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Singular Bilinear Integrals
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b2530   International Strategic Relations and China’s National Security: World at the Crossroads

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Jefferies, Brian, 1956– Title: Singular bilinear integrals / by Brian Jefferies (University of New South Wales, Australia). Description: New Jersey : World Scientific, 2017. | Includes bibliographical references. Identifiers: LCCN 2016048677 | ISBN 9789813207578 (hardcover : alk. paper) Subjects: LCSH: Vector-valued measures. | Integrals. | Bilinear forms. | Ideal spaces. | Vector valued functions. Classification: LCC QA325 .J44 2017 | DDC 518/.54--dc23 LC record available at https://lccn.loc.gov/2016048677

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

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To the memory of Igor Kluv´anek

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Preface

The idea for this monograph germinated at the “Vector Measures, Integration and Applications” conference held in Eichst¨att (Germany) in September 2008. Three topics concerning bilinear integration inspired by the 1980 survey paper “Applications of Vector Measures” of I. Kluv´ anek [82] were treated in the conference talk [65]. Bilinear integration treats the problem of integrating a function with values in an infinite dimensional vector space with respect to a measure with values in another infinite dimensional vector space. The concept of decoupled bilinear integrals has proved to be common to diverse applications of vector integration in quantum physics, stochastic analysis, scattering and operator theory. Decoupling is required when the classical theories of bilinear integration of R. Bartle [11] and I. Dobrakov [39,40] cannot be applied, as is the case in the applications just mentioned. The term singular is used somewhat loosely in the title to describe the situation where the classical theory of bilinear integration does not work. The study of decoupled bilinear integration affords the opportunity to touch upon the diverse and interesting subjects mentioned that lend themselves to the techniques of functional analysis. An ingenue to mathematics may be entranced by the idea that Science depends on a reliable notion of the measurement of phenomena. Especially in quantum physics, this intuition is proved valid once we are forced to consider the spectral theory of differential operators. In quantum physics the results of physical observations are determined by the values of the self adjoint spectral measure associated with the operator determined by an observable quantity, so the study of vector measures and vector integration lies at the foundations of scientific enterprise. Although the theory of integration of scalar quantities with respect to vii

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vector valued measures and the integration of vector valued quantities with respect to scalar valued measures seems to be largely settled, difficulties arise when integrating vector or operator valued functions with respect to vector or operator valued measures. Unfortunately, such problems regularly arise in quantum physics where spectral measures are fundamental in terms of physical observation. This monograph treats the mathematics that evolves from the integration of vector valued functions with respect to spectral measures that features in the representation of solutions in a number of problems in mathematical physics. Thanks are due to my collaborators L. Garcia-Raffi, S. Okada and P. Rothnie in this work. Chillingham, 2016

Brian Jefferies

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Contents

Preface 1.

Introduction

1

1.1 1.2

6

1.3

1.4

1.5

1.6 2.

vii

Vector measures . . . . . . . . . . . . . . . . . . . . . . . . Integration of scalar functions with respect to a vector valued measure . . . . . . . . . . . . . . . . . . . . . . . . . . Integration of vector valued functions with respect to a scalar measure . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Pettis integral . . . . . . . . . . . . . . . . . . 1.3.2 The Bochner integral . . . . . . . . . . . . . . . . Tensor products . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Injective and projective tensor products . . . . . . 1.4.2 Grothendieck’s inequality . . . . . . . . . . . . . . Semivariation . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Semivariation in Lp -spaces . . . . . . . . . . . . . 1.5.2 Semivariation of positive operator valued measures Bilinear integration after Bartle and Dobrakov . . . . . .

Decoupled bilinear integration 2.1 2.2 2.3 2.4

10 12 13 14 15 17 21 24 26 32 35 41

Bilinear integration in tensor products Order bounded measures . . . . . . . . The bilinear Fubini theorem . . . . . . Examples of bilinear integrals . . . . .

ix

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44 53 54 59

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3.

Singular Bilinear Integrals

Operator traces 3.1 3.2 3.3 3.4

3.5 4.

4.2 4.3 4.4

5.4 5.5

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Background on probability and discrete processes 4.1.1 Conditional probability and expectation . 4.1.2 Discrete Martingales . . . . . . . . . . . . 4.1.3 Discrete stopping times . . . . . . . . . . Stochastic processes . . . . . . . . . . . . . . . . Brownian motion . . . . . . . . . . . . . . . . . . 4.3.1 Some properties of Brownian paths . . . Stochastic integration of vector valued processes .

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CLR inequality 7.1 7.2 7.3

101 104 106 109 111 112 114 115 123

Time-dependent scattering theory . . . . . . . . . . Stationary state scattering theory . . . . . . . . . . Time-dependent scattering theory for bounded Hamiltonians and potentials . . . . . . . . . . . . . Bilinear integrals in scattering theory . . . . . . . . Application to the Lippmann-Schwinger equations

Evolution processes . . . . . . Measurable functions . . . . . Progressive measurability . . Operator bilinear integration Random evolutions . . . . . .

72 74 75 83 91 94 94 101

. . . . 123 . . . . 125 . . . . 128 . . . . 131 . . . . 138

Random evolutions 6.1 6.2 6.3 6.4 6.5

7.

Trace class operators . . . . . . . . . . . . . . . . The Hardy-Littlewood maximal operator . . . . . The Banach function space of traceable functions Traceable operators on Banach function spaces . 3.4.1 Lusin filtrations . . . . . . . . . . . . . . 3.4.2 Connection with other generalised traces Hermitian positive operators . . . . . . . . . . . .

Scattering theory 5.1 5.2 5.3

6.

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Stochastic integration 4.1

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143 148 150 157 167 171

Asymptotic estimates for bound states . . . . . . . . . . . 171 Lattice traces for positive operators . . . . . . . . . . . . . 175 The CLR inequality for dominated semigroups . . . . . . 184

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Contents

8.

Operator equations 8.1 8.2 8.3

8.4

Operator equations . . . . . . . . . . . . . . . . . . . . . Double operator integrals . . . . . . . . . . . . . . . . . Traces of double operator integrals . . . . . . . . . . . . 8.3.1 Schur multipliers and Grothendieck’s inequality 8.3.2 Schur multipliers on measure spaces . . . . . . . The spectral shift function . . . . . . . . . . . . . . . . .

191 . . . . . .

193 202 209 212 214 221

Bibliography

229

Index

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Chapter 1

Introduction

In classical measure theory, as given in [59] for example, the variation of a real or complex valued set function is an essential tool used to estimate the size of its range and space of integrable functions. For a finitely additive set function μ : A → C defined on an algebra A of subsets of a nonempty set Σ, the variation V (μ) : A → [0, ∞] of μ is given by  |μ(A ∩ B)|, A ∈ P. V (μ)(A) = sup P

B∈P

The supremum is over all finite partitions P of Σ be elements of the algebra A. The set function μ has uniformly bounded range on the algebra A if and only if μ has finite total variation V (μ)(Σ). Once we move to infinite dimensional spaces and replace the modulus |·| by a norm  · , we find that many simple vector valued set functions with uniformly bounded range fail to have finite total variation or its variation may only take the values zero or infinity, see Example 1.1 below. A simple example appears immediately in a Hilbert space H with The sequence {en /n}∞ an orthonormal basis {en }∞ n=1 . n=1 is summable in H but not absolutely summable. The total variation V (m)(N) of  the countably additive set function m : A −→ n∈A en /n, A ⊂ N, is ∞ V (m)(N) = n=1 1/n = ∞. By contrast, the total semivariation 1  ∞ ∞    1 2  ξn   = m(N) = sup   n2 ξ∈H,ξ≤1 n=1 n n=1 of m is finite. For a measure m with values in a Banach space X, that is, a countably additive set function m : E → X defined on a σ-algebra E of subsets of a nonempty set Σ, the semivariation m : E → [0, ∞) defined in Section 1.1 below is always finite and features in the consideration of the 1

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integration of scalar valued functions with respect to the vector measure m considered in Section 1.2. The study of vector valued integration received an impetus from the spectral theory of selfadjoint operators on a Hilbert space H lying at the heart of the mathematical foundations of quantum mechanics [134]. Any selfadjoint operator T : D(T ) → H with dense domain D(T ) in H has a representation  λ dPT (λ). T = σ(T )

The spectrum σ(T ) of T is the set of all real numbers λ for which λI − T is not invertible in the collection L(H) of bounded linear operators on H and the spectral measure PT : B(σ(T )) → L(H) associated with T is an operator valued measure σ-additive on the Borel σ-algebra B(σ(T )) of σ(T ) ⊂ R for the strong operator topology of L(H), that is, for all pairwise disjoint Borel ∞ sets Bn ∈ B(σ(T )), n = 1, 2, . . . , and every h ∈ H, the sum n=1 PT (Bn )h converges in the norm of H and ∞  ∞   Bn h = PT (Bn )h. PT n=1

n=1

The operator valued measure PT has values in the selfadjoint projections so that PT (A ∩ B) = PT (A)PT (B) for A, B ∈ B(σ(T )) and PT (σ(T )) = I, the identity operator on H [121, Definition 12.17]. In the case of a compact selfadjoint operator T , the integral becomes the sum  T = λ.PT ({λ}) λ∈σ(A)

in which PT ({λ}) is the orthogonal projection onto the finite dimensional subspace of H generated by eigenvectors of T for the eigenvalue λ. The strong operator topology of L(H) is defined by the collection {ph : h ∈ H } of seminorms given by ph (T ) = T h,

T ∈ L(H),

for each h ∈ H. Because the strong operator topology is not metrisable unless H is finite dimensional, we shall consider vector measures taking values in a locally convex Hausdorff topological vector space E over C, briefly a locally convex space or lcs, whose topology is defined by a family of seminorms [123, Section II.4]. The next section is a brief guide to the basic facts concerning measures with values in a locally convex space such as the space L(H) of operators on a Hilbert space H endowed with the strong operator topology.

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Integration of vector valued functions with respect to scalar measures is considered in Section 1.3. The distinction between variation and semivariation features again with the Bochner integral and the Pettis integral. Bilinear integration treats the integration of an X-valued function with respect to a Y -valued measure m with a given bilinear map from X × Y into Z. Typically X, Y and Z are infinite dimensional Banach spaces, but we shall also be concerned with the bilinear maps (T, x) −→ T x and (x, T ) −→ T x with T a continuous linear map between Banach spaces E and F and x ∈ E. For bilinear integration, the X-semivariation of the Y -valued measure m in Z with respect to the bilinear map (·, ·) is a central concept. In this work, we shall be mainly treating bilinear integration in tensor products, where there exists a tensor product topology τ for which the τ Y and (x, y) −→ vector space Z is equal to the complete tensor product X ⊗ x ⊗ y, x ∈ X, y ∈ Y , is the associated bilinear map. Various aspects of the τY X-semivariation of a Y -valued measure in a tensor product space X ⊗ are considered in Section 4. The fundamental work of R. Bartle [11] treats the integration of an X-valued function with respect to a Y -valued measure m in Z by approximating in the X-semivariation of m. In the case where one of the vector spaces X or Y is finite dimensional, the integrals described in Sections 2 and 3 below are obtained. A further refinement of Bartle’s approach due to I. Dobrakov is described in Section 5. The situation where there are sufficiently many sets with finite X-semivariation may be thought of as the case of regular bilinear integration. If H is an infinite dimensional Hilbert space and P is a spectral measure acting on H, then except in trivial cases, P fails to have finite Hsemivariation with respect to the bilinear map (x, T ) −→ T x, x ∈ H, T ∈ L(H). The applications of bilinear integration to quantum physics and operator theory considered here give rise to the singular bilinear integrals mentioned in the title in which X-semivariation plays no direct role. By taking our integrals to have values in the tensor products such as τ H or H⊗ τ L(H) with different choices of the tensor product τ Y , L(H)⊗ X⊗ topology τ , there is a type of decoupling between the values of the vector valued function that is being integrated with respect to a vector valued measure. Such a decoupling is already a feature of stochastic integration in probability theory where Brownian paths almost surely have infinite variation on any interval. A general treatment of bilinear integration in tensor products is given

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in Chapter 2, based on joint work of the author with S. Okada [70]. The reminder of the monograph is devoted to applications of decoupled bilinear integration to diverse areas of analysis and mathematical physics. For a bounded linear operator T : L2 ([0, 1]) → L2 ([0, 1]) with

1 an integral kernel k, we can try to view the trace tr(T ) of T as the integral 0 k(x, x) dx, n just as the trace of a matrix {aij }ni,j=1 is the sum i=1 aii of the diagonal elements. For a trace class operator on L2 ([0, 1]), this is in fact the case for a specific choice of integral kernel k. If we set Φk (x) = k(x, ·) ∈ L2 ([0, 1]), x ∈ [0, 1], then tr(T ) may be viewed as a type of bilinear integral  1  1

Tk , dm =

Φk , dm 0

0

with respect to the vector measure m : B → χB , B ∈ B([0, 1]). Chapter 3 is based on [66] and explores the lattice (order) ideal of traceable operators T on a Banach function space X for which the bilinear in

tegral Σ T, dm exists. By contrast, on a Hilbert space, the collection of trace class operators is an operator ideal. The two classes coincide for hermitian positive operators T , that is, selfadjoint operators T whose spectrum σ(T ) is a subset of [0, ∞). The ideas here resurface in Chapter 7 in the proof of the CLR inequality in quantum physics. The topics of this monograph are heavily weighed in the direction of functional analysis and measure theory. A brief guide to probability theory and stochastic processes is given in Chapter 4. The final Section 4.4 is devoted to general stochastic integration which may be viewed as a type of bilinear integration that has been treated in Chapter 2. The connection between stationary state and time dependent scattering theory is treated in Chapter 5 based on joint work of the author with L. Garcia-Raffi [49]. In order to treat this subject, it is necessary to integrate operator valued functions with respect to spectral measures, so the tools developed in Chapter 2 may be applied. The treatment of random evolutions in Chapter 6 requires the integration of operator valued functions with respect to operator valued measures to obtain a version of the Feynman-Kac formula. Integrals of this type are of the decoupled variety treated in Chapter 2. This material is from [64]. The treatment of progressive measurability in Section 6.3 is from [68]. The Feynman-Kac formula reappears in Chapter 7 in the proof of the CLR inequality for dominated semigroups on a Hilbert space L2 (μ). As mentioned in Chapter 6, we can associate a σ-additive evolution process with a dominated semigroup on L2 (μ) and the Feynman-Kac formula with

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respect to this process is a feature of our proof of the CLR inequality in the semiclassical approximation of quantum physics. Double operator integrals are a type of singular bilinear integral and the solution of certain operator equations are also expressed as bilinear integrals—the integral of a resolvent with respect to a spectral measure. Chapter 8 touches upon trace class perturbations of selfadjoint operators in scattering theory where we employ double operator integrals. Grothendieck’s inequality and our discussion of Lusin filtrations in Chapter 3 are used to prove Peller’s characterisation [104] of the class of double operator integrable functions for the space of bounded linear operators. We end with an elementary construction of Krein’s spectral shift function as the boundary value of a harmonic function in the upper half plane.

Prerequisites. The subject of ‘assumed knowledge’ is a fraught consideration for an author of what may be viewed as a research monograph. Topics that may have been routinely discussed in graduate courses in mathematics in some countries 30 years ago may have fallen out of favour, despite lying at the foundations of future scientific discoveries. As should be clear from the preceding discussion, many routine facts from functional analysis are employed in our analysis. H.H. Schaefer [123] provides a succinct account of topological vector spaces. Basic facts about Banach lattices may be found in [96]. Classical measure theory is exposed in [59] and [16] provides a connection with probability theory. A comprehensive but not exhaustive study of measure theory is given in the volumes of D. Fremlin [45, 46] that complements and is complemented by L. Schwartz [125]. Basic facts about complex analysis are found in [120] and the spectral theorem for selfadjoint operators and distribution theory are treated in [121]. The first few chapters of [38] and [86] contain most of what we need concerning vector valued measures. Other topics that arise relating to operator theory [76], harmonic analysis [54], stochastic processes [28] and scattering theory [136] are treated in the following chapters. In view of the diverse range of mathematical topics listed, it should be apparent that this monograph promotes the view that the study of integration and measure in infinite dimensions continues to provide insights into mathematical analysis and its applications, just as it did as the mathematical foundations of quantum physics were laid by J. von Neumann in 1932 [134].

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Vector measures

The collection of all continuous linear functionals on a lcs E is denoted by E  . The ordered pair E, E  forms a dual pair for which there is a bilinear map (x, ξ) −→ x, ξ given by x, ξ = ξ(x) for ξ ∈ E  , x ∈ E. The notation is convenient because it is analogous to the inner product (h, g) for vectors h, g belonging to a Hilbert space H, which we take to be linear in the first variable and antilinear in the second. Given an E-valued function Φ defined over some set Σ, or a collection Σ of sets, we use the notation Φ, x to denote the function σ → Φ(σ), x , σ ∈ Σ. Similarly, if Φ takes its values in the space L(E, F ) of continuous linear maps from E into the lcs F , then for each x ∈ E and y  ∈ F  , Φx : σ → Φ(σ)x and Φx, y  : σ → Φ(σ)x, y  , for all σ ∈ Σ. The Hahn-Banach Theorem establishes that the collection of all seminorms x −→ | x, ξ |, x ∈ E, for ξ ∈ E  separates the points of E and so defines a locally convex Hausdorff topology σ(E, E  ) on E called the weak topology. Let (Σ, E) be a measurable space, that is, E is a σ-algebra of subsets of a set Σ. The term measure space is used for the triple (Σ, E, μ) with (Σ, E) a measurable space and μ : E → [0, ∞] an extended real valued measure, meaning that the equality   ∞ ∞   An = μ(An ) μ n=1

n=1

of extended real numbers holds for any pairwise disjoint sets An ∈ E, n = 1, 2, . . . . If a measure μ has infinite values, then the family {A ∈ E : μ(A) < ∞} of subsets of Ω constitutes a δ-ring, that is, a ring D of sets closed under the operations of symmetric difference Δ and intersection ∩ such that ∞ An ∈ D n=1

for all An ∈ D, n = 1, 2, . . . . By restricting analysis to measures defined on δ-rings, extended real numbers can be avoided. A vector measure m : E → E with values in a locally convex space E is a set function that is countably additive (σ-additive) in the locally convex topology of E. By this we mean that for any pairwise disjoint sets An ∈ E, n = 1, 2, . . ., the equality  ∞ ∞   m An = m(An ) n=1

n=1

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holds. The sum on the right-hand side converges in the topology of E. In the case that E = R or E = C, m is called a scalar measure. In the special case of an operator valued measure M : E → Ls (X) acting on a Banach space X, σ-additivity is always assumed to be for the strong operator topology of L(X): for any pairwise disjoint sets An ∈ E, n = 1, 2, . . ., the equality  ∞ ∞   An x = M (An )x M n=1

n=1

holds in X for each x ∈ X. The union ∪∞ n=1 An is the same however the pairwise disjoint family {An : n = 1, 2, . . . } of sets is indexed, so the sums above are also independent of the ordering of the index, giving rise to the notion that a sequence of vectors xn , n = 1, 2, . . ., in a lcs E is unconditionally summable if there exists x ∈ E with the property that for every neighbourhood U of 0 in E,  there is a finite set K of natural numbers such that x − j∈J xj ∈ U, for any finite set J of natural numbers containing K. The term weakly unconditionally summable is used in the case that the sequence is unconditionally summable for the weak topology of E. A basic result is that as far as unconditional summability is concerned, the weak and norm topologies of a normed space are equivalent. Theorem 1.1 (Orlicz-Pettis [38, I.4.4]). Let E be a lcs. A sequence of vectors xn , n = 1, 2, . . ., in E is weakly unconditionally summable if and only if it is unconditionally summable in any topology consistent with the duality between E and E  . If m is countably additive in the weak topology of E, then by the OrliczPettis lemma, it is automatically countably additive in the original topology of E. Terms such as ‘m-a.e.’ have the same meaning as for scalar measures: off an m-null set, by which we mean a set N ∈ E such that m(A) = 0 for all A ∈ E contained in N . A scalar measure μ : E → C has bounded range on E [38, I.1.19], so the variation |μ| of μ is a finite nonnegative measure on E defined by

 |μ(Aj ∩ A)| |μ|(A) = sup j

for all A ∈ E. The supremum is taken over the family ΠE of all pairwise disjoint subsets A1 , . . . , Ak of Σ belonging to E, and all k = 1, 2, . . . . The same definition is adopted in the case that E is an algebra of subsets of Σ and m : E → C is an additive set function with bounded range.

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Singular Bilinear Integrals

Suppose that m : E → E is a vector measure and p is a continuous seminorm on E. The p-semivariation of m is the set function p(m) : E → [0, ∞) defined by p(m)(A) = sup{| m, ξ |(A)} for all A ∈ E. The supremum is taken over all elements ξ belonging to the polar Up◦ = {ξ : | x, ξ | ≤ 1, ∀x ∈ E, p(x) ≤ 1} of Up . An application of the uniform boundedness principle to the associated family of scalar measures shows that for every continuous seminorm p on E, the p-semivariation takes finite values, see Proposition 1.1 below. However,  the p-variation of m defined by Vp (m)(A) = sup{ j p(m(Aj ∩ A))} may take the value infinity. The supremum here is again taken over ΠE . Example 1.1. Let 1 < p < ∞. Let B([0, 1]) be the Borel subsets of [0, 1] and let m : B([0, 1]) → Lp ([0, 1]) be the vector measure defined by m(A) = χA , for every A ∈ B([0, 1]). Denote the Lebesgue measure on B([0, 1]) by λ. For any Borel set A contained in [0,1] such that λ(A) > 0 and any n = 1, 2, . . . , we can find disjoint Borel subsets A1 , . . . , An of A such that λ(Ai ) = λ(A)/n for all i = 1, . . . , n. Then n n   m(Ai )p = (λ(A)/n)1/p = n1−1/p λ(A)1/p . i=1

i=1

Plainly this means that the  · p -variation of m is infinite on A. We may also describe Vp (m) as the smallest positive measure μ such that p(m(A)) ≤ μ(A), for every A ∈ E. In the case that E is a Banach space with norm  · , the semivariation of m is written as m and the variation of m as V (m). The normed space E is finite dimensional if and only if the norms m → m(Σ) and m → V (m)(Σ) are equivalent. More accurately, a result of Dvoretsky-Rogers shows that if E is infinite and V (m)(Σ) < ∞ for every E-valued measure m : E → E, then E is finite dimensional [123, IV.10.7, Corollary 3]. The same notation is adopted in the case that E is an algebra of subsets of Σ and m : E → Y is an additive set function with bounded range. For two vector measures m : E → X and n : E → Y with values in locally convex spaces X and Y , we write n  m if every m-null set is an n-null set. If X, Y are normed spaces, then limm(A)→0+ n(A) = 0. It is clear that a set A ∈ E is m-null if and only if p(m)(A) = 0 for every continuous seminorm p on X. The p-semivariation of a vector measure is related to its range by the following estimates.

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Introduction

9

Proposition 1.1 ([38, Proposition I.1.11]). Let m : E → E be a vector measure and let p be a continuous seminorm on E. Then for every A ∈ E, we have sup{p(m(B)) : B ⊆ A, B ∈ E} ≤ p(m)(A) ≤ 4 sup{p(m(B)) : B ⊆ A, B ∈ E}. Let E be a lcs. A family Λ of E-valued measures on E is called uniformly countably additive if for any sequence of sets An , n = 1, 2, . . ., in E decreasing to the empty set and every continuous seminorm p on E, we have supμ∈Λ p(μ)(An ) → 0 as n → ∞. The family Λ is bounded if supμ∈Λ p(μ)(Ω) < ∞ for every continuous seminorm p on E. For a vector measure m and a continuous seminorm p, the uniform countable additivity and boundedness of | m, ξ |, ξ ∈ Up◦ , implies the following result [86, Theorem II.1.1]. Theorem 1.2 (Bartle-Dunford-Schwartz). Let Λ be a bounded and uniformly countably additive family of scalar measures μ : E → C. Then there exists a finite positive measure λ on E such that λ(A) ≤ supμ∈Λ |μ|(A) for all A ∈ E and λ(A) → 0, A ∈ E, implies that supμ∈Λ |μ|(A) → 0. Corollary 1.1. Let m : E → E be a vector measure. For every continuous seminorm p on E, there exists a finite positive measure λp on E such that λp (A) ≤ p(m)(A), for every A ∈ E, and λp (A) → 0, A ∈ E, implies that p(m)(A) → 0. Theorem 1.3 (Nikodym Boundedness Theorem, [38, I.3.1]). Let

mι ι∈I be a family of vector measures mι : E → E, ι ∈ I, such that

mι (A) ι∈I is a bounded subset of E for every A ∈ E. Then for every continuous seminorm p on E, supι∈I p(mι )(Σ) < ∞. Theorem 1.4 (Vitali-Hahn-Saks, [38, I.4.8]). Let mn , n = 1, 2, . . . , be E-valued measures on E such that mn (A), n = 1, 2, . . . , converges in E for every A ∈ E. Then mn n∈N is a bounded and uniformly countably additive family of vector measures. In particular, the set function m : E → E defined by m(A) = limn→∞ mn (A) for each A ∈ E is countably additive. Corollary 1.2. Let mn , n = 1, 2, . . . , be E-valued measures on E such that the vectors mn (A), n = 1, 2, . . . , converge in E for every A ∈ E. For every

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continuous seminorm p on E, there exists a finite positive measure λp on E such that λp (A) → 0, A ∈ E, implies that supn∈N p(mn )(A) → 0. Further important properties of vector measures result from studying their ranges [86, Theorem IV.6.1]. A lcs is said to be complete if every Cauchy net converges. It is quasicomplete if every bounded Cauchy net converges. A Cauchy sequence {xn }∞ n=1 in E is necessarily bounded, so it converges in a quasicomplete space E. Theorem 1.5. Let E be a quasicomplete lcs. If m : E → E is a vector measure, then the range m(E) of m is relatively weakly compact. Of course, set functions are usually defined by an intuitive procedure on an elementary family of sets and then extended, by some method, to a more complicated family. The following extension theorem of Carath´eodoryHahn-Kluv´anek [38, I.V.2] does the job for vector measures. For σadditivity on an algebra A, attention is restricted to a countable family of pairwise disjoint sets from A whose union also belongs to A. Theorem 1.6. Let E be a quasicomplete lcs, let A be an algebra of subsets of Σ and let E be the σ-algebra generated by A. A σ-additive set function m : A → E is the restriction to A of a unique vector measure m ˜ defined on E if and only if the range m(A) of m is relatively weakly compact, and in this case, m(E) ˜ ⊆ m(A). It follows from this theorem that if μ : E → C is a scalar measure defined on the σ-algebra E of Σ and m : S → E is a vector measure with values in the quasicomplete lcs E, then there exists a unique vector measure μ ⊗ m : E ⊗ S → E defined on the σ-algebra E ⊗ S generated by E and S such that (μ ⊗ m)(E × S) = μ(E)m(S) for every E ∈ E and S ∈ S. 1.2

Integration of scalar functions with respect to a vector valued measure

Let E be a σ-algebra of subsets of a set Σ. Let m : E → E be a vector measure with values in a lcs E. A scalar function f : Σ → C is m-integrable in E if it is integrable with respect to the scalar measure m, y  for every y  ∈ E  , and for each A ∈ E, there exists a vector f m(A) ∈ E such that  

f m(A), y = f d m, y  , for all y  ∈ E  . A

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The set function f m : E → E is σ-additive in the original topology of E by the Orlicz-Pettis lemma. We shall often adopt the more conventional notations   f dm and f (γ) dm(γ) A

A

for f m(A), A ∈ E, and write m(f ) = f m(Σ) for the definite integral of f with respect to the vector measure m. It is often convenient and natural to regard integration as forming the product f.m of a function f with respect to a measure m. When we consider the bilinear integral of an X-valued function f with respect to a Y -valued measure m, using analogous notation, the integral f ⊗ m has values in the Y completed with respect to a suitable topology. tensor product X ⊗ The case of integration of a scalar function f : Σ → C with respect to a measure m : E → Ls (X) taking values in the space of operators acting on a Banach space X is of special significance. Then, the dual space of Ls (X) with the strong operator topology may be identified with X ⊗X  [123, Corollary IV.3.4] and for each x ∈ X and x ∈ X  , the function f is mx, x -integrable and for each A ∈ E, there exists f m(A) ∈ L(X)

such that f m(A)x, x = A f d mx, x , for all x ∈ X and x ∈ X  . The indefinite integral f m of an m-integrable f with respect to an Evalued measure m is a vector measure, so the p-semivariation of f m with respect to the continuous seminorm p defines the seminorm p(m) on the space L1 (m) of m-integrable functions, that is, p(m)(f ) = p(f m)(Ω) for every f ∈ L1 (m). According to [86, Lemma II.2.2], the equality p(m)(f ) = sup{| m, ξ |(|f |) : ξ ∈ Up◦ }, f ∈ L1 (m), is valid. Note that this implies that p(m)(Ω) = sup{p(m(f ))}, where the supremum is taken over all S-simple functions f with f ∞ ≤ 1. The following convergence theorem for vector measures together with the analogues of the Beppo Levi convergence theorem and the monotone convergence theorem are proved in [86, II.4]. Theorem 1.7. (Dominated convergence) Let E be a quasicomplete lcs and let m : S → E be a vector measure. If {fn }∞ n=1 is a sequence of m-integrable functions converging m-a.e. to a function f , and if there is an m-integrable function g with |fn | ≤ g m-a.e., for all n = 1, 2, . . . , then f is m-integrable, m(fn ) → m(f ) and p(m)(f − fn ) → 0 as n → ∞, for every continuous seminorm p on E. It follows that every bounded S-measurable function is m-integrable [86, II.3 Lemma 1] and the space L1 (m) of functions integrable with respect to

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a vector measure m : S → E is a vector lattice. An m-integrable function f is said to be m-null if its indefinite integral is (identically) the zero vector measure. Two m-integrable functions f, g are m-equivalent if the function |f − g| is m-null. We also say that f and g are equal m-almost everywhere (m-a.e.). The class of all m-integrable functions m-equivalent to an m-integrable function f is denoted by [f ]m . Denote by τ (m) the topology on L1 (m) determined by the family of seminorms f → p(m)(f ), f ∈ L1 (m), for every continuous seminorm p on E. The quotient space of L1 (m) modulo the subspace of all m-null functions is denoted by L1 (m). The resulting topology, denoted again by τ (m), turns L1 (m) into a lcs under the corresponding notions of pointwise addition and scalar multiplication almost everywhere. Now L1 (m) = {[f ]m : f ∈ L1 (m)}, so if we put p(m)([f ]m ) = p(m)(f ), for every f ∈ L1 (m) and every continuous seminorm p on E, the resulting system of seminorms defines the topology τ (m) on L1 (m). It is clear that the topology τ (m) is analogous to the usual L1 -norm topology of the classical Lebesgue space. In the case where E = C, the space L1 (m) is the standard Lebesgue space with the topology defined by its norm. As in the case of a scalar measure, for 1 ≤ r < ∞, we may define Lr (m) = {[f ]m : f |f |r−1 ∈ L1 (m) }, pr (m)(f ) = p(m)(|f |r ),

[f ]m ∈ Lr (m),

and give Lr (m) the topology defined by the family of seminorms [f ]m −→ pr (m)(f ),

[f ]m ∈ Lr (m)

for every continuous seminorm p on E. For r = ∞, [f ]m ∞ = inf{M : |f | ≤ M m-a.e.} and L∞ (m) is the Banach space of classes [f ]m of S-measurable functions f for which [f ]m ∞ < ∞ equipped with the norm  · ∞ . 1.3

Integration of vector valued functions with respect to a scalar measure

In infinite dimensional vector spaces such as linear spaces of functions or operators, there is an unavoidable distinction between ‘strong’ integrals and ‘weak’ integrals. It is fair to say that, mostly, estimates that facilitate the

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convergence of strong integrals are routine whereas, say, proving that certain classes of integrals converge weakly may be relevant to the foundations of mathematics itself. 1.3.1

The Pettis integral

A definition similar to integration with respect to vector measures may be adopted for the integration with respect to a scalar measure μ : E → C, of functions f : Σ → E with values in a locally convex space E. The function f is said to be scalarly μ-integrable if for all y  ∈ E  , the scalar function f, y  is μ-integrable. Then f is μ-integrable if it is scalarly integrable, and for each A ∈ E, there exists a vector f μ(A) ∈ E such that 

f μ(A), y  = f, y  dμ, for all y  ∈ E  . A

As

before, f μ : E → E is σ-additive, and we write μ(f ) = f μ(Σ) and f dμ = f μ(A), for every A ∈ E. The special case of the locally convex A space space E = L(X) of operators acting on a Banach space X has a direct translation, which goes as follows. A function f : Σ → L(X) is said to be scalarly μ-integrable if for all x ∈ X and x ∈ X  , the scalar function f x, x is μ-integrable. Then f is μ-integrable if it is scalarly integrable, and for each A ∈ E, there exists a vector f μ(A) ∈ L(X) such that  

f μ(A)x, x = f x, x dμ, for all x ∈ X, x ∈ X  . A

We shall sometimes need to distinguish integrability in the weak sense from stronger forms of integrability, which we shall describe shortly; a vector valued function integrable in the above sense is said to be Pettis μintegrable [38, II.3.2]. A survey of the Pettis integral and a list of references up to 2002 has been given by K. Musial [98] and M. Talagrand gives an earlier treatment [130] of deep measure theoretic aspects of the Pettis integral. The definition of the Pettis integral is natural because it is analogous to the integration of a scalar valued function with respect to a vector measure. However, it is not innocuous—for example, the statement “for every Banach space X, every bounded and scalarly Lebesgue measurable function f : [0, 1] → X is Pettis integrable with respect to Lebesgue measure” is independent of ZFC, the usual axioms of set theory with the Axiom of Choice [98, Theorem 8.2]. Even for a separable Hilbert space H, in which

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scalar integrablity and

∞ Pettis integrability are equivalent, the existence of Pettis integrals like 0 f (t) dt t for certain H-valued functions f is related to fundamental problems in harmonic

∞ analysis and operator theory [69]; for these examples, it is clear that 0 f (t)H dt t = ∞. 1.3.2

The Bochner integral

Now suppose that (X,  ·X ) is a Banach space and μ is a scalar measure on E. A function f : Σ → X is said to be strongly μ-measurable if it is the limit μ-almost everywhere of X-valued E-simple functions, that is, functions that are finite sums of functions xχE for any x ∈ X and E ∈ E. The integral of X-valued E-simple functions is defined in the obvious way. For strongly μ-measurable functions, f X : γ → f (γ)X , γ ∈ Σ, is μ-measurable. We say that a strongly μ-measurable function f is Bochner μ-integrable if the integral Σ f X d|μ| is finite. It turns out that for such functions, there exist X-valued E-simple functions fk , k = 1, 2,

. . . , with the property that fk → f μ-a.e. as k → ∞ and the integrals Σ fk − fj X d|μ| converge to zero as k, j → ∞ [38, II.2.2]. Hence, f is Pettis μ-integrable and

f dμ → f μ(A) in X as k → ∞, uniformly for all A ∈ E. A k Let 1 ≤ p < ∞. Then Lp (Σ, E, μ; X) denotes the vector space of μequivalence classes [f ]μ of μ-strongly measurable functions f : Σ → X such that f pX is μ-integrable. It is a Banach space under the norm  [f ]μ p =

Σ

f pX d|μ|

1/p .

In most circumstances, we write f instead of [f ]μ . The subscript is omitted in the case that μ is Lebesgue measure. For p = ∞, the space L∞ (Σ, E, μ; X) denotes the Banach space of (equivalence classes of) strongly μ-measurable functions f : Σ → X for which f X is μ-essentially bounded. The norm is the μ-essential bound f ∞ of f X , that is, f ∞ = inf{c : f X ≤ c μ-a.e.}. The same terminology and notation is adopted in the case that μ is a nonnegative measure. If Σ is a nonempty rectangle in Euclidean space Rn , E is the Borel σ-algebra BΣ of Σ and μ is the n-dimensional Lebesgue measure, we use the notation Lp (Σ) in place of Lp (Σ, E, μ), and Lp (Σ; X) in place of Lp (Σ, E, μ; X).

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Introduction

1.4

Tensor products

Our brief discussion of topological tensor products is taken from [123, III.6, IV.9]. A lively treatment of the historical background may be found in [38, VIII.5]. We start with the algebraic definition of a tensor product. Given two vector spaces E and F over the same set of scalars R or C, the vector space of all bilinear forms on E × F is denoted by B(E, F ). For each x ∈ E and y ∈ F , the evaluation map ux,y : f −→ f (x, y), f ∈ B(E, F ), is a linear form on B(E, F ) and so an element of the algebraic dual B(E, F )∗ of the space B(E, F ) of bilinear forms. The mapping χ : (x, y) −→ ux,y of E × F into B(E, F )∗ is itself bilinear. The vector space generated by its range χ(E × F ) is denoted by E ⊗ F . The map χ is called the canonical bilinear map of E × F into the tensor product E ⊗ F of E and F . Denoting the element ux,y of E ⊗ F by x ⊗ y for each x ∈ E and y ∈ F , each element  of E ⊗ F is a finite sum j λj (xj ⊗ yj ) with the sum over the empty set being 0. If G is another vector space over the same set of scalars, the mapping u −→ u ◦ χ is an isomorphism of the space L(E ⊗ F, G) of all linear maps from the vector space E ⊗F into G onto the space B(E, F ; G) of all bilinear maps from E × F into G [123, III.6.1]. Consequently, the algebraic dual (E ⊗ F )∗ of the tensor product E ⊗ F of E and F may be identified with the vector space of all bilinear forms on E × F . Now suppose that E and F are locally convex spaces. The tensor product equipped with a topology τ is written as E ⊗τ F . A locally convex topology τ on E ⊗ F is compatible with the tensor product if the following conditions hold: a) the canonical map χ : E × F → E ⊗τ F is separately continuous; b) for every ξ ∈ E  and η ∈ F  , the linear functional defined by ξ ⊗ η : x ⊗ y −→ x, ξ . y, η ,

x ∈ E, y ∈ F,

belongs to (E ⊗τ F ) ; c) if G1 ⊂ E  is equicontinuous on E and G2 ⊂ F  is equicontinuous on F , then G1 ⊗ G2 = {ξ ⊗ η : ξ ∈ G1 , η ∈ G2 } is an equicontinuous family of linear functionals on E ⊗τ F . In the context of normed vector spaces X and Y , a tensor product norm  · τ on X ⊗ Y has the property that there exists C > 0 such that (T1)

x ⊗ yτ ≤ Cx y for all x ∈ X and y ∈ Y , and

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(T2)

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X  ⊗Y  may be identified with a linear subspace of the continuous τ Y ) of X ⊗τ Y and x ⊗ y   ≤ Cx  y   for dual (X ⊗τ Y ) = (X ⊗   all x ∈ X and y ∈ Y  .

