕u 𝜕u , ) + 2( 2 , 1 ), 𝜕z1 𝜕z2 𝜕z2 𝜕z1

([p2 , p∗1 ]u1 , u2 ) = 2(

𝜕u1 𝜕u2 𝜕u 𝜕u , ) + 2( 1 , 2 ), 𝜕z1 𝜕z2 𝜕z2 𝜕z1

314 � 16 The 𝜕-complex on the Segal–Bargmann space

([p2 , p∗2 ]u2 , u2 ) = 4(u2 , u2 ) + 4(

𝜕u2 𝜕u2 𝜕u 𝜕u , ) + 4( 2 , 2 ). 𝜕z1 𝜕z1 𝜕z2 𝜕z2

So we obtain 2

∑ ([pk , p∗j ]uj , uk )

j,k=1

2 2 󵄨󵄨 𝜕u 2 𝜕u 󵄨󵄨󵄨 󵄨󵄨󵄨 𝜕u 𝜕u 󵄨󵄨󵄨 󵄨 = ∫ (|u1 |2 + 4|u2 |2 + 󵄨󵄨󵄨 1 + 2 2 󵄨󵄨󵄨 + 󵄨󵄨󵄨 1 + 2 2 󵄨󵄨󵄨 ) e−|z| dλ 󵄨󵄨 𝜕z1 𝜕z2 󵄨󵄨 󵄨󵄨 𝜕z2 𝜕z1 󵄨󵄨 2 ℂ

for u = ∑2j=1 uj dzj ∈ dom(◻̃ D,1 ). Again, (16.7) is satisfied.

(c) We easily see that for p∗1 (z1 , z2 ) = zk1 and p∗2 (z1 , z2 ) = zm 2 , where k, m ∈ ℕ, and we ∗ get [p2 , p1 ] = 0 and 2

‖u‖2 ≤ C ∑ ([pk , p∗j ]uj , uk ), j,k=1

where C > 0. (d) Let p∗1 (z1 , z2 ) = az1 + bz2

and

p∗2 (z1 , z2 ) = cz1 + dz2

for a, b, c, d ∈ ℝ. A computation shows that ([p1 , p∗1 ]u1 , u1 ) = (a2 + b2 )‖u1 ‖2

and

([p1 , p∗2 ]u2 , u1 ) = (ac + bd)(u2 , u1 )

and

([p2 , p∗2 ]u2 , u2 ) = (c2 + d 2 )‖u2 ‖2 ,

([p2 , p∗1 ]u1 , u2 ) = (ac + bd)(u1 , u2 ).

Given a, b ∈ ℝ, all nonzero, we suppose that c = μb and d = −μa for μ ∈ ℝ. Then we have ac + bd = 0, and we get 2

‖u‖2 ≤ C ∑ ([pk , p∗j ]uj , uk ), j,k=1

where C = min{(a2 + b2 ), (c2 + d 2 )}−1 . (e) If we take p∗1 (z1 , z2 ) = az21 + bz22

and

p∗2 (z1 , z2 ) = cz21 + dz22 ,

for a, b, c, d ∈ ℝ, then we get 2

∑ ([pk , p∗j ]uj , uk ) = 2‖au1 + cu2 ‖2 + 2‖bu1 + du2 ‖2

j,k=1

16.5 Notes

� 315

2 2 󵄩󵄩 𝜕u 𝜕u 󵄩󵄩󵄩 󵄩󵄩󵄩 𝜕u 𝜕u 󵄩󵄩󵄩 󵄩 + 󵄩󵄩󵄩2a 1 + 2c 2 󵄩󵄩󵄩 + 󵄩󵄩󵄩2b 1 + 2d 2 󵄩󵄩󵄩 . 󵄩󵄩 𝜕z1 𝜕z1 󵄩󵄩 󵄩󵄩 𝜕z2 𝜕z2 󵄩󵄩

Given a, b, c, all nonzero, we choose d such that ac = −bd. Then 2‖au1 + cu2 ‖2 + 2‖bu1 + du2 ‖2 = 2(a2 + b2 )‖u1 ‖2 + 2(c2 + d 2 )‖u2 ‖2 , which implies that 2

‖u‖2 ≤ C ∑ ([pk , p∗j ]uj , uk ), j,k=1

where C = min{2(a2 + b2 ), 2(c2 + d 2 )}−1 .

16.5 Notes Most of the material of this chapter was taken from [44] and [45]. Weighted Bergman spaces on Hermitian manifolds, which carry a similar duality of differentiation and multiplication as the Segal–Bargmann space, are studied in [48] and [49].

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Index A2 (Ω) 1 Tp0,1 (bΩ) 91 Tp1,0 (bΩ) 91 Δ(0,0) 210 φ Δφ

(0,q)