The topology defined by a tensor product norm is clearly compatible with the tensor product as defined above. The tensor product X ⊗ Y equipped with the topology defined by the norm  · τ is written here as X ⊗τ Y and τ Y . The term cross-norm is used if C = 1. its completion is X ⊗ Example 1.2. Let (Σ, E, μ) be a measure space, 1 ≤ p < ∞ and let X be a Banach space. For an element f ∈ Lp (Σ, E, μ) and a vector x ∈ X, it is natural to identify f ⊗ x with the μ-equivalence class of the X-valued function σ −→ f (σ)x defined for μ-almost all σ. Then f ⊗ xLp (Σ,E,μ;X) = f p x. Moreover, if 1 < q ≤ ∞ is the dual index satisfying 1/p + 1/q = 1 and g ∈ Lq (Σ, E, μ) and x ∈ X  , then g ⊗ x Lq (Σ,E,μ;X  ) = gq x . For every f ∈ Lp (Σ, E, μ; X), including elements of the tensor product Lp (Σ, E, μ) ⊗ X ⊂ Lp (Σ, E, μ; X), we can write

f, g ⊗ x =

and because  Σ

Σ

f, x .g(σ) dμ(σ)

| f, x |.|g(σ)| dμ(σ) ≤ f Lp(Σ,E,μ;X) gq x ,

the tensor product g ⊗ x defines a continuous linear function on Lp (Σ, E, μ) ⊗ X so that the restriction of the norm  · Lp (Σ,E,μ;X) to Lp (Σ, E, μ) ⊗ X is a tensor product norm. Each element of Lp (Σ, E, μ; X) can be approximated in norm by X-valued E-simple functions, so Lp (Σ, E, μ)⊗X is actually dense in the Banach space Lp (Σ, E, μ; X). Hence Lp (Σ,E,μ;X) X may be identified the complete tensor product Lp (Σ, E, μ)⊗ p with L (Σ, E, μ; X). It is convenient to abbreviate to the simpler notation p X so that we can write Lp (Σ, E, μ)⊗ p X. Lp (Σ, E, μ; X) = Lp (Σ, E, μ)⊗ For the case p = 2 and a Hilbert space H, the identification 2H L2 (Σ, E, μ; H) = L2 (Σ, E, μ)⊗ can be realised as a Hilbert space tensor product. The tensor product norm  · Lp (Σ,E,μ;X) clearly depends on the special nature of the Lp -space of pth-Bochner integrable functions. In the next section we consider two important examples of tensor product norms defined for any normed vector spaces X and Y .

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Introduction

1.4.1

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Injective and projective tensor products

The projective tensor product arises in many applications of bilinear integration in tensor products. Suppose that E and F are locally convex spaces. The finest locally convex topology of E ⊗ F for which the canonical bilinear map χ : E × F → E ⊗ F is continuous is called the projective tensor product topology on E ⊗ F . We use E ⊗π F to denote the tensor product E ⊗ F equipped with the projective topology. If U is a neighbourhood base of zero in E and V is a neighbourhood base of zero in F , then the family of balanced, convex hulls {bco(U ⊗ V ) : U ∈ U, V ∈ V} is a neighbourhood base of zero for the projective topology. Here the notation U ⊗ V means the set of all x ⊗ y with x ∈ U , y ∈ V . The dual space (E ⊗π F ) can be identified with the space B(E, F ) of all continuous bilinear forms on E × F , so that the equicontinuous subsets of (E ⊗π F ) are mapped to the equicontinuous subsets of bilinear forms on E × F [123, III.6.2]. Suppose that p is a continuous seminorm on E and q is a continuous seminorm on F such that U = {x ∈ E : p(x) < 1} and V = {y ∈ F : q(y) < 1} are the neighbourhoods of zero associated with the seminorms p and q respectively. Then gauge r of bco(U ⊗ V ) is given by    p(xj )q(yj ) : u = xj ⊗ yj r(u) = inf j

j

with the infimum taken over such finite representations of u ∈ E ⊗ F . Moreover, r(x ⊗ y) = p(x)q(y) for all x ∈ E and y ∈ F [123, III.6.3]. The seminorm r = p ⊗ q is called the tensor product of the seminorms p and q. For normed vector spaces X and Y , the tensor product of their respective norms is a norm defining the projective topology of X ⊗π Y . The completion of the projective tensor product E ⊗π F is written as π F . Occasionally we shall employ the following representation of eleE⊗ π F in the case that E and F are metrisable locally convex ments of E ⊗ spaces or normed vector spaces.

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Theorem 1.8 ([123, III.6.4]). Let E and F be metrisable locally convex π F . Then there exists an absolutely summable sespaces and let u ∈ E ⊗ ∞ of scalars and null sequences {xj }∞ quence {λj }∞ j=1 j=1 ⊂ E, {yj }j=1 ⊂ F such that u=

∞ 

λj xj ⊗ yj .

(1.1)

j=1

π F . If E and F are normed vector The sum converges absolutely in E ⊗ spaces, then ⎧ ⎫ ∞ ⎨ ⎬ uπ = inf |λj |xj E yj F ⎩ ⎭ j=1

where the infimum is over all representations (1.1). Example 1.3. Let (Σ, E, μ) be a measure space and let X be a Banach space. In Example 1.2 we saw how the identification pX Lp (Σ, E, μ; X) = Lp (Σ, E, μ)⊗ is valid for a tensor product norm in the case 1 ≤ p < ∞. For the case p = 1, π X [123, III.6.5]. If X = L1 (Ω, S, ν) we have L1 (Σ, E, μ; X) = L1 (Σ, E, μ)⊗ for another measure space (Ω, S, ν), then the identities π L1 (Ω, S, ν) = L1 (Σ, E, μ; L1 (Ω, S, ν)) L1 (Σ, E, μ)⊗ = L1 (Ω, S, ν; L1 (Σ, E, μ)) obtain and in case μ and ν are σ-finite measures π L1 (Ω, S, ν) = L1 (Σ × Ω, E ⊗ S, μ ⊗ ν). L1 (Σ, E, μ)⊗ The case of general measure spaces is worked out in [45, 253Yi]—some care needs to be exercised defining the ‘product measure’. In the case 1 < p < ∞, there is no tensor product norm  · τ defined by a method that works for all normed vector spaces E and F such that τ X for every Banach space X [38, p. 253]. Lp (Σ, E, μ; X) = Lp (Σ, E, μ)⊗ The tensor product norm given in Example 1.2 is specific to the space Lp (Σ, E, μ). π L2 (Ω, S, ν) has special sigThe projective tensor product L2 (Σ, E, μ)⊗ nificance in operator theory. We examine it more closely in Proposition 2.4 in Chapter 2 and Section 3.1 of Chapter 3.

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Let E and F be nonzero locally convex spaces over the scalars C. An  element u = j xj ⊗ yj of the tensor product E ⊗ F may be viewed as a separately continuous bilinear form fu : Eσ × Fσ → C given by 

xj , x yj , y  , x ∈ E  , y  ∈ Y  , (1.2) fu (x , y  ) = j

that is, x −→ fu (x , y  ), x ∈ E  , is continuous for the weak topology σ(E  , E) for each fixed y  ∈ F  and y  −→ fu (x , y  ), y  ∈ F  , is continuous for the weak topology σ(F  , F ) for each fixed x ∈ E  . The collection Be (Eσ , Fσ ) of all separately continuous bilinear forms on Eσ × Fσ is equipped with the topology e of uniform convergence on all sets S × T with S equicontinuous in E  and T equicontinuous in F  . The relative topology of Be (Eσ , Fσ ) on the subspace E ⊗F is called the injective  F , is sometimes written as tensor product topology . Its completion E ⊗ ˆ . If E and F are complete ˇ whereas E ⊗ π F is abbreviated to E ⊗F E ⊗F  F may be locally convex spaces, then Be (Eσ , Fσ ) is itself complete, so E ⊗   identified with the closure of E ⊗ F in the space Be (Eσ , Fσ ) of separately continuous bilinear forms [123, IV.9.1].  In the case that E and F are normed vector spaces and u = j xj ⊗yj ∈ E ⊗ F , the norm ⎧  ⎫  ⎨ ⎬  u = sup  xj , x yj , y   : x ∈ BE  , y  ∈ BF  ⎩ ⎭  j

defines the injective tensor product topology with respect to the closed unit balls BE  and BF  of the spaces E  , F  dual to E and F , respectively. As mentioned earlier, the dual space (E ⊗π F ) of the projective tensor product E ⊗π F of two nonzero locally convex spaces E and F can be identified with the space B(E, F ) of all continuous bilinear forms on E × F . The characterisation of (E ⊗ F ) leads to an important operator ideal that arises in the theory of double operator integrals we consider in Chapter 8. The dual (E ⊗ F ) may be identified with a subspace of B(E, F ) = (E ⊗π F ) which we now identify. Theorem 1.9 ([123, IV.9.2]). The continuous bilinear form v ∈ B(E, F ) represents an element of the dual space (E ⊗ F ) if and only if there exist closed, equicontinuous sets S ⊂ Eσ , T ⊂ Fσ and a Radon measure γ : B(S × T ) → [0, ∞) such that  (ju)(x , y  ) dγ(x , y  ), u ∈ E ⊗ F,

u, v = S×T

for the embedding j : E ⊗ F → B(Eσ , Fσ ) defined by formula (1.2).

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Bilinear forms v ∈ B(E, F ) representing an element of (E ⊗ F ) are called integral and a linear map u ∈ L(E, Fσ ) is called 1-integral if there exists an integral bilinear form v such that y, ux = v(x, y) for all x ∈ E, y ∈ F . The term integral operator is used in a different sense in Chapter 3, so the prefix ‘1’ serves to distinguish between the two concepts. A 1-integral map u ∈ L(E, Fσ ) has the representation 

x, x y  dγ(x , y  ) (1.3) ux = S×T

with γ : B(S × T ) → [0, ∞) a Radon measure, as described above. The integral is a Pettis integral because the strong measurability of the integrand may not be valid. Example 1.4. Let (Σ, E, μ) and (Ω, S, ν) be σ-finite measure spaces and E = L1 (Σ, E, μ), F = L1 (Ω, S, ν). Suppose that the continuous linear map u : L1 (Σ, E, μ) → L∞ (Ω, S, ν) is 1-integral. The space L∞ (Ω, S, ν) is the space dual to L1 (Ω, S, ν) with the duality  f (ω)g(ω) dν(ω), f ∈ L1 (Ω, S, ν), g ∈ L∞ (Ω, S, ν).

f, g = Ω

Then the linear map u has the representation (1.3) for some Radon measure γ on the compact product set S × T in Eσ × Fσ . Let η : B(S) → [0, ∞) be the Radon measure defined by the marginal measure A −→ γ(A, T ), A ∈ B(S). Each element x of E = L1 (Σ, E, μ) uniquely defines a continuous function x −→ x, x , x ∈ S. Let v1 : E → L∞ (S, B(S), η) be the corresponding embedding and j : L∞ (S, BS, η) → L1 (S, BS, η) the natural inclusion. Then u = v2 ◦ j ◦ v1 for the bounded linear map v2 : L1 (S, B(S), η) → L∞ (Ω, S, ν) defined by  f (x )y  dγ(x , y  ), f ∈ L1 (S, B(S), η). v2 f = S×T

The integral converges as a Pettis integral in Fσ because T is a compact subset of L∞ (Ω, S, ν) for the weak*-topology, or more simply, the RadonNikodym Theorem ensures that the measure  B −→ f (x ) χB , y  dγ(x , y  ), B ∈ B(T ), S×T

has a unique density v2 f . Consequently, u admits the factorisation L1 (μ) v1 ↓ L∞ (η)

u −→

−→ j

L∞ (ν) ↑ v2 L1 (η)

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which means that u is strictly 1-integral according to the definition given in [36, p. 95]. The injective norm of the tensor product E ⊗ F = L1 (Σ, E, μ) ⊗ L1 (Ω, S, ν) is the relative norm of the space Be (Eσ , Fσ ) of separately continuous bilinear forms on Eσ × Fσ . A bounded linear map u : L1 (μ) → L∞ (ν) defines a linear functional  

yj , uxj , w = xj ⊗ yj ∈ L1 (μ) ⊗ L1 (ν) w −→ j 1

j



1

on L (μ) ⊗ L (ν). The number j yj , uxj can be expressed as tr(uv) for the finite rank linear map v : L∞ (ν) → L1 (μ) given by  xj yj , f , f ∈ L∞ (ν). vf = j

The trace tr(uv) is well-defined because uv is a finite rank operator on L∞ (ν) and tr(uv) is the trace in any matrix representation of the action of the continuous linear map uv, see [36, p. 125] for example. Because 

wL1 (μ)⊗ L1 (ν) = wBe (Eσ ,Fσ ) = vL(L∞ (ν),L1 (μ))

for w = j xj ⊗ yj ∈ L1 (μ) ⊗ L1 (ν), it follows that the bounded linear map u : L1 (μ) → L∞ (ν) is 1-integral if and only if there exists M > 0 such that |tr(uv)| ≤ M vL(L∞ (ν),L1 (μ))

(1.4)

for every finite rank operator v : L∞ (ν) → L1 (μ) [36, Theorem 6.16 (a)]. The equivalence of the factorisation given above and the bound (1.4) is an ingredient of Theorem 8.9 in Chapter 8. 1.4.2

Grothendieck’s inequality

A discussion of Grothendieck’s inequality and its many applications is given in the survey [112]. In this section, attention is limited to some consequences needed to investigate double operator integrals in Chapter 8. π ∞ is the completion of the tensor The projective tensor product ∞ ⊗ ∞ ∞ product  ⊗  with respect to the norm ⎧ ⎫ n n ⎨ ⎬  xj ∞ yj ∞ : u = xj ⊗ yj , xj , yj ∈ ∞ . uπ = inf ⎩ ⎭ j=1

j=1

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Another distinguished norm on ∞ ⊗ ∞ is given by ⎧ ⎛ ⎞ 12 ⎛ ⎞ 12 ⎫ ⎪ ⎪ n n ⎨ ⎬   2⎠ 2⎠ ⎝ ⎝ sup | xj , ξ | . sup | yj , η | γ2 (u) = inf ⎪ ⎪ η∈ 1 ,η1 ≤1 ⎩ξ∈ 1 ,ξ1 ≤1 j=1 ⎭ j=1 n where the infimun runs over all possible representations u = j=1 xj ⊗ yj for xj , yj ∈ ∞ , j = 1, . . . , n and n = 1, 2, . . . . Then γ2 may also be viewed as the norm of factorisation through a Hilbert space: γ2 (u) = inf{sup xi . sup yj } i

j

where the infimum runs over all Hilbert spaces H and all xj , yj ∈ H for  which u ∈ ∞ ⊗ ∞ has the finite representation u = i,j (xi , xj )ei ⊗ ej with respect to the standard basis {ej }j of ∞ . Another way of viewing γ2 (u) is ⎧⎛  ⎫ ⎞ 12  ⎞ 12    ⎛  ⎪ ⎪ n n ⎨      ⎬     γ2 (u) = inf ⎝ |xj |2 ⎠  . ⎝ |yj |2 ⎠      ⎪ ⎪ ⎩ j=1   j=1  ⎭ ∞ ∞ n over representations u = j=1 xj ⊗ yj , xj , yj ∈ ∞ , because   ⎛ ⎞ 12  n  n     2⎠  ⎝ | xj , ξ | =  sup  αj xj  sup  2 ξ∈ 1 ,ξ1 ≤1  j |αj | ≤1  j=1 j=1 ∞ ⎛ ⎞ 12 n  = sup ⎝ |xj (k)|2 ⎠ k

j=1

⎛ ⎞ 12   n     ⎝  = |xj |2 ⎠     j=1 

.



Proposition 1.2. Let ϕ : N × N → C be a function that defines a Schur multiplier Mϕ : L(2 ) → L(2 ), that is, in matrix notation Mϕ ({aij }i,j∈N ) = {ϕ(i, j)aij }i,j∈N . The following conditions are equivalent. (i) Mϕ L(L( 2 )) ≤ 1. (ii) There exist a Hilbert space H and functions x : N → B1 (H), y : N → B1 (H) with values in the closed unit ball B1 (H) of H such that ϕ(n, m) = (x(n), x(m)), n, m ∈ N.

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(iii) For all finite subsets E, F of N, the bound          ϕ(i, j)e ⊗ e i j  i∈E,j∈F ∞

≤1 ⊗ γ2 ∞

holds. Proof. Suppose first that ϕ is zero off a finite set E × F . Then the bound (i) is equivalent to the condition that        ≤1  ϕ(i, j)a α(i)β(j) ij    i∈E,j∈F for all linear maps a : 2 (E) → 2 (F ) with norm a ≤ 1 and matrix {aij } with respect to the standard basis and all α ∈ B1 (2 (E)), β ∈ B1 (2 (F )), that is, ϕ belongs to the polar C1◦ of the set C1 of all matrices {α(i)aij β(j)}(i,j)∈E×F with a, α, β as described. According to [112, Remark 23.4], the set C1 is itself the polar C2◦ of the set C2 of all matrices   ≤ 1. Then (i) holds {ψij }(i,j)∈E×F with  i∈E,m∈F ψ(i, j)ei ⊗ ej 

∞ ⊗ γ2 ∞

if and only if ϕ belongs to C2◦◦ = C2 , which is exactly condition (iii). Conditions (ii) and (iii) are equivalent by the definition of the norm γ2 . The passage to all of N × N follows from a compactness argument. Remark 1.1. a) The argument above uses the factorisation of the norm γ2∗ dual to γ2 described in [112, Proposition 3.3] and [112, Remark 23.4]—this only relies on the Hahn-Banach Theorem. b) The representation (8.14) below is the measure space version of the implication (ii) =⇒ (i) above. The necessity of the condition (8.14) in the general measure space setting is proved using complete boundedness arguments in [129, Theorem 3.3], see also [78, 131]. One version of Grothendieck’s inequality from [112] is that the norm γ2 and the projective tensor product norm are equivalent on ∞ ⊗ ∞ with γ2 (u) ≤ uπ ≤ KG γ2 (u),

u ∈ ∞ ⊗ ∞ .

The constant KG is Grothendieck’s constant. The projective tensor product version of Proposition 1.2 follows, with the same notation.

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Proposition 1.3. Let E, F be finite subsets of N and let ϕ : N × N → C be a function vanishing off E × F . Then        1   ϕ(i, j)e ⊗ e ≤ MϕL(L( 2 )) i j  KG  ∞ ∞ i∈E,j∈F

⊗π

        = ϕ(i, j)ei ⊗ ej  .  i∈E,j∈F ∞ ∞

⊗ γ2

Passing to infinite sets, a bounded function ϕ : N × N → C with Mϕ L(L( 2 )) < ∞ necessarily has a representation ϕ(i, j) =

∞ 

a(i, k)β(j, k),

i, j ∈ N,

k=1

∞ with k=1 a(·, k)∞ β(·, k)∞ < ∞, as in Peller’s representation (8.12). The following formulation may be viewed as the dual version of Proposition 1.3, see [112, Equation (3.11)]. Proposition 1.4 ([112, Theorem 2.4]). Let n = {aij }ni,j=1 be scalars such that      n   aij αi βj  ≤ sup |αi | sup |βj |  i j i,j=1 

1, 2, . . . and let

for all scalars αi , βj , i, j = 1, . . . , n. Then there exists K > 0 independent of n, such that for any Hilbert space H and any xi ∈ H, yj ∈ H, i, j = 1, . . . , n, the bound     n     a (x , y ) ij i j  ≤ K sup xi H sup yj H  i j  i,j=1 holds. The smallest such constant K valid for all H and n = 1, 2, . . . is Grothendieck’s constant KG . 1.5

Semivariation

In the context of bilinear integration, Bartle [11] worked with a concept related to semivariation originally introduced in [50]; it is needed in the proof of the bounded convergence theorem for bilinear integrals.

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Introduction

Let m : S → Y be a vector measure defined on the σ-algebra S of subsets of a set Ω. The X-semivariation βX : S → [0, ∞] of m in X ⊗τ Y is defined by ⎧  ⎫  ⎬ k ⎨    βX (m)(A) = sup  xj ⊗ m(Aj ) (1.5)  ⎭. ⎩  j=1 τ

The supremum is taken over all pairwise disjoint sets A1 , . . . , Ak from S contained in A ∈ S and vectors x1 , . . . , xk from X, such that xj  ≤ 1 for all j = 1, . . . , k and k = 1, 2, . . . . A similar notion applies if the canonical bilinear map (x, y) → x ⊗ y from X × Y into X ⊗ Y is replaced by some continuous bilinear map (x, y) → xy into a locally convex space Z; see [25]. If X = C, then the C-semivariation of m in Y coincides with the usual notion of semivariation of a vector valued measure mentioned in Section 1.2. Two variations of this theme are relevant to the applications considered in this work. If E and F are Banach spaces and m : S → L(E, F ) is an additive set function, then ⎧  ⎫  ⎬ k ⎨    m(A )x (1.6) βE (m)(A) = sup  j j  ⎩  ⎭ j=1 F

is the E-semivariation of m in F , with the Aj as above and xj E ≤ 1 for j = 1, . . . , k. The relevant bilinear map from E × L(E, F ) into F is (x, T ) → T x. Similarly, the L(E, F )-semivariation βL(E,F ) (m) of an Evalued additive set function m in F is associated with the bilinear map (T, x) → T x, T ∈ L(E, F ), x ∈ E. For F = C, the Hahn-Banach Theorem ensures that E  -semivariation βE  (m) is equal to the variation V (m) of m with respect to the norm of E. Unlike the scalar semivariation for a vector measure, there is no guarantee that these bilinear semivariations have values other than 0 or ∞, as happens in some simple examples. Suppose that E is a Banach space and F = C. Then an operator valued measure ν : S → L(E, C) is a finitely additive set function with values in the dual Banach space E  . Denote by V (ν) the variation of the E  -valued set function ν. It then follows from the Hahn-Banach Theorem that V (ν) = βE (ν) on S. Example 1.5. Let 1 < p < ∞ and 1/p + 1/q = 1. Let S = B([0, 1]). The Lp ([0, 1])-valued measure m : A −→ χA on S defines a measure ν : S → L(Lq ([0, 1]), C). Since V (ν)(A) = ∞ for every set A with positive Lebesgue

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measure [38, Example I.1.16], it follows that the Lq ([0, 1])-semivariation βLq ([0,1]) (ν) of ν in C has only values 0 or ∞. If for all sets Ak ∈ S decreasing to the empty set, we have βX (m)(Ak ) → 0 as k → ∞, then we say that the X-semivariation βX (m) is continuous. A study of continuity for semivariation has been conducted by Dobrakov [39– 41]. If βX (m) is continuous, then βX (m)(Ω) < ∞. In fact, an equivalent formulation for the continuity of βX (m) is that the set of X ⊗ Y -valued measures φ⊗m as φ ranges over all X-valued S-simple functions with values in the closed unit ball of X, is bounded and uniformly countably additive for the norm  · τ . The result of Bartle-Dunford-Schwartz, Theorem 1.2, ensures that there exists a finite nonnegative measure λ on S such that λ ≤ βX (m) and limλ(E)→0+ βX (m)(E) = 0; see [41, Lemma 2]. In the paper [11], continuity of the semivariation is called, unhelpfully, the *property. Another of Dobrakov’s results [39, *-Theorem] implies that if X, Y and τ Y are Banach spaces for which X⊗τ Y contains no subspace isomorphic X⊗ to c0 , then the X-semivariation β(m) of m in X ⊗τ Y is continuous once it is finite. τ Y are normed vector spaces, then a Y -valued meaIf X,Y and X ⊗ sure m with finite variation V (m) : S → [0, ∞) necessarily has finite Xsemivariation in X ⊗τ Y , by virtue of the separate continuity (T1) of the map X × Y → X ⊗ Y . Moreover, the X-semivariation of m in X ⊗τ Y is continuous. 1.5.1

Semivariation in Lp -spaces

Let (Γ, E, μ) be a σ-finite measure space. In this section we examine conditions guaranteeing the finiteness of semivariation for Lp -space valued measures. Here, attention is limited to the case in which X is a Banach space, Y is the Banach space Lp (Γ, E, μ) and τ is the relative topology of Lp (Γ, E, μ; X) on Lp (Γ, E, μ) ⊗ X. In this case, if  · p is the norm of Lp (Γ, E, μ; X) and BX is the closed unit ball of X, then for each E ∈ S, the X-semivariation βX (m)(E) of m : S → Lp (Γ, E, μ) in Lp (Γ, E, μ; X) p,C is actually the usual is denoted by S → m p,X (S), S ∈ S. Thus, m semivariation m of m as an Lp (Γ, E, μ)-valued measure. As mentioned in [11], every vector measure with finite variation has finite X-semivariation. It turns out that for Lp -spaces, there is a simple sufficient condition which guarantees finite semivariation, which is not as restrictive as the condition of finite variation.

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27

We first introduce some terminology from vector lattices in the context of Lp -spaces. The order relation f ≥μ 0, f ∈ Lp (μ), also written as 0 ≤μ f , is taken to mean that f is the equivalence class of a function greater than or equal to zero μ-almost everywhere. The set of elements f ≥μ 0 of Lp (μ) is written Lp (μ)+ . We shall drop the subscript μ when it is clear from the context below. The spaces Lp (μ), 1 ≤ p ≤ ∞ are complex vector lattices [124, Definition II.11.1] with respect to these order relations. A subset A of Lp (μ) is said to be order bounded if there exists u ≥μ 0 such that u ≥μ |f | for all f ∈ A. Every order bounded subset A of Lp (μ) has a supremum, which is to say that Lp (μ; R) is said to be Dedekind complete [124, Proposition II.8.3, Exercise II.23]. Definition 1.1. Let 1 ≤ p ≤ ∞. A bounded linear operator T from Lp (μ) to Lp (μ) is said to be positive if T f ≥μ 0 for every f ∈ Lp (μ; R) such that f ≥μ 0. A bounded linear operator T : Lp (μ) → Lp (μ) is said to be a regular operator if it can be written as a linear combination of positive operators. The restriction TR of a bounded linear operator T : Lp (μ) → Lp (μ) to Lp (μ; R) may be written as TR = T1 + iT2 for bounded real linear operators T1 , T2 acting on Lp (μ; R) and T is regular if and only if both T1 and T2 map order bounded intervals in Lp (μ; R) into order bounded intervals, in which case the positive linear operators T1± , T2± on Lp (μ; R) are defined by setting, for each j = 1, 2, Tj+ f = sup{Tj u : 0 ≤μ u ≤μ f },

Tj− f = sup{−Tj u : 0 ≤μ u ≤μ f },

for every f ∈ Lp (μ; R) with f ≥μ 0. Then TR = T1+ − T1− + i(T2+ − T2− ). The modulus |T | : Lp (μ) → Lp (μ) of a regular operator T is defined by the formula |T |u = sup0≤μ |f |≤μ u |T f |, for all u ≥μ 0 [124, Proposition IV.1.6]. The following result is formulated in terms of Banach lattices [138, Section 83] such as Lp (μ) with μ a σ-finite measure. Lemma 1.1. Let X be a Dedekind complete Banach lattice, T an algebra of subsets of a set Γ and m : T → X an additive set function. Then m has order bounded range if and only if there exists a nonnegative additive set function ν : T → X such that |m(A)| ≤ ν(A) for each A ∈ T , in the order of X. If m is σ-additive and the norm of X is order continuous, then ν may be chosen σ-additive.

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Proof. We look in the direction in which m is assumed to have order bounded range and X is a Dedekind complete real Banach lattice. The implication in the other direction is clear. Let L = sim(T ) be the set of all real valued T -simple functions.

Then L is a Riesz space and the map Im : L → X defined by Im h = Γ h dm, h ∈ L, is linear. Moreover, Im is order bounded because m has order bounded range. By [138, Theorem + − and Im from L into M such 83.3], there exist positive linear operators Im + − that Im = Im − Im . Then the additive map ν : T → X defined by ν(A) = + − (χA ) + Im (χA ), for all A ∈ T has the required properties. In the case Im that m is σ-additive and the norm of X is order continuous, then the map Im is sequentially order continuous, so [138, Lemma 84.1] shows that ν is σ-additive. The complex case is straightforward. The measure ν constructed above is denoted by |m| as it is the smallest measure dominating m. Proposition 1.5. Suppose that 1 ≤ p < ∞ and m : S → Lp (Γ, E, μ) is an order bounded measure. Then for every Banach space X, the semivariation m p,X (Γ) of m on Γ is finite and continuous. Proof. Let ν : S → Lp (Γ, E, μ) be a pointwise positive vector measure for which |m(A)| ≤ ν(A) a.e. for all A ∈ S; such a vector measure exists by Lemma 1.1. Let k = 1, 2, . . . , and suppose that xj ∈ X satisfy xj  ≤ 1 for all j = 1, . . . , k. Let Ej , j = 1, . . . , k, be pairwise disjoint sets belonging to S. Then       k  k    p   p     m(E x ) (γ) dμ(γ) ≤ x  m(E ) (γ) dμ(γ) j j j j   Γ

Γ

j=1



  k Γ

j=1

p  p ν(Ej )(γ) dμ(γ) ≤ (ν(Γ)(γ)) dμ(γ) = ν(Γ)pp . Γ

j=1

A typical way to produce measures which are not order bounded follows; see also Example 1.8. Example 1.6. Let 1 ≤ p ≤ 2, 1/p + 1/q = 1 and let φ : [0, 1] → Lp [0, 1] be defined for all x, y ∈ [0, 1] by φ(x)(y) =

∞  k=1

kχ[1/(k+1),1/k) (x)eiky .

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1 Then φ is Pettis integrable in Lp [0, 1] because 0 | φ(x), g | dx < ∞ for every g ∈ Lq [0, 1] [38, II.3.7]. Let m : S → Lp (Γ, E, μ) be the indefinite Pettis integral of φ. If there existed a nonnegative measure ν : S → Lp (Γ, E, μ) such that |m(A)| ≤ ν(A) for all A ∈ S, then for each j = 1, 2, . . . , j 

j    |m [1/(k + 1), 1/k) | =

k=1

k=1

1 1 ≤ ν(Γ), k+1

which is impossible. Thus, by Lemma 1.1, m cannot be order bounded in Lp (Γ, E, μ). The situation for Hilbert spaces is accommodated by Grothendieck’s inequality. Proposition 1.6. Let H be a Hilbert space and m : S → L2 (Γ, E, μ) a measure. Let m : S → [0, ∞) be the semivariation of m in L2 (Γ, E, μ). Then the measure m has finite H-semivariation m 2,H in L2 (Γ, E, μ; H). Moreover, there exists a constant C > 0 independent of H and m and a finite measure 0 ≤ ν ≤ m such that limν(E)→0 m(S) = 0 and m 2,H (S) ≤ Cm(S), for all S ∈ S. Proof. Let n = 1, 2, . . . and suppose that x1 , . . . , xn belong to the closed unit ball of H. Suppose that E1 , . . . , En are pairwise disjoint sets from S. Then,    n   n    2    m(E x ) (ω) dμ(ω) = (xj , xk )H m(Ej ), m(Ek ) L2 (Γ,E,μ) . j j   Ω

H

j=1



k,j=1

 Let aj,k be the complex number m(Ej ), m(Ek ) L2 (Γ,E,μ) for each j, k = 1, . . . , n. By Grothendieck’s inequality Proposition 1.4 or [93, Theorem 2.b.5],  n    n          (1.7) aj,k (xj , xk )H  ≤ KG sup  aj,k sj tk ,  k,j=1

k,j=1

where the supremum on the right is over all complex numbers sj , tk , j, k = 1, . . . , n such that |sj | ≤ 1 and |tk | ≤ 1 for all j, k = 1, . . . m, and KG is n Grothendieck’s constant. But the sum k,j=1 aj,k sj tk is equal to   n n n     m(Ej ), m(Ek ) L2 (Γ,E,μ) sj tk = m(Ej )sj , m(Ek )tk k,j=1



j=1

k=1

 = m(f ), m(g) L2 (Γ,E,μ) ,

L2 (Γ,E,μ)

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n n for the scalar S-simple functions f = j=1 sj χ Ej and g = k=1 tk χEk . By the Cauchy-Schwarz inequality,  m(f ), m(g)  ≤ m(f )2 m(g)2 . Since supu∞ ≤1 m(u)2 ≤ M = 2 sup{x2 : x ∈ bco(m(S))} [86, Lemma IV.6.1], the right-hand side of equation (1.7) is bounded by KG M 2 . Here we have appealed to the fact that the vector measure   2 m is bounded on S, so M < ∞. It follows that Ω  nj=1 xj m(Ej ) (ω)H dμ(ω) is bounded √ by KG M 2 , that is, the H-semivariation of m is bounded by KG M . Any measure equivalent to m is also equivalent to H-semivariation m 2,H in 2 L (Γ, E, μ; H). Remark 1.2. Let H be a Hilbert space. If E is any L(H)-valued spectral measure and h ∈ H, the identity    ∞ ∞   2 2 E(fn )hH = E |fn | h, h n=1

n=1

ensures that the H-valued measure Eh has bounded 2 -semivariation in 2 = ⊕∞ 2 (H)—the Hilbert space tensor product H⊗ j=1 H with norm ∞ 2 2 u 2 (H) = j=1 uj H . The remarkable consequence of Grothendieck’s inequality in Proposition 1.6 is that the same holds true when the measure Eh is repaced by any H-valued measure. In the case of Hilbert spaces, the following related result gives a convenient condition for which the operator semivariation of an operator valued measure is finite. In this case, the bilinear map (u, v) → u ⊗ v, u ∈ L(H), v ∈ L(L2 (Γ, E, μ)), has its values in L(L2 (Γ, E, μ; H)). Proposition 1.7. There exists a positive number C such that for every Hilbert space H, every σ-finite measure space (Γ, E, μ) and every operator valued measure M : S → L(L2 (Γ, E, μ)), the following bound holds for the L(H)-semivariation βL(H) (M ) of M in L(L2 (Γ, E, μ; H)): βL(H) (M )(B) ≤ C sup{V (M  h)(B) : h ∈ H, hH ≤ 1},

for all B ∈ S.

Of course, the supremum on the right-hand side of the inequality may be infinite. Proof. The number βL(H) (M )(B) is the supremum of all numbers   k (Ak ⊗ [M (Bk )])φ2 for Ak ∈ L(H) with Ak  ≤ 1, {Bk } pairwise disjoint subsets of B ∈ S and φ ∈ L2 (Γ, E, μ) ⊗ H with φ2 ≤ 1. Suppose  that φ = j hj χEj for finitely many hj ∈ H and {Ej } pairwise disjoint

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elements of E satisfying φ22 =

 j

hj 2H μ(Ej ) ≤ 1. Then,

  2  (Ak ⊗ [M (Bk )])φ2 =

Ak hj , Ak hj  M (Bk )χEj , M (Bk )χEj k,k ,j,j 

k

  

hj H M (Bk )χEj , hj H M (Bk )χEj skj tk j  , ≤ KG sup  k,k ,j,j 

by Grothendieck’s inequality [93, Theorem 2.b.5]. The supremum is over all scalars |skj | ≤ 1 and |tk j  | ≤ 1.   Let uk = j skj hj χEj and vk = j  tk j  hj  χEj be elements of L2 (Γ, E, μ). Then uk 2 ≤ 1 and vk 2 ≤ 1 and so we have   2   (Ak ⊗ [M (Bk )])φ2 ≤ KG sup 

M (Bk )uk , M (Bk )vk  k

{uk },{vk } k,k

 

M (Bk )uk , M (Bk )uk  ≤ 4KG sup  {uk } k,k

 2 M (Bk )uk 2 , = 4KG sup  {uk }

k



by the polarisation identity. The norm  k M (Bk )uk 2 is given by     !  M (Bk )uk 2 = sup  M (Bk )uk , h  k

h2 ≤1

k

  = sup 

uk , M (Bk ) h  ≤ sup V (M  h)(B). h2 ≤1

k

h2 ≤1

Let Y be a Banach space and 1 ≤ p < ∞. A vector measure m : S → Y is said to have finite p-variation if there exists C > 0 such that for every n = 1, 2, . . . and every finite family of pairwise disjoint sets Ej , j = 1, . . . , n, n the inequality j=1 m(Ej )p ≤ C holds. Proposition 1.8. Let 1 ≤ p < ∞ and let m : S → Lp (Γ, E, μ) be a measure. Let U be a σ-algebra of subsets of a set Λ and ν : U → [0, ∞) a p,Lp (ν) finite measure. If the measure m has finite Lp (ν)-semivariation m p in L (μ ⊗ ν), then m has finite p-variation. Proof. We may suppose that U contains infinitely many disjoint non-νnull sets, otherwise m necessarily has finite p-variation and finite Lp (ν)semivariation.

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Let E1 , . . . , Em be pairwise disjoint sets with positive ν-measure. For each j = 1, . . . , n, set fj = χEj /ν(Ej )1/p . Let F1 , . . . , Fm be pairwise disjoint sets belonging to S. Then fj p = 1 and p    1/p  n    m p,Lp (ν) (Γ) ≥ f (λ)m(F )(γ) dν(λ) dμ(γ) j j   Γ

=

Λ

  n Γ j=1

=

  n Γ j=1

j=1

|m(Fj )(γ)|p

 Λ

 1/p |fj (λ)|p dν(λ) dμ(γ)

⎛ ⎞1/p 1/p n  |m(Fj )(γ)|p dμ(γ) =⎝ m(Fj )pp ⎠ . j=1

For the case p = 2, we know from Proposition 1.6 that m has finite 2-variation, as follows from the fact that the inclusion of 1 in 2 is an absolutely summing operator [38, p. 255]. The following observation is used in Example 1.8 below. Example 1.7. Let X be an infinite dimensional Banach space. If {λj }∞ j=1 ∞ is a sequence of positive numbers such that j=1 λ2j < ∞, then there exists an unconditionally summable sequence {xj }∞ j=1 in X such that xj  = λj [93, Theorem 1.c.2 p. 16]. ∞ 2 Now let 1 ≤ p < 2. We can choose {λj }∞ j=1 such that j=1 λj < ∞ and ∞ p λ = ∞. It follows that there exists an unconditionally summable j=1 j ∞ p p in X such that x  = ∞. For X = L (Ω, S, μ), sequence {xj }∞ j j=1 j=1  the vector measure m(E) = j∈E xj therefore has infinite p-variation, and so it has infinite Lp (ν)-semivariation in Lp (ν ⊗ μ), by Proposition 1.8. For the case p > 2, an Lp (μ)-valued vector measure with infinite Lp (ν)semivariation in Lp (ν ⊗ μ) is constructed in [72]. 1.5.2

Semivariation of positive operator valued measures

In this section, we see that operator valued measures taking their values in the Banach lattice of positive operators on an Lp -space have bounded L(X)-semivariation for any Banach space X and the same holds true for operator valued measures dominated by a positive measure. The bounded (S, Q)-processes on Lp -spaces considered in Section 6.5 have this property, so Theorems 1.10 and 6.6 below see use in Chapter 6 on random evolutions.

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Introduction

Let (Γ, E, μ) be a σ-finite measure space and 1 ≤ p ≤ ∞. The strongly closed subspace of L(Lp (Γ, E, μ)) consisting of all positive operators T is written as L+ (Lp (Γ, E, μ)), that is, T f ≥ 0 for all f ∈ Lp (Γ, E, μ) with f ≥ 0 μ-a.e.. Where it is convenient, the space Lp (Γ, E, μ) is abbreviated to Lp (μ) and Lp+ (μ) denotes the nonnegative elements of Lp (μ). Let (Ω, S) be a measurable space. An operator valued measure M : S → L+ (Lp (μ)) is called a positive operator valued measure and we write M ≥ 0 in this case. For p = ∞, we suppose that M f is weak* σ-additive in L∞ (μ) for each f ∈ L∞ (μ). An operator valued measure N : S → L(Lp (μ)) is dominated by a positive operator valued measure M if for every S ∈ S, the inequality |N (S)| ≤ M (S) holds in the Banach lattice of regular operators on Lp (μ), that is, |N (S)f | ≤ M (S)f,

μ-a.e. for every f ∈ Lp+ (μ).

Theorem 1.10. Let X be a Banach space and suppose that the operator valued measure N : S → L(Lp (μ)) is dominated by an operator valued measure M ≥ 0. Then N has finite L(X)-semivariation βL(X) (N ) and βL(X) (N )(S) ≤ M (S),

S ∈ S.

If 1 ≤ p < ∞ and f ∈ Lp (μ), then the vector measure N f has continuous X-semivariation in Lp (Γ, E, μ; X). Proof. We first establish the result for the special case N = M ≥ 0. For l ∈ N and 1 ≤ i ≤ l let Ai ∈ L(X) with Ai L(X) ≤ 1. For the same finite set of i’s let {Bi }li=1 be pairwise disjoint subsets of B ∈ S. For k ∈ N k let g = j=1 xj χGj be an X-valued E-simple function with xj ∈ X and Gj ∈ E pairwise disjoint for j = 1, . . . , k. Also assume that gLp(μ;X) ≤ 1. Then taking pointwise estimates, we have p   p            =  (Ai ⊗ [M (Bi )])g   (Ai xj )[M (Bi )χGj ](γ) dμ(γ)   p Γ   i i,j L(L (μ;X)) X ⎞p ⎛   ⎝ ≤ Ai xj X [M (Bi )χGj ](γ)⎠ dμ(γ) Γ

i,j

Γ

i,j

⎞p ⎛   ⎝ ≤ xj X [M (Bi )χGj ](γ)⎠ dμ(γ).

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The right-hand side is equal to  ⎛ ⎞p       ⎝ ⎠ M (Bi ) xj X χGj     p  i j L (μ)  p  p            ≤ M (Bi ) x  χ j X Gj     p   p i j L(L (μ)) L (μ)  p     = M (Bi ) gpLp(μ;X)   p L(L (μ))

i

≤ M (B)pL(Lp (μ)) . The last equality follows from the observation that if A, B ∈ S with A ⊆ B then M (A)L(Lp (μ)) ≤ M (B)L(Lp (μ)) . This is easily seen by noting that M (B)f − M (A)f = M (B\A)f ≥ 0 holds for all f ≥ 0. Since X-valued E-simple functions are dense in Lp (μ; X), by Definition 6.2 this implies βL(X) (M )(B) ≤ M (B)L(Lp (μ;X)) . Next we show that for an arbitrary y ∈ Lp (μ), the Lp (μ)-valued measure M y has continuous X-semivariation. This follows immediately from Proposition 1.5 if we can show that M y is dominated by a positive vector measure. However, decomposing y first into its real and imaginary parts and then into its positive and negative parts gives us |M y| ≤ 2M |y|. This is easily seen since, for all A ∈ S, |M (A)y| = |M (A)y1 + iM (A)y2 |   = M (A)y + − M (A)y − + iM (A)y + − iM (A)y −  ≤

1 M (A)(y1+

+

y1− )

1

+

M (A)(y2+

+

2 y2− )

2

= M (A)(|y1 | + |y2 |) ≤ 2M (A)|y|. Since M is a positive operator valued measure it follows that M |y| ≥ 0. This completes the proof for the positive case. To prove that the dominated measure N has finite L(X)-semivariation it suffices to note that |N (A)χG | ≤ M (A)χG for all A ∈ S and G ∈ E and repeat the argument for the positive case. To prove N y has continuous X-semivariation for each y ∈ Lp (μ) it suffices to note that |N y| ≤ 2M |y|, that is, the vector measure N y is dominated by a positive measure. The result then follows from Proposition 1.5.

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Introduction

1.6

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35

Bilinear integration after Bartle and Dobrakov

Suppose that S is a σ-algebra of subsets of a set Σ and m : S → E is a vector measure with values in a lcs E. If fn : Σ → C, n = 1, 2, . . . , are mintegrable functions converging pointwise to a function f : Σ → C m-almost

everywhere and limn→∞ A fn dm converges in E for each set A ∈ S, then the Vitali-Hahn-Saks Theorem ensures that f is m-integrable and   lim fn dm = f dm, A ∈ S. (1.8) n→∞

A

A

Moreover, the indefinite integrals fn m, n = 1, 2, . . . , converge to f m in semivariation, that is, p(fn m − f m)(Σ) = p(m)(f − fn ) → 0 as n → ∞ for every continuous seminorm p on E, see [86, II.5 Theorem 2] and [43, Theorem IV.10.9]. For a set A ∈ S, the p-semivariation p(m)(A) of m on A is given by p(m)(A) = sup{| m, ξ |(A) : ξ ∈ Up◦ } = sup{m(ϕχA ) : ϕ S-simple, ϕ∞ ≤ 1}    cj m(Aj ) , = sup  j

where the last supremum is over all pairwise disjoint subsets A1 , . . . , Ak of A and all scalars |cj | ≤ 1 for j = 1, . . . , k and k = 1, 2, . . . . In the case of bilinear integration, these convergence results suggest that semivariation may also be used to control the convergence of integrals of a vector valued function with respect to a vector valued measure. Let X and Y be Banach spaces and suppose that  · τ is a norm tensor product on X ⊗ Y . The X-semivariation βX (m) in X ⊗τ Y of a Y -valued measure m : S → Y is defined by equation (1.5). Following R. Bartle [11] and I. Dobrakov [39], we shall employ the extended real valued set function βX (m) : S → [0, ∞] to control the integration of X-valued functions with τY . respect to m in the completed tensor product space X ⊗ Similar remarks apply to other cases of bilinear integration. If E, F are Banach spaces and m : S → L(E, F ) is σ-additive for the strong operator topology, then the E-semivariation βE (m) : S → [0, ∞] of m in F is defined τY , E = X by formula (1.6). In the preceding case, we may take F = X ⊗ ˜ = βX (m). and m(A)x ˜ = x ⊗ m(A) for x ∈ X and A ∈ S, so that βX (m) In order to integrate L(E, F )-valued functions with respect to an Evalued measure and the bilinear map (T, x) → T x, T ∈ L(E, F ), x ∈ E,

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then the L(E, F )-semivariation βL(E,F ) (m) : S → [0, ∞] of m in F can be employed. The salutary Example 1.5 shows that these set functions may not be helpful for controlling the integration process. On the other hand, suppose that the X-semivariation βX (m) in X ⊗τ Y of a Y -valued measure m : S → Y is not only finite on the σ-algebra S, but also continuous as defined in Section 1.5. For any X-valued S-simple function ϕ : Σ → X given by ϕ=

n 

xj χAj

j=1

for pairwise disjoint sets Aj ∈ S and xj ∈ X, j = 1, . . . , n and n = 1, 2, . . . , the integral ϕ ⊗ m is defined by linearity in the usual way as (ϕ ⊗ m)(A) =

n 

xj ⊗ m(Aj ∩ A),

A ∈ S,

j=1

independently of the representation of ϕ. Then ϕ ⊗ m : S → X ⊗ Y is a finitely additive set

function. It is sometimes convenient to write (ϕ⊗m)(A)

as A ϕ ⊗ dm or A ϕ(σ) ⊗ m(dσ) for A ∈ S. Under the assumption that βX (m) is continuous, Theorem 9 of [11] may be adopted as a definition of integrability. Definition 1.2 (Integration in the sense of Bartle). Let S be a σalgebra of subsets of a nonempty set Σ. Let X and Y be Banach spaces and suppose that  · τ is a norm tensor product on X ⊗ Y . Suppose that the X-semivariation βX (m) in X ⊗τ Y of the Y -valued measure m : S → Y is continuous. τ Y if A function f : Σ → X is said to be (Bartle) m-integrable in X ⊗ there exist X-valued simple S-functions ϕn , n = 1, 2, . . . , converging to f τ Y as n → ∞ in X ⊗ m-almost everywhere such that (ϕn ⊗ m)(A) converges

for each A ∈ S. The notation (f ⊗ m)(A) and A f ⊗ dm is used to denote the limit for A ∈ S. According to [11, Theorem 10], there is an analogue of the convergence result (1.8) for Bartle bilinear integrals. Theorem 1.11. Let X, Y and m : S → Y be as in Definition 1.2. If fn : Σ → X, n = 1, 2, . . . , are functions that are (Bartle) m-integrable τ Y converging m-a.e. to f : Σ → X and for every > 0 there exists in X ⊗  

δ > 0 such that  A fn ⊗ dmτ < for all A ∈ S such that βX (m)(A) < δ,

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Introduction

then f is (Bartle) m-integrable and   fn ⊗ dm = f ⊗ dm lim n→∞

A

A

τ Y uniformly for A ∈ S, so that limn→∞ f ⊗ m − fn ⊗ mτ (Σ) = 0. in X ⊗ The Vitali Convergence Theorem is proved in [11, Theorem 10] without the assumption that βX (m) is continuous or that m is countably additive. There almost everywhere convergence is replaced by the analogue of ‘convergence in measure’ with respect to the subadditive set function βX (m). In the case of Banach spaces E and F and an operator valued measure m : S → L(E, F ) that is σ-additive for the strong operator topology, the E-semivariation βE (m) of m defined by formula (1.6) has the property that mL(E,F ) ≤ βE (m) for the scalar semivariation mL(E,F ) with respect to the uniform operator norm of L(E, F ), for we can choose xj = cj x with xE ≤ 1 and |cj | ≤ 1 in (1.6). Hence, if the E-semivariation βE (m) of m is continuous, then m is necessarily σ-additive for the uniform operator topology—a condition that is rarely satisfied in applications. Unless the E-semivariation βE (m) of the operator valued measure m is controlled by a measure and so continuous, the Vitali Convergence Theorem of [11, Theorem 10] is difficult to apply. In a series of papers [39–41], I. Dobrakov circumvented this obstruction by the elementary device of approximating by simple functions based on sets where βE (m) is finite. Further expositions of Dobrakov’s approach have appeared in [42, 103]. The papers [39–41] are formulated in terms of the δ-rings S1 of sets A ∈ S for which βE (m)(A) < ∞ and the σ-ring it generates. For the basic Example 1.5, S1 consists of subsets of [0, 1] with Lebesgue measure zero. In order to avoid examples with a trivial set of integrable functions, we shall suppose that the operator valued measure m has σ-finite E-semivariation βE (m) in F , that is, there exist pairwise disjoint sets Σj ∈ S, j = 1, 2, . . . , such that Σ = ∪∞ j=1 Σj and βE (m)(Σj ) < ∞ for each j = 1, 2, . . . . The same terminology is used when the vector measure m : S → Y is said to have σ-finite X-semivariation βX (m) in X ⊗τ Y . Then the equality σ(S1 ) = S holds. Definition 1.3 (Integration in the sense of Dobrakov). Let S be a σ-algebra of subsets of a nonempty set Σ. Let X and Y be Banach spaces and suppose that  · τ is a norm tensor product on X ⊗ Y . Suppose that the X-semivariation βX (m) in X ⊗τ Y of the Y -valued measure m : S → Y is σ-finite. Let S1 = {A ∈ S : βX (m)(A) < ∞}.

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τ Y if A function f : Σ → X is said to be (Dobrakov) m-integrable in X ⊗ there exist X-valued S1 -simple functions ϕn , n = 1, 2, . . . , converging to f τ Y as n → ∞ in X ⊗ m-almost everywhere such that (ϕn ⊗ m)(A) converges

for each A ∈ S. The notation (f ⊗ m)(A) and A f ⊗ dm is used to denote the limit for A ∈ S. That the definite integral f ⊗ m is independent of the approximating sequence of simple functions is proved in [39, Theorem 2], with a clarification in [42, Theorem 7]. According to [39, Theorem 2], there is an analogue of the convergence result (1.8) for Dobrakov bilinear integrals. Theorem 1.12. Let X, Y and m : S → Y be as in Definition 1.3. If fn : Σ → X, n = 1, 2, . . . , are functions that are (Dobrakov) m τ Y converging m-a.e. to f : Σ → X and {fn ⊗ m : n ∈ N} integrable in X ⊗ τ Y -valued measures, then is a uniformly countably additive collection of X ⊗ f is (Dobrakov) m-integrable and   lim fn ⊗ dm = f ⊗ dm n→∞

A

A

τ Y uniformly for A ∈ S, so that limn→∞ f ⊗ m − fn ⊗ mτ (Σ) = 0. in X ⊗ As mentioned in Section 1.5, if βX (m) is continuous, then S1 = S, so Theorem 1.12 is a genuine improvement of Theorem 1.11 and the two notions of bilinear integration coincide [39, Example 4, p. 535]. In [39, Example 7 ], an example is given of a bounded and Dobrakov integrable function f : Σ → E and a uniformly countably additive operator valued measure m : S → L(E, F ) such that βE (m) is not continuous and f is not even βE (m)-measurable in the sense of [11]. For the remainder of this section, we use the term m-integrable instead of (Dobrakov) m-integrable. The connection with the Bochner integral is noted in [39, Theorem 6] from the obervation that the inequality βX (m) ≤ V (m) holds between Xsemivariation βX (m) in X ⊗τ Y of the Y -valued measure m : S → Y and the variation V (m) : S → [0, ∞] of m. τ Y be Banach spaces, with τ a tensor Proposition 1.9. Let X,Y and X ⊗ product topology norm on X ⊗ Y . Suppose that m : S → Y is a measure for which there exist sets Σk ∈ S, k = 1, 2, . . . , increasing to Σ such that the total variation V (m)(Σk ) of m on Σk is finite for each k = 1, 2, . . . , that is, m has σ-finite variation in Y .

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39

Introduction

If f : Σ → X is Bochner integrable with respect to the σ-finite measure τ Y and V (m), then f is m-integrable in X ⊗  V (f ⊗ m) ≤ f  dV (m). Σ

The nature of bilinear integration is unlike the cases in which either the measure or function is scalar; for example, bounded vector valued functions need not be integrable with respect to a vector valued measure [39, Example 7 ]. Example 1.8. A measure m : S → Y can have σ-finite variation without having continuous X-semivariation, or even finite X-semivariation. Let 1 ≤ p < 2, and X = Y = Lp ([0, 1]). We give X ⊗ Y the relative τ Y = Lp ([0, 1]2 ). topology τ of Lp ([0, 1]2 ), so that X ⊗ ∞ Suppose that {yj }j=1 is an unconditionally summable sequence in ∞ Lp ([0, 1]) such that j=1 yj p = ∞ [93, Theorem 1.c.2 p. 16]. Let Ω = N and let S be the family of all subsets of Ω. Define the Y -valued measure  m : S → Y by m(A) = k∈A yk for every A ∈ S. Let Ej , j = 1, 2, . . . , be pairwise disjoint subsets of [0, 1] with positive Lebesgue measure |Ej |. Set fj = χEj /|Ej |1/p for each j = 1, 2, . . . . Then fj p = 1, j = 1, 2, . . . , and as k → ∞, we have  1  k  1 k    k p  fj yj (t)Lp ([0,1]) dt = |yj (t)|p dt = yj p → ∞. 0

j=1

j=1

0

j=1

The vector measure m has infinite X-semivariation but σ-finite variation. It is easy to check from Definition 1.3 that a function G : Ω → Lp ([0, 1]) is m-integrable in Lp ([0, 1]2 ) if and only if {G(j)⊗ yj }∞ j=1 is unconditionally summable in Lp ([0, 1]2 ), because Gχ{1≤j≤n} are S1 -simple functions converging pointwise to G in Lp ([0, 1]) as n → ∞. This is obviously guaranteed ∞ by the condition j=1 G(j)p fj p < ∞ of Proposition 1.9. However, the bounded Lp ([0, 1])-valued function j → fj is not m-integrable. is an unconditionExample 1.9. Let 1 ≤ p < 2 and suppose that {yj }∞  j=1 ally summable sequence in Lp ([0, 1]) such that j yj p = ∞. Then, as

1 above, there exist positive functions fj on [0, 1] such that 0 fj (t)p dt = 1  and j yj fj does not converge in Lp ([0, 1]2 ). Now let {aj } be an absolutely summable sequence of positive scalars  and set gj = aj fj . The Lp ([0, 1])-valued measure m : A → j∈A gj has  finite variation V (m) : A → j∈A aj . Set xj = yj /aj . Then {aj xj }∞ j=1 is

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 unconditionally summable in Lp ([0, 1]), but j xj ⊗ gj does not converge in Lp ([0, 1]2 ). In other words, the function j → xj is Pettis integrable with respect to V (m) in Lp ([0, 1]), but it is not m-integrable in Lp ([0, 1]2 ). Of course, a sequence which is Bochner integrable with respect to V (m) is necessarily m-integrable. The following example is a version of [39, Example 7 ] adapted to tensor products. Example 1.10. Let 1 ≤ p < 2 and 0 < a < 1/p−1/2. According to Orlicz’s Lemma [93, Theorem 1.c.2 p. 16], there exists an unconditionally summable p 1/2 ln(j + 1)). For each sequence {zj }∞ j=1 in L ([0, 1]) such that zj p = 1/(j −a j = 1, 2, . . . , set yj = j zj . Then ∞ 

yj pp =

j=1

Neverthless, {j Let m(A) =

a

∞  j=1

1 = ∞. j ap j p/2 ln(j + 1)p

yj } ∞  j=1

p = {zj }∞ j=1 is unconditionally summable in L ([0, 1]). j∈A yj for every subset A of N.

Claim 1.1. The function f : j → j a 1 with values in Lp ([0, 1]), is mintegrable in Lp ([0, 1]2 ), but it is not (Bartle) m-integrable in Lp ([0, 1]2 ). Proof. We already know that f is m-integrable, because {f (j)yj }∞ j=1 is simple unconditionally summable in Lp ([0, 1]). Let φ be an Lp [0, 1]-valued   function.  Then  a for each  j ∈ N, we have f (j) − φ(j)p ≥ f (j)p − φ(j)p  = j −φ(j)p . Let J be an integer greater than (maxj φ(j)p + 1)1/a . Then for all j ≥ J, the inequality f (j) − φ(j)p ≥ 1 holds. Let A = {j : f (j) − φ(j)p ≥ 1}. The argument used in Example 1.8 now shows that the Lp ([0, 1])-semivariation of m in Lp ([0, 1]2 ) of the set A  is greater than j≥J yj pp = ∞. Consequently, f cannot be approximated in Lp ([0, 1])-semivariation of m by simple functions, so it cannot be (Bartle) m-integrable in Lp ([0, 1]2 ). Both the Bartle and Dobrakov approaches to bilinear integration utilise semivariation to control the convergence of the integrals—we call these regular bilinear integrals. In the next chapter, we see how the use of semivariation can be avoided for bilinear integrals with values in tensor products of vector spaces as is required for bilinear integration with respect to spectral measures employed in subsequent chapters.

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Chapter 2

Decoupled bilinear integration

τY The approach considered in Section 1.6 to bilinear integration in X ⊗ of an X-valued function f : Σ → X with respect to a Y -valued measure m : S → Y is adequate for Banach spaces X and Y once we know that m τY . has σ-finite X-semivariation in X ⊗ Example 1.5 gives a particularly simple X  -valued measure m for which the X-semivariation βX (m) = V (m) of m in C has only the values 0 or ∞. It is worthwhile looking at this example in greater detail because it is representative of many vector measures arising in applications. For p = 2 in Example 1.5, the L2 ([0, 1])-valued measure m is equal to Q1 for the spectral measure Q : B(R) → L(L2 ([0, 1])) such that Q(B) is the selfadjoint projection operator h −→ χB .h, h ∈ L2 ([0, 1]), for each B ∈ B(R) and 1 is the constant function equal to one on the interval [0, 1]. The spectral measure Q is associated with the position operator of a quantum particle on the line. According to the Spectral Theorem for selfadjoint operators, any spectral measure P : B(R) → L(H) in a separable Hilbert space H is unitarily equivalent to a spectral measure similar to Q except for a discrete part, so like the vector measure m of Example 1.5, the variation V (P h) will have values that are either zero or infinity for h ∈ H, except in trivial cases. On the other hand, for any vector h ∈ H and pairwise disjoint Borel subsets

41

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B1 , B2 , . . . of R, with respect to the inner product (·, ·) of H we have 2 ∞ ∞ P (∪∞ n=1 Bn )hH = (P (∪n=1 Bn )h, P (∪n=1 Bn )h) ∞  (P (Bn )h, P (Bm )h) =

=

=

n,m=1 ∞ 

(P (Bn ∩ Bm )h, h)

n,m=1 ∞ 

P (Bn )h2H .

n=1

Hence, the H-valued measure P h : B −→ P (B)h, B ∈ B(R), necessarily has finite 2-variation. More generally, any Hilbert space valued measure has finite 2-variation by Propositions 1.6 and 1.8. Given an operator valued function f : Σ → L(X, Y ) and a vector measure m : S → X, leaving measurability conditions aside for the moment, we

wish to consider the integral Σ f dm ∈ Y in generality sufficient to treat scattering theory in Chapter 5. For an element y  ∈ Y  of the space Y  dual to Y , the X  -valued function  y ◦ f : Ω → X  is defined by

x, (y  ◦ f )(ω) = f (ω)x, y  ,

x ∈ X, ω ∈ Ω.

It is reasonable to expect that the identity " #  f dm, y  =

y  ◦ f, dm Σ

Σ

ought to hold for each y  ∈ Y  , in which the right-hand side is the integral of the X  -valued function y  ◦f acting on the range of the X-valued measure m. However, the total variation V (m)(Ω) of m satisfies the equation 

s, dm V (m)(Ω) = sup s∞ ≤1

Σ

by the Hahn-Banach Theorem. The supremum is taken over all X  -valued n S-simple functions s = k=1 xk χBk with xk  ≤ 1 and pairwise disjoint Bk ∈ S for k = 1, . . . , n and n = 1, 2, . . . , where  n 

s, dm =

m(Bk ), xk . Σ

k=1

As mentioned above, in the case that m = P h for a spectral measure P , the variation V (m) may have only the values 0 and ∞. Because there may

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be so few sets on which m has finite variation, the approaches considered in Section 1.6 are unsuited to vector measures m of this type. In the special case of the integration of X-valued functions with re τ Y , we can exploit spect to Y -valued measures in the tensor product X ⊗ the product structure of the tensor product to avoid consideration of X τ Y of the measure. semivariation in X ⊗ Expanding on this approach, suppose that an L(X, Y )-valued function has an integral with respect to an X-valued measure that takes its values in τ X. A careful choice of the tensor product topology τ facilitates L(X, Y )⊗ a continuous bilinear extension J1 of the product map T ⊗ x → T x to τ X so that the indefinite integral f m may be realised as J1 ◦ (f ⊗ L(X, Y )⊗ m). Similarly, the integral mf of an X-valued function f with respect to an L(X, Y )-valued measure may be realised as J2 ◦ (f ⊗ m) for the continuous τ L(X, Y ). bilinear extension J2 of the product map x ⊗ T → T x to X ⊗ The intermediary decoupled integral f ⊗ m may exist even if m has diminished variational bounds such as the case where X is equal to a Hilbert space H and m = P h with respect to a spectral measure P and h ∈ H. A basic example arises in the Hilbert space H = L2 (μ) for a σ-finite measure μ and the vector measure m : A → χA . For a suitable choice of τ H, functions k for which Φk : x → k(x, ·), x ∈ Σ, the tensor product H⊗ τ H determine a bounded integral operator Tk : H → H is m-integrable H⊗ with integral kernel k and the scalar 

Tk , dm = J1 ((Φk ⊗ m)(Σ)) Σ

is a generalised trace of the operator Tk . The choice τ = π for the projective tensor product topology π is associated with the operator ideal of trace

class operators and Σ Tk , dm is the usual trace of Tk . This example from operator theory is expanded upon in Chapter 3. Like the measure P h, a Gaussian random measure m : B([0, T ]) → L2 (P ) is orthogonally scattered [95] with finite 2-variation. For an adapted E-valued simple process X, the decoupled integral X ⊗ m has values in L2 (P ⊗ P, E). For a suitable Banach space E, the stochastic integral X.m can be written as J ◦ (X ⊗ m) for a continuous extension of the product map J : f ⊗ g → f.g of an E-valued random function f and a random scalar function g. The bilinear naure of stochastic integration is more fully explored in Chapter 4. In the next section, decoupled bilinear integrals f ⊗ m are studied in more detail.

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2.1

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Bilinear integration in tensor products

τ Y be Banach spaces, with τ a norm tensor product Let X,Y and X ⊗ topology on X ⊗ Y as in Section 1.5. Let m : S → Y be a vector measure defined on the σ-algebra S of subsets of a set Ω. As usual, the integral of X-valued functions with respect to m is first defined for elementary functions. An X-valued S-simple function is a function φ for which there exist k = 1, 2, . . . , sets Ej ∈ S and vectors cj ∈ X, j = 1, . . . , k, such that k φ = j=1 cj χEj . The integral φ ⊗ m of φ with respect to the Y -valued  measure m is defined by (φ⊗ m)(A) = kj=1 cj ⊗ [m(A∩Ej )], for all A ∈ S. Then (φ ⊗ m)(A) ∈ X ⊗ Y. We shall make some restrictive assumptions concerning the spaces X and Y and their tensor product X ⊗ Y allowing us to integrate a class of functions more general than the simple functions. As mentioned in Section 1.5, property (T2) of a norm tensor product topology τ ensures that X  ⊗Y  may be identified with a linear subspace of the continuous dual (X ⊗τ Y ) = τ Y ) of X ⊗τ Y . With suitable modifications of the definitions, the (X ⊗ results of this section work also for the tensor product X ⊗τ Y of two locally convex spaces X, Y . Definition 2.1. The topology τ determined by a tensor product norm  · τ is said to be completely separated if the subspace X  ⊗ Y  of (X ⊗τ Y ) τ Y of the normed space X ⊗τ Y , that is, if separates the completion X ⊗   u ∈ X ⊗τ Y and u, x ⊗ y = 0 for all x ∈ X  and y  ∈ Y  , then u = 0. An equivalent formulation of the condition that τ is completely separated, is that if {sn }n∈N , is any τ -Cauchy sequence in X ⊗ Y for which lim sn , x ⊗ y  = 0,

n→∞

for all x ∈ X  and y  ∈ Y  ,

then limn→∞ sn τ = 0. For a completely separated tensor product topol τ Y of X ⊗τ Y is naturally identified with a ogy τ , the completion X ⊗   subspace of the completion of X ⊗ Y in the topology σ X ⊗ Y, X  ⊗ Y  . If one of the Banach spaces X and Y has the approximation property, then the projective tensor product topology on X ⊗ Y is completely sep  Y may arated [88, 43.2 (7)]. Because the injective tensor product X ⊗ be identified with the closure of X ⊗ Y in the space of separately continuous bilinear forms Be (Xσ , Yσ ) [123, IV.9.1] and X  ⊗ Y  clearly separates Be (Xσ , Yσ ), it follows that is a completely separated tensor product topology.

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We begin by stating an elementary but useful condition [87, 18.4 (7)]. Proposition 2.1. Let τ be a norm tensor product topology on X ⊗ Y . If the  closed unit ball of X ⊗τ Y in the norm  · τ is closed for the topology σ X ⊗ Y, X  ⊗ Y  , then τ is completely separated. For example, it follows from the Hahn-Banach theorem and Proposition 2.1 that for 1 ≤ p ≤ ∞, the Lp (Γ, E, μ; X)-topology on Lp (Γ, E, μ) ⊗ X is completely separated. The following ubiquitous convergence result is a variant of Vitali’s convergence theorem [43, III.3.6]. Lemma 2.1. Let (Ω, S) be a measurable space and μ : S → C, a scalar measure. Suppose that fk , k = 1, 2, . . . , are μ-integrable scalar functions converging μ-almost everywhere to a scalar function f , with the property that the sequence {fk μ(A)}∞ k=1 converges for each A ∈ S. Then f is μintegrable and fk μ(A) → f μ(A) uniformly for A ∈ S, as k → ∞. Proof. The measures fk μ, k = 1, 2, . . . , are uniformly countably additive on S, by the Vitali-Hahn-Saks theorem [38, I.5.6]. An appeal to Egorov’s measurability theorem [43, III.5.12], ensures that there exists an increasing family of sets Ωj ∈ S such that ∪j Ωj is a set of full μ-measure, and for each j = 1, 2, . . . , the functions fk converge to f uniformly on Ωj as k → ∞. Let > 0 and choose j so large that |fk μ|(Ω\Ωj ) < for all k = 1, 2, . . . . Then for K large enough, |fk − fl | < on Ωj , for all k, l ≥ K. Hence, for all A ∈ S and k, l ≥ K, |fk μ(A) − fl μ(A)| ≤ |fk μ|(Ω \ Ωj ) + |fl μ|(Ω \ Ωj ) + (|fk − fl |.|μ|)(Ωj ) < 2 + |μ|(Ω). Because is any positive number, limk→∞ fk μ(A) converges uniformly for all A ∈ S, the function f is integrable, and limk→∞ fk μ(A) = f μ(A), for all A ∈ S. A similar result holds for strongly measurable Pettis integrable with values in a Banach space X. For an X-valued S-simple function φ, the integral φ ⊗ m is σ-additive in X ⊗τ Y by property (T1) of a tensor product topology, see Section 1.5. The following lemma is needed for Definition 2.2 to make sense. Lemma 2.2. Let τ be a completely separated norm topology on X ⊗ Y . Suppose that sk , k = 1, 2, . . . , are X-valued S-simple functions for which

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{(sk ⊗ m)(A)}∞ k=1 is τ -Cauchy in X ⊗ Y for each A ∈ S and sk → 0 m-a.e. Then (sk ⊗ m)(A) → 0 in X ⊗τ Y for each A ∈ S. Proof. For each x ∈ X  and y  ∈ Y  , the scalars 

(sk ⊗ m)(A), x ⊗ y  = sk , x d m, y  ,

k = 1, 2, . . . ,

A

converge as k → ∞ for every A ∈ S, and sk , x → 0 m, y  -a.e. An appeal to Lemma 2.1, shows that limk→∞ (sk ⊗ m)(A), x ⊗ y  = 0 is true for all x ∈ X  and y  ∈ Y  . But we know that (sk ⊗ m)(A), k = 1, 2, . . . , is already τ -Cauchy in X ⊗ Y , so the fact that τ is completely separated tells us that limk→∞ (sk ⊗ m)(A) = 0 in τ . Our bilinear integral is defined by adopting the conclusion of [11, Theorem 9], a translation to the bilinear context of “Dunford’s second integral”, or in modern parlance, the Pettis integral for strongly measurable functions. Definition 2.2. Let (Ω, S) be a measurable space and X, Y Banach spaces. Suppose that τ is a completely separated norm tensor product topology on X ⊗ Y . Let m : S → Y be a Y -valued measure. τ Y if there exist A function φ : Ω → X is said to be m-integrable in X ⊗ X-valued S-simple functions φk , k = 1, 2, . . . , such that φk → φ m-a.e. as k → ∞, and {(φk ⊗ m)(A)}∞ k=1 converges in X ⊗τ Y for each A ∈ S. Let  (φ ⊗ m)(A) = φ(ω) ⊗ dm(ω) A

denote this limit. Sometimes, we write m(φ) for the integral (φ ⊗ m)(Ω). To check that φ ⊗ m is well-defined, suppose that we have some other X-valued S-simple functions φj , j = 1, 2, . . . , such that φj → φ m-a.e. as j → ∞ and the sequence {(φj ⊗ m)(A)}∞ j=1 converges in X ⊗τ Y for each   A ∈ S. Then [φj − φj ] → 0 m-a.e. as j → ∞ and {([φj − φj ] ⊗ m)(A)}∞ j=1 τ Y , for each A ∈ S, as j → ∞. By Lemma 2.2, we must converges in X ⊗ have (φ ⊗ m)(A) = limj→∞ (φj ⊗ m)(A) = limj→∞ (φj ⊗ m)(A), for each set A ∈ S. The set function φ ⊗ m is the setwise limit of σ-additive set functions φk ⊗ m, k = 1, 2, . . . , so by the Vitali-Hahn-Saks theorem [38, I.5.6], it is itself σ-additive for the topology τ . It is easy to see that the map (f, m) → f ⊗ m is bilinear, in the obvious sense. Let m : S → Y be a measure. If A ∈ S and the restriction of m to the σ-algebra S ∩ A = {E ∩ A : E ∈ S} has σ-finite X-semivariation, then we say that m has σ-finite X-semivariation on A.