211, 281

𝜕-Neumann operator 100, 127, 133, 218, 246, 249, 250, 254, 259, 264 ϵ-net 215, 218 s-number 34 Sp 53 𝒦(H1 , H2 ) 26 ℒ(H1 , H2 ) 26 𝒱-elliptic form 79 𝜕-complex 94, 126, 209, 250 𝜕-complex 291 σ-algebra 151 ◻ 95, 100 ◻φ 127 i𝜕𝜕 91 adjoint operator 38, 65, 126, 258 algebra 254 annihilation operator 17 approximation to the identity 108 Arzelà–Ascoli theorem 215, 217 Baire category theorem 68 Banach algebra 153, 195 Bargmann transform 19 basic estimate 100, 142, 293, 307, 312 Bergman kernel 5, 10, 50, 257, 264, 274, 289, 299 Bergman projection 6, 254 Bergman space 1 Bessel’s inequality 7, 32 biholomorphic map 12 Borel set 151 boundedness 215 Brascamp–Lieb inequality 189 canonical solution operator 38, 50, 52, 63, 103, 129, 207, 257–259, 270 Cauchy–Riemann equation 26 Cayley transform 167, 168, 173 closable operator 65 closed graph 92, 93 closed graph theorem 68, 69, 167 https://doi.org/10.1515/9783111182926-018

closed operator 65, 165 closed range 100 closure of an operator 65 coercive 218, 219 commutator 135, 260 compact operator 26, 27, 29, 38, 158, 232, 233, 235, 246, 250, 261, 262, 267, 270, 296 compact resolvent 271, 272 compact set 247, 300, 302, 311 complex Laplacian 95, 127 complex measure 151 connection 134 continuous form 79 continuous spectrum 166 core of the operator 78 creation operator 17 curvature 134 cutoff function 199 decoupled 285 derivative of a distribution 83 diamagnetic inequality 203 diamagnetic property 267 difference quotient 196 Dirac Delta distribution 82 Dirac operator 212, 271 Dirichlet form 127, 219, 233 discrete spectrum 181 distribution 82 doubling measure 265, 272 dual norm 5 electric potential 202 elliptic partial differential operator 198, 202 equicontinuity 215 essential range 153, 166 essential spectrum 181, 272 essential supremum 153 essentially bounded 153 essentially self-adjoint operator 76, 176, 180 exact sequence 88 exhaustion function 92 Fefferman–Phong Lemma 268 Fock space 15, 46, 275 Friedrichs extension 79, 190 Friedrichs’ Lemma 110

322 � Index

fundamental form 134 fundamental solution 83 Gårding’s inequality 218–220, 233, 250 gauge invariance 206, 227 Gauß–Green Theorem 86, 90 Gelfand transform 156, 195 Gram–Schmidt process 6 Hahn–Banach Theorem 21, 137 Hankel operator 261 Helffer–Sjöstrand formula 195 Hermitian form 79 higher-order reflection 115 Hilbert–Schmidt operator 34, 54, 57, 63, 246 Hölder’s inequality 216, 252 holomorphic vector bundle 134 Hörmander L2 -estimate 142, 189 interior multiplication 135, 211, 281 invariant subspace 182 Jensen’s inequality 265 Kähler manifold 134 Kato’s inequality 203, 268 lattice 254 Levi form 91 lower semicontinuous function 186 magnetic field 202, 207, 227, 271 meager set 68 Minkowski’s inequality 108 Muckenhoupt characteristic constant 264 Nakano vanishing theorem 135 non-coercive 97 normal operator 154 normal subalgebra 153 nowhere dense set 68 Ohsawa–Takegoshi extension theorem 141 open-mapping theorem 68, 72 operator norm 26 orthogonal projection 3, 151 orthonormal 6 orthonormal basis 8 parallelogram rule 2, 30

Parseval’s equation 7 Pauli operator 212, 271 plurisubharmonic 92, 126, 130, 132, 143, 145, 148, 149, 212, 233, 234, 246, 249 point spectrum 166 polar decomposition 159 polarization identity 14, 168 positive definite matrix 198 positive operator 158, 176 precompact 215–218 property (P)̃ 249 property (P) 249, 250, 289 pseudoconvex 91, 99, 106, 120, 137, 142, 144, 145, 249, 250, 257, 259, 261 pseudodifferential operator 254 quadratic form 101 Radon–Nikodym theorem 162 Rellich–Kondrachov Lemma 220, 239 reproducing property 5 resolution of the identity 151, 161 resolvent of an operator 154 resolvent set 154, 165 restriction of a distribution 83 reverse Hölder class 264 Riemann mapping 14 Riesz representation theorem 4, 178 Ruelle’s Lemma 188, 189 Schatten-class 34, 52, 54, 62 Schrödinger operator 126, 207, 271 Schrödinger operator with magnetic field 196, 202, 208 Segal–Bargmann space 15 self-adjoint 76 self-adjoint operator 6, 29, 54, 65, 79, 95, 172–175, 177, 179, 184, 186, 188, 272 semibounded operator 190 sesquilinear form 79 set of the second category 68 sgn-function 111 simple function 153 Sobolev space 85, 218 spectral decomposition 156 spectral theorem 151, 156 spectrum 153, 165, 172, 298 spectrum of a bounded operator 154 square root 159

Index � 323

Stirling’s formula 41 Stone–WeierstraßTheorem 254 strictly positive operator 187 strictly pseudoconvex 92 strongly elliptic 219 subharmonic 207, 208 support of a distribution 83 symbolic calculus 156 symmetric operator 76, 180 tangential Cauchy–Riemann operator 91 test function 82 total variation 151 totally bounded set 26 trace 117 trivial line bundle 135

twist factor 137 twisted 𝜕-complex 136 uncertainty principle 19 uniform boundedness principle 69, 152 unitary operator 20 weak null-sequence 70 weak solution 198 weighted Sobolev space 239 Weyl sequence 182 Weyl spectrum 182 Witten Laplacian 126, 196, 210 Zorn’s lemma 8

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