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The following result is a useful consequence of σ-finite X-semivariation. τ Y be Banach spaces, where τ is a comTheorem 2.1. Let X, Y and X ⊗ pletely separated norm tensor product topology on X ⊗ Y . Suppose that the vector measure m : S → Y has σ-finite X-semivariation in X ⊗τ Y . If fj , j = 1, 2, . . . , are m-integrable functions such that fj converges m a.e. to an X-valued function f , and {fj ⊗ m(A)}∞ n=1 converges in X ⊗τ Y τ Y and {(fj ⊗ m)(A)}∞ for each A ∈ S, then f is m-integrable in X ⊗ n=1 τ Y to (f ⊗ m)(A), uniformly for A ∈ S. converges in X ⊗ τ Y . As usual, μτ : S → [0, ∞) Proof. Let  · τ denote the norm of X ⊗ τ Y . Because the denotes the semivariation of a measure μ : S → X ⊗ ∞ τ Y , they are uniformly vector measures {fj ⊗m}j=1 converge setwise in X ⊗ bounded by the Nikodym boundedness theorem and uniformly countably additive by the Vitali-Hahn-Saks theorem, so there exists a nonnegative measure ν : S → [0, ∞) such that limν(E)→0 fj ⊗ mτ (E) = 0, uniformly for j = 1, 2, . . . . Moreover, ν may be chosen with the property that the bound 0 ≤ ν(E) ≤ supj fj ⊗ mτ (E) holds for all E ∈ S [38, I.2.5]. Let N = {ω ∈ Ω : limj→∞ fj (ω) = f (ω)}. Then N is an m-null set, so by Corollary 2.1, fj ⊗ mτ (N ) = 0 for all j = 1, 2, . . . . It follows from the inequality above that N is a ν-null set. Let > 0 and choose δ > 0 such that for every set E ∈ S with the property that ν(E) < δ, the inequality fj ⊗ mτ (E) < /4 holds for all j = 1, 2, . . . . Let B be the closed unit ball of X. There exist increasing sets Ωk , k = 1, 2, . . . , belonging to S on which the (B,  · τ )-semivariation is finite, and whose union is Ω. The σ-additivity of the measure ν guarantees that for some K ∈ N, we have ν(ΩcK ) < δ/2. An appeal to Egorov’s theorem [43, III.5.12] ensures that there exists a set Bδ such that ν(Bδc ) < δ/2 and fk − f X → 0 uniformly on Bδ . Let Aδ = Bδ ∩ ΩK . Then ν(Acδ ) < δ, fk − f X → 0 uniformly on Aδ as k → ∞ and β(B,r) (Aδ ) < ∞. Choose K = 1, 2, . . . such that sup f (ω) − fk (ω)X
0 and every continuous seminorm p on X, there exist Xvalued S-simple functions φk , k = 1, 2, . . . , such that p(φk (ω)) ≤ p◦φ∞ + for all ω ∈ Ω and k = 1, 2, . . . , the functions φk converge to φ m-a.e. and (φk ⊗ m)(A) → (φ ⊗ m)(A) as k → ∞ for each A ∈ S. Proof. As φ is m-integrable, there exist X-valued S-simple functions ψk , k = 1, 2, . . . , such that ψk → φ m-a.e. and (ψk ⊗ m)(A) → (φ ⊗ m)(A) as m → ∞ for each A ∈ S. Let be a positive number and p a continuous seminorm on X. For each k = 1, 2, . . . , let ∞ {ω : ψj (ω) ≤ φ∞ + }. φk = ψk χAk , Ak = j=k

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By the triangle inequality, φk (ω) → φ(ω) as k → ∞ at all points ω ∈ Ω at which ψk (ω) → φ(ω) as k → ∞, and ∪∞ k=1 Ak is a set of full m-measure. The equality (φk ⊗ m)(A) = (ψk ⊗ m)(A ∩ Ak ) is valid for all k = 1, 2, . . . and all A ∈ S. By the Vitali-Hahn-Saks theorem, {ψk ⊗m}∞ k=1 is a uniformly countably additive family of X ⊗τ Y -valued measures, so that for every τ -continuous seminorm r on X ⊗ Y , lim sup r [(ψk ⊗ m)(A ∩ Aj ) − (ψk ⊗ m)(A)] = 0.

j→∞

k

Hence, for each A ∈ S, (φk ⊗ m)(A) → (f ⊗ m)(A) as k → ∞. Another standard property of vector integrals is that continuous linear maps can be dragged inside the integral to act on the integrand—a property which takes the following form in the present context. Suppose that Xj , Yj , j = 1, 2 are locally convex spaces and τ1 is a tensor product topology on X1 ⊗ Y1 , and τ2 is a tensor product topology on X2 ⊗ Y2 . The tensor product of two linear maps S : X1 → X2 and T : Y1 → Y2 , is the linear map S ⊗ T : X1 ⊗ Y1 → X2 ⊗ Y2 defined for each x ⊗ y ∈ X1 ⊗ Y1 by (S ⊗ T )(x ⊗ y) = (Sx) ⊗ (T y). There is no guarantee that S ⊗ T is continuous from τ1 to τ2 if S and T are continuous. However, if S ⊗ T : X1 ⊗τ1 Y1 → X2 ⊗τ2 Y2 is continuous, then the same symbol S ⊗ T denotes the associated continuous linear map between the τ1 Y1 and X2 ⊗ τ2 Y2 . completions X1 ⊗ Proposition 2.3. Suppose that Xj , Yj , j = 1, 2, are locally convex spaces and τ1 is a completely separated tensor product topology on X1 ⊗ Y1 , and τ2 is a completely separated tensor product topology on X2 ⊗ Y2 . Let m : S → Y1 be a measure and suppose that S : X1 → X2 and T : Y1 → Y2 are continuous linear maps whose tensor product S ⊗ T : X1 ⊗τ1 Y1 → X2 ⊗τ2 Y2 is continuous. τ1 Y1 , then Sφ is T m-integrable in If φ : Ω → X1 is m-integrable in X1 ⊗ τ2 Y2 and X2 ⊗   φ ⊗ dm = [Sφ] ⊗ d[T m], for every A ∈ S. (S ⊗ T ) A

A

Proof. Let φk , k = 1, 2, . . . , be X1 -valued S-simple functions satisfying the assumptions of Definition 2.2. Then Sφk → Sφ m-a.e. as k → ∞, because S is continuous. The continuity of T guarantees that T m := T ◦ m

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is a Y2 -valued measure. Since S ⊗ T is τ1 -τ2 -continuous, the sequence {(S ⊗ T )([φk ⊗ m](A))}∞ k=1 converges in X2 ⊗τ2 Y2 for each A ∈ S. A glance at Definition 2.2 is enough to complete the proof. 2.2

Order bounded measures

We return to the setting of Subsection 1.5.1 where the semivariation of Lp valued measures was examined. Let (Γ, E, μ) be a σ-finite measure space. The modulus |m| : S → Lp+ (μ) of an order bounded measure m : S → Lp (μ) was defined in Lemma 1.1. For any Banach space X, if X ⊗ Lp (μ) is equipped with the relative τ Lp (μ) may be identified with Lp (μ; X). topology τ of Lp (μ; X), then X ⊗ According to Proposition 1.5, an order bounded measure m : S → Lp (μ) has finite and continuous X-semivariation m p,X in Lp (μ; X) for any Banach space X, so integration in Lp (μ; X) in the sense of Definitions 1.2, 1.3 and 2.2 coincide for the vector measure m. Theorem 2.3. Suppose that 1 ≤ p < ∞ and m : S → Lp (Γ, E, μ) is an order bounded measure and f : Ω → X is strongly m-measurable. If f X is |m|-integrable, then f is m-integrable in Lp (μ; X) and the inequality          ≤ f ⊗ dm (γ) f  d|m| (γ) (2.3) X   A

A

X

holds for all A ∈ S and for μ-almost every γ ∈ Γ. Proof. First we establish the estimate for simple functions. Let l ∈ N. l Suppose that h = j=1 xj χEj is an X-valued S-simple function with xj ∈ X and Ej ∈ S pairwise disjoint for j = 1, . . . , l. Then for each A ∈ S and μ-almost every γ ∈ Γ,    l            h ⊗ dm (γ) =  xj (m(Ej ∩ A))(γ)   A  j=1  X X



l 

xj X (|m|(Ej ∩ A))(γ)

j=1



 hX d|m| (γ).

= A

The positivity of the vector measure |m| is crucial here. Next we prove that f is m-integrable in Lp (X). By assumption, f is strongly m-measurable and so there exists a sequence {ψj }∞ j=1 of X-valued

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S-simple functions such that ψj → f m-almost everywhere as j → ∞. Now let  ψj (ω) if ψj (ω)X ≤ 2f (ω)X fj (ω) = 0 if ψj (ω)X > 2f (ω)X . Then each fj is an X-valued S-simple function such that fj → f m-almost everywhere and further fj (ω)X ≤ 2f (ω)X for all ω ∈ Ω. Thus to ensure integrability it suffices to show that the sequence { A fj ⊗ dm}∞ j=1 converges in Lp (μ; X) for each A ∈ S. Let j, k ∈ N. By construction, fj (·) − f (·)X ≤ 3f (·)X and by assumption f X is |m|-integrable so, making use of inequality (2.3) and dominated convergence for vector measures [86, II.4], we have       (fj − fk ) ⊗ dm ≤ fj − fk X d|m|   Lp (X)

A

A





fj − f X d|m| +

≤ A

f − fk X d|m| A

→0 as j, k → ∞. Thus f is m-integrable in Lp (μ; X). Finally, we establish that inequality (2.3) holds for the function f . We



know that limj→∞ A fj ⊗ dm = A f ⊗ dm in Lp (μ; X) for each A ∈ S. By taking an appropriate subsequence, if necessary, we may assume that              fj ⊗ dm (γ) →  f ⊗ dm (γ)   A

A

X

X

for μ-almost every γ ∈ Γ also. Since f X is assumed to be |m|-integrable, dominated convergence for vector measures again ensures that   fj X d|m| → f X d|m| A

A

in L (μ) as j → ∞ for all A ∈ S.  By taking a further subsequence, if necessary, we may assume that A fj X d|m| (γ) → A f X d|m| (γ) for μ-almost every γ ∈ Γ as well. This guarantees that inequality (2.3) holds for the function f . p

2.3

The bilinear Fubini theorem

A decoupling strategy to integrate an L(X, Y )-valued function with respect to an X-valued measure was outlined at the beginning of the chapter. Although we shall be examining such integrals in greater detail in

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Chapter 6 when we consider scattering theory, a simple formulation of Fubini’s Theorem is facilitated by the underlying bilinear structure. Let X, Y be Banach spaces. A locally convex space E is said to be bilinear admissible for X, Y if a) E contains the vector space L(X, Y ) ⊗ X as a dense subspace, b) the composition map J : L(X, Y ) × X → Y defined by T ∈ L(X, Y ), x ∈ X,

J(T, x) = T x,

has a continuous linear extension JE : E → Y from L(X, Y ) ⊗ X to E. c) for x ∈ X, x ∈ X  and y  ∈ Y  , the linear functional defined by x ⊗ y  ⊗ x : T ⊗ u −→ T x, y  u, x ,

T ∈ L(X, Y ), u ∈ X,

is continuous on L(X, Y ) ⊗ X for the relative topology of E. d) the family of all linear functionals x ⊗ y  ⊗ x for x ∈ X, x ∈ X  and y  ∈ Y  separates points of E. If Y = X, then we merely say E is bilinear admissible for X. If τ is a completely separating tensor product topology on Ls (X, Y ) ⊗ X for the space L(X, Y ) endowed with the strong operator topology, then we may τ X, the completion of the linear space L(X, Y ) ⊗ X in take E = L(X, Y )⊗ the locally convex topology τ . Sometimes the quasicompletion [87, 23.1] is taken. Remark 2.2. If the Banach space X has the approximation property [123, πX III.9], then X ⊗X  separates points of the projective tensor product X  ⊗ [88, 43.2(12)] and this is precisely the property needed to define the trace of a nuclear operator on X [94]. In Example 2.1, the separation property is 1 what we need to define the generalised trace 0 Φϕ dm when X is a Banach function space such as Lp ([0, 1]), 1 ≤ p < ∞. Definition 2.3. Suppose that the locally convex space E is bilinear admissible for the Banach spaces X and Y . Let (Ω, S) be a measurable space. A function f : Ω → L(X, Y ) is said to be m-integrable in E for a vector measure m : S → X, if for each x ∈ X, x ∈ X  , y  ∈ Y  , the scalar function

f x, y  is integrable with respect to the scalar measure m, x and for each S ∈ S, there exists an element (f ⊗ m)(S) of E such that    (2.4)

(f ⊗ m)(S), x ⊗ y ⊗ x = f x, y  d m, x S 







for every x ∈ X, x ∈ X and y ∈ Y .

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If f is m-integrable in E, then f m(S) ∈ Y is defined for each S ∈ S by   f m(S) = JE (f ⊗ m)(S) .



We also denote f m(S) by S f dm or S f (ω) dm(ω). Because the linear space X ⊗ Y  ⊗ X  separates points of E, the vector (f ⊗ m)(S) ∈ E is well-defined for each S ∈ S. The same definition is adopted if S is generated by the δ-ring S0 [16, Definition 1.2.13] and m : S0 → X is a vector measure on S0 . The existence of the E-valued set function f ⊗ m is formulated here in τ Y for a suitable completely the weak sense. In the case that E = L(X, Y )⊗ separating tensor product topology τ on L(X, Y ) ⊗ Y , the existence of the E-valued measure f ⊗ m is usually verified along the lines of Definition 2.2. In the case that X is the set of scalars, f m is the indefinite (Pettis) integral of a Y -valued function with respect to a scalar measure m as in Subsection 1.3.1. We shall use the term Bochner integral to distinguish the stronger integration process when f is approximated in the norm of Y , see Subsection 1.3.2. Definition 2.3 facilitates a simple version of Fubini’s Theorem in the operator context. In the following statement, an L(X, Y )-valued function is said to be μ-integrable, if it is Pettis μ-integrable in L(X, Y ) for the strong operator topology, as defined in Subsection 1.3.1. Theorem 2.4. Suppose that the locally convex space E is bilinear admissible for the Banach spaces X and Y . Let (Ω, S) be a measurable space and (Γ, E, μ) a σ-finite measure space, and m : S → X a vector measure. Suppose that f : Ω × Γ → L(X, Y ) is (m ⊗ μ)-integrable in E. If (i) for m-almost all ω ∈ Ω, the L(X, Y )-valued function f (ω, · ) is μ-integrable, and (ii) for μ-almost all γ ∈ Γ, the L(X, Y )-valued function f ( · , γ) is m-integrable in E, then the function

 ω −→

f (ω, γ) dμ(γ) Γ

is m-integrable in E, the function  γ −→ f (ω, γ) dm(ω) Ω

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is integrable in Y with respect to μ, and the equalities     f d(m ⊗ μ) = f (ω, γ) dμ(γ) dm(ω) Ω×Γ Ω Γ    f (ω, γ) dm(ω) dμ(γ) = Γ

Ω

hold.

Proof. Let Φ(ω) = Γ f (ω, γ) dμ(γ) for all ω ∈ Ω for which f (ω, ·) is μintegrable and Φ(ω) = 0 otherwise. For each x ∈ X, y  ∈ Y  , x ∈ X  and S ∈ S, we have 

Φ(ω)x, y  = f (ω, γ)x, y  dμ(γ), Γ

so that 

 



f (ω, γ)x, y  dμ(γ) d m, x S Γ !   = f ⊗ (m ⊗ μ) (S × Γ), x ⊗ y  ⊗ x

Φ(ω)x, y  d m, x =

S

by the scalar version of Fubini’s Theorem. It follows that Φ is m-integrable in E and    Φ(ω) ⊗ dm(ω) = f ⊗ (m ⊗ μ) (S × Γ), S

  

hence Ω Φ(ω) dm(ω) = JE f ⊗ (m ⊗ μ) (Ω × Γ) = Ω×Γ f d(m ⊗ μ). A similar appeal to the scalar version of Fubini’s Theorem applies to the other iterated integral.

Because we are dealing with vector valued integrals, the existence of the integrals (i) and (ii) almost everywhere is not ensured by the integrablity of f with respect to the product measure m ⊗ μ. Sometimes we shall need a simple modification of Theorem 2.4 in which condition b) of the definition of a bilinear admissible space E is replaced by b ) there exists a lcs F containing L(X, Y ) ⊗ X and embedded in E such that the composition map J : L(X, Y ) × X → Y defined by J(T, x) = T x,

T ∈ L(X, Y ), x ∈ X,

has a continuous linear extension JF : F → Y from L(X, Y ) ⊗ X to F .

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Then Definition 2.3 can be modified by writing   f m(S) = JF (f ⊗ m)(S) if (f ⊗ m)(S) ∈ F for S ∈ S. Theorem 2.5. Suppose that the lcs E, F are as just described. Let (Ω, S) be a measurable space and (Γ, E, μ) a σ-finite measure space, and m : S → X a vector measure. Suppose that f : Ω × Γ → L(X, Y ) is (m ⊗ μ)-integrable in E and (m ⊗ μ)(Ω × Γ) ∈ F . If (i) for m-almost all ω ∈ Ω, the L(X, Y )-valued function f (ω, · ) is μintegrable, and (ii) for μ-almost all γ ∈ Γ, the L(X, Y )-valued function f ( · , γ) is mintegrable in E and Ω f (ω, γ) ⊗ dm(ω) ∈ F , then the function  ω −→

f (ω, γ) dμ(γ) Γ

is m-integrable in E, the function  f (ω, γ) dm(ω) γ −→ Ω

is integrable in Y with respect to μ, and the equalities     f d(m ⊗ μ) = f (ω, γ) dμ(γ) dm(ω) Ω×Γ Ω Γ    f (ω, γ) dm(ω) dμ(γ) = Γ

Ω

hold. Example 2.1. Let (Ω, S, μ) be a finite measure space. The space of all μ-equivalence classes of S-measurable scalar functions is denoted by L0 (μ). It is equipped with the topology of convergence in μ-measure and vector operations pointwise μ-almost everywhere. Any Banach space X that is a subspace of L0 (μ) with the properties that (i) X is an order ideal of L0 (μ), that is, if g ∈ X, f ∈ L0 (μ) and |f | ≤ |g| μ-a.e., then f ∈ X, and (ii) if f, g ∈ X and |f | ≤ |g| μ-a.e., then f X ≤ gX ,

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is called a Banach function space (based on (Ω, S, μ)). We suppose that X contains constant functions and m : S −→ χS , S ∈ S, is σ-additive in X, for example, X is σ-order continuous, see [99, Corollary 3.6]. If X is reflexive and μ is non-atomic, then it follows from [99, Corollary 3.23] that the values of the variation V (m) of m are either zero or infinity. Suppose that ϕ : Ω × Ω → C is a jointly measurable function and Tϕ : X → X is a bounded linear operator such that  ϕ( · , t)f (t) dμ(t) Tϕ f = Ω

for a dense set of f ∈ X. Suppose also that Φϕ (s) = [ϕ(s, · )] ∈ X  for μ-almost all s ∈ Ω, that is, there exists Ks > 0 such that      ϕ(s, t)f (t) dμ(t) ≤ Ks f X , f ∈ X.   Ω

Such operators are called Carleman operators in [55]. A bounded linear operator Tϕ for which there exists a bilinear (X, C)admissible space E such that Φϕ is m-integrable in E, is a type of generalised trace class operator and Ω Φϕ dm = JE Ω Φϕ ⊗ dm is the trace of Tϕ . For example, if X has the approximation property [123, III.9] and π X, then Tϕ is a nuclear operator and Φ dm is actually the E = X ⊗ Ω ϕ trace of Tϕ [94], see Example 2.3 for the Hilbert space case. There are closed property. subspaces of p , 1 ≤ p < ∞, p = 2, without the approximation

1 As we shall see in the next chapter, the value 0 Φϕ dm = 12 is obtained for the Volterra integral operator Tϕ on L2 ([0, 1]) with a judicious choice of the auxiliary spaces E and F . The Volterra integral operator is HilbertSchmidt but not trace class on L2 ([0, 1]). 1 The integral 0 Φϕ dm is a type of singular bilinear integral referred to in the title of this work, because the diagonal {(t, t) : t ∈ [0, 1]} has measure

1 zero in [0, 1]2 , so that the integral 0 ϕ(t, t) dt is not well-defined for a general integral kernel ϕ associated with the operator Tϕ . As we shall see in the next chapter, the auxiliary spaces E, F determine the averaging

process of ϕ around the diagonal and the density of the measure A −→ A Φϕ dm, A ∈ B([0, 1]), with respect to Lebesgue measure represents the average t −→ ϕ(t, ˜ t) of ϕ around the diagonal.

2.4

Examples of bilinear integrals

There are no surprises in this section. We apply the preceding theory to some natural examples to show that the expected class of integrable

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functions is obtained, and their definite integrals give the expected operators. Nevertheless, the examples also illustrate the difficulty with applying the classical theories of bilinear integration in the context of the integration of operator valued functions with respect to operator valued measures. The following example shows that the X-semivariation of a Y -valued measure in X ⊗τ Y may take only the values zero and infinity. Example 2.2. Let m : B([0, 1]) → L2 ([0, 1]) be the vector measure m(B) = χB , B ∈ B([0, 1]). Then the L1 [0, 1]-semivariation of m in L1 ([0, 1]; L2 ([0, 1])) is infinite on any Borel set E with positive Lebesgue measure |E|. For, let n be any positive integer and suppose that Ej , j = 1, . . . , n, are pairwise disjoint sets with Lebesgue measure |Ej | = |E|/n, j = 1, . . . , n, — the range of the Lebesgue measure on the Borel σ-algebra B(E) of E is the interval [0, |E|]. Let fj = χEj /|Ej | for each j = 1, . . . , n. The n L1 ([0, 1]; L2 ([0, 1]))-norm of j=1 fj ⊗ m(Ej ) is  1  n   n     fj ⊗ m(Ej )1 = fj (x)m(Ej )2 dx = |E|1/2 n1/2 . 0

j=1

j=1

Because n is any positive integer, the L1 [0, 1]-semivariation of m in the space L1 ([0, 1]; L2 ([0, 1])) is infinite on E. Of course, the L2 ([0, 1])semivariation of m in L1 ([0, 1]; L2 ([0, 1])) and the L1 [0, 1]-semivariation of m in L1 ([0, 1]2 ) are finite; see Proposition 1.5. The only L1 [0, 1]-valued functions which are m-integrable in the space L ([0, 1]; L2 ([0, 1])) in the sense of Definition 1.2 or Definition 1.3, are the null functions. Nevertheless, it is natural to consider the integration in L1 ([0, 1]; L2 ([0, 1])) of functions with values in L1 [0, 1], with respect to the L2 ([0, 1])-valued measure m: elements of L1 ([0, 1]; L2([0, 1])) are associated with the space H2,1 of Hille-Tamarkin operators [124, p. 282], the bounded linear operators T : L2 ([0, 1]) → L2 ([0, 1]) with integral kernel k : [0, 1] × 1 [0, 1] → C such that T u(x) = 0 k(x, y)u(y) dy for u ∈ L2 ([0, 1]) and  1  12 2 |k(x, y)| dy dx < ∞. 1

0

Because the L1 [0, 1]-semivariation of m in L1 ([0, 1]; L2 ([0, 1])) is infinite, the conditions of Theorem 2.2 do not hold and bounded operator valued functions need not be integrable. The following example of trace class operators shall assume importance in Chapter 3.

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Example 2.3. Let m : B([0, 1]) → L2 ([0, 1]) be the vector measure defined by m(B) = χB for every B ∈ B([0, 1]). Then the L2 ([0, 1])-semivariation of m in the projective tensor product L2 ([0, 1]) ⊗π L2 ([0, 1]) (see [88, Section 41.2]) is infinite on any Borel set A with positive Lebesgue measure |A|. To see this, let n be any positive integer and suppose that Aj , j = 1, . . . , n, are pairwise disjoint subsets of A, with Lebesgue measure |A|/n. Let φj = (n/|A|)1/2 χAj for each j = 1, . . . , n. Then Φ : L2 ([0, 1]) ⊗π L2 ([0, 1]) → C, defined by Φ(f ⊗ g) = (f |g) for every f ∈ L2 ([0, 1]) and every g ∈ L2 ([0, 1]), is continuous but    n n φj ⊗ m(Aj ) = (φj |m(Aj )) = |A|1/2 n1/2 . Φ j=1

j=1

Because n is any positive integer, the L2 ([0, 1])-semivariation of m in the space L2 ([0, 1]) ⊗π L2 ([0, 1]) is infinite on A. Nevertheless, Proposition 2.4 below demonstrates that it is natural to π L2 ([0, 1]) of funcconsider the integration in the vector space L2 ([0, 1])⊗ 2 2 tions with values in L ([0, 1]), with respect to the L ([0, 1])-valued mea π L2 ([0, 1]) sure m. Because the L2 ([0, 1])-semivariation of m in L2 ([0, 1])⊗ is infinite, bounded vector valued functions need not be integrable; see Example 1.8. Remark 2.3. The only L2 ([0, 1])-valued functions which are (Dobrakov) π L2 ([0, 1]) or (Bartle) m-integrable in the tensor product space L2 ([0, 1])⊗ are the null functions. The space C2 (L2 ([0, 1])) of Hilbert-Schmidt operators acting on L ([0, 1]) is endowed with the Hilbert-Schmidt norm, [88, Section 42.4]. Let K denote the isometric isomorphism from L2 ([0, 1]2 ) onto C2 (L2 ([0, 1])), which sends an element k of L2 ([0, 1]2 ) to the Hilbert-Schmidt operator Tk : L2 ([0, 1]) → L2 ([0, 1]) with kernel k, [135, Theorem 6.11], that 1 is, (Tk φ)(x) = 0 k(x, y)φ(y) dy for almost all x ∈ [0, 1] and for all φ ∈ L2 ([0, 1]). For all φ, ψ ∈ L2 ([0, 1]) and k ∈ L2 ([0, 1]2 ), we have the equality  k(x, y)φ(y)ψ(x) dxdy. ([Kk]φ , ψ) = (k , φ ⊗ ψ) = 2

[0,1]2

By appealing to the representation given in Theorem 1.8, the projec π L2 ([0, 1]) may be identified with a linear tive tensor product L2 ([0, 1])⊗ 2 2 subspace of L ([0, 1] ) in the obvious way.

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π L2 ([0, 1]) under K is the Then the image C1 (L2 ([0, 1])) of L2 ([0, 1])⊗ 2 space of nuclear operators on L ([0, 1]). The nuclear and trace class operators on the Hilbert space L2 ([0, 1]) are the same. If we equip C1 (L2 ([0, 1]) with the nuclear norm, [88, 42.5.(8)], then K induces an isom π L2 ([0, 1]) onto C1 (L2 ([0, 1])). Further properties of etry from L2 ([0, 1])⊗ trace class operators are mentioned in Section 3.1 below. These observations indicate that for practical reasons the projective tensor product π L2 ([0, 1]) is a worthy object of study. L2 ([0, 1])⊗ The following proposition illustrates our claim that, in the present context, we have written down the ‘right’ definition of integration of vector valued functions with respect to vector valued measures in tensor product spaces. Subsequent chapters lend more evidence for this claim. In the following proposition, given a function k ∈ L2 ([0, 1]2 ) and a point x ∈ [0, 1], let [k(x, · )] denote the equivalence class in L2 ([0, 1]) containing k(x, · ). Moreover let Q : B([0, 1]) → L(L2 ([0, 1])) denote the measure given by Q(A)φ = χA φ for every A ∈ B([0, 1]) and φ ∈ L2 ([0, 1]). Then Q is a spectral measure; that is, Q(A ∩ B) = Q(A)Q(B) for all A, B ∈ B([0, 1]), and Q([0, 1]) is the identity operator. Proposition 2.4. Let m : B([0, 1]) → L2 ([0, 1]) be the vector measure given by m(B) = χB , B ∈ B([0, 1]). A function f : [0, 1] → L2 ([0, 1]) is π L2 ([0, 1]) if and only if there exists a function m-integrable in L2 ([0, 1])⊗ 2 k : [0, 1] → C such that (i) k is the kernel of a trace class operator; and (ii) the set {x ∈ [0, 1] : f (x) = [k(x, · )] in L2 ([0, 1])} is a set of full measure. If f is m-integrable and A ∈ B([0, 1]), then [f ⊗ m](A) is equal to the equivalence class in L2 ([0, 1]2 ) of the function (x, y) ∈ [0, 1]2 . (x, y) → χA (x)k(x, y),   Moreover, the equality K [f ⊗ m](A) = Q(A)Kk is valid for each A ∈ B([0, 1]). Proof. Suppose first that the conditions (i) and (ii) are satisfied. According to [88, 42.5.(5)], the operator Tk has a representation Tk φ =

∞  j=1

ηj (φ , gj )hj ,

φ ∈ L2 ([0, 1]),

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∞ where {η}∞ j=1 is an absolutely summable scalar sequence, and {gj }j=1 and ∞ 2 {hj }j=1 are orthonormal sequences in the Hilbert space L ([0, 1]). By the ∞ Beppo Levi convergence theorem, j=1 |ηj | |gj (x)| |hj (y)| < ∞, so that we ∞ can define ξ(x, y) = j=1 ηj gj (x)hj (y), for almost all (x, y) ∈ [0, 1]2 . Then the function ξ belongs to L2 ([0, 1]2 ). The operators Tk and Tξ , defined by the kernels k and ξ, respectively, are equal. However, L2 ([0, 1]) ⊗ L2 ([0, 1]) separates elements of L2 ([0, 1]2 ), so k = ξ almost everywhere on [0, 1]2 . Given j = 1, 2, . . . , the functions gj and hj may be expressed as gj = ∞ ∞ l=1 φjl and hj = n=1 ψjn , both almost everywhere in [0,1] and with respect to the norm topology of L2 ([0, 1]), for some scalar valued sequences ∞ ∞ {φjl }∞ l=1 and {ψjn }n=1 of B([0, 1])-simple functions with l=1 φjl 2 ≤ 2  ∞ and n=1 ψjn 2 ≤ 2, respectively. It then follows from the condition (ii)  that for almost all x ∈ [0, 1] we have ∞ j,l,n=1 |λj | |φjl (x)| ψjn 2 < ∞, and ∞ so the identity f (x) = j,l,n=1 ηj φjl (x)ψjn holds in the norm topology of L2 ([0, 1]).

Because A (φjl (x)ψjn )⊗ dm(x) = χA φjl ⊗ψjn as elements of the Banach π L2 ([0, 1]) for all j, l, n ∈ N and because the triple sequence space L2 ([0, 1])⊗ χ {ηj [ A φjl ] ⊗ ψjn }∞ j,l,n=1 is summable in the projective tensor topology for every A ∈ S, the function f is m-integrable according to Definition 2.2 and  ∞  f ⊗ dm = ηj [χA φjl ] ⊗ ψjn , A ∈ B([0, 1]). A

j,l,n=1

π L2 ([0, 1]). Conversely suppose that f is m-integrable in L2 ([0, 1])⊗ π L2 ([0, 1]) is expressed as a Then the element (f ⊗ m)(A) of L2 ([0, 1])⊗ 1 2 function k ∈ L ([0, 1]) so that K( 0 f ⊗ dm) is a trace class operator with kernel k. In terms of the inner product ( · , · ) of L2 ([0, 1]), we have   1  1  f ⊗ dm , φ ⊗ ψ = (f , φ) d(m, ψ) 0

0

so that  $  1 (f , φ) d(m , ψ) = K 0

0

1

%   f ⊗ dm φ, ψ =

k(x, y)φ(y)ψ(x) dxdy,

[0,1]2

for all φ, ψ ∈ L2 ([0, 1]). On taking ψ to be the characteristic function of a Borel set, we see that for each φ ∈ L2 ([0, 1]), the equality (f (x) | φ) =

1 2 k(x, y)φ(y) dy holds for almost all x ∈ [0, 1]. The 0  separability  of L ([0, 1]) ensures that (ii) holds. Finally the formula K [f ⊗ m](A) = Q(A)Kk is valid for each A ∈ B([0, 1]).

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Example 2.4. Let m be the vector measure defined in Proposition 2.4. Let k(x, y) =

∞ 

nχ[1/(n+1),1/n) (x) χ[1/(n+1),1/n) (y)

n=1

1 for all x, y ∈ [0, 1]. Then 0 k(x, y)2 dy ≤ 1 for all x ∈ [0, 1], but a straightforward calculation shows that k is not the kernel of a trace class operator. The function f : [0, 1] → L2 ([0, 1]) defined by f (x) = [k(x, · )] for all x ∈ [0, 1] is therefore a bounded L2 ([0, 1])-valued function which is not π L2 ([0, 1]). m-integrable in L2 ([0, 1])⊗ On the other hand, replacing L2 ([0, 1]) by L1 ([0, 1]) in Proposition 2.4, if k ∈ L1 ([0, 1]2 ), then by Fubini’s Theorem the function x → f (x, ·) has values in L1 ([0, 1]) for almost all x ∈ [0, 1]. Combined with the observation π L1 ([0, 1]) ≡ L1 ([0, 1]2 ) [123, Section 6.5], the following that L1 ([0, 1])⊗ statement follows easily. Proposition 2.5. Let m : B([0, 1]) → L1 ([0, 1]) be the vector measure given by m(B) = χB , B ∈ B([0, 1]). A function f : [0, 1] → L1 ([0, 1]) is π L1 ([0, 1]) if and only if there exists a function m-integrable in L1 ([0, 1])⊗ 1 2 k ∈ L ([0, 1] ) such that the set {x ∈ [0, 1] : f (x) = [k(x, · )] in L1 ([0, 1])} is a set of full measure. If f is m-integrable and A ∈ B([0, 1]), then [f ⊗ m](A) is equal to the equivalence class in L1 ([0, 1]2 ) of the function (x, y) → χA (x)k(x, y), (x, y) ∈ [0, 1]2 . Remark 2.4. Let (Σ, S) be a measurable space and X, Y Banach spaces. Bilinear integration of a function φ : Σ → X with respect to a measure π Y is more singular than m : S → Y in the projective tensor product X ⊗  Y because the bibilinear integration in the injective tensor product X ⊗ linear map    (x , y ) −→ φ, x d m, y  , x ∈ X  , y  ∈ Y  , Σ

is separately continuous, so m necessarily has finite X-semivariation in  Y , see [47, p. 327]. Consequently, the integral in the sense of Definition X⊗ 2.2 coincides with Bartle’s bilinear integral for the injective tensor product topology. Bilinear integration in injective tensor products is considered in [102] and [47] gives an application to generalised Carleman operators [55].

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We shall need the following ‘uniform’ version of the Lebesgue dominated convergence theorem for vector valued functions. Lemma 2.4. Let (Γ, T , ν) be a finite measure space, 1 ≤ p < ∞ and X a Banach space. Suppose that I is an index set and fn,ι : Γ → X, n = 1, 2, . . . , ι ∈ I, are strongly ν-measurable functions for which there exists a nonnegative function g ∈ Lp (ν) with the property that for each n = 1, 2, . . . and ι ∈ I, the bound fn,ι(γ) ≤ g(γ) holds for ν-almost all γ ∈ Γ. Suppose that for every > 0, limn→∞ [supι∈I ν({fn,ι − fι  ≥ })] = 0. Then for each ι ∈ I, the X-valued function fι belongs to Lp (ν; X) and $ %  p lim sup fn,ι (γ) − fι (γ) dν(γ) = 0. n→∞

ι∈I

Γ

Proof. The usual argument applies: for each N > 0,   fn,ι (γ) − fι (γ)p dν(γ) ≤ fn,ι (γ) − fι (γ)p dν(γ) Γ

{g≥N }

+ p ν({g ≤ N }) + 2p N p ν({fn,ι − fι  ≥ })  p ≤2 g(γ)p dν(γ) {g≥N }

+ ν(Γ) + 2p N p ν({fn,ι − fι  ≥ }). p

Choosing small enough, then N large enough, and then n sufficiently large, we can ensure that supι∈I Γ fn,ι(γ) − fι (γ)p dν(γ) is as small as we like. Proposition 2.6. Let (Γ, T , ν) be a measure space, 1 ≤ p < ∞ and X a Banach space. Suppose that I is an index set and fn,ι : Γ → X, n = 1, 2, . . . , ι ∈ I, are strongly ν-measurable functions for which there exists a nonnegative function g ∈ Lp (ν) with the property that for each n = 1, 2, . . . and ι ∈ I, the bound fn,ι (γ) ≤ g(γ) holds for ν-almost all γ ∈ Γ. Suppose that for ν-almost all γ ∈ Γ, lim [sup{fn,ι(γ) − fι (γ)}] = 0.

n→∞ ι∈I

Then for each ι ∈ I, the X-valued function fι belongs to Lp (ν; X) and $ %  p lim sup fn,ι (γ) − fι (γ) dν(γ) = 0. n→∞

ι∈I

Γ

Proof. Let > 0 and choose a set Γ of finite ν-measure so large that   fn,ι (γ) − fι (γ)p dν(γ) ≤ 2p g(γ)p dν(γ) < , Γc

Γc

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for all ι ∈ I and n = 1, 2, . . . . By the preceding result, it is enough to prove that lim [sup ν({γ ∈ Γ : fn,ι (γ) − fι (γ) ≥ δ})] = 0,

n→∞ ι∈I

(2.5)

for every > 0 and δ > 0. If for every countable subfamily J of I, (2.5) is true when the index set I is replaced by J, then it is true for the index set I itself, so we may assume from the outset that I is itself countable. The argument is standard: for each δ > 0, the set ∞ 

{γ ∈ Γ : fn,ι (γ) − fι (γ)X < δ}

m=1 ι∈I n≥m

has full ν-measure in Γ . Let α > 0. Then for some n0 = 1, 2, . . . , we have    ∞ {γ ∈ Γ : fn,ι (γ) − fι (γ)X ≥ δ} < α. (2.6) ν ι∈I n=n0

For every ι ∈ I and n ≥ n0 , the set Sn,ι (δ) := {γ ∈ Γ : fn,ι(γ) − fι (γ)X ≥ δ} is contained in the set in the argument of the measure ν in (2.6), so ν(Sn,ι (δ)) < α, for every ι ∈ I and n ≥ n0 . But α is any positive number, so (2.5) follows. The following result is an analogue of the Fubini-Tonelli theorem in which the integral with respect to a product measure is replaced by a bilinear integral—the existence of the ‘iterated integrals’ guarantees the existence of the bilinear integral. Theorem 2.6. Let (Γ, T , ν) be a σ-finite measure space, 1 ≤ p < ∞, (Ω, S) a measurable space and m : S → X a measure with values in a Banach space X. Let f : Ω × Γ → C be an (S ⊗ T )-measurable function with the property that (i) f (ω, ·) ∈ Lp (ν), for m-almost all ω ∈ Ω, (ii) f (·, γ) is m-integrable in X for ν-almost all γ ∈ Γ, and (iii) there exist g ∈ Lp (ν) such that for every A ∈ S, the bound     f (ω, γ) dm(ω)X ≤ g(γ) A

holds for ν-almost all γ ∈ Γ.

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p Then ω → f (ω, ·), ω& ∈ Ω, is m-integrable ' in L (ν; X), and for ν-almost all γ ∈ Γ, the equality Ω f (ω, ·) ⊗ dm(ω) (γ) = Ω f (ω, γ) dm(ω) holds.

Proof. We prove the result first in the case that ν is a finite measure and f ≥ 0. Suppose that sn , n = 1, 2, . . . , are (S ⊗ T )-simple functions increasing to f . Each function sn (·, γ), n = 1, 2, . . . , is bounded and so m-integrable, for each γ ∈ Γ. Then, for every A ∈ S and n = 1, 2, . . . ,

 A sn (ω, γ) dm(ω) ≤ 4g(γ) for ν-almost all γ ∈ Γ, because the inequality μ(u)X ≤ 4u∞ sup μ(A)X A∈S

is valid for any vector measure μ : S → X with values in a Banach space X, and any bounded μ-measurable function u.

Now for ν-almost all γ ∈ Γ, A sn (ω, γ) dm(ω) → A f (ω, γ) dm(ω), uniformly for all A ∈ S, by dominated convergence for vector ' mea&

s (ω, ·) dm(ω) : γ → sures, Theorem 1.7. The X-valued functions A n

A sn (ω, γ) dm(ω), n = 1, 2, . . . , therefore converge in X, uniformly for A ∈ S, at ν-almost all points γ ∈ Γ, and are dominated in norm by 4g. Dominated convergence in Lp (ν; X), ' Proposition 2.6, ensures that the X&

valued functions A sn (ω, ·) dm(ω) , n = 1, 2, . . . , converge in Lp (ν; X), uniformly for all A ∈ S. An appeal to (i) and dominated convergence shows that sn (ω, ·) → f (ω, ·) in Lp (ν), for m-almost all ω ∈ Ω. Unfortunately, the Lp (ν)-valued functions sn (ω, ·), n = 1, 2, . . . , need not be Lp (ν)-valued S-simple functions, so we cannot apply our definition of bilinear integration without further analysis. Let B ∈ S ⊗ T . If Bk belongs to the algebra generated by product sets from S and T and χBk ΔB → 0 m ⊗ ν-a.e. as k → ∞, then by Fubini’s theorem, for ν-almost all γ ∈ Γ, χBk ΔB (ω, γ) → 0 for m-almost all ω ∈ Ω. By dominated convergence for vector mea

→ A χB (ω, γ) dm(ω) in X, uniformly for all sures, A χBk (ω, γ) dm(ω)

A ∈ S. The bound A χBk (ω, γ) dm(ω) ≤ m(A), k = 1, 2, . . . , and the finiteness of ν ensures that the X-valued functions A χBk (ω, ·) dm(ω), k = 1, 2, . . . , converge in Lp (ν; X), uniformly for A ∈ S by Proposition 2.6. Moreover, by another appeal to dominated convergence and Fubini’s theorem, χBk (ω, ·) → χB (ω, ·) in Lp (ν) for m-almost all ω ∈ Ω. p &But

χBk , k = 1, 2, . '. . , are L (ν)-valued S-simple functions for which χ (ω, ·) ⊗ dm(ω) (γ) = A χBk (ω, γ) dm(ω), k = 1, 2, . . . , so the funcA Bk tion ω → χB (ω, ·), ω ∈ Ω, is necessarily m-integrable in Lp (ν; X) and the

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equality

$

%  χB (ω, ·) ⊗ dm(ω) (γ) = χB (ω, γ) dm(ω)

A

A

holds for ν-almost all γ ∈ Γ. Now let λ be a scalar measure equivalent to m. The convergence of strongly measurable vector valued functions λ-a.e. implies convergence in measure, so by the argument above, we can find scalar valued simple functions φn , n = 1, 2, . . . , based on product sets S × T , with S ∈ S and T ∈ T , such that λ({ω ∈ Ω : sn (ω, ·) − φn (ω, ·)Lp (ν) ≥ 1/n}) < 1/n and      [sn (ω, ·) − φn (ω, ·)] ⊗ dm(ω) ≤ 2−n , n = 1, 2, . . . .   A

p

By passing to subsequences, if necessary, we may suppose that sn (ω, ·) − φn (ω, ·)Lp (ν) → 0 for m-almost all ω ∈ Ω, as n

→ ∞. Then φn (ω, ·) → f (ω, ·) in Lp (ν) for m-almost all ω ∈ Ω and A φn (ω, ·) ⊗ dm(ω), n = 1, 2, . . . , converges in Lp (ν; X), uniformly for A ∈ S. Because the functions ω → φn (ω, ·), ω ∈ Ω, are actually Lp (ν)-valued S-simple functions for n = 1, 2, . . . , it follows that the function ω → f (ω, ·) is m-integrable and $ %   f (ω, ·) ⊗ dm(ω) (γ) = lim φn (ω, γ) dm(ω) = f (ω, γ) dm(ω), n→∞

Ω

Ω

Ω

for ν-almost all γ ∈ Γ. We now consider the case of a complex valued function f . If the inequality (iii) holds, then it also holds for the real and imaginary parts of f , and their positive and negative parts, possibly with g replaced by 4g, so the result follows for all complex valued functions f in the case that ν(Γ) < ∞. Now suppose that ν is σ-finite and let Γk ∈ T , k = 1, 2 . . . , be increasing sets with finite ν-measure, whose union is Γ. If conditions (i)-(iii) hold, then ω → Q(Γk )f (ω, ·) is m-integrable in Lp (ν) for each k = 1, 2, . . . . Here Q is the spectral measure of multiplication by characteristic functions acting on Lp (ν). By (iii), for ν-almost all γ ∈ Γ, we have $  %     [Q(B)f (ω, ·)] ⊗ dm(ω) (γ)   ≤ χB (γ)g(γ), A ∈ S, B ∈ T . A X

In particular, the functions A [Q(Γn )f (ω, ·)]⊗dm(ω), n = 1, 2, . . . , converge in Lp (ν; X), uniformly for A ∈ S, as n → ∞. Let λ be a measure equivalent to m and choose Lp (ν)-valued S-simple functions φn such that λ({ω ∈ Ω : Q(Γn )f (ω, ·)−φn (ω, ·)Lp (ν) ≥ 1/n}) < 1/n and      [Q(Γn )f (ω, ·) − φn (ω, ·)] ⊗ dm(ω) ≤ 2−n , A ∈ S, n = 1, 2, . . . .   A

p

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Then limn→∞ A [φn (ω, ·)] ⊗ dm(ω) = limn→∞ A [Q(Γn )f (ω, ·)] ⊗ dm(ω) in Lp (ν; X), uniformly for A ∈ S. On passing to a subsequence, if necessary, as n → ∞, the functions φn (ω, ·) converge in Lp (ν), for m-almost all ω ∈ Ω, to limn→∞ Q(Γn )f (ω, ·) = f (ω, ·). It follows that f (ω, ·) is m-integrable in Lp (ν) and for ν-almost all γ ∈ Γ, % $  f (ω, ·) ⊗ dm(ω) (γ) = lim [χΓn (γ)f (ω, γ)] dm(ω) n→∞ Ω Ω  = f (ω, γ) dm(ω). Ω

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Operator traces

The study of traces and the operator ideal of trace class operators has a venerable tradition with applications to quantum physics and scattering theory [127], algebraic geometry [9] and non-commutative geometry [21]. The trace of a linear map T : H → H on a Hilbert space H with finite di mension n = 1, 2, . . . , is the number nj=1 ajj for any matrix representation {ajk }nj,k=1 of the linear map T with respect to a basis of H. By analogy with the finite dimensional case, if Tk : L2 ([0, 1]) → L2 ([0, 1]) is a linear operator with an integral kernel k, then one might hope that  1 tr(Tk ) = k(x, x) dx (3.1) 0

if Tk is a trace class operator as in Section 3.1 below. However, {(x, x) : x ∈ [0, 1] } is a set of measure zero in [0, 1]2 and if k = k1 almost everywhere on [0, 1]2 , then Tk = Tk1 , so the right-hand side of equation (3.1) is not well-defined. It turns out that formula (3.1) is valid if the kernel k is continuous on [0, 1] × [0, 1] or otherwise has a specific representation such as equation (3.3) below. The of formula (3.1) may be viewed as a bilinear inte 1 right-hand side 1 gral 0 Tk , dm := 0 Φk dm of the type considered in Example 2.1 of the preceding chapter. Unless Tk is a positive operator in the sense of complex Hilbert spaces, that is, (Tk u, u) ≥ 0 for all u ∈ L2 ([0, 1]), the convergence 1 of the bilinear integral 0 Tk , dm is not sufficient to guarantee that Tk is a trace class operator. As we shall see, the collection of bounded linear op1 erators Tk for which the bilinear integral 0 Tk , dm converges constitutes a lattice ideal C1 (L2 ([0, 1])) of regular operators on L2 ([0, 1]) that includes 71

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the operator ideal C1 (L2 ([0, 1])) of trace class operators on L2 ([0, 1]). In Chapter 7, the bilinear integral Σ Tk , dm is featured in the proof of the CLR inequality for dominated semigroups on a Hilbert space L2 (Σ, E, μ). 3.1

Trace class operators

The singular values {λj }∞ j=1 of a compact linear operator T : H → H on a Hilbert space H are the eigenvalues of the compact selfadjoint operator ∞ 1 (T ∗ T ) 2 . The operator T is called trace class if T 1 = j=1 λj < ∞, or ∞ equivalently, j=1 |(T hj , hj )| < ∞ for any orthonormal set {hj }∞ j=1 in H. The collection C1 (H) of trace class operators on H is an operator ideal and Banach space with the norm ·1 . The references [53,127] are encyclopedias of trace class operators. In the case of an infinite dimensional separable Hilbert space H, the trace of T ∈ C1 (H) is defined to be tr(T ) =

∞ 

(T hj , hj )

j=1

with respect to any orthonormal basis {hj }∞ j=1 of H [127, Theorem 3.1]. Lidskii’s equality asserts that tr(A) is actually the sum of the eigenvalues of the compact operator T [127, Theorem 3.7]. Suppose that Tk : L2 ([0, 1]) → L2 ([0, 1]) is a bounded linear operator with an integral kernel k : [0, 1] × [0, 1] → C, that is, k is a measurable 1 function such that for every f ∈ L2 ([0, 1]), the integral 0 |k(x, y)f (y)| dy

1 is finite for almost all x ∈ [0, 1] and (Tk f )(x) = 0 k(x, y)f (y) dy for almost all x ∈ [0, 1]. Any trace class operator T : L2 ([0, 1]) → L2 ([0, 1]) is an integral operator with distinguished kernel k for which (3.1) is valid due to the representation ∞  λj φj (h, ψj ), h ∈ L2 ([0, 1]), (3.2) T : h −→ j=1

1 with respect to the L2 -inner product (f, g) = 0 f (x)g(x) dx, f, g ∈ L2 ([0, 1]), and the singular values {λj }∞ j=1 of T . The sets {φj } and {ψj } of vectors are orthonormal in L2 ([0, 1]), so the representation (3.1) is valid for the distinguished integral kernel k defined by k(x, y) =

∞  j=1

λj φj (x)ψj (y)

(3.3)

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for all x, y ∈ [0, 1] for which the right-hand side is absolutely convergent. In particular, if Tk : L2 ([0, 1]) → L2 ([0, 1]) is a trace class linear operator with a continuous integral kernel k on [0, 1]2 , then (3.1) holds [127, Theorem 3.9]. C. Brislawn observed in [18] that if T : L2 ([0, 1]) → L2 ([0, 1]) is a trace class linear operator and k0 is any integral kernel of T , then the kernel k = lim→0+ ϕ ∗ k0 has the property that T = Tk and equation (3.1) holds. Here ϕ (x) = −2 ϕ(x/ ), x ∈ R2 , > 0, for some nonnegative

function ϕ on 2 2 R that is zero outside [−1, 1] and with the property that R2 ϕ(x) dx = 1 and ϕ has an integrable radially decreasing majorant: the characteristic function ϕ = χ[− 12 , 12 ]2 will do. The convolution u ∗ v of u, v ∈ L1 (Rn ) is defined for almost all x ∈ Rn by the formula  u(x − y)v(y) dy. u ∗ v(x) = Rn

The convolution u∗v of a function u ∈ L1 (R2 ) with v ∈ L1 ([0, 1]2 ) is defined almost everywhere in [0, 1]2 by setting v equal to zero outside [0, 1]2 . Then the map k0 −→ lim→0+ ϕ ∗ k0 is a smoothing operator for which the value of k = lim→0+ ϕ ∗ k0 at a point (x, x) of the diagonal is defined by averages in [0, 1]2 about (x, x) for almost every x ∈ [0, 1]. A related idea appears in [53, Theorem 8.4]. It is clear that this idea need not be confined to trace class operators. Example 3.1 ([18, Example 3.2]). The Volterra operator T is defined by  x f (y) dy, x ∈ [0, 1], for f ∈ L2 ([0, 1]). (T f )(x) = 0

Then T is defined by the integral kernel k0 = χ{y0 In the formula above, the function f is put equal to zero outside the square [0, 1]2 and Cr = [−r, r] × [−r, r] for r > 0. The maximal function M (f ) is equivalent to the maximal function obtained by averaging over centred disks [54, Exercise 2.1.3], but for the purposes of the present discussion it is convenient to emphasise the product structure of the unit square. According to Lebesgue’s differentiation theorem [54, Corollary 2.1.16], if f ∈ L1 ([0, 1]2 ) we have

f (x + t) dt lim Cr = f (x) (3.5) r→0+ |Cr | for almost all x ∈ [0, 1]2 , so that |f | ≤ M (f ) almost everywhere and the set Lf of Lebesgue points x ∈ [0, 1]2 of f where

|f (x + t) − f (x)| dt =0 lim Cr r→0+ |Cr |

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has full measure in [0, 1]2 . Let φ : ]−1, 1[ → [0, ∞[ be a continuous function with compact sup1 port and −1 φ(t) dt = 1. For the function ϕ : R2 → R defined by ϕ(x, y) = φ(x)φ(y), for x, y ∈ ]−1, 1[ and zero outside ]−1, 1[2 , we set ϕ (x) = −2 ϕ(x/ ), x ∈ R2 , > 0. Then a variant of Lebesgue’s differentiation theorem for an integrable function f shows that ϕ ∗ f → f in Lp ([0, 1]2 ) for 1 ≤ p < ∞ and almost everywhere as → 0+ [54, Corollary 2.1.17]. We are interested in the class of bounded linear operators Tk : L2 ([0, 1]) → L2 ([0, 1]) with a distinguished kernel k : [0, 1]2 → C for which |k| also defines a bounded linear operator T|k| : L2 ([0, 1]) → L2 ([0, 1]) (absolute integral operators) and the intersection Lk ∩diag of the Lebesgue set Lk of k with the diagonal diag = {(x, x) : x ∈ [0, 1]} has full linear measure. Because constant functions belong to L2 ([0, 1]), the kernel k necessarily belongs to L1 ([0, 1]2 ), so we first look at a subspace of L1 ([0, 1]2 ) consisting of functions f for which Lf ∩diag has full linear measure. 3.3

The Banach function space of traceable functions

Let (Σ, B, μ) be a σ-finite measure space. The space of all μ-equivalence classes of scalar functions measurable with respect to B is denoted by L0 (μ). It is equipped with the topology of convergence in μ-measure over sets of finite measure and vector operations pointwise μ-almost everywhere. The following definition has already been mentioned in Example 2.1. Definition 3.1 (Banach function space). Any Banach space X that is a subspace of L0 (μ) with the properties that (i) X is an order ideal of L0 (μ), that is, if g ∈ X, f ∈ L0 (μ) and |f | ≤ |g| μ-a.e., then f ∈ X, and (ii) if f, g ∈ X and |f | ≤ |g| μ-a.e., then f X ≤ gX , is called a Banach function space (based on (Σ, B, μ)). The set of f ∈ X with f ≥ 0 μ-a.e. is written as X+ . The map J : [0, 1] → [0, 1]2 defined by J(x) = (x, x), x ∈ [0, 1], maps [0, 1] homeomorphically onto diag. For f ∈ L1 ([0, 1]2 ), the extended real 1 number ρ(f ) ∈ [0, ∞] is defined by ρ(f ) = f 1 + 0 M (f ) ◦ J(x) dx with M (f ) given by equation (3.4).

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Proposition 3.1. The space L1 (ρ) = {f ∈ L1 ([0, 1]2 ) : ρ(f ) < ∞} with norm ρ is a Banach function space continuously embedded in L1 ([0, 1]2 ). Proof. Properties (i) and (ii) follow from the observation that M (f ) ≤ M (g) everywhere if |f | ≤ |g| almost everywhere on [0, 1]2 . According to [96, Proposition 2.6.2], it is enough to prove that L1 (ρ) has the RieszFischer property. Suppose that fj ≥ 0 almost everywhere for j = 1, 2, . . . ∞ ∞ and j=1 ρ(fj ) < ∞. Then monotone convergence ensures that j=1 fj converges almost everywhere in [0, 1]2 and in L1 ([0, 1]2 ) to a nonnegative ∞ integrable function f and M (f ) ≤ j=1 M (fj ) everywhere on [0, 1]2 and ∞ so ρ(f ) ≤ j=1 ρ(fj ). Consequently, the function space L1 (ρ) possesses the Riesz-Fischer property. The inequality f 1 ≤ ρ(f ) ensures that the inclusion of L1 (ρ) in L1 ([0, 1]2 ) is continuous. Suppose that f ∈ L1 (ρ). By [54, Corollary 2.1.12], there exists C > 0 independent of f such that sup>0 |(ϕ ∗ f )(x)| ≤ CM (f )(x) for every x ∈ [0, 1]2 , so if we let f˜ = lim sup→0+ (ϕ ∗ f ) on [0, 1]2 , then f˜ = f almost everywhere on [0, 1]2 by [54, Corollary 2.1.17], |f˜ ◦ J| ≤ CM (f ) ◦ J and  1  1 ˜ |f (x, x)| dx ≤ C M (f ) ◦ J(x) dx < ∞, 0

0

1 so in this sense, elements of L (ρ) possess an integrable trace 0 |f˜(x, x)| dx

1 on diag ⊂ [0, 1]2 . However, the mapping f −→ 0 f˜(x, x) dx, f ∈ L1 (ρ), is only sublinear, so next we examine a subspace for which the lim sup can be replaced by a genuine limit almost everywhere on diag. If u and v are two real valued functions defined on [0, 1], the tensor product u ⊗ v : [0, 1]2 → R of u and v is defined by (u ⊗ v)(x, y) = u(x)v(y), x ∈ [0, 1]. A similar notation is used for the equivalence classes of functions so that [u ⊗ v] ◦ J := [u.v]. Then L∞ ([0, 1]) ⊗ L∞ ([0, 1]) denotes the linear space of all finite linear combinations of elements u ⊗ v with u, v ∈ L∞ ([0, 1]). Each element f of the finite tensor product L∞ ([0, 1]) ⊗ L∞ ([0, 1]) is essentially bounded on [0, 1]2 and M (f ) ≤ f ∞ , so f ∈ L1 (ρ) and f ◦ J ∈ L∞ ([0, 1]). ρ L∞ ([0, 1]) denote the norm closure of L∞ ([0, 1]) ⊗ Let L∞ ([0, 1])⊗ ∞ L ([0, 1]) in the Banach function space L1 (ρ). 1

Proposition 3.2. Let f ∈ L1 ([0, 1]2 ). Then f belongs to the Banach space ρ L∞ ([0, 1]) L∞ ([0, 1])⊗ ρ L∞ ([0, 1]) if and only if ϕ ∗ f → f in L1 (ρ) as → 0+. If f ∈ L∞ ([0, 1])⊗ 1 then (ϕ ∗ f ) ◦ J converges a.e. on [0, 1] and in L ([0, 1]) as → 0+.

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Proof. By an application of the Cauchy-Schwarz inequality and the L2 bound for the Hardy-Littlewood maximal operator [54, Theorem 2.1.6], there exists C > 0 such that if u, v ∈ L2 ([0, 1]), then  1  1 M (u ⊗ v)(x, x) dx ≤ M (u)(x)M (v)(x) dx 0

0

≤ Cu2 v2 .

(3.6)

Here M (u) and M (v) are the one-dimensional maximal functions of u and v defined as in formula (3.4). Suppose first that f = u ⊗ v for u, v ∈ L∞ ([0, 1]). Then ϕ ∗ f = (φ ∗ u) ⊗ (φ ∗ v) because ϕ = φ ⊗ φ and so  1  1 M (ϕ ∗ f − f )(x, x) dx ≤ C M (φ ∗ u − u)(x)M (v)(x) dx 0

0



+ 0

1

M (φ ∗ v − v)(x)M (u)(x) dx

≤ C  (φ ∗ u − u2 v2 + φ ∗ v − v2 u2 ) →0 as → 0+ by the Cauchy-Schwartz intequality and the L2 -bound for the Hardy-Littlewood maximal operator. Consequently, ϕ ∗ f → f in L1 (ρ) as → 0+ when f is a linear combination of products of functions belonging to L∞ ([0, 1]). There exists C > 0 such that ϕ ∗ f 1 ≤ Cf 1 for every f ∈ L1 ([0, 1]2 ) and  1  1 M (ϕ ∗ f )(x, x) dx ≤ C M (f )(x, x) dx, > 0. (3.7) 0

0

To check the inequality (3.7), suppose that ψ = π −1 χD1 for the unit disk D1 centred at zero in R2 and let ϕ˜ be the least decreasing radial majorant of ϕ. Because ϕ is continuous with compact support, ϕ˜ is integrable on R2 . Then ϕ˜ ∗ ψδ is a radial function for which  ∞ r(ϕ˜ ∗ ψδ )(re1 ) dr = ϕ˜ ∗ ψδ L1 (R2 ) 2π 0

= ϕ˜ L1 (R2 ) ψδ L1 (R2 ) = ϕ ˜ L1 (R2 ) . As in the proof of [54, Theorem 2.1.10], there exists C  > 0 such that sup,δ>0 ϕ˜ ∗ ψδ ∗ |f | ≤ C  M (f ). Because the maximal function (3.4) is equivalent to the maximal function for centred disks, there exists C ≥ 1

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such that M (ϕ ∗ f ) ≤ M (ϕ˜ ∗ f ) ≤ CM (f ) from which the inequality (3.7) follows. Consequently, the linear map f −→ ϕ ∗ f , f ∈ L1 (ρ), is continuous ρ L∞ ([0, 1]), then ϕ ∗ on L1 (ρ) for each > 0 so that if f ∈ L∞ ([0, 1])⊗ f → f in L1 (ρ) as → 0+. Because ϕ ∗ f ∈ C([0, 1]2 ) and C([0, 1]) ⊗ C[0, 1]) is dense in C([0, 1]2 ) in the uniform norm, it follows that ϕ ∗ f ∈ ρ L∞ ([0, 1]) for each > 0, and the limit of ϕ ∗ f in L1 (ρ) as L∞ ([0, 1])⊗ ρ L∞ ([0, 1]). → 0+ also belongs to L∞ ([0, 1])⊗ Let T∗ f = sup>0 |ϕ ∗ f |◦ J for f ∈ L1 (ρ). Then T∗ : L1 (ρ) → L1 ([0, 1]) is uniformly continuous. An argument similar to the proof of [54, Theorem 2.1.14] shows that (ϕ ∗ f ) ◦ J converges almost everywhere and in L1 ([0, 1]) as → 0+ for each f ∈ L1 (ρ). ρ L∞ ([0, 1]) and set f˜ = lim→0+ ϕ ∗ f wherever the Let f ∈ L∞ ([0, 1])⊗ 2 limit exists in [0, 1] and zero elsewhere. Writing f # for the corresponding function with φ replaced by χ[− 12 , 12 ] , it follows from equation (3.5) that f # = f˜ almost everywhere on [0, 1]2 and f # ◦ J = f˜ ◦ J almost everywhere on [0, 1], because the last equality certainly holds when f belongs to the ∞ ∞ 1 ˜ dense subspace

L ([0, 1]) ⊗ L ([0, 1]). In particular, f ◦ J ∈ L ([0, 1]) and the integral B f˜ ◦ J(x) dx, B ∈ B([0, 1]), does not depend on the choice of the function ϕ. Example 3.2. For a continuous function f on [0, 1]2 equal to zero on R2 \ [0, 1]2 , the continuous functions ϕ ∗ f converge uniformly to f on compact subsets of ]0, 1[2 [54, Theorem 1.2.19 (2)], so that f ∈ ρ L∞ ([0, 1]) and f˜ = f . Hence, C([0, 1]2 ) and C([0, 1])⊗C([0, 1]) L∞ ([0, 1])⊗ ρ L∞ ([0, 1]). are dense in L∞ ([0, 1])⊗ Functions belonging to W 1,1 (R2 ) or the space Lα,p (R2 ) of Bessel potentials on R2 also admit a trace on diag(R2 ) if p, αp > 1, see [1, Section 6.2]. A result of T. Carleman [22] shows there exists a continuous periodic function φ : R → C with period one such that  p ˆ |φ(n)| =∞ n∈Z

for all p < 2. If k(x, y) = φ(x−y), then k is a continuous

1 kernel, M (k)◦J ≤ φ∞ and k(x, x) = φ(0) for all x ∈ [0, 1] and so 0 k(x, x) dx = φ(0), although the Hilbert-Schmidt operator Tk with kernel k is not a trace class operator. Because k ≤ φ∞ and a constant function is the kernel of a finite rank operator, the trace class operators do not form a lattice ideal in

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the Banach lattice of Hilbert-Schmidt operators despite being an operator ideal in L(L2 ([0, 1])). Example 3.3. The kernel χ{y 0} is itself a regular family of sets whose associated maximal function is equivalent to the one defined for cubes by formula (3.4). Furthermore, if g ∈ L∞ ([0, 1]) ⊗ L∞ ([0, 1]), then (ϕ ◦ T ) ∗ g → g in L1 (ρ) as → 0+, hence ϕ ∗ (g ◦ T −1 ) converges to g ◦ T −1 in L1 (ρ) as → 0+ as well. Taking g to be the characteristic functions of squares and T to be rotation through π/4 gives χ{y 0 such that sup>0 |(ϕ ∗ f0 )(x)| ≤ CM (f0 )(x) for every x ∈ [0, 1]2 . Because ρ([f0 ]) < ∞, f is μ-integrable. The statement now follows from the argument of [85, Proposition 2.13], as follows. Let K denote the space of (B ⊗ B)-simple functions on [0, 1] × [0, 1] and for any function f : [0, 1] × [0, 1] → C, let ⎧ ⎫ ∞ ∞ ⎨ ⎬  μ(|fj |) : f = fj . ρK (f ) = inf ⎩ ⎭ j=1

j=1

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The infimum is taken over all sums with fj ∈ K, j = 1, 2, . . . , and ∞  μ(|fj |) < ∞, such that f (x) = ilarly, let

j=1

∞ j=1

ρB×B (f ) = inf

fj (x) wherever the sum converges absolutely. Sim-

⎧ ∞ ⎨ ⎩

|cj |μ(Aj × Bj ) : f =

j=1

∞ 

cj χAj ×Bj

j=1

⎫ ⎬ ⎭

,

∞ where f (x) = j=1 cj χAj ×Bj for every x ∈ [0, 1] × [0, 1] such that the sum ∞ χ j=1 |cj | Aj ×Bj (x) is finite. The infimum of ρK (f ) is taken over a larger collection of functions than for ρB×B (f ), so ρK (f ) ≤ ρB×B (f ). A complex valued function may be decomposed into its real and imaginary parts, so suppose that f is real valued and ρK (f ) < ∞. Monotone convergence shows that μ(|f |) ≤ ρK (f ). Writing f = f + − f − in its positive f + = f ∨ 0 and negative f − = (−f ) ∨ 0 parts, there exist monotonically increasing sequences fj± , j = 1, 2, . . . , of (B ⊗ B)-simple functions such that   ∞ ∞  + − f= (fj+1 − fj+ ) + f1+ − (fj+1 − fj− ) + f1− j=1

j=1

pointwise on [0, 1] × [0, 1] and ∞ ∞   + − μ(fj+1 − fj+ ) + μ(fj+1 − fj− ) + μ(|f1 |). μ(|f |) = j=1

j=1

Consequently, ρK (f ) ≤ μ(|f |) and the equality ρK (f ) = μ(|f |) follows. If f is a (B ⊗ B)-measurable function with μ(|f |) = ∞, then clearly ρK (f ) = ∞ too. The representation is established once we show that ρK (f ) = ρB×B (f ).  The vector space H of all simple functions f = nj=1 cj χAj ×Bj with cj ∈ C, Aj ∈ B, Bj ∈ B, j = 1, . . . , n and n = 1, 2, . . . , is dense for both the seminorms ρK and ρB×B , so it suffices to establish the equality for all f ∈ H. We may also suppose that the rectangles Aj × Bj , j = 1, 2, . . . , n, are pairwise disjoint in the representation of f ∈ H as above. Then ρK (f ) = n j=1 |cj |μ(Aj )ν(Bj ) ≥ ρB×B (f ), so ρK (f ) = ρB×B (f ). ρ L∞ ([0, 1]), is uniquely The linear map J ∗ f = f˜ ◦ J, f ∈ L∞ ([0, 1])⊗ ∞ ρ L∞ ([0, 1]). determined by the Banach function space L ([0, 1])⊗ Corollary 3.1. There exists a unique continuous linear map ρ L∞ ([0, 1]) → L1 ([0, 1]) J ∗ : L∞ ([0, 1])⊗ such that J ∗ (u ⊗ v) = u.v for every u, v ∈ L∞ ([0, 1]).

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It follows from Theorem 1.8, that the projective tensor product π L2 ([0, 1]) L2 ([0, 1])⊗ is the set of all sums ∞ ∞   φj ⊗ ψj a.e., with φj 2 ψj 2 < ∞. k= j=1

(3.8)

j=1

π L2 ([0, 1]) is given by The norm of k ∈ L2 ([0, 1])⊗ ⎧ ⎫ ∞ ⎨ ⎬ kπ = inf φj 2 ψj 2 ⎩ ⎭ j=1

where the infimum is taken over all sums for which the representation (3.8) π L2 ([0, 1]) is actually the completion holds. The Banach space L2 ([0, 1])⊗ of the algebraic tensor product L2 ([0, 1]) ⊗ L2 ([0, 1]) with respect to the projective tensor product norm [123, Section 6.1]. The estimate (3.6) establishes the following result. π L2 ([0, 1]) emProposition 3.4. The projective tensor product L2 ([0, 1])⊗ ρ L∞ ([0, 1]). beds onto a proper dense subspace of L∞ ([0, 1])⊗ There is a one-to-one correspondence between the space of trace class π L2 ([0, 1]), so that the trace operators acting on L2 ([0, 1]) and L2 ([0, 1])⊗ π L2 ([0, 1]) given, for class operator Tk has an integral kernel k ∈ L2 ([0, 1])⊗ example, by formula (3.3). If the integral kernel k defined by equation (3.8) has the property that k(x, y) =

∞ 

φj (x)ψj (y)

j=1

 for all x, y ∈ [0, 1] such that the sum ∞ j=1 |φj (x)ψj (y)| is finite, then k is the integral kernel of a trace class operator Tk and the equality  1 ∞  1  φj (x)ψj (x) dx = k(x, x) dx tr(Tk ) = j=1

0

0

holds. Moreover, if K is any integral kernel of the trace class operator Tk , ˜ then K(x, x) = k(x, x) for almost all x ∈ [0, 1], see [18, Theorem 3.1] or the proof of Theorem 3.3 below in the case of L2 (μ) for a σ-finite measure μ. The representation of Proposition 3.3 for elements of the Banach space ∞ ρ L∞ ([0, 1]) may be viewed as a substitute for the representation L ([0, 1])⊗ π L2 ([0, 1]). (3.8) of an element of the projective tensor product L2 ([0, 1])⊗

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Example 3.4. Another way to view the trace tr(Tk ) of a trace class operator Tk : L2 ([0, 1]) → L2 ([0, 1]) with an integral kernel k is as a type of bilinear integral with respect to the L2 ([0, 1])-valued vector measure n χ m : B −→ χB , B ∈ B([0, 1]). For example, if k = j=1 Bj ⊗ fj 2 for Borel subsets Bj of [0, 1] and fj ∈ L ([0, 1]), j = 1, . . . , n and Φk : [0, 1] → L2 ([0, 1]) is the L2 ([0, 1])-valued simple function defined by n Φk (x) = j=1 χBj (x).fj , x ∈ [0, 1], then 

Φk , dm = B

n   j=1

 fj (x) dx =

B∩Bj

k(x, x) dx B

1]). and B Φk ⊗ dm = (χB ⊗ 1).k ∈ L2 ([0, 1]) ⊗ L 2 ([0, 1]) for B ∈ B([0,

By Proposition 3.4 the bilinear integrals B Φk , dm and B Φk ⊗ dm π L2 ([0, 1]) where Tk is a trace class also makes sense for k ∈ L2 ([0, 1])⊗ operator and  1

Φk , dm tr(Tk ) = 0

independently of the integral kernel k representing the operator Tk . Example 3.5. On the other hand, if k ∈ L1 ([0, 1]2 ), then by Fubini’s Theorem, the function Φk (x) = f (x, ·) has values in L1 ([0, 1]) for almost

1 π L1 ([0, 1]) ≡ all x ∈ [0, 1] and 0 Φk ⊗ dm = k is an element of L1 ([0, 1])⊗ 1 2 L ([0, 1] ), see Proposition 2.5 above.

1 Now suppose that the bilinear integral 0 Φk ⊗ dm

≡ k belongs to the ρ L∞ ([0, 1]) of L1 ([0, 1]2 ). Then Φk , dm is defined subspace L∞ ([0, 1])⊗ B for each Borel set B contained in [0, 1] by the formula   1   J∗

Φk , dm = Φk ⊗ dm dλ, (3.9) 0

B

B



where the linear map J is defined immediately preceding Corollary 3.1 ρ L∞ ([0, 1]) consists of above. In this sense, the linear space L∞ ([0, 1])⊗ 1 2 the traceable elements of L ([0, 1] ), that is, the kernel k of the operator Tk is associated with a “generalised trace” of Tk whenever k belongs to ρ L∞ ([0, 1]). L∞ ([0, 1])⊗ Example 3.6. Now let us recast Example 3.1 in the present setting of vector integration theory. We have a bounded linear operator Tk : L2 ([0, 1]) → L2 ([0, 1]) whose integral kernel k need not be the integral kernel of a trace class operator acting on L2 ([0, 1]).

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In the notation of Section 2.3, suppose that X = L2 ([0, 1]), Y = C, ρ L∞ ([0, 1]). Then E = L1 ([0, 1]2 ) and F = L∞ ([0, 1])⊗ L2 ([0, 1]) ≡ L(X, Y ) = X  . Suppose that Φk takes values in L(X, C) ≡ L2 ([0, 1]) = X, that is, k is a Carleman kernel [60, Section 11]. Then the vector measure m : B([0, 1]) → X has infinite variation on any Borel set with positive Lebesgue measure. The inclusion L2 ([0, 1]) ⊗ L2 ([0, 1]) ⊂ F is valid and the map

1 f −→ 0 (J ∗ f )(x) dx, f ∈ F , is a continuous linear extension of the duality map  1 u(x)v(x) dx. u, v ∈ L2 ([0, 1]). u ⊗ v −→ 0

1 In the situation of Example 3.1, we have 0 Φk ⊗ dm ∈ F and the singular

1 bilinear integral 0 Φk , dm defined by formula (3.9 equals 12 . As a matter of notation, if T : L2 ([0, 1]) → L2 ([0, 1]) has an integral ρ L∞ ([0, 1]), then the integral 1 Φk , dm kernel k belonging to L∞ ([0, 1])⊗ 0 exists and is independent of any integral kernel k representing T , so it

1 makes sense to write 0 T, dm instead of the more cumbersome notation

1

Φk , dm and we shall do so forthwith. 0 3.4

Traceable operators on Banach function spaces

It is clear that the ideas of the preceding section are concerned mainly with the order properties of the Banach function space L2 ([0, 1]), although the smoothing operators k −→ ϕ ∗ k, k ∈ L1 ([0, 1]2 ), > 0, depend on the group structure of R2 . For a σ-finite measure space (Σ, B, μ), the same result is achieved by taking the maximal function with respect to a suitable ( filtration En n∈N for which B = n En , that is, En n∈N is an increasing family of σ-algebras contained in the σ-algebra B and the smallest σ-algebra containing En for every n ∈ N is B. The filtration determined by dyadic partitions En n∈N of R localised to [0, 1] gives the results of Section 3.3 above, that is, for each n = 1, 2, . . . , the algebra En is the collection of all finite unions of sets [(k − 1)/2n , k/2n ), k = 1, . . . , 2n . The right-hand endpoint constitutes a negligible set. Let X be a Banach function space based on the σ-finite measure space (Σ, B, μ), as in Definition 3.1. A continuous linear operator T : X → X is called positive if T : X+ → X+ . The collection of all positive continuous

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linear operators on X is written as L+ (X). If the real and imaginary parts of a continuous linear operator T : X → X can be written as the difference of two positive operators, it is said to be regular. The modulus |T | of a regular operator T is defined by |T |f = sup |T g|, |g|≤f

f ∈ X+ .

The collection of all regular operators is written as Lr (X) and it is given the norm T −→ |T |, T ∈ Lr (X) under which it becomes a Banach lattice [96, Proposition 1.3.6]. A continuous linear operator T : X → X has an integral kernel k if k : Σ × Σ → C is a Borel measurable function such that T = Tk for the operator given by  k(x, y)f (y) dμ(y), μ-almost all x ∈ Σ, (Tk f )(x) = Σ

in the sense that, for each f ∈ X, we have Σ |k(x, y)f (y)| dμ(y) < ∞ for μ-almost all x ∈ Σ and the map x −→ Σ k(x, y)f (y) dμ(y) is an element of X. If Tk ≥ 0, then k ≥ 0 (μ ⊗ μ)-a.e. on Σ × Σ [96, Theorem 3.3.5]. A continuous linear operator T is an absolute integral operator if it has an integral kernel k for which T|k| is a bounded linear operator on L2 (μ). Then |Tk | = T|k| [96, Theorem 3.3.5] and the kernel k is (μ ⊗ μ)-integrable on any product set A× B with finite measure. The collection of all absolute integral operators is a lattice ideal in Lr (X) [96, Theorem 3.3.6]. n Suppose that T ∈ L(X) has an integral kernel k = j=1 fj ⊗ χAj that is an X-valued simple function with μ(Aj ) < ∞. Then it is natural to view   n  

T, dm := fj dμ = k(x, x) dμ(x) Σ

j=1

Aj

Σ

as a bilinear integral. Our aim is to extend the integral to a wider class of absolute integral operators acting on the Banach function space X. Suppose that for each n = 1, 2, . . . , the collection Pn of sets belonging to the σ-algebra B is a countable partition of Σ into sets with finite measure such that Pn+1 is a refinement of Pn for each n = 1, 2, . . . , that is, every element of Pn is the union of elements of Pn+1 . Then the σ-algebra En generated by the partition Pn of Σ is the collection of all unions of elements ( of Pn , so that En ⊂ En+1 for n = 1, 2, . . . . Suppose that B = n En , the smallest σ-algebra containing all En , n = 1, 2, . . . . It follows that B is countably generated. The filtration En n∈N is denoted by E. Suppose that k ≥ 0 is a Borel measurable function defined on Σ × Σ that is integrable on every set U × V for U, V ∈ P1 . For each x ∈ Σ,

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the set Un (x) is the unique element of the partition Pn containing x. For each n = 1, 2, . . . , the conditional expectation kn = E(k|En ⊗ En ), as in Subsection 4.1.1 below, can be represented for μ-almost all x, y ∈ Σ as   1 E(k|En ⊗ En )(x, y) = k(s, t) dμ(s)dμ(t) μ(Un (x))μ(Un (y)) Un (x) Un (y)

 U×V k d(μ ⊗ μ) χ (3.10) = U×V (x, y). μ(U )μ(V ) U,V ∈Pn

The point here is that the formula above defines a distinguished element of the conditional expectation E(k|En ⊗ En ) that possesses a trace on the diagonal of Σ × Σ, that is, B −→ E(χB |En ⊗ En )(x, y), B ∈ B, x, y ∈ Σ, is a regular conditional measure [16, Definition 10.4.1]. Let N be the set of all x ∈ Σ for which there exists n = 1, 2, . . . such that μ(Un (x)) = 0. Then μ(Um (x)) = 0 for all m > n because Pm is a refinement of Pn if m > n. Moreover, N is μ-null because N ⊂

∞   {U ∈ Pn : μ(U ) = 0}. n=1

If 0 ≤ k1 ≤ k2 (μ ⊗ μ)-a.e., then E(k1 |En ⊗ En )(x, y) ≤ E(k2 |En ⊗ En )(x, y),

n = 1, 2, . . . ,

for all (x, y) ∈ N c × N c . In particular, E(k1 |En ⊗ En )(x, x) ≤ E(k2 |En ⊗ En )(x, x), for all x ∈ N c and the representation E(k|En ⊗ En )(x, x) =

 U∈Pn

U×U

n = 1, 2, . . . ,

k d(μ ⊗ μ)

μ(U )2

χU (x)

on the diagonal is valid μ-almost everywhere. Although diag(Σ × Σ) = {(x, x) : x ∈ Σ} may be a set of (μ ⊗ μ)-measure zero, the application of the conditional expectation operators k −→ E(k|En ⊗ En ), n = 1, 2, . . . , has the effect of regularising k. By an appeal to the Martingale Convergence Theorem [16, Theorem 10.2.3], the function kn converges (μ ⊗ μ)-a.e. to k as n → ∞. For any Borel measurable function f : Σ × Σ → C that is integrable on every set U × V for U, V ∈ P1 , let ME 2 (f )(x, y) = sup E(|f ||En ⊗ En )(x, y), n∈N

x, y ∈ Σ,

(3.11)

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be the maximal function associated with the martingale E(|f ||En ⊗En ) n∈N . The maximal function associated with dyadic partitions Pn of [0, 1) into intervals [(k − 1)/2n , k/2n ), k = 1, 2, . . . , 2n of length 2−n is equivalent to the maximal function considered in Section 3.3 [54, Exercise 2.1.12]. Let C1 (E, X) denote the collection of absolute integral operators Tk : X → X whose integral kernels k have the property that E(|f ||E1 ⊗ E1 ) takes finite values and  ME 2 (k)(x, x) dμ(x) < ∞. Σ

Where convenient, if k is the integral kernel of T , the maximal function ME 2 (k) is also written as ME 2 (T ). The map J : Σ → Σ × Σ defined by J(x) = (x, x) for x ∈ Σ maps Σ bijectively onto diag(Σ × Σ). Theorem 3.1. The space C1 (E, X) is a lattice ideal in Lr (X), that is, if S, T ∈ Lr (X), |S| ≤ |T | and T ∈ C1 (E, X), then S ∈ C1 (E, X). Moreover, C1 (E, X) is a Dedekind complete Banach lattice with the norm  · C1 (E,X) defined by  T C1(E,X) = |T | + ME 2 (T ) ◦ J dμ, T ∈ C1 (E, X). (3.12) Σ

Proof. If S, T ∈ Lr (X), |S| ≤ |T | and T ∈ C1 (E, X), then S is an absolute integral operator by [96, Theorem 3.3.6]. If k1 is the integral kernel of S and k2 is the integral kernel of T , then by [96, Theorem 3.3.5], the inequality |k1 | ≤ |k2 | holds (μ ⊗ μ)-a.e. . Then |ME 2 (k1 )(x, x)| ≤ |ME 2 (k2 )(x, x)| for all x ∈ Σ, so that   ME 2 (k1 ) ◦ J dμ ≤ ME 2 (k2 ) ◦ J dμ < ∞. Σ

Σ

Hence S ∈ C1 (E, X) and SC1 (E,X) ≤ T C1(E,X) . To show that C1 (E, X) is complete in its norm, suppose that   ∞   |Tj | + ME 2 (Tj ) ◦ J dμ < ∞ j=1

Σ

∞

for Tj ∈ C1 (E, X). Then T = j=1 Tj in the space of regular operators on ∞ X. The inequality |T | ≤ j=1 |Tj | ensures that T is an absolute integral ∞ operator with kernel k by [96, Theorem 3.3.6] and |k| ≤ j=1 |kj | (μ ⊗ μ)a.e. Suppose first that X is a real Banach function space. Each positive part Tj+ of Tj , j = 1, 2, . . . has an integral kernel kj+ . By monotone convergence, there exists a set of full μ-measure on which E(k + |En ⊗ En )(x, x) ≤

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∞

E(kj+ |En ⊗ En )(x, x) for each n = 1, 2, . . . . Taking the supremum and applying the monotone convergence theorem pointwise and under the sum ∞ + + 2 2 shows

that M+E (k )(x, x) ≤ j=1 ME (kj )(x, x) for μ-almost−all x ∈ Σ and Σ ME 2 (k ) ◦ J dμ < ∞. Applying the same argument to T and then the real and imaginary parts of T ensures that T ∈ C1 (E, X) and    ∞   |Tj | + |T | + ME 2 (T ) ◦ J dμ ≤ ME 2 (Tj ) ◦ J dμ . j=1

Σ

j=1

Σ

Dedekind completeness is inherited from Lr (X) [96, Theorem 1.3.2] and L1 (μ) [96, Example v) p. 9]. As in Section 3.3, we may define k˜ = lim supn→∞ E(k|E

n ⊗ En ) for the integral kernel k of an operator T ∈ C1 (E, X) so that Σ k˜ ◦ J dμ ≤ for any integral T C1 (E,X) . The same function k˜ : Σ × Σ → R is obtained

˜ kernel

k associated with the operator T . The integral Σ k ◦ J dμ is denoted as Σ T, dm , which is the notation used in [66] and is consistent with the notation following Example 3.6, provided that dyadic partitions of [0, 1] are used. We next see when the limsup can be replaced by a genuine limit, as is the case for trace class operators on L2 (μ). In order that there are sufficiently many finite rank operators T : X → X with an integral kernel, we assume that both X and the K¨othe dual  |f g| dμ < ∞ for all f ∈ X} X × = {g ∈ L0 (μ) : Σ

of X be order dense in L0 (μ), see [96, Theorem 3.3.7]. Then X ∩ X × is order dense in L0 (μ). We suppose that the filtration E = {En }∞ n=1 is based on partitions Pn , n = 1, 2, . . . , consisting of sets of finite positive measure whose characteristic functions belong to X ∩ X × . Hence, if n = 1, 2, . . . and A ∈ Pn , then i) 0 < μ(A) < ∞, ii) χ

A ∈ X and iii) A |f | dμ < ∞ for all f ∈ X. Suppose also that the conditional expectation En : f −→ E(f |En ), f ∈ X, is a bounded linear operator on the Banach function space X for each n = 1, 2, . . . and En → Id in the strong operator topology as n → ∞, as is the case for X = Lp (μ), 1 ≤ p < ∞.

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It is a simple matter to check that   f.E(g|En ) = E(f |En ).g, Σ

Σ

f ∈ X, g ∈ X × ,

so E(g|En ) → g in the weak topology σ(X × , X) for each g ∈ X × . Suppose also that Σ ME 2 (f ⊗ g) ◦ J dμ < ∞ for all f ∈ X and g ∈ X × . Then by dominated convergence, En (f ).En (g) → f.g μ-a.e. and in L1 (μ) for every f ∈ X and g ∈ X × . The closure in C1 (E, X) of the collection of all finite rank operators T : X → X with integral kernels of the form k=

n 

fj ⊗ gj ,

(3.13)

j=1

for fj ∈ X, gj ∈ X × for j = 1, . . . , n and n = 1, 2, . . . , is denoted by E X ×. X⊗ The following statement is the martingale analogue of Proposition 3.2, proved along the same lines. We assume below that the conditional expectation operators En , n = 1, 2, . . . , enjoy the properties mentioned above. Proposition 3.5. Suppose that k is the integral kernel of the operator T ∈ C1 (E, X) and Tn has the integral kernel E(k|En ⊗ En ) for n = 1, 2, . . . . If E X × then Tn → T in the strong operator topology as n → ∞ and T ∈ X⊗ E(k|En ⊗ En ) ◦ J converges a.e. on Σ and in L1 (μ) as n → ∞. Proof. Suppose first that T : X → X is a finite rank linear operator with a kernel k given by formula (3.13). Then kn = E(k|En ⊗ En ) converges to k (μ ⊗ μ)-a.e. as n → ∞, by the Martingale Convergence Theorem [16, Theorem 10.2.3]. First we need to check that Tn f → T f in X as n → ∞ for each f ∈ X by noting that E(u ⊗ v|En ⊗ En ) = E(u|En ) ⊗ E(v|En ),

u ∈ X, v ∈ X × ,

for each n = 1, 2, . . . , and appealing to the convergence properties of the conditional expectation operator En as n → ∞. Clearly E(k|En ⊗ En ) ◦ J converges a.e. on Σ and in L1 (μ) as n → ∞ because k is a sum of product functions and Σ ME 2 (f ⊗ g) ◦ J dμ < ∞ for all f ∈ X and g ∈ X × . For a general operator T ∈ C1 (E, X) with kernel k, the equality En T En = Tn holds, so Tn → T in the strong operator topology on L(X). The inequality |En T En | ≤ En |T |En in the Banach lattice of regular operators and the uniform boundedness principle together with the inequality ME 2 (En ⊗ En (k)) ≤ ME 2 (k),

n = 1, 2, . . . ,

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ensures that the map T −→ En T En , T ∈ C1 (E, X), is uniformly bounded on the Banach lattice C1 (E, X) for n = 1, 2, . . . . Consequently, the convergence a.e. of E(k|En ⊗ En ) ◦ J on Σ and in L1 (μ) as n → ∞ follows by approximation of k by elements of X ⊗ X × .



E X × , we have T, dm = limn→∞ E(k|En ⊗ En ) ◦ J dμ. For T ∈ X ⊗ Σ Σ The space C1 (E, X) is itself a Banach lattice and a lattice ideal in the Banach lattice of regular operators on X. The closure of X ⊗ X × in C1 (E, X) with respect to the locally convex Hausdorff topology associated with the seminorms  T −→ |T |f  + ME 2 (T ) ◦ J dμ, T ∈ C1 (E, X), Σ

for f ∈ X could also be used, but it is not a metrisable topology if X is infinite dimensional. We state the analogue of Proposition 3.3 in the present setting. E X × has a representative kernel Proposition 3.6. Every element T ∈ X ⊗ k : Σ × Σ → R for which there exist numbers cj ∈ C and sets Aj , Bj from B, j = 1, 2, . . . , such that ∞ 

|cj |(μ(Aj ).μ(Bj ) + μ(Aj ∩ Bj )) < ∞

j=1

∞ χ and k(x) = j=1 cj Aj ×Bj (x) for every x ∈ Σ × Σ such that the sum ∞ ∞ ˜ χ χ j=1 |cj | Aj ×Bj (x) is finite. In particular, f ◦ J = j=1 cj Aj ∩Bj = f ◦ J almost everywhere. Remark 3.1. The limit k˜ ◦ J is independent μ-a.e. of the filtration E if the kernel k is sufficiently ‘regular’, as in the following cases. π L2 (μ)—see the proof of Theorem 3.3 below. a) k ∈ L2 (μ)⊗ b) μ is a finite regular Borel measure on a compact Hausdorff space K ˜ x) = k(x, x) for μand k : K × K → C is continuous. Then k(x, almost all x ∈ K, because C(K) ⊗ C(K) is dense in C(K × K) by the Stone-Weiertstrass Theorem. c) Functions belonging to the closure of L∞ (μ) ⊗ L∞ (μ) in L∞ (μ ⊗ μ). d) k = χ{x 0 such that for every x ∈ Σ0 , the following two conditions hold: (i) for every U ∈ UE (x), there exists V ∈ UE  (x) such that U ⊂ V and μ(V ) ≤ c1 μ(U ), (ii) for every V ∈ UE  (x), there exists U ∈ UE (x) such that V ⊂ U and μ(U ) ≤ c2 μ(V ). Then for each f ∈ L1 (μ ⊗ μ), the inequalities c22 ME 2 (f ) ≤ ME 2 (f ) ≤ c21 ME 2 (f ) hold on Σ0 × Σ0 for the maximal functions defined by equation (3.11) with respect to either filtration E and E  . Consequently, for any other filtration E  such that E and E  satisfy (i) and (ii), the maximal functions ME 2 (f ) and ME  (f ) are equivalent and so C1 (E, X) = C1 (E  , X). If we have a given strict Lusin μ-filtration E, then the Hausdorff topological space (Σ0 , τE ) is a Lusin space for which B is the associated Borel σ-algebra on Σ0 [125, Theorem 5, p. 101], so there exists a metric dE on Σ0 whose topology is stronger than τE and (Σ0 , dE ) is complete and separable. Then B is also the Borel σ-algebra for the metric dE [125, Corollary 2, p. 101]. In this case there also exists a metric maximal function MdE (f ) defined by

f d(μ ⊗ μ) B (x)×Br (y) , x, y ∈ Σ0 , MdE (f )(x, y) = sup r μ(Br (x))μ(Br (y)) r>0 comparable to the maximal function ME 2 if (i) and (ii) are satisfied with respect to the neighbourhood base of balls in the metric dE .

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Connection with other generalised traces

An axiomatic treatement of traces on operator ideals is given in [108, 109] with recent updates in [110, 111]. The starting point is the Calkin theorem [110, Theorem 2.2] which asserts that the collection of all operator ideals on a separable Hilbert space H is in one-to-one correspondence with symmetric sequence ideals. The correspondence is obtained from the singular values of operators in the ideal. A trace on an operator ideal U(H) then corresponds to a unitarily invariant linear functional on U(H) or, equivalently, a symmetric linear functional on the corresponding sequence ideal [110, Theorem 6.2]. A particular example that has assumed importance recently because of noncommutative geometry is the Dixmier trace defined on the Marcinkiewicz operator ideal. The Dixmier trace is an example of a singular trace because it vanishes on all finite rank operators, see [21] for example. By contrast, for the purposes of this chapter, the emphasis with the Hardy-Littlewood maximal function approach to traces is on the Banach lattice of all absolute

integral operators T on a Banach function space X, so that T ≥ 0 implies Σ T, dm ≥ 0—just what is needed in the proof of the Cwikel-Lieb-Rozenbljum inequality for dominated semigroups in Chapter 7. A result of D. Lewis [92] shows that on an infinite dimensional Hilbert space, the collection of all Hilbert-Schmidt operators is the only Banach operator ideal isomorphic to a Banach lattice, despite the observation that a symmetric sequence ideal is itself a Riesz space. For a choice of Banach limit ω ∈ (∞ ) , the map  T −→ ω({E(k|En ⊗ En ) ◦ J}∞ T ∈ C1 (E, X), n=1 ) dμ, Σ

is continuous and linear on C1 (E, X), so there may be many possible choices of a continuous trace on the whole Banach lattice C1 (E, X) depending on ω. As Proposition 3.5 indicates, all such choices of a continuous trace agree E X × of the lattice ideal C1 (E, X) in Lr (X). on the closed subspace X ⊗ The Selberg trace formula also relates regularised traces (geometric information) to asymptotic estimates for eigenvalues (spectral information) of a Laplacian, see [9] for a survey of this deep subject. 3.5

Hermitian positive operators

Suppose that (Σ, B, μ) is a σ-finite measure space. On a complex Hilbert space H, we call a bounded linear operator T : H → H hermitian positive

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if (T u, u) ≥ 0 for all u ∈ H. A trace condition ensures that an hermitian positive bounded linear operator on L2 (μ) is trace class. Theorem 3.3. Let Tk : L2 (μ) → L2 (μ) be an hermitian positive integral operator with kernel k and let E = En n∈N be a Lusin μ-filtration for which for each n = 1, 2, . . . . The operator Tk is E(|k||En ⊗ En ) has finite values

trace class if and only if Σ ME 2 (k)(x, x) dμ(x) < ∞. If Tk is trace class, then the formula  ˜ x) dμ(x) tr(Tk ) = k(x, (3.14) Σ

holds with respect to the integral kernel k˜ of the operator Tk defined by k˜ = lim E(k|En ⊗ En ), n→∞

wherever the limit exists. Moreover, Σ ME 2 (k)(x, x) dμ(x) ≤ 4tr(Tk ). Versions of Theorem 3.3 are well-known from the work of J. Weidmann [135], Gohberg-Krein [51], C. Brislawn [18, 19] and [127, Theorem 2.12] for the continuous case. In these works, the term integral operator is replaced by Hilbert-Schmidt operator —where, at least locally, the integral kernel k belongs to L2 (μ ⊗ μ). Nuclear operators between Lp -spaces and Banach spaces have been studied recently along similar lines in [23, 24, 31–33, 44]. The assumption in Theorem 3.3 that Tk is just an integral operator with kernel k is a significant relaxation of the condition that k ∈ L2 (μ ⊗ μ), because we do not even assume that Tk is locally a compact linear operator—this conclusion is actually a consequence of Theorem 3.3 where the compactness of the bounded linear operator Tk follows from the trace

condition Σ ME 2 (k)(x, x) dμ(x) < ∞ and positivity on L2 (μ). The simple example of the Volterra integral operator described in Example 3.1 shows that the assumption that Tk is an hermitian positive operator cannot be omitted. The essential ingredients of the proof of Theorem 3.3 below are the notion of a Lusin μ-filtration mentioned above, the Martingale Convergence Theorem [16, Theorem 10.3.13] and the non-commutative Fatou lemma [127, Theorem 2.7 (d)] asserting that the closed unit ball of C1 (H) is sequentially closed in the weak operator topology on L(H), given here for later reference. Proposition 3.8. Let H be a separable Hilbert space. If An ∈ C1 (H) converges weakly to an operator A ∈ L(H) as n → ∞ and supn An 1 < ∞, then A ∈ C1 (H) and A1 ≤ supn An 1 .

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Given an integral operator Tk : L2 (μ) → L2 (μ) with kernel k as in Theorem 3.3 and a Lusin μ-filtration E, it is a simple matter [60, Lemma 7.4] to divide up the sets in the partition P1E so that k is integrable on U ×V for all sets U, V belonging to P1E —we shall assume this has already been done so that kn = E(k|En ⊗ En ) is a finite function for each n = 1, 2, . . . , as has been assumed in Theorem 3.3. Proof of Theorem 3.3. The conditional expectation operator En : f −→ E(f |En ) is a contraction on L2 (μ) given by

 U f dμ χ , (En f )(x) = E(f |En )(x) = U (x) μ(U ) E U∈Pn

so that En is an absolute integral operator with kernel  χU (x)χU (y) . en (x, y) = μ(U ) E U∈Pn

Hence en (x, y) = en (y, x) for all x, y ∈ Σ0 , a set of full measure in Σ. The background of conditional expectation operators in probability theory is outlined in Section 4.1.1 below. Applying the Fubini-Tonelli Theorem, for each n = 1, 2, . . . , a glance at formula (3.10) shows that



  k(x2 , x1 ) dμ(x1 )dμ(x2 ) V U χ f dμ (En Tk En f )(x) = V (x) μ(U )μ(V ) U E U,V ∈Pn

= (Tkn f )(x) 2

for all f ∈ L (μ) and x ∈ Σ0 . The interchange of integrals is valid provided that k is integrable on sets U × V with U, V ∈ PnE , guaranteed by the assumption above about the Lusin μ-filtration E. Consequently, Tkn = En Tk En is a bounded linear operator on L2 (μ) with norm bounded by the norm of Tk (compare [24, Theorem 3.9] in the Hilbert-Schmidt setting). Let PnE = {Un,1 , Un,2 , . . . } be an enumeration of the partition PnE . If E E ), then the conditional Pn,m is its truncation to length m and En,m = σ(Pn,m expectation operators En,m : f −→ E(f |En,m ) converge to En in the strong operator topology of L(L2 (μ)) as m → ∞. Each operator En,m Tk En,m has finite rank and (En,m Tk En,m u, u) = (Tk En,m u, En,m u) ≥ 0 for k = 1, 2, . . . and u ∈ L2 (μ). Because finite rank operators are trace class, the equality  kn,m (x, x) dμ(x) En,m Tk En,m 1 = tr(En,m Tk En,m ) = Σ

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2 holds for kn,m = E(k|En,m ) by the elementary version of formula (3.14). The first equality is valid because the finite rank operator En,m Tk En,m is hermitian positive, so the trace norm of En,m Tk En,m is just the sum of its eigenvalues. It is worthwhile looking at the simple proof from linear algebra of formula (3.14) in the basic case of finite rank operators. The formula

 U×V k d(μ ⊗ μ) χ kn,m = U×V μ(U )μ(V ) E U,V ∈Pn,m

expresses the matrix of the finite rank operator En,m Tk En,m as +

, U ×V k d(μ ⊗ μ) 1

1

μ(U ) 2 μ(V ) 2

E U,V ∈Pn,m 1

E with respect to the finite orthonormal subset O = {μ(U )− 2 χU : U ∈ Pn,m } 2 of L (μ). The conditional expectation operator En,m is the orthogonal projection onto the subspace spanned by O. Then

 U×U k d(μ ⊗ μ) tr(En,m Tk En,m ) = μ(U ) E U∈Pn,m  = kn,m (x, x) dμ(x), Σ

which is just formula (3.14) in this elementary setting. The representation



 k(x2 , x1 ) dμ(x1 )dμ(x2 ) χU (x) U U kn,m (x, x) = μ(U )2 E U ∈Pn,m  χU (x)μ(U )−2 (Tk χU , χU ) = E U ∈Pn,m

shows that 0 ≤ kn,m (x, x) ≤ kn (x, x) ≤ ME 2 (k)(x, x) for all x ∈ Σ0 and the inequality  ME 2 (k)(x, x) dμ(x) tr(En,m Tk En,m ) ≤ Σ

holds for all n, m = 1, 2, . . . . Taking m → ∞ and applying the noncommutative Fatou lemma, Proposition 3.8, each operator Tkn is trace class and  ME 2 (k)(x, x) dμ(x). tr(Tkn ) ≤ Σ

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Again tr(Tkn ) is equal to the trace norm Tkn 1 of Tkn because each operator Tkn is positive. By the Martingale Convergence Theorem, En → I in the strong operator topology of L(L2 (μ)), so Tkn → Tk in the strong operator topology as n → ∞. Another application of the non-commutative Fatou lemma shows that Tk is trace class and  ME 2 (k)(x, x) dμ(x). tr(Tk ) ≤ Σ

The operator Tk has the representation (3.2) and ME 2 (k) ◦ J ≤

∞ 

λj ME (φj )ME (ψj )

j=1

and Σ ME (φj )ME (ψj ) dμ ≤ 4 for j = 1, 2, . . . by the Cauchy-Schwartz in2 equality and the L2 -norm equivalence of ME (f ) and

f the L -norm equivalence of ME (f ) and f [16, Corollary 10.3.11]. Then Σ ME 2 (k)(x, x) dμ(x) ≤ π L2 (μ) [107, 4Tk 1 = 4tr(Tk ) by the isometry between C1 (H) and L2 (μ)⊗ Theorem 8.3.3]. Furthermore, by the Martingale Convergence Theorem and dominated convergence   ∞  λj lim E(φj |En ).E(ψj |En ) dμ k˜ ◦ J dμ = Σ

=

j=1 ∞ 

Σ n→∞

λj (φj , ψj ) = tr(Tk ).

j=1

Remark 3.2. i) If the σ-algebra B is not itself countably generated, there may exist a smaller countably generated σ-algebra Bk for which the integral kernel k is (Bk ⊗ Bk )-measurable. If not, then Tk is certainly not a trace class operator because a function with a representation (3.3) is measurable with respect to the σ-algebra generated by a countable family of measurable product functions. A Lusin μ-filtration may then be constructed for (Σ, Bk ). ii) In the case of a Radon measure μ on a Hausdorff space Σ, a continuous kernel k is the kernel of a Hilbert-Schmidt operator on each product K × K of a compact set K with itself (provided that K has positive measure), be

˜ x) dμ = k(x, x) dμ cause k is uniformly bounded on K ×K. Then K k(x, K is equal to the trace of the positive operator Q(K)Tk Q(K) by Proposition 3.7. Now we may appeal to the inner regularity of μ [127, Theorem 2.12].

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Example 3.7. Suppose that T : L2 ([0, 1]) → L2 ([0, 1]) is the operator of convolution with respect to a bounded periodic function ϕ : R → C with period one. Then  1 ME 2 (ϕ ◦ u)(x, x) dx ≤ ϕ∞ 0

for the mapping u : (x, y) −→ x − y. The Fourier transformˆ: L2 ([0, 1]) → 2 diagonalises the normal operator T . By the Riemann-Lebesgue lemma the eigenvalues ϕ(n) ˆ of T satisfy ϕ(n) ˆ → 0 as n → ∞, so T is necessarily compact. In fact, T is a HilbertSchmidt operator so ϕˆ ∈ 2 . However, it is possible to choose a continuous ϕ for which ϕˆ ∈ / p for any p < 2, see [127, p. 24] and [22].

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Chapter 4

Stochastic integration

In this Chapter, we describe how the approach to the stochastic integration of Banach space valued processes developed by J.A.M.van Neerven, M. Veraar and L. Weis [133] is related to bilinear integration as described in Definition 2.2. So far, our treatment of bilinear integration has involved mainly functional analysis and measure theory. We start with a brief discussion of probability theory and stochastic processes. Although we do not need all the material found in [75], it provides an account of stochastic processes and stochastic integration in greater depth than provided here. 4.1

Background on probability and discrete processes

Probability theory uses the language of measure theory. Although probability and stochastic analysis is now part of the toolbag of most analysts, we give a brief r´esum´e in this section. A probability measure space (Ω, F , P ) is a measure space for which P : F → [0, 1] is a probability measure. Thus, F is a σ-algebra of subsets of Ω and P is a nonnegative σ-additive set function for which P (Ω) = 1. The set Ω is called the sample space and F is the collection of events, so that P (A) is the probability that the event A ∈ F occurs. Some basic examples illustrate the ideas. Example 4.1. i) Suppose that a fair coin is tossed n consecutive times for some n = 1, 2, . . . . The possible outcomes is the set of all ordered ntuples such as HT HH . . . T H for which H denotes an outcome of heads in a particular toss and T for tails. Representing H by 1 and T by 0, we can take the sample space Ω to be the Cartesian product {0, 1}n = {0, 1}×· · ·×{0, 1} of the two-point set {0, 1} with itself n times. 101

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The probability P {ω} of any particular outcome ω ∈ Ω is the collection of all subsets of Ω and  1 #A P (A) = = n 2n 2

1 2n ,

so F is

ω∈A

for any subset A of Ω. The number of elements of the finite set A is denoted by #A. The set function P is clearly σ-additive. Because Ω has 2n elements, it follows that P (Ω) = 1. ii) Suppose that a fair coin is tossed infinitely many times. The possible outcomes is the set of all infinite sequences such as HT HH . . . , so that we can take the sample space Ω to be the Cartesian product ∞ n=1 {0, 1}. Now the probability P {ω} of any particular outcome ω ∈ Ω is zero. Here we take F to be the Borel σ-algebra of Ω (in the product topology, where {0, 1} has the discrete topology) and P is the product ⊗∞ n=1 μ of the measures μ with μ(A) = #A/2 for A ⊂ {0, 1}. Let (Ω, F , P ) be a probability measure space. A random variable ξ : Ω → R is a function which is measurable with respect to the σ-algebras F on Ω and the Borel σ-algebra B(R) on R. More accurately, it is the equivalence class [ξ] ∈ L0 (P ) of all measurable functions equal to ξ P almost everywhere, as in Section 1.2, but we rarely need to make this distinction. Notation: The set {ξ ∈ B} = {ω : ξ(ω) ∈ B} = ξ −1 (B) is an element of the σ-algebra F for each B ∈ B(R). The σ-algebra σ(ξ) = {ξ −1 (B) : B ∈ B(R)} is the smallest σ-algebra for which the random variable ξ is measurable. Similarly, if {ξi : i ∈ I } is a collection of random variables, then σ(ξi : i ∈ I ) is the σ-algebra generated by {ξi : i ∈ I }, that is, the smallest σ-algebra containing all events {ξi ∈ B} for i ∈ I and B ∈ B(R). The distribution Pξ of a random variable ξ : Ω → R is the Borel measure defined by Pξ (B) = P {ξ ∈ B} for all B ∈ B(R). The distribution function Fξ of ξ is defined by Fξ (x) = P {ξ ≤ x},

x ∈ R.

Note: If f : R → R is a function, then    f dFξ = f dPξ = f ◦ ξ dP R

R

Ω

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in the sense that if one of the integrals exists, then they all do and are equal to each other. A random variable ξ has an absolutely continuous distribution if there exists fξ : R → [0, ∞) such that  P {ξ ∈ B} = fξ (x) dx, B ∈ B(R). B

Then fξ is called the density of ξ. By the Radon-Nikodym Theorem [60, Theorem 19.23], this is the same as saying that the distribution Pξ of ξ is zero on Borel sets of Lebesgue measure zero in R. The expectation of an integrable random variable ξ is  ξ dP. Eξ = Ω

If ξ is square integrable, then its variance is defined by var ξ = Ω (ξ − Eξ)2 dP . Example 4.2. A random variable X : Ω → R is said to be normally distributed if there exist μ ∈ R and σ > 0 such that  2 e−(x−μ) /(2σ) dx PX (B) = (2πσ)−1/2 B

for all B ∈ B(R). Then  EX = X dP Ω  = x dPX by the change of variables formula, R  2 −1/2 xe−(x−μ) /(2σ) dx = (2πσ) R   2 2 −1/2 = μ(2πσ) e−(x−μ) /(2σ) dx + (2πσ)−1/2 te−t /(2σ) dt = μ.

R

R

Similarly, varX = σ. We shall also have occasion to consider random vectors ξ : Ω → Rd , for which we can write ξ = (ξ1 , . . . , ξd ) for real valued random variables ξ1 , . . . , ξd . Then Pξ1 ,...,ξd (B) = P {(ξ1 , . . . , ξd ) ∈ B},

B ∈ B(Rd ),

is the joint distribution of the random variables ξ1 , . . . , ξd . It is a Borel probability measure on Rd .

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Sometimes we consider random variables ξ with a discrete distribution ∞ ∞ so that we can write Pξ = j=1 cj δxj , where cj ≥ 0, j=1 cj = 1 and cj = P {ξ = xj } is the mass at xj . The Dirac measure at xj has been denoted by δxj . Then  P {ξ ∈ B} = P {ξ = xj }, B ∈ B(Rd ). xj ∈B

4.1.1

Conditional probability and expectation

Let A, B ∈ F be two events with P (B) = 0. The conditional probability P (A|B) of A given B is defined by the formula P (A|B) =

P (A ∩ B) . P (B)

This idea recurs frequently. The event B gives us extra information with which to judge the probability of the occurrence of the event A. Hence it makes sense to call two sets A, B ∈ F independent if P (A ∩ B) = P (A)P (B). A finite collection of events A1 , . . . , An ∈ F is called independent if P (Aj1 ∩ · · · ∩ Ajk ) = P (Aj1 ) · · · P (Ajk ) for any subset {j1 , . . . , jk } of indices {1, . . . , n}. Two random variables ξ, η are independent if the sets {ξ ∈ A} and {η ∈ B} are independent for each A, B ∈ B(R). Similarly, the random variables ξ1 , . . . , ξn are independent if for all B1 , . . . , Bn ∈ B(R), the events {ξ ∈ B1 }, . . . , {ξ ∈ Bn } are independent. A collection of random variables is independent if any finite subfamily is. Example 4.3. Suppose that a fair coin is tossed infinitely many times with 1 representing heads and 0 representing tails at each toss. Let ξn : Ω → {0, 1} be the random variable representing the outcome of the nth toss for n = 1, 2, . . . . Then 1 = P {ξj1 = n1 } · · · P {ξjk = nk } 2k (4.1) for all distinct indices j1 , . . . , jk = 1, 2, . . . , all n1 , . . . , nk ∈ {0, 1} and all k = 1, 2, . . . . The infinite collection {ξn : n = 1, 2, . . . } of random variables is therefore independent. P ({ξj1 = n1 } ∩ · · · ∩ {ξjk = nk }) =

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On a technical note, as mentioned above, P is the infinite product ⊗∞ n=1 μ of probability measures. This means that P is the additive set function such that P (E) is defined by Equation (4.1) for any event E = {ξj1 = n1 } ∩ · · · ∩ {ξjk = nk }. Some work needs to be done to prove that P is σ-additive on the algebra generated by the sets E [60, Theorem 22.8, p. 433]. Two integrable random variables ξ, η are uncorrelated if E(ξη) = E(ξ)E(η). Independent integrable random variables are necessarily uncorrelated. Two σ-algebras G, H of sets belonging to F (sub-σ-algebras of F) are independent if for every A ∈ G and B ∈ H, the events A and B are independent. A random variable ξ is independent of a σ-algebra G if σ(ξ) and G are independent σ-algebras. A similar definition applies for n random variables ξ1 , . . . , ξn being independent of m σ-algebras G1 , . . . , Gm . The existence of the conditional expectation of an integrable random variable with respect to a sub-σ-algebra is an immediate consequence of the Radon-Nikodym Theorem [60, Theorem 19.23]. First we look at the significance of the concept in some simple examples. Let (Ω, F, P ) be a probability measure space and let B ∈ F be an event that occurs with nonzero probability. The conditional expectation E(ξ|B) of a random variable ξ ∈ L1 (P ) given B is the number

ξdP E(ξ|B) = B . P (B) The conditional expectation E(ξ|B) is just the expected value of ξ : B → R with respect to the probability P  = P/P (B) defined on subsets of B. Suppose that ξ ∈ L1 (P ) and η is a discrete random variable with values belonging to the set {y1 , y2 , . . . }. The conditional expectation E(ξ|η) of ξ given the random variable η is the random variable E(ξ|η) : Ω → R defined by E(ξ|η)(ω) = E(ξ|{η = yj }),

if η(ω) = yj ,

for ω ∈ Ω and j = 1, 2, . . . . If instead we think of E(ξ|η) as E(ξ|σ(η)), the conditional expectation with respect to the σ-algebra generated by η, then we arrive at the following generalisation. Let (Ω, F , P ) be a probability measure space and let G be a σ-algebra contained in F and ξ ∈ L1 (P ). The conditional expectation E(ξ|G) of ξ given G is a random variable such that

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1) E(ξ|G) is G-measurable and

2) A E(ξ|G) dP = A ξ dP for all A ∈ G.

The measure μ : G → R defined by μ(A) = A ξ dP for all A ∈ G is absolutely continuous with respect to P on G, that is, μ(A) = 0 for all A ∈ G such that P (A) = 0, so by the Radon Nikodym theorem, such a random variable E(ξ|G) exists and any two versions are equal P -a.e. If η is a discrete random variable, then E(ξ|η) satisfies conditions 1) and 2) above with respect to the σ-algebra G = σ(η). Hence, E(ξ|σ(η)) = E(ξ|η). Theorem 4.1. (Properties of conditional expectation) Let (Ω, F , P ) be a probability measure space and let G be a σ-algebra contained in F . Up to a set of P -measure zero, we have Linearity 1) for each ξ, η ∈ L1 (P ) and a, b ∈ R, E(aξ + bζ|G) = aE(ξ|G) + bE(ζ|G); Preservation of expectation 2) E(E(ξ|G)) = E(ξ), for each ξ ∈ L1 (P ); The random variable ξ is “known” 3) E(ξζ|G) = ξE(ζ|G), for each G-measurable ξ with ξ, ζ ∈ L1 (P ) ; Independence 4) E(ξ|G) = E(ξ), if ξ ∈ L1 (P ) is independent of G; Tower property 5) E(E(ξ|G)|H) = E(ξ|H), for each ξ ∈ L1 (P ) and σ-algebra H ⊂ G; Positivity 6) E(ξ|G) ≥ 0 for each ξ ∈ L1 (P ) with ξ ≥ 0 a.e. The mapping ξ → E(ξ|G) is a selfadjoint projection on L2 (P ). We have already appealed to this general property in Section 3.5 for the elementary case of the σ-algebra associated with a countable partition of sets, where it is simple to verify directly.

4.1.2

Discrete Martingales

Let (Ω, F , P ) be a probability space. As is usual in probability theory, we shall often fail to mention the underlying probability space. A filtration on Ω is a sequence F1 ⊂ F2 ⊂ · · · ⊂ F of σ-algebras. A sequence ξ1 , ξ2 , . . . of

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random variables is adapted to a filtration F1 , F2 , . . . , if ξn is Fn -measurable for each n = 1, 2, . . . . Example 4.4. Let ξ1 , ξ2 , . . . be random variables and Fn = σ(ξ1 , . . . , ξn ), n = 1, 2, . . . . Then F1 , F2 , . . . is a filtration to which the random variables ξ1 , ξ2 , . . . are adapted. Let F1 , F2 , . . . be a filtration. A sequence ξ1 , ξ2 , . . . of random variables is called a martingale with respect to the filtration F1 , F2 , . . . if 1) ξn is integrable for each n = 1, 2, . . . ; 2) the sequence ξ1 , ξ2 , . . . of random variables is adapted to F1 , F2 , . . . ; 3) E(ξn+1 |Fn ) = ξn almost surely for each n = 1, 2, . . . . The sequence ξ1 , ξ2 , . . . of random variables is called a supermartingale if instead of 3) we have E(ξn+1 |Fn ) ≤ ξn ,

a.s.

for each n = 1, 2, . . . and a submartingale if E(ξn+1 |Fn ) ≥ ξn ,

a.s.

for each n = 1, 2, . . . . Notice that from 3) and properties of conditional expectation, we have E(ξn+1 ) = E(E(ξn+1 |Fn )) = E(ξn ),

n = 1, 2, . . . ,

so that for a martingale ξ1 , ξ2 , . . . , the expectation E(ξn ) = E(ξ1 ) of ξn , n = 1, 2, . . . , is necessarily constant. Example 4.5. i) Let η1 , η2 , . . . be independent integrable random variables with zero mean and set ξn = η1 + · · · + ηn and Fn = σ(η1 , . . . , ηn ) for each n = 1, 2, . . . . Then ξ1 , ξ2 , . . . is adapted to F1 , F2 , . . . and E(|ξn |) ≤ E(|η1 |) + · · · + E(|ηn |) < ∞ for each n = 1, 2, . . . . To see that ξ1 , ξ2 , . . . is a martingale, we note that E(ξn+1 |Fn ) = E(ηn+1 |Fn ) + E(ξn |Fn ) = E(ηn+1 |Fn ) + ξn

(because ξn is Fn -measurable)

= E(ηn+1 ) + ξn (because ηn+1 is independent of ξj for 1 ≤ j ≤ n) = ξn

(because ηn+1 has zero mean).

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ii) Let ξ ∈ L1 (P ) and let F1 , F2 , . . . be any filtration. If we set ξn = E(ξ|Fn ) for n = 1, 2, . . . , then ξ1 , ξ2 , . . . is a martingale. iii) Let P be Lebesgue measure on [0, 1) and let ξ ∈ L1 (P ). Let Φn be the partition [(k − 1)/2n , k/2n ), k = 1, . . . , 2n of [0, 1) into disjoint intervals and let Fn be the algebra generated by Φn for each n = 1, 2, . . . . Then   ξn = 2−n χA ξ dP, n = 1, 2, . . . , A∈Φn

A

is a martingale with respect to F1 , F2 , . . . . Moreover, ξn → ξ in L1 (P ) as n → ∞. Martingales of this type were used in Section 3.5 above. Gambling. The most compelling way to view a discrete parameter martingale is as a mathematical model of gambling over an infinite number of rounds of a game. Let η1 , η2 , . . . be integrable random variables where ηn represents the winning or losses per unit stake in the nth round. Then ξn = η1 + · · · + ηn will be the total winnings or losses per unit stake after n rounds, n = 1, 2, . . . . As we saw in Example i) above, if η1 , η2 , . . . are independent with zero mean, then ξ1 , ξ2 , . . . is a martingale. We would like to allow the possibility that the winnings or losses per unit stake ηn depend on the preceding outcomes η1 , η2 , . . . , ηn−1 , possibly by using some gambling strategy that depends on what happened before the nth bet. Let F1 , F2 , . . . be the filtration defined by Fn = σ(η1 , . . . , ηn ) for n = 1, 2, . . . . Set F0 = {∅, Ω} and ξ0 = 0. For each n = 1, 2, . . . , the σ-algebra Fn−1 represents the accumulated knowledge about the game after n − 1 rounds. Then Martingale: Submartingale: Supermartingale:

E(ξn |Fn−1 ) = ξn−1 E(ξn |Fn−1 ) ≥ ξn−1 E(ξn |Fn−1 ) ≤ ξn−1

a.s. a.s. a.s.

fair game; favourable to you; unfavourable to you.

So, for a submartingale say, for almost all trials ω ∈ Ω, the expected total outcome E(ξn |Fn−1 )(ω) after the nth round, given accumulated knowledge Fn−1 about the preceding rounds, exceeds the total outcome ξn−1 (ω) after the (n− 1)th round. In this sense, the game is favourable to the player because using knowledge of the preceding outcomes increases the players winnings. It is natural to call α1 , α2 , . . . a gambling strategy, if for each n = 1, 2, . . . , the random variable αn is Fn−1 -measurable, that is, for a given

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trial ω ∈ Ω, a bet of size αn (ω) is made in the nth round, depending on the accumulated knowledge Fn−1 gained in the preceding rounds. So, let α1 , α2 , . . . be a gambling strategy and let ζn = α1 η1 + · · · + αn ηn = α1 (ξ1 − ξ0 ) + · · · + αn (ξn − ξn−1 ) be the total winnings at the nth round of the game. This finite sum is, in fact, a stochastic integral of the type we shall consider in greater generality later on. Now suppose that we only have a finite sum K of money to bet within any round: |αn | ≤ K for n = 1, 2, . . . . Then appealing to the properties of conditional expectation, we have E(ζn |Fn−1 ) = E(ζn−1 + αn (ξn − ξn−1 )|Fn−1 ) = ζn−1 + αn E((ξn − ξn−1 )|Fn−1 ) = ζn−1 + αn (E(ξn |Fn−1 ) − ξn−1 ) because ζn−1 , αn and ξn−1 are all Fn−1 -measurable. Consequently, if ξ1 , ξ2 , . . . is a martingale (respectively, a submartingale or supermartingale), then so is ζ1 , ζ2 , . . . . This interpretation underlies our consideration of stochastic integration later. Although mathematical finance has spurred on much development in stochastic modelling, recent history has demonstrated that the concealment of past information in, say, complex financial derivatives can lead to catastrophic failures in financial markets. At this time of life, the author prefers rational central planning as a means of organising complex social systems, despite having fallen out of favour in current political thinking. 4.1.3

Discrete stopping times

Let F1 , F2 , . . . be the filtration. A stopping time τ : Ω → {1, 2, . . . }∪{∞} (with repect to the filtration F1 , F2 , . . . ) is a random variable such that {τ = n} ∈ Fn for each n = 1, 2, . . . . Example 4.6. Suppose that we toss a coin with stakes $1, starting with $5 and stopping when we win $10 or lose the lot. Then the number τ of tosses at which we stop is a stopping time. To see this, let ξn be the total winnings at the nth toss of the coin. Then τ = min{n : ξn = 0 or 10} is the first hitting time of the set {0, 10} and we have {τ = n} = {0 < ξ1 < 10} ∩ · · · ∩ {0 < ξn−1 < 10} ∩ {ξn = 0 or 10} ∈ Fn .

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The random variable ξτ defined by ξτ (ω) = ξτ (ω) (ω) for ω ∈ Ω has values 0 or 10 with equal probability. If τ is a stopping time and ξ1 , ξ2 , . . . is a sequence of random variables adapted to the filtration F1 , F2 , . . . , then the sequence ξτ ∧n , n = 1, 2, . . . , stopped at τ is given by for all ω ∈ Ω and n = 1, 2, . . . .

ξτ ∧n (ω) = ξτ (ω)∧n (ω),

Then ξτ ∧n , n = 1, 2, . . . is also adapted to the filtration F1 , F2 , . . . . Moreover, if ξ1 , ξ2 , . . . is a martingale (resp. submartingale, supermartingale), then so is the stopped sequence ξτ ∧n , n = 1, 2, . . . , as may be seen by defining the gambling strategy  1 if τ ≥ n αn = 0 if τ < n, and observing that ξτ ∧n = α1 (ξ1 −ξ0 )+· · ·+αn (ξn −ξn−1 ) for n = 1, 2, . . . . Because ξτ ∧n , n = 1, 2, . . . , is a martingale, we necessarily have E(ξτ ∧n ) = E(ξτ ∧1 ) = E(ξ1 ),

for all n = 1, 2, . . . .

Suppose that P {τ < ∞} = 1 so that τ ∧ n  τ a.s. as n → ∞. Taking n → ∞, we would hope that the expected value E(ξτ ) of the random variable ξτ defined by ξτ (ω) = ξτ (ω) (ω) for all ω ∈ Ω is equal to the expected value E(ξ1 ) of the initial winnings. The following example illustrates that some care needs to be exercised. Example 4.7 (St. Petersburg lottery). Suppose that a coin is tossed and initially $1 is bet on heads. After that, the strategy is to double the stakes if tails occurs or quit with a win. Let ηj = 1 for a heads and ηj = −1 for a tails on the jth toss. The gambling strategy is then  n−1 if η1 = · · · = ηn−1 = −1, 2 αn = 0 otherwise and τ = min{n : ηn = 1} is the associated stopping time. If we set ζn = η1 + 2η2 + · · · + 2n−1 ηn for n = 1, 2, . . . , then ζτ ∧n is the winnings after the nth round and ζτ ∧n , n = 1, 2, . . . , is a martingale. We know that P (τ < ∞) = 1, because the probability of all tails is zero. But ζτ = 1 because −1 − 2 − · · · − 2n−1 + 2n = 1 and E(ζτ ∧n ) = E(ζ1 ) = 0 for all n = 1, 2, . . . , so that 1 = E(ζτ ) = lim E(ζτ ∧n ) = E(ζ1 ) = 0. n→∞

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The problem is that although the martingale ζτ ∧n , n = 1, 2, . . . , converges almost surely to ζτ as n → ∞ (because P (τ < ∞) = 1), it does not converge in L1 (P ) because the stakes are increasing exponentially fast. Theorem 4.2 (Optional Stopping Theorem). Let ξn , n = 1, 2, . . . be a martingale and τ a stopping time with respect to a filtration F1 , F2 , . . . , such that i) τ < ∞ a.s., ii) ξτ is integrable and iii) E(ξn 1{τ >n} ) → 0 as n → ∞. Then E(ξτ ) = E(ξ1 ). Proof. Writing ξτ = ξτ ∧n + (ξτ − ξn ) 1{τ >n} , we see that     E(ξτ ) = E(ξτ ∧n ) + E ξτ 1{τ >n} − E ξn 1{τ >n} . The last term approaches zero as n → ∞ by condition iii). The equality E(ξτ ∧n ) = E(ξ1 ) holds for each n = 1, 2, . . . , because ξτ ∧n , n = 1, 2, . . . , is a martingale. By assumption i), we have 1{τ >n}   0 a.s., so by monotone convergence and assumption ii), it follows that E ξτ 1{τ >n} → 0 as n → ∞. Hence E(ξτ ) = E(ξ1 ). 4.2

Stochastic processes

From now on, we suppose that (Ω, F, P ) is a complete probability space. This means that: Ω is a set, F a σ-algebra of sets of Ω, P a probability measure on Ω such that each subset of a P -null set also belongs to F . Recall that if (Ω, F , P ) is not complete, we may extend F to F by including all subsets of null sets, and define P on F in the obvious way, so that (Ω, F , P ) is complete. Suppose Γ is an index set, {Xγ : γ ∈ Γ} a family of random variables. The σ-algebra σ{Xγ : γ ∈ Γ} is the smallest σ-algebra of subsets of Ω such that Xγ is measurable for all γ ∈ Γ. This is the σ-algebra generated by the random variables Xγ , γ ∈ Γ. A filtration is a family {Ft : t ∈ R+ } of sub-σ-algebras of F such that Fs ⊆ Ft , for all s < t. A filtration {Ft : t ∈ R+ } is called a standard filtration if

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(1) Ft = Ft+ := ∩s>t Fs for all t > 0 (right continuity); (2) F0 contains all the P -null sets (completeness). %  $ j j+1 : j ∈ N and null Example 4.8. Let Fn be generated by , 2n 2n sets. Set Ft = F[t] . This is complete but not right continuous. A (d-dimensional) stochastic process is a function X : I × Ω −→ Rd where I is an interval in R+ and for every t ∈ I, the function X(t, ·) is measurable. We say X is measurable if X is B(I) ⊗ F -measurable. If 0 ∈ I, then we say x ∈ Rd that is the initial value of X if X(0, ·) = x a.s. We denote it by {Xt } where Xt = X(t, ·). Given an increasing family {Ft : t ∈ I} of σ-algebras the process is adapted to Ft if Xt is Ft -measurable for all t ∈ I. A stochastic process X is said to be left continuous (respectively, right continuous) if the function t −→ X(t, ω) is left continuous (respectively, right continuous) on I for every ω ∈ Ω. Two stochastic processes X = {Xt } and Y = {Yt } are versions of each other if P (Xt = Yt ) = 1

for all t ∈ I.

They are indistinguishable if P (Xt = Yt

for all t ∈ I) = 1.

If Xt and Yt are indistinguishable, they must be versions of each other, but versions are not necessarily indistinguishable. Remark. If Xt and Yt are continuous versions of each other, they are indistinguishable. Almost always we work with I = R+ and d = 1. 4.3

Brownian motion

The strange spontaneous movements of small particles suspended in liquid were first studied by Robert Brown, an English botanist, in 1828, although they had apparently been noticed much earlier by other scientists. L. Bachelier gave the first mathematical description of the phenomenon in 1900, going so far as to note the Markov property of the process. In 1905 A. Einstein and, independently and around the same time, M. V. Smoluchowski developed physical theories of Brownian motion which

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could be used, for example, to determine molecular diameters. The mathematical theory of Brownian motion was invented in 1923 by N. Wiener, and accordingly the Brownian motion that we will be working with is frequently called the Wiener process. A process B = {Bt : t ∈ R+ } is called a Brownian motion in R if (i) For 0 ≤ s < t < ∞, Bt − Bs is a normally distributed random variable with mean zero and variance t − s. Recall that X is normally distributed with variance σ and mean zero if  |x|2 1 e− 2σ dx. P (X ∈ A) = √ 2πσ A (ii) For all 0 ≤ t0 < t1 . . . < t < ∞ and  = 1, 2, . . . , the set {Bt0 ; Btk − Btk−1 ;

k = 1, . . . ,  }

is a finite collection of independent random variables. Note that by (i) Bt − Bs depends only on t − s (temporal homogeneity). A Brownian motion in Rd is a d-tuple B = {Bt } = {Bt1 , Bt2 . . . Btd : t ∈ R+ } where each {Bti } is a Brownian motion on R and the B i ’s are independent. We say B starts at x if B0 = x a.s. If B0 = x a.s., then  2 1 − |x−y| 2t e dy Pt (x, A) = P (Bt ∈ A) = (2πt)d/2 A for t > 0, x ∈ Rd and A ∈ B(Rd ). It follows that Pt (x0 + x, x0 + A) = Pt (x, A),

for all x0 ∈ Rd

(spatial homogeneity).

Property (ii) is called independence of increments of B. A more exact idea of the modulus of continuity of a typical Brownian path is given by the following theorem of A. Khintchine and P. L´evy Theorem 4.3 (Local law of the iterated logarithm). For each t ≥ 0, ⎧ ⎫ ⎨ ⎬ Bt+h (ω) − Bt (ω) P ω : lim sup . = 1 = 1.   ⎩ ⎭ h→0+ 2h log log 1 h

Corollary 4.1. Let 0 < < 12 . For P -almost all ω ∈ Ω, the following condition holds: there exists C > 0 such that for each t > 0, there exists δt, > 0 such that |Bu (ω) − Bt (ω)| ≤ C |u − t| ,

for all |u − t| < δt, .

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Canonical construction of Brownian motion (Wiener’s construction) Let Ω = C([0, 1]) and B(C[0, 1]) be the σ-algebra generated by the cylinders sets {ω ∈ Ω : ω(t1 ) ∈ B1 , . . . , ω(tn ) ∈ Bn } for given B1 , . . . , Bn ∈ B(R), 0 ≤ t1 < t2 < . . . < tn ≤ 1 and n = 1, 2, . . . . Consider the stochastic process (Wt )t∈[0,1] on Ω such that (1) Wt (ω) = ω(t) (2) W0 = 0 a.e. (3) Wt is Ft -measurable and Wt − Ws is independent of Fs for every 0 < s < t.

−x2 ) dx. (4) P {Wt − Ws ∈ A} = [2π(t − s)]−1/2 A exp( 2(t−s) This is called the canonical 1-dimensional Brownian motion and the measure P is called Wiener measure. The work here goes into proving that P is actually σ-additive on B(C([0, 1])). We may take Ft to be σ-algebra {Ws : s ≤ t} duly completed. Wiener proved that the Wiener measure exists and may be constructed as a limit of induced measures of random walks.

4.3.1

Some properties of Brownian paths

The ensemble of Brownian travellers moves according to subtle and remarkable principles. Some feeling for the nature of the process may be obtained from consideration of the following facts.

4.3.1.1

Unbounded variation

Let B be standard Brownian motion. For almost all ω, the sample paths t → Bt (ω) are of unbounded variation on any interval, see Theorem 4.3.

4.3.1.2

Non-differentiability

An immediate consequence of the local law of the iterated logarithm, Theorem 4.3, is that for a fixed t, and almost all ω, Bt (ω) is not differentiable at t. It can be proved that in fact Bt (ω) is nowhere differentiable for almost all ω. (And thus, in the sense of Wiener measure P on C([0, ∞)), ‘almost all’ continuous functions are nowhere differentiable.)

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Law of the iterated logarithm P

  Bt (ω) =1 =1 ω : lim sup √ 2t log log t t→∞

(A. Khintchine). This shows that the Brownian traveller wanders off to infinity, and gives an idea of the speed with which she does so. However, she still returns to any neighbourhood of the origin infinitely many times. 4.4

Stochastic integration of vector valued processes

As mentioned above, the total winnings of a gambling game is a type of stochastic integral. For continuous time processes, a stochastic integral is a random process of the form  (4.2) Yt dXt . Unfortunately, the most important examples of processes like a Brownian motion process do not have the property that t −→ Xt (ω) has finite variation for almost all sample points ω, so the interpretation of the ‘integral’ above is problematic from the point of view of measure theory. Rather than repeat the accepted theory of stochastic integration of random processes that may be found in the monograph [75], we shall interpret the stochastic integral (4.2) as a type of singular bilinear integral of a random process Y with respect to a random measure dXt . Our approach has the advantage of being able to treat processes Y with values in a Banach space E that has the necessary geometric property (UMD spaces). Let (Σ, S, μ), (Ω, E, P) be probability measure spaces. Let L0 (Ω, E, P) be the space of real valued random variables with respect to P equipped with convergence in P-measure. A mapping W : S → L0 (Ω, E, P) is called a Gaussian random measure on (Σ, S, μ) if (a) for all disjoint sets A1 , A2 , . . . from S, we have W

∞  n=1

∞   An = W (An ), n=1

where the sum converges with probability one;

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(b) the random variables W (A1 ), . . . , W (An ) are independent for all disjoint sets A1 , . . . , An belonging to S and all n = 1, 2, . . . ; (c) for every A ∈ E, the random variable W (A) has a normal distribution with mean zero and variance μ(A). Example 4.9. Let Σ = R+ with μ Lebesgue measure on R+ . If Wt is Wiener process, then we write W ((s, t]) = Wt − Ws for 0 ≤ s < t. For each t > 0, let Ft be the σ-algebra generated by the random variables {Ws }0≤s≤t . According to property (i) of Brownian motion, the random variable W ((s, t]) is Gaussian with mean zero and variance t − s. Property (ii) of Brownian motion establishes that the random variables W (A1 ), . . . , W (An ) are independent for all disjoint half-open intervals A1 , . . . , An and all n = 1, 2, . . . . For any finite union A of half-open intervals, W (A) is defined by additivity and it can be checked directly that W (A) is a Gaussian random variable with mean zero and variance μ(A). The extension of W to the δ-ring of all Borel sets with finite Lebesgue measure is facilitated by the following lemma concerning Gaussian vectors. Lemma 4.1 ([117], Lemma 2.1). Let Xn be symmetric Gaussian random vectors such that Xn → X in probability P. Then X is a symmetric Gaussian random vector and, for every 0 < p < ∞, EXn − Xp → 0 as n → ∞. Thus, a Gaussian random measure W is a vector measure in any of the spaces Lp (Ω, E, P) for 0 ≤ p < ∞. Let E be a Banach space and 1 ≤ p < ∞. The Banach space of all equivalence classes of strongly P-measurable functions f : Ω → E for which

p p p Ω f  dP < ∞ is denoted by L (Ω, E, P; E) or just L (P; E). The norm is given by 1/p  p f  dP , f ∈ Lp (P; E). f −→ Ω

The relative topology of Lp (P; E) on Lp (Ω, E, P) ⊗ E is completely separated , so we may consider the integral of an E-valued function f : Σ → E with respect to W in Lp (P; E), in the sense of Definition 2.2. Then for each ξ ∈ E  , the scalar function f, ξ : σ → f (σ), ξ , σ ∈ Σ, is W -integrable and by the Itˆo isometry [75], the random variable Σ f, ξ dW is Gaussian with variance Σ | f, ξ |2 dμ.

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By [117, Corollary 4.2], an E-valued function f : Σ → E, integrable with respect to W in Lp (P; E) as in Definition 2.2 is necessarily integrable with respect to the Gaussian random measure W in the sense of [117, Definition 2.1]. The converse statement follows from the proof of [117, Theorem 4.1] and [117, Corollary 2.1]. It follows from the lemma above, that if f : Σ → E is integrable with respect to W in Lp (P; E) for some 1 ≤ p < ∞, then it is integrable with respect to W in Lp (P; E) for every 1 ≤ p < ∞. A function f : Σ → E is integrable with respect to W in Lp (P; E) for some 1 ≤ p < ∞, if and only if it is strongly measurable and stochastically integrable with respect to W as in [132, Definition 2.1]. It follows from [117, Proposition 6.1] that W has finite E-semivariation in Lp (P; E) if and only if E is a Banach space of type 2. The domain of the function f just considered does not depend on sample points ω ∈ Ω, that is, f is not a random process, so the integral   ˆ p Lp (P) ≡ Lp (P; E) f dW = f ⊗ dW ∈ E ⊗ Σ

Σ

is a decoupled bilinear integral of the type considered in Chapter 2. We shall now see that when f is a progressively measurable random E

valued process, it is possible to decouple the stochastic integral Σ f dW in the manner described in Chapter 2 by using special features of the Gaussian random measure W . At the same time, we can deal with vector valued random processes f for free. Our purpose is to emphasise connections with other areas of analysis featuring singular bilinear integrals, so we do not strive for the utmost generality in dealing with stochastic integration. The treatment in [75] suffices for practical applications such as financial mathematics and stochastic modelling. Let (Ω , E  , P  ) be another probability measure space. Then strongly measurable functions ϕ : Σ → Lp (P  ; E) may also be integrated with respect to W in the space Lp (P  ⊗ P; E), in the sense of Definition 2.2. If ϕ : Σ → Lp (P  ; E) is strongly measurable, then an appeal to [117, Corollary 4.2] shows that ϕ is W -integrable in Lp (P  ⊗ P, E), if and only if the E-valued function t −→ ϕ(t), g , t ∈ Σ, is W -integrable in Lp (P ; E) for every g ∈ Lq (P ) and for every set A ∈ S, there exists (ϕ ⊗ W )(A) ∈ Lp (P  ⊗ P, E) such that (ϕ ⊗ W )(A), g = A ϕ(t), g dW (t) P-a.e. in Lp (P ; E) for each g ∈ Lq (P  ). Here 1/p + 1/q = 1 and · , g denotes integration against g in the P -variable. Roughly speaking, if we apply the pointwise multiplication map to the element (ϕ ⊗ W )(A) of Lp (P ⊗ P, E), then we obtain the stochastic integral

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of a strongly measurable function ϕ : Σ → Lp (P ; E) with respect to the Gaussian random measure W . ˆ π L2 (P) denotes the projective tensor prodFor example, if L2 (P, E)⊗ 2 2 uct of L (P, E) with L (P) [88, 41.2], then an application of the CauchySchwarz inequality shows that there exists a unique continuous linear map ˆ π L2 (P) → L1 (P, E) J : L2 (P, E)⊗ such that for every f, g ∈ L2 (P) and x ∈ E, the equality J((x.g) ⊗ f )(ω) = x.g(ω)f (ω) holds for almost every ω ∈ Ω, that is, J is the pointwise multiplication map that restricts a product function to the diagonal of the Cartesian product Ω × Ω of the sample space Ω with itself—a situation reminiscent of taking the trace of a linear operator as considered in Chapter 3. Requiring that the bilinear integral (ϕ⊗W )(A) belongs to the projective ˆ π L2 (P) is a restrictive assumption. As we saw in tensor product L2 (P, E)⊗ Chapter 3, ‘generalised traces’ of bounded linear operators still make sense when we move away from projective tensor products into certain Banach function spaces. It is remarkable that under mild assumption on the Banach space E, the multiplication map J is continuous for the relative topology of L2 (P ⊗ P, E) into L2 (P, E), provided it is restricted to a certain class of definite integrals (ϕ ⊗ W )(A) with respect to a Gaussian random measure W . The remainder of this chapter is devoted to the proof of this simple but powerful observation. Let W be a Gaussian random measure on R+ . For each t > 0, let Ft be the σ-algebra generated by the random variables {W ((0, s])}0≤s≤t . A function φ : R+ → L2 (P)⊗ E is said to be an elementary progressively measurable function if there exist times 0 < t1 < · · · < tN , vectors xmn ∈ E and sets Amn ∈ Ftn−1 , n = 1, . . . , N , m = 1, . . . , M such that φ(t) =

M N  

xmn χ

n=1 m=1

Amn



(tn−1 ,tn ]

(t),

t ∈ R+ .

Then φ has values in every space Lp (P)⊗E for 1 ≤ p ≤ ∞, φ is W -integrable in Lp (P) ⊗ E ⊗ Lp (P) for every 1 ≤ p < ∞ and we have  M N     φ ⊗ dW = (xmn χ ) ⊗ W ((tn−1 , tn ]) . (4.3) R+

n=1 m=1

Amn

Let X of L∞ (P) ⊗ E ⊗ Lp (P) consisting of all

denote the linear subspace ∞ vectors R+ φ ⊗ dW with φ : R+ → L (P) ⊗ E an elementary progressively

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measurable function. For each 1 ≤ p < ∞, let J : L∞ (P) ⊗ E ⊗ Lp (P) → Lp (P, E) be the linear map defined by J(g ⊗ x ⊗ f )(ω) = xg(ω).f (ω) for almost all ω ∈ Ω. Definition 4.1. A Banach space E is called a UMD space (or, E has the unconditional martingale difference property) if for any 1 < p < ∞, there exists Cp > 0 such that for any E-valued martingale difference {ξj }nj=1 and n = 1, 2, . . . , the inequality   p   n  n p       E j ξj  ξj    ≤ Cp E    j=1  j=1   E

E

holds for every j ∈ {±1}, j = 1, . . . , n. The following result is from [50, Theorems 2 and 2 ]. The simplified proof presented below comes from [133, Lemma 3.4]. It is given here to spell out the connection with bilinear integration in tensor products. Theorem 4.4. Let E be a UMD space and 1 < p < ∞. The multiplication map J is continuous from X into Lp (P, E) for the relative topology of the Banach space Lp (P ⊗ P, E) on X.

Proof. Suppose that R+ φ ⊗ dW is given by formula (4.3). For each n = M 1, . . . , N , let ξn = m=1 (xmn χ ) and define Amn   dn = ξn .W ((tn−1 , tn ]) ⊗ 1 Then

en = ξn ⊗ W ((tn−1 , tn ]). /

0 N φ ⊗ dW ⊗ 1. n=1 en = R+ φ ⊗ dW and n=1 dn = J R+

N



For n = 1, . . . , N , let r2n−1 := 12 (dn + en ) and r2n := 12 (dn − en ). We now show that {rj }2N j=1 is a martingale difference sequence with respect to the filtration {Gj }2N j=1 , where G2n = σ(Ftn ⊗ Ftn )    G2n−1 = σ Ftn−1 ⊗ Ftn−1 , W ((tn−1 , tn ]) ⊗ 1 +    1 ⊗ W ((tn−1 , tn ]) . Clearly rj is G2k -measurable for j = 1, . . . , 2k and k = 1, 2, . . . , N . Denote the expectation with respect to P⊗P by EP⊗P . Since ξn is Ftn−1 -measurable, we have EP⊗P (dn +en |G2n−2 ) = EP⊗P (dn |Ftn−1 ⊗Ftn−1 )+EP⊗P (en |Ftn−1 ⊗Ftn−1 ) = 0,

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so that EP⊗P (r2n−1 |G2n−2 ) = 0. On the other hand,   dn + en = (ξn ⊗ 1). (W (tn ) − W (tn−1 )) ⊗ 1 + 1 ⊗ (W (tn ) − W (tn−1 )) , so r2n−1 is G2n−1 -measurable and we have 2EP⊗P (r2n |G2n−1 ) = EP⊗P (dn − en |G2n−1 )   = (ξn ⊗ 1).EP⊗P W ((tn−1 , tn ]) ⊗ 1   −1 ⊗ W ((tn−1 , tn ]) |G2n−1 ) = 0. The last equality follows from the observation that   f = W ((tn−1 , tn ]) ⊗ 1 and

  g = 1 ⊗ W ((tn−1 , tn ])

are independent of Ftn−1 ⊗ Ftn−1 and G2n−1 =   i.i.d. random variables σ Ftn−1 ⊗ Ftn−1 , f + g . Consequently, EP⊗P (f − g|G2n−1 ) = EP⊗P (f |G2n−1 ) − EP⊗P (g|G2n−1 ) 1 1 = (f + g) − (f + g) = 0, 2 2 and it follows that {rj }2N is a martingale difference sequence. j=1 2N N 2N N Because n=1 dn = j=1 rj and n=1 en = j=1 (−1)j+1 rj , the inequalities N  N   N    p   p   p       en  ≤ EP⊗P  dn  ≤ Cp EP⊗P  en  Cp−1 EP⊗P        n=1

E

n=1

E

n=1

E

follow by appealing to the UMD property, so that p    p         φ ⊗ dW  ≤ E J φ ⊗ dW  Cp−1 EP⊗P     R+  R+ E Ep      ≤ Cp EP⊗P  φ ⊗ dW  .   R+ E

Hence, J : X → Lp (P, E) is continuous for the relative topology of Lp (P ⊗ P, E) on the linear space , + X= R+

φ ⊗ dW : φ elementary progressively measurable .

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Consequently, if φ : R+ → Lp (P, E) is the limit a.e. of elementary progressively measurable functions φn , n = 1, 2, . . . , such that ∞  φn ⊗ dW A

n=1

converges in L (P ⊗ P, E) for each A ∈ B(R+ ), then φ is W -integrable in Lp (P⊗P, E) in the sense of Definition 2.2. Moreover, if E is a UMD Banach space, then for t > 0, the stochastic integral of φ with  respect to W on the 

interval (0, t] may be represented by J (0,t] φ ⊗ dW and it belongs to the space Lp (P, E). We leave our discussion of stochastic integration at this point. The treatment in [133] is aimed at applications to stochastic partial differential equations and Lp -regularity. In [75], the scalar case E = C is treated with the Gaussian random measure W replaced by a continuous semimartingale X. Here the L2 theory is applied utilising a judicious choice of stopping times and the Itˆo calculus is obtained. p

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Chapter 5

Scattering theory

Scattering theory can be considered as a part of the more general perturbation theory in physics [76]. The main idea is that detailed information about an unperturbed selfadjoint operator H0 in a Hilbert space H enables us to draw conclusions about another selfadjoint operator H provided that H0 and H differ little from one to another in an appropriate sense. One of the earliest successes of abstract scattering theory dealt was with the case where H − H0 is a trace class operator acting on the Hilbert space H. We return to this topic in Section 8.4 below where we consider Krein’s spectral shift function. In the mathematical formulation of quantum mechanics, H0 is the free Hamiltonian and H = H0 + V is the total Hamiltonian associated with the interaction potential V obtained from the model in classical mechanics that we seek to quantise. The operator sum ‘H0 + V ’ is usually interpreted as a form sum so that H is a selfadjoint operator acting in H. The spectrum σ(H) ⊂ R of H must be bounded below, otherwise particle interactions may need to be taken into account. We briefly discuss in the following sections the two main approaches to the mathematical formulation of quantum mechanical scattering theory— the time-dependent and the time-independent or stationary scattering theory.

5.1

Time-dependent scattering theory

In the time-dependent scattering theory, we consider the time evolution of an incident particle (wave packet) under the influence of the interaction 123

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with a scattering centre or with another particle by the evolution equation ∂u = Hu, u(0) = f. ∂t The behaviour of the state u for large times is studied in terms of the free equation i∂u0 /∂t = H0 u0 . With the appropriate assumptions on the potential V , for every vector f orthogonal to the eigenvectors of H, there (±) exist vectors f0 orthogonal to the eigenvectors of the free Hamiltonian H0 such that i

lim u(t) − u0 (t) = 0,

t→±∞ (±)

(±)

if u0 (0) = f0 . The initial data f and f0 f = lim e t→±∞

are related by the equality

itH −itH0

e

(±)

f0

(±)

because u(t) = e−itH f and u0 (t) = e−itH0 f0 operators

for ±t ≥ 0. The wave (a)

W± = W± (H, H0 ) = lim eitH e−itH0 P0 t→±∞

(5.1)

encode this property provided that the limits in the strong operator topol(a) ogy exist. The operator P0 is the orthogonal projection onto the abso(a) lutely continuous subspace H0 of the operator H0 . The scattering operator S = W+∗ W− connects the asymptotic behaviour of a quantum system as (−) (+) t → −∞ and t → ∞ in terms of the free problem, that is, S : f0 → f0 . In the context of quantum computation where H is a finite dimensional Hilbert space, the scattering operator represents the quantum computation associated with a potential V . In the time-independent or stationary scattering theory, one studies solutions of the time-independent Schr¨ odinger equation with a parameter that belongs to the continuous part of the spectrum of the total Hamitonian operator H. These solutions lie outside the Hilbert space and are characterized by certain asymptotic properties partly motivated by physical considerations. The observables, in particular the S-operator, are obtained from the asymptotic properties of such solutions [7]. It has been well known for a long time that these two methods are mathematically very different. The connections between them has been a problem studied since the seventies. Of fundamental importance is the task to establish conditions for which the final objects of the calculations (the S operator) are identical in both cases. This question is not easy to answer because of the nature of the calculations in the stationary scattering

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formalism. This theory uses mathematical manipulations that must first be interpreted in some sense before they can be made rigorous so that it is possible to compare with the time-dependent method, which is a very well-developed mathematical theory. The recent developments in scattering theory can be found, for example, in [137] and references therein. Specially important is the work developed by M. Sh. Birman and D. R. Yafaev in stationary scattering theory and for the time-dependent theory, the work developed by Werner O. Amrein, Vladimir Georgescu, J.M. Jauch and K. B. Shina (see for example [5, 6, 8]). We can cite also the book from Berthier [12] and the work of J. Derezi´ nski and C. G´erard (see for example [35] and references therein).

5.2

Stationary state scattering theory

In this chapter we focus our attention on the passage from time-dependent to the stationary formalism. Our starting point is the paper from W.O. Amrein, V. Georgescu, J.M. Jauch [7]. The principal problem to solve in this passage is the following: the basic quantities in the time-dependent theory (e.g. the wave operator) will be expressed in terms of a Bochner integral of certain operators over the time available. These formulas have been known for a long time. Operators in the Bochner integral can be expressed as a spectral integrals via the Spectral Theorem. Then the passage is achieved if we are able to interchange the two integrals and evaluate the time integral. The main problem of a mathematical nature is under which conditions we can interchange the two integrals and verify that the conditions are in fact satisfied for the integrals that we encounter in scattering theory. In order to develop this alternative approach, we have to change the definition of wave operators W± replacing the unitary groups by the corresponding resolvents R0 (z) = (H0 − z)−1 and R(z) = (H − z)−1 ([137]). In this stationary approach, in place of the limits in time, one has to study the boundary values in a suitable topology of the resolvents as the spectral parameter z tends to the real axis. An important advantage of the stationary approach is that it gives convenient formulas for the wave operators and the scattering matrix. The temporal asymptotics of the time-dependent Schr¨odinger equation is closely related to the asymptotics at large distances of solutions of the stationary

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Schr¨odinger equation: −ΔΨ + V (x)Ψ = λΨ.

(5.2)

In other words, from the physical point of view, in the time-dependent formalism we consider that the particle being scattered have to behave as a free particle in the far past and in the far future (t → ±∞). In the time-independent formalism we consider that the particle being scattered has to behave as a free particle far away from the scattering centre, where the influence of the potential is negligible and the total Hamitonian is practically the free Hamitonian. In terms of boundary values of the resolvent, the scattering solution, or eigenfunction of the continuous spectrum, can be constructed using the Lippmann-Schwinger equations (see for example [137]). More precisely, as in [7], suppose that the potential V is a real valued measurable function and H = H0 + V and H0 are both self adjoint on the domain D(H0 ) of H0 . The strong operator limits in the formula (5.1) (a) for the wave operators W± are assumed to exist with P0 = I, that is, H0 has absolutely continuous spectrum. Furthermore, we suppose that range(W+ ) = range(W− ) is the orthogonal complement of the closed linear subspace spanned by the eigenvectors of H. Suppose that F0 is the spectral measure of H0 and F is the spectral measure of H, so that by the Spectral Theorem for selfadjoint operators [121, Theorem 13.30]   λ dF0 (λ), H = λ dF (λ). H0 = R

R

Typical formulas that relate quantities in the stationary and timedependent approaches are given in [7, (30), (31)]:   W− = lim i R(λ − i ) dF0 (λ) = lim −i dF (λ)R0 (λ − i ), (5.3) ↓0 ↓0 R  R W+ = lim −i R(λ + i ) dF0 (λ) = lim i dF (λ)R0 (λ − i ), (5.4) ↓0

↓0

R

R

and [7, (34), (35)]:

 (W− − I) D(H0 ) = lim − R(λ − i )V dF0 (λ) ↓0  R = lim dF (λ)V R0 (λ − i ), ↓0

R

(5.5)

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 (W+ − I) D(H0 ) = lim − R(λ + i )V dF0 (λ) ↓0  R = lim dF (λ)V R0 (λ − i ). ↓0

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(5.6)

R

The limits in formulas (5.5) and (5.6) are taken in the strong operator topology of L(D(H0 ), H) and in terms of the spectral measure F0 and resolvent R0 of the free Hamiltonian, we obtain [7, (38), (39)]:  (W− − I) D(H0 ) = lim − R0 (λ − i )V W− dF0 (λ) ↓0  R dF0 (λ)V W− R0 (λ − i ), (5.7) = lim ↓0 R  (W+ − I) D(H0 ) = lim − R0 (λ + i )V W+ dF0 (λ) ↓0  R dF0 (λ)V W+ R0 (λ − i ). (5.8) = lim ↓0

R

The formulas (5.5) and (5.7) correspond to the ‘Lippmann-Schwinger equations’ ψ(ω, λ) = ψ0 (ω, λ) − R(λ − i0)V ψ0 (ω, λ)

(5.9)

ψ(ω, λ) = ψ0 (ω, λ) − R0 (λ − i0)V ψ(ω, λ)

(5.10)

for the scattering solutions ψ(ω, λ) of the stationary Schr¨odinger equation (5.2), see [7, p. 427]. The mathematical difficulty with the ‘Lippmann-Schwinger equations’, as written in equations (5.9) and (5.10) above, is how to interpret the limits R(λ − i0) and R0 (λ − i0) when the function ψ(ω, λ) does not belong to the underlying Hilbert space H (λ > 0 belongs to the absolutely continuous spectrum of H = −Δ + V and ψ(ω, λ) is not an eigenfunction of the operator H). In stationary state scattering theory, the problem of interpreting the limit is circumvented by appealing to the specific structure of the Schr¨odinger operator H = −Δ + V and specifying the asymptotic behaviour of the scattering solutions ψ(ω, λ) as |x| → ∞, see [137] and Section 5 below. In contrast, formulas (5.5) and (5.7) are applicable in a much wider context. Each of the formulas (5.3)-(5.8) involve the integration of an operator valued function Φ with respect to a spectral measure P in which the values of Φ act on the values of P or the values of P act on the values of Φ by operator composition—a type of bilinear integration. In Section 5.4 below, we discuss in greater detail why regular bilinear integration cannot

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deal with the situation described above. Our exposition concentrates on formula (5.5). A similar approach suffices to deal with the other formulas. To see how formula (5.5) reduces to the interchange of integrals, first note that for each ψ ∈ D(H0 ), the left-hand side of (5.5) can be written as  ∞ (W− − I)ψ = lim i e−t eitH V e−itH0 ψ dt, (5.11) ↓0

0

see [7, (16)]. The integral is a Bochner integral in the Hilbert space H. For each > 0, we have    ∞  ∞ e−t eitH V e−itH0 ψ dt = eitH V e−i(λ−i)t d(F0 ψ)(λ) dt. 0

R

0

(5.12) On the right-hand side of (5.5), we have     ∞ −i(λ−i)t itH R(λ − i )V dF0 (λ) = i e e dt V d(F0 ψ)(λ), R

R

0

so we need to show that    ∞ eitH V e−i(λ−i)t d(F0 ψ)(λ) dt 0

 

R



e

= R

0

−i(λ−i)t itH

e

 dt V d(F0 ψ)(λ)

for each > 0. The values of the L(D(H0 ), H)-valued function   ∞ e−i(λ−i)t eitH dt V, λ ∈ R, λ −→ 0

act on the D(H0 )-valued measure F0 ψ, giving rise to the bilinear integration process that has been determined in earlier chapters. The main feature here is that the integrals are ‘decoupled’, so it is easy to see that a type of Fubini Theorem is valid for vector valued integrals. 5.3

Time-dependent scattering theory for bounded Hamiltonians and potentials

The second basic example from [65, Section 4] is relevant to the connection between stationary-state and time-dependent scattering theory [7] where H0 represents the free Hamiltonian operator and V represents an interaction potential. Attention is first retricted to bounded operators. It is well known that quantum computation relies on calculations within a finite dimensional complex inner product space H whose dimension represents the number of qubits available [79].

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Consequently the dynamics of quantum computations is determined by matrix Hamiltonians acting on H, so that the case of bounded Hamiltonian operators and bounded interaction potentials treats the dynamics of quantum computations and their asymptotic limit. A quantum computer is particularly adapted to Monte-Carlo type approximations of Feynman path integrals, so a quantum computer may be capable of analysing, say, the Riemann hypothesis or colliding supermassive black holes at the level of quantum gravity. At the time of writing, quantum computers do not exist and neither Feynman path integrals nor quantum gravity nor the Riemann hypothesis is well understood, so we must leave such speculation to physically inclined Philosophers of Science and Mathematics and noncommutative geometers [29]. Example 5.1. Let (H, ( · | · )) be a separable Hilbert space. Suppose H0 and V are bounded selfadjoint operators. Then H = H0 + V is also a bounded selfadjoint operator and the function f : R+ × R → L(H) defined for > 0 by f (t, σ) = eitH V e−i(σ−i)t for t ≥ 0 and σ ∈ R is uniformly bounded in L(H). Let P be the spectral measure associated with the selfadjoint operator H0 and h ∈ H. Lebesgue measure on R+ is denoted by λ. We would like to verify the identities   ∞ f (t, σ) d(λ ⊗ (P h))(t, σ) = e−t eitH V e−itH0 h dt R+ ×R

0

  =



e R

−t itH

e

Ve

−itσ

 dt d(P h)(σ)

0

that help to establish the connection between stationary-state and timedependent scattering theory in the case of unbounded selfadjoint operators H0 and V [7]. The H-valued measure λ ⊗ (P h) : S → H is given by λ ⊗ (P h)(A × B) = λ(A)(P h)(B),

A ∈ Bf (R+ ), B ∈ B(R).

Here Bf (R+ ) is the collection of Borel subsets of R+ with finite Lebesgue measure and S is the δ-ring generated by the collection {A × B : A ∈ Bf (R+ ), B ∈ B(R)} of product sets in R+ ×R. We check that the projective tensor product πH E = L(H)⊗ of H and L(H) with the uniform operator norm  is bilinear  admissible for H in the sense of Section 2.3 above and f is λ ⊗ (P h) -integrable in E,

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with the appropriate modification for integration with respect to a vector measure defined on a δ-ring. Because a Hilbert space H necessarily has the approximation property [125, III.9], H ⊗ H ⊗ H separates points of E [88, 43.2(12)]. Let B(t) = {σ : (t, σ) ∈ B } be the section at t ≥ 0 of the Borel subset B of R+ × R. We check that the function  itH   −itH0  −t ΦB e V ⊗ e P (B(t))h , t ≥ 0,  : t −→ e is Bochner integrable in E. The function t −→ P (B(t))h, t ≥ 0, is strongly measurable so by Lusin’s Theorem on each interval [0, T ], there exists a Borel set Σ of arbitrarily large measure on which it is continuous in H. Then ΦB  is continuous in E on Σ because H and H0 are assumed to be bounded selfadjoint operators so that the unitary groups eitH and e−itH0 , t ∈ R, are continuous for the uniform operator topology, as is easily seen from the exponential power series expansion. Consequently, ΦB  is strongly measurable in E. Because  ∞  ∞     V h B , Φ E dt ≤ e−t eitH V  . e−itH0 P (B(t))h dt ≤ 0 0 it follows that Φ is Bochner integrable in E. For every u, v, w ∈ H, we have #  ∞ " ∞      ΦB dt, u ⊗ v ⊗ w = e−t eitH V uv e−itH0 P (B(t))hw dt  0 0     =

f (t, σ), u ⊗ v d λ ⊗ (P h)w (t, σ). B   According to Definition 2.2, the function f is λ ⊗ (P h) -integrable in E and for every Borel subset B of R+ × R, we have  ∞  f d(λ ⊗ (P h)) = ΦB  dt. 0

B

Furthermore, for each t ≥ 0, the L(H)-valued function f (t, · ) is (P h)integrable in E the vector measure P h and for each B ∈ B(R), we have  f (t, σ) ⊗ d(P h) = e−t eitH V ⊗ e−itH0 P (B)h ∈ E, B

so that  R

 f (t, σ) d(P h)(σ) = J

R

f (t, σ) ⊗ d(P h) = e−t eitH V e−itH0 h.

For each σ ∈ R, the function f ( · , σ) is Bochner integrable for the uniform norm of L(H) and the L(H)-valued function  ∞ f (t, σ) dt, σ ∈ R, σ −→ 0

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is (P h)-integrable in E. For every Borel subset C of R, we have    ∞  ∞ + ×C f (t, σ) dt d(P h) = J ΦR dt.  C

0

0

The scalar version of Fubini’s Theorem and the assumption that H ⊗ H ⊗ H separates points of the bilinear admissible space E ensure the identity    ∞    ∞ f (t, σ) dt d(P h) = f (t, σ) d(P h) dt R

0

0

R

relevant to scattering theory. We can still make sense of Example 5.1 if H0 and V are unbounded π H will no operators, but it is clear that the auxiliary space E = L(H)⊗ longer suffice, because the unitary groups eitH and e−itH0 , t ∈ R, are only continuous for the strong operator topology—it is too much to expect that the function ΦB  defined in the example above will be Bochner integrable in π H. We first need to consider the approximation the Banach space L(H)⊗ of L(H)-valued functions in the strong operator topology. 5.4

Bilinear integrals in scattering theory

In this section we return to the situation considered in Example 5.1, but with the more physically realistic case of unbounded selfadjoint operators. Let (H, ( · | · )) be a separable Hilbert space. Suppose H0 : D(H0 ) → H and V : D(V ) → H are selfadjoint operators with respective dense domains D(H0 ) ⊂ D(V ). We suppose that H = H0 + V is also a selfadjoint operator on D(H0 ). For example, this is the situation for the free Hamiltonian 2 Δ of a particle moving in R3 subject to a Coulomb potential H0 = − 2m V (x) = c/|x|, x ∈ R3 \ {0}, c = 0. The function f : R+ × R → L(D(H0 ), H) given by f (t, σ) = eitH V e−i(σ−i)t for t ≥ 0 and σ ∈ R is defined on D(H0 ). Let P0 be the spectral measure associated with the selfadjoint operator H0 and h ∈ D(H0 ). As above, Lebesgue measure on R+ is denoted by λ. The spectral measure P0 commutes with the selfadjoint operator H0 in the sense that for each Borel subset B of R, the inclusion P0 (B)D(H0 ) ⊂ D(H0 ) is valid and H0 P0 (B)h = P0 (B)H0 h for every h ∈ D(H0 ). We give D(H0 ) the graph 1  norm h −→ h2H + H0 h2H 2 , h ∈ D(H0 ), under which it is itself a Hilbert space.

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Now we seek a suitable space E, bilinear admissible for D(H0 ), H, for  which f is λ ⊗ (P h) -integrable in E with respect to the D(H0 )-valued measure λ ⊗ (P h). Let X , Y be Banach spaces. For y ∗ ∈ Y ∗ , we have 1 2 1 n 2  n        Tj xj , y ∗  =  xj , Tj∗ y ∗    j=1   j=1  ≤

n 

xj X .Tj∗ y ∗ X ∗ ,

j=1

for all Tj ∈ L(X , Y) and xj ∈ X , j = 1, . . . , n and all n = 1, 2, . . . . Hence, if we let ⎧ ⎫ n n ⎨ ⎬  xj X .Tj∗ y ∗ X ∗ : u = Tj ⊗ xj uτ = sup inf (5.13) ⎩ ⎭ y ∗ ≤1 j=1

n

j=1

over all representations u = j=1 Tj ⊗ xj , n = 1, 2 . . . , of u ∈ L(X , Y) ⊗ X , then the inequality JuY ≤ uτ holds for the product map Ju = n j=1 Tj xj by the Hahn-Banach Theorem. The completion of the linear τ X with respect to the norm ·τ is written as L(X , Y)⊗ τX. space L(X , Y)⊗ n n Lemma 5.1. For each u = j=1 Tj ⊗xj ∈ L(X , Y)⊗X , let Tu = j=1 xj ⊗ n ∗ ∗ π X ∗ ) denote the linear map y ∗ −→ Tj∗ ∈ L(Y ∗ , X ⊗ j=1 xj ⊗ (Tj y ), ∗ ∗ y ∈Y . Then the linear mapping k : u −→ Tu , u ∈ L(X , Y) ⊗ X , is the restric τ X onto a closed linear tion to L(X , Y) ⊗ X of an isometry of L(X , Y)⊗ π X ∗ ) in the uniform norm. subspace of L(Y ∗ , X ⊗ Proof. For u ∈ L(X , Y) ⊗ X , we have uτ = Tu L(Y ∗ ,X ⊗  π X ∗). Lemma 5.2. Let X , Y be Banach spaces with Y ∗ norm separable. If B is a bounded, absolutely convex subset of L(X , Y) ⊗ X with τ -closure B, then π X ∗ ). kB is a closed subset of Ls (Y ∗ , X ⊗ π X ∗ ) can be Proof. Every element v of the closure of kB in Ls (Y ∗ , X ⊗ ∞ ∞ ∗ ∗ ∗ represented as v = j=1 λj xj ⊗ Tj in Ls (Y , X ⊗π X ), where j=1 |λj | < n ∞, the finite sum j=1 λj xj ⊗ Tj∗ belongs to B for each n = 1, 2, . . . and Tj∗ → 0 in Ls (Y ∗ , X ∗ ), xj → 0 in X as j → ∞. To see this, we note that Y ∗ is separable, so the relative topology of π X ∗ ) on kB coincides with the relative topology of X ⊗ πF , Ls (Y ∗ , X ⊗ ∗ ∗ where F is the metrisable locally convex space L(Y , X ) endowed with

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the topology of pointwise convergence on a countable dense subset of the unit ball of Y ∗ [88, (1) p. 138]. Then the given representation for v follows πF . from Theorem 1.8 applied to X ⊗ Because    ∞ ∞      λ T ⊗ x ≤ sup |λj |xj X Tj∗ y ∗ X ∗ j j j  ∗ ≤1 y  j=n j=n τ

≤ sup{xj X Tj L(X ,Y) } j

∞ 

|λj | → 0

j=n

π X ∗ ). as n → ∞, it follows that v ∈ kB, hence kB is closed in Ls (Y ∗ , X ⊗ A Banach space X has the approximation property if X ∗ ⊗ X is dense in Lκ (X ), with κ the topology of precompact convergence [123, Section III.9]. n Each element j=1 x∗j ⊗ xj of X ∗ ⊗ X defines the finite rank operator x −→

n 

x, x∗j xj ,

x ∈ X.

j=1

τ X is bilinear admissible Lemma 5.3. The Banach space E = L(X , Y)⊗ for the Banach spaces X and Y provided that X has the approximation property. Proof. Properties a)-c) of bilinear admissibility clearly hold, so it remains to establish property d): the family of all linear maps x⊗y ∗ ⊗x∗ : T ⊗x −→

T x, y ∗ x, x∗ for x ∈ X , x∗ ∈ X ∗ and y ∗ ∈ Y ∗ separates points of E. Let Be ((X ⊗ Y ∗ )s , Xs∗ ) denote the linear space of all separately continuous bilinear forms on (X ⊗Y ∗ )×X ∗ for the topologies of simple convergence σ(X ⊗ Y ∗ , L(X , Y)) and σ(X ∗ , X ). It is equipped with the topology e of biequicontinuous convergence [88, p. 167]. Because X has the approximation property, the canonical linear map π X → Be ((X ⊗ Y ∗ )s , Xs∗ ) ψ : Ls (Y ∗ , X ∗ )⊗ is one-to-one [88, 43.2 (12)]. The topology τ has a fundamental system of neighbourhoods of zero π X , so E embeds in Be ((X ⊗ closed for the weaker topology of Ls (Y ∗ , X ∗ )⊗ Y ∗ )s , Xs∗ ) too [88, 18.4 (4)]. Because (X ⊗ Y ∗ ) × X ∗ separates the space Be ((X ⊗ Y ∗ )s , Xs∗ ) of bilinear forms, it follows that X ⊗ Y ∗ ⊗ X ∗ separates points of E.

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Remark 5.1. The

approximation property for X is needed to make sense of the integral Ω f dm of an L(X , Y)-valued function f with respect to an X -valued measure m. Similarly, the approximation property is needed to define the trace of a nuclear operator on X in the case Y = C. All Banach spaces of practical interest, including Hilbert spaces, possess the approximation property, see [94]. The following result improves Theorem 4.4 of [49] in the sense that we obtain a stronger form of integrability in the conclusion (c) below. Theorem 5.1. Let (H, ( · | · )) be a separable Hilbert space and let A : D(A) → H be a selfadjoint operator with spectral measure PA . Let E denote τ D(A) where the Hilbert space D(A) is endowed with the space L(D(A), H)⊗ the graph norm. Let (Γ, E, μ) be a σ-finite measure space. Suppose that the measurable function u : R × Γ → C has the property that for every h ∈ D(A), the function u( · , γ) is PA h-integrable in D(A). Let f : Γ → L(D(A), H) be a strongly μ-measurable in Ls (D(A)H) function for which there exist positive μ-measurable functions α, β, v, on Γ, with the following properties: (i) |u(σ, γ)| ≤ v(γ) for every σ ∈ R and γ ∈ Γ, (ii) f (γ)h ≤ α(γ)Ah + β(γ)h for every h ∈ D(A), γ ∈ Γ, and (iii) Γ v(γ)(α(γ) + β(γ)) dμ(γ) < ∞. Then for each h ∈ D(A),

(a) the function γ −→ f (γ) S u(σ, γ) d(PA h)(σ), γ ∈ Γ, is Bochner μintegrable in H for each S ∈ B(R), (b) the function γ −→ u(σ, γ)f (γ)g, γ ∈ Γ, is Bochner μ-integrable in H for each σ ∈ R and g ∈ D(A), and

(c) the L(D(A), H)-valued function σ −→ T u(σ, γ)f (γ) dμ(γ), σ ∈ R, is strongly measurable in L(D(A), H) and (PA h)-integrable in E with respect to the D(A)-valued measure PA h, for each set T ∈ E. Moreover the equality   S

 u(σ, γ)f (γ) dμ(γ) d(PA h)(σ) T    = f (γ) u(σ, γ) d(PA h)(σ) dμ(γ) T

S

holds for every S ∈ B(R) and T ∈ E.

(5.14)

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Proof. Property (a) follows from the strong μ-measurability of f in L(D(A), H) and the norm estimates       f (γ) u(σ, γ) d(PA h)(σ) dμ(γ)   Γ S        α(γ) A u(σ, γ) d(PA h)(σ) ≤  dμ(γ) Γ S      + β(γ)  u(σ, γ) d(PA h)(σ)  dμ(γ) Γ S  ≤ PA hsv(D(A)) v(γ)(α(γ) + β(γ)) dμ(γ) Γ

< ∞. Here PA hsv(D(A)) is the semivariation norm of the D(A)-valued measure PA h. The estimates (i)-(iii) also give Property (b). Now let B be an element of the σ-algebra B(R) ⊗ E with B(γ) = {σ ∈ R : (σ, γ) ∈ B} its section at γ ∈ Γ. We check that the L(D(A), H) ⊗ D(A)-valued function  u(σ, γ) d(PA h)(σ), γ ∈ Γ, ΦB : γ −→ f (γ) ⊗ B(γ)

τ D(A) and is integrable in the Banach space E = L(D(A), H)⊗       ΦB (γ) dμ(γ) ≤ PA hsv(D(A)) v(γ)(α(γ) + β(γ)) dμ(γ).   Γ

Γ

τ

Firstly, the inequality   v(γ)f (γ)∗ L(H,D(A)) dμ(γ) = v(γ)f (γ)L(D(A),H) dμ(γ) Γ Γ  = sup v(γ)f (γ)ψ(γ)H dμ(γ) ψ∞ ≤1

 ≤

Γ

v(γ)(α(γ) + β(γ)) dμ(γ) Γ

follows from assumption (ii), where the supremum is taken over D(A)valued E-simple functions ψ. It follows that   ΦB (γ)τ dμ(γ) ≤ PA hsv(D(A)) v(γ)(α(γ) + β(γ)) dμ(γ). Γ

Γ

We are not assuming that f is strongly μ-measurable for the uniform norm of L(D(A), H), so some caution is needed.

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Now k ◦ ΦB : Γ → L(D(A), H ⊗ H) defines a linear map LB : D(A) → L1 (μ, H ⊗ H) by LB g = [k ◦ ΦB g],

g ∈ D(A).

π H)) The bounded linear operator LB is the limit in L(D(A), L1 (μ, H⊗ 1 of (Pn ⊗ I)LB , n = 1, 2, . . . , for conditioning operators Pn : L (μ, H) → L1 (μ, H), n = 1, 2, . . . , on finitely generated σ-algebras. Then (Pn ⊗I)LB = [k ◦ ΦB,n ] for  ΦB,n : γ −→ (Pn f )(γ) ⊗ u(σ, γ) d(PA h)(σ), γ ∈ Γ, B(γ)

and Pn f is an L(D(A), H)-valued E-simple function for each n = 1, 2, . . . , so ΦB,n has values in Ls (D(A), H) ⊗ D(A). Moreover,  (Pn ⊗ I)LB L(D(A),L1(μ,H⊗ ≤ ΦB τ dμ,  π H)) Γ

H) ⊗ D(A) for each n = 1, 2 . . . , and B (Pn ⊗ I)LB dμ ∈ L(D(A),   (Pn ⊗ I)LB dμ = LB dμ (5.15) lim n→∞

C

C

π H) for each C ∈ E. in Ls (D(A), H⊗ We want to show that for each C ∈ E, the linear map  LB g dμ, g ∈ D(A), g −→ C   belongs to the uniform closure kE of k L(D(A), H) ⊗ D(A) in the space π H) of linear operators. An appeal to Lemma 5.2 and equaL(D(A), H⊗ tion (5.15) shows that this is true on any set Γn = {ΦB τ ≤ n} and π H) has finite varibecause the vector measure LB μ : E → Ls (D(A), H⊗

ation V (LB μ)(Γ) = Γ ΦB (γ)τ dμ(γ) in the uniform operator norm of π H), we have LB μ : E → kE. The equality LB μ = k ◦ (ΦB .μ) L(D(A), H⊗ ensures that ΦB is μ-integrable in E. Because the Banach space E is bilinear admissible for D(A), H, the integral      u(σ, γ)f (γ) ⊗ d (PA h) ⊗ μ (σ, γ) ∈ E B

is uniquely defined by the scalar equation         u(σ, γ)f (γ) ⊗ d (PA h) ⊗ μ (σ, γ) u ⊗ v ⊗ w B    = u(σ, γ)(f (γ)u|v) d((PA h|w) ⊗ μ)(σ, γ) B     = ΦB dμ u ⊗ v ⊗ w Γ

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for each B ∈ B(R) ⊗ E, u, w ∈ D(A) and v ∈ H. Moreover, statement (c) holds and    u(σ, γ)f (γ) dμ(γ) ⊗ d(PA h)(σ) S T      = u(σ, γ)f (γ) ⊗ d (PA h) ⊗ μ (σ, γ), S×T

for all S ∈ B(R) and T ∈ E. The equality (5.14) of the iterated integrals follows from the scalar version of Fubini’s Theorem and the bilinear admissability of E. Corollary 5.1. Let (H, ( · | · )) be a separable Hilbert space and H0 : D(H0 ) → H be a selfadjoint operator with spectral measure P0 . Let τ D(H0 ) E = L(D(H0 ), H)⊗ where the Hilbert space D(H0 ) is endowed with the graph norm. Suppose that V : D(V ) → H is a selfadjoint operator with dense domain D(H0 ) ⊂ D(V ). We suppose that H = H0 + V is also a selfadjoint operator on D(H0 ), > 0 and the function f : R+ × R → L(D(H0 ), H) is given by f (t, σ) = eitH V e−i(σ−i)t ,

t ∈ R+ , σ ∈ R.

Then for each h ∈ D(H0 ), (a) the function t −→ e−t eitH V e−iH0 t g, t ∈ R+ , is Bochner integrable in H for each g ∈ D(H0 ), (b) the function t −→ f (t, σ)g, t ∈ R+ , is Bochner μ-integrable in H for each σ ∈ R and g ∈ D(H0 ), and

(c) the L(D(H0 ), H)-valued function σ −→ T f (t, σ) dt, σ ∈ R, is strongly measurable in L(D(H0 ), H) and (P0 h)-integrable in E with respect to the D(H0 )-valued measure P0 h, for each set T ∈ B(R+ ). Moreover the equality     f (t, σ) dt d(P0 h)(σ) = e−t eitH V e−iH0 t P0 (S)h dt S

T

T

holds for every S ∈ B(R) and T ∈ B(R+ ). Proof. Statements (a) and (b) follow directly from the fact that e−itH and e−itH0 are continuous unitary semigroups on H and D(H0 ). The selfadjoint operator V is closed and has domain containing D(H0 ). By the Closed Graph Theorem it is bounded in the graph norm of the Hilbert space D(H0 ). Conditions (i)-(iii) of Theorem 5.1 are satisfied for the function u(t, σ) = e−t e−iσt and f : t −→ eitH V , t ∈ R+ , σ ∈ R, where we take v(t) = e−t , t ∈ R+ , and α, β are constants.

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Application to the Lippmann-Schwinger equations

The ‘Lippmann-Schwinger equations’ (5.9) and (5.10) for the scattering solutions ψ(ω, λ) of the stationary Schr¨odinger equation (5.2), are valid for stationary scattering theory under rather stringent conditions. For example, in the stationary Schr¨odinger equation (5.2) in Rd , suppose that V is a uniformly bounded potential for which there exist C > 0 and ρ > d such that |V (x)| ≤ C(1 + |x|)−ρ ,

x ∈ Rd .

(5.16)

Let Sd−1 = {x ∈ Rd : |x| = 1 } be the unit sphere in Rd . Then for any λ > 0 and ω ∈ Sd−1 , equation (5.2) has a unique solution ψ( · ; ω, λ) with asymptotics ψ(x; ω, λ) = e

√ i λω,x



ei λ|x| + a (d−1)/2 + o(|x|−(d−1)/2 ) |x|

as |x| → ∞ in Rd . The solution ψ( · ; ω, λ) is associated with the incoming plane wave √

ψ0 (x; ω, λ) = ei

λω,x

,

x ∈ Rd .

Then equations (5.9) and (5.10) are satisfied and if we define ψ − (x, ξ) = ψ(x; ω, λ), ψ+ (x; ξ) = ψ(x; −ω, λ) √ for ξ = λω and x ∈ Rd , then the transformations  ψ ± (x; ξ)f (x) dx, f ∈ L2 (Rd ), (F ± f )(ξ) = (2π)−d/2 Rd

define isometries from the absolutely continuous subspace H(a) of H = −Δ + V onto L2 (Rd ) such that the wave operators (5.1) have the representation W± = (F ± )∗ F0 with respect to the Fourier transform F0 on L2 (Rd ) [137, equation (1.23)] and F ± H = |ξ|2 F ± . The analysis can be pushed through to ρ > (d + 1)/2 in the bound (5.16) by utilising an appropriate function space for asymptotic estimates [137, Section 3.5] and to the case of short-range potentials ρ > 1 by averaging over ω ∈ Sd−1 [137, Section 2.3].

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Under the conditions for which Theorem 4.4 is valid, from equations (5.5) and (5.7), for each h ∈ D(H0 ), we have  R(λ − i )V d(F0 h)(λ), (5.17) W− F0 (B)h = F0 (B)h − lim ↓0 B  W− F0 (B)h = F0 (B)h − lim R0 (λ − i )V W− d(F0 h)(λ), (5.18) ↓0

B

for every Borel subset B of R+ = [0, ∞), so equations (5.17) and (5.18) are equalities between H-valued measures. The H-valued measures B −→ W− F0 (B)h and B −→ F0 (B)h are absolutely continuous on R+ . Roughly speaking, d(F0 ψ0 ( · ; ω, λ )) = δ(λ − λ )ψ0 ( · ; ω, λ ), dλ d(W− F0 ψ0 ( · ; ω, λ )) = δ(λ − λ )ψ( · ; ω, λ ), dλ although the plane wave ψ0 ( · ; ω, λ ) is not an element of H and the measures are not differentiable in H, but it is clear that equations (5.5) and (5.7) are the integrated versions of the ‘Lippmann-Schwinger equations’ (5.9) and (5.10), valid under very general conditions. Theorem 5.2. Let H0 : D(H0 ) → H be a positive selfadjoint operator in the Hilbert space H with absolutely continuous spectrum. Let V : D(V ) → H be a symmetric operator such that D(H0 ) ⊆ D(V ) and H = H0 + V is selfadjoint on D(H0 ). Suppose also that the limits (5.1) exist. Let F0 be the spectral measure of H0 and F the spectral measure of H and R0 (z) = (H0 − z)−1 , R(z) = (H − z)−1 the corresponding resolvents. Then for each > 0, the two L(D(A), H)-valued functions λ −→ R(λ − i )V and λ −→ R0 (λ − i )V W− , λ ∈ R, are (F0 ψ)-integrable in the Banach τ D(H0 ) and the equalities space L(D(H0 ), H)⊗  W− ψ − ψ = − lim R(λ − i )V d(F0 ψ)(λ) ↓0 R  = − lim R0 (λ − i )V W− d(F0 ψ)(λ). ↓0

R

holds for each ψ ∈ D(H0 ). Proof. We first show how the first equation of (5.5) follows from Theorem 5.1. According to equations (5.11) and (5.12), we can write    ∞ itH −i(λ−i)t e V e d(F0 ψ)(λ) dt (5.19) W− ψ − ψ = lim i ↓0

0

R

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for every ψ ∈ D(H0 ). The function u : R × R+ → C defined by u(λ, t) = ie−i(λ−i)t ,

λ ∈ R, t ≥ 0,

is a uniformly bounded continuous function on R × R+ , so u( · , t) is (F0 ψ)integrable and F0 (H0 ψ)-integrable in H for each t > 0. Let B(t)ψ = eitH V ψ for every t ∈ R and ψ ∈ D(H0 ) and v(t) = e−t , t ≥ 0. Then the L(D(H0 ), H)-valued function t −→ B(t), t ∈ R+ , is strongly continuous in Ls (D(H0 ), H) and |u(λ, t)| ≤ v(t) for all λ ∈ R, t ∈ R+ . By assumption the domain D(V ) of the symmetric operator V contains the domain D(H0 ) of the selfadjoint operator H0 , so by the Closed Graph Theorem, there exists c > 0 such that V ψ2 ≤ c2 (H0 ψ2 + ψ2 ) for every ψ ∈ D(H0 ), so also B(t)ψ2 ≤ c2 (H0 ψ2 + ψ2 ) for every ψ ∈ D(H0 ), because eitH is a unitary operator for every t ≥ 0. Condition (ii) of Theorem 5.1 is easily verified. Now for each t ≥ 0, Borel subset S of R and ψ ∈ D(H0 ), the equality  B(t) u(λ, t) d(F0 ψ)(λ) = ie−t eitH V e−itH0 F0 (S)ψ S

holds, so the function t −→ B(t) S u(λ, t) d(F0 ψ)(λ), t ≥ 0, is continuous in H and   ∞

0

B(t) S

u(λ, t) d(F0 ψ)(λ) dt ≤ cψD(H0 ) / ,

so it is Bochner integrable in H, verifying conclusion (1) of Theorem 5.1. Similarly, the function t −→ u(λ, t)B(t)ψ, t > 0, is Bochner integrable in H for each ψ ∈ D(H0 ) and λ ∈ R, because u(λ, t)B(t)ψ = ie−t eit(H−λ) V ψ and  ∞

0

u(λ, t)B(t)ψ dt = −R(λ − i )V ψ,

so conclusion (2) of Theorem 5.1 is also verified directly. It follows from equation (5.19) and Theorem 5.1, that for each > 0, the L(D(H0 ), H)-valued function λ −→ R(λ − i )V , λ ∈ R, is (F0 ψ)-integrable τ D(H0 ) and the equalities in L(D(H0 ), H)⊗    ∞ itH −i(λ−i)t W− ψ − ψ = lim i e V e d(F0 ψ)(λ) dt ↓0 R 0 = − lim R(λ − i )V d(F0 ψ)(λ) ↓0

R

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hold for each ψ ∈ D(H0 ). Under our assumptions, we can also verify that    ∞ itH0 −i(λ−i)t e V W− e d(F0 ψ)(λ) dt W− ψ − ψ = lim i ↓0

0

141

(5.20)

R

for every ψ ∈ D(H0 ) [7, (20)]. The intertwining property HW− = W− H0 and the assumption that D(H) = D(H0 ) ⊆ D(V ) ensures that W− D(H0 ) ⊂ D(V ), so that the righthand side of equation (5.20) makes sense and λ −→ R0 (λ − i )V W− , λ ∈ R, is actually an L(D(H0 ), H)-valued function. Now if we define B(t)ψ = eitH0 V W− ψ for every ψ ∈ D(H0 ) and t ≥ 0, and leave the functions u, v as above, then  ∞ u(λ, t)B(t)ψ dt = −R0 (λ − i )V W− ψ, 0

and an application of Theorem 5.1 ensures that L(D(H0 ), H)-valued function λ −→ R0 (λ − i )V W− , λ ∈ R, is (F0 ψ)-integrable in τ D(H0 ) and Ls (D(H0 ), H)⊗  W− ψ − ψ = − lim R0 (λ − i )V W− d(F0 ψ)(λ). ↓0

R

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Chapter 6

Random evolutions

The Feynman-Kac formula is used in the theory of stochastic processes to represent solutions of the heat equation with a source or sink term and finds widespread applications in mathematical finance and mathematical physics. I. Kluv´anek pointed out in [81] that the Feynman-Kac formula may also be viewed as a means to represent the superposition of two general evolutions and is readily interpreted in terms of integration with respect to operator valued measures.

6.1

Evolution processes

The concepts of direct interest in modelling of evolving quantum systems are the dynamical group t −→ e−itH , t ∈ R, representing the evolution of quantum states and the spectral measure Q representing the observation of the position of the system in the underlying classical configuration space Σ of the system. These lead to the operator valued set functions described in [64]. Suppose that Ω is a nonempty set. A family S of subsets of Ω is called a semi-algebra of sets if Ω ∈ S and for every A ∈ S and B ∈ S, there exist a positive integer n and pairwise disjoint set Xj ∈ S, j = 0, 1, . . . , n, such that A ∩ B = X0 ,

A\B =

n 

Xj

j=1

* and the union kj=0 Xj belongs to S for every k = 0, 1, . . . , n. The example of a semi-algebra to keep in mind is the collection of all subintervals [a, b), 0 ≤ a < b ≤ 1, of [0, 1). 143

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Let (Σ, E) be a measurable space. For each s ≥ 0, suppose that Ss is a semi-algebra of subsets of a nonempty set Ω such that Ss ⊆ St for every 0 ≤ s < t. For every s ≥ 0, there are given functions Xs : Ω → Σ with the property that Xs−1 (B) ∈ St for all 0 ≤ s ≤ t and B ∈ E. It follows that the cylinder sets E = {Xt1 ∈ B1 , . . . , Xtn ∈ Bn } := {ω ∈ Ω : Xt1 (ω) ∈ B1 , . . . , Xtn (ω) ∈ Bn } =

Xt−1 (B1 ) 1

∩ ···∩

(6.1)

Xt−1 (Bn ) n

belong to St for all 0 ≤ t1 < . . . < tn ≤ t and B1 , . . . , Bn ∈ E. Let X be a Banach space. The vector space of all continuous linear operators T : X → X is denoted by L(X ). It is equipped with the uniform operator topology, but measures with values in L(X ) of practical interest are only σ-additive for the strong operator topology. A semigroup S of operators acting on X is a map S : [0, ∞) → L(X ) such that S(0) = IdX , the identity map on X and S(t + s) = S(t)S(s) for all s, t ≥ 0. The semigroup S represents the evolution of state vectors belonging to X . If limt→0+ S(t)x = x for every x ∈ X , then S is called a C0 -semigroup of operators. An L(X )-valued spectral measure Q on E is a map Q : E → L(X ) that is σ-additive in the strong operator topology and satisfies Q(Σ) = IdX and Q(A ∩ B) = Q(A)Q(B) for all A, B ∈ E. In quantum theory, Q is typically multiplication by characteristic functions associated with the position observables, but the spectral measures associated with momentum operators also appear. Now suppose that Mt : St → L(X ) is an additive operator valued set function for each t ≥ 0. The system (Ω, St t≥0 , Mt t≥0 ; Xt t≥0 )

(6.2)

is called a time homogeneous Markov evolution process if there exists a L(X )-valued spectral measure Q on E and a semigroup S of operators acting on X such that for each t ≥ 0, the operator Mt (E) ∈ L(X ) is given by Mt (E) = S(t − tn )Q(Bn )S(tn − tn−1 ) · · · Q(B1 )S(t1 )

(6.3)

for every cylinder set E ∈ St of the form (6.1) and the process is called an (S, Q)-process. The basic ingredients are the semigroup S describing the evolution of states and the spectral measure Q describing observation of states represented by vectors in X. An explicit proof that formula (6.13)

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actually does define an additive set function has been written, for example, in [85, Proposition 7.1]. In many significant examples the family St is a σ-algebra of subsets of a nonempty set Ω and Mt : St → L(X ) is σ-additive for the strong operator topology of L(X ), that is, the sum ⎛ ⎞ ∞ ∞   Ej ⎠ x = Mt (Ej )x, x ∈ X , Mt ⎝ j=1

j=1

converges in the norm of X for all pairwise disjoint Ej ∈ St , j = 1, 2, . . . . Then we say that the process (6.2) is σ-additive. It is well-known that σ-additivity fails spectacularly for the unitary group t −→ eitΔ . Example 6.1. Let (Σ, B) be a measurable space. Suppose that (Ω, S, P x x∈Σ ; Xt t≥0 ) is a Markov process with P x (X0 = x) = 1 for each x ∈ Σ [28, p. 6]. Then the transition functions pt (x, dy) are given by pt (x, B) = P x (Xt ∈ B) for x ∈ Σ, B ∈ B and t ≥ 0. For any signed measure μ : B → R and t ≥ 0, the measure S(t)μ : B → R is given by  pt (x, B) dμ(x), for B ∈ B, (S(t)μ)(B) = Σ

and the spectral measure Q is given by Q(B)μ = χB .μ, B ∈ B. Then Mt (E)μ defined by equation (6.13), is the measure  P x ({Xt ∈ B} ∩ E) dμ(x), B −→ Σ

B ∈ B, so Mt is an operator valued measure acting on the space X of signed measures with the total variation norm. In general, the semigroup S is not a C0 -semigroup on X . For a Feller process [28, p. 50], S is a weak*continuous semigroup of operators on the space of signed Borel measures M (Σ) = C0 (Σ) . For the case of Brownian motion in Rd considered previously in Section 4.3, we have |x−y|2

e− 2t dy, pt (x, dy) = (2πt)d/2

t > 0, x ∈ Rd .

Example 6.2. Let (Σ, E, μ) be a measure space and let Tt : Σ → Σ, t ∈ R, be a group of measure preserving maps. Then S(t) : f −→ f ◦ T−t for f ∈

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L2 (μ) and t ∈ R, defines a continuous unitary group S of linear operators on L2 (μ). Observe that if δx is the unit point mass at x ∈ Σ, then δx ◦ T−t = δx ◦ Tt−1 = δTt x . Suppose that the spectral measure Q is defined by Q(B)f = χB .f for B ∈ B and f ∈ L2 (μ). The measure Mt defined by equation (6.13) is given by Mt = S(t)(Q ◦ σ −1 ). Here σ : x −→ Tt x, for t ≥ 0 and x ∈ Σ. The process Xt t≥0 is given by the evaluation maps Xt (ω) = ω(t) for t ≥ 0 and ω ∈ Ω = σ(Σ). If we take Σ to be the phase space of a system in classical mechanics with μ the Liouville measure, then the dynamical flow t −→ Tt x, t ∈ R, x ∈ Σ, of the system satisfies the assumptions above. The operator valued path space measure Mt is concentrated on the sample space Ω of all classical paths in the time interval [0, t]. The viewpoint of operator valued measures of the process (6.2) deals with the essential mathematical representations of the concepts of interest to physics: the dynamical group represented by S and observations, represented by the spectral measure Q. It is due to I. Kluv´anek [81–85]. There are two results relevant to the proof of Theorem 7.1 in the next chapter for dominated semigroups. The modulus semigroup is the smallest semigroup of operators such that |S(t)u| ≤ |S|(t)u for all u ∈ L2+ (μ). If the modulus semigroup exists, then S is called a dominated semigroup of operators on L2 (μ). Theorem 6.1 ([71, Theorem 3.1]). Let (Σ, E, μ) be a σ-finite measure space. For a semigroup S of operators on L2 (μ) and Q(B)f = χB f , B ∈ E, f ∈ L2 (μ), the operator valued, additive set function Mt : St → L(L2 (μ)) of the Markov evolution process (6.2) has uniformly bounded range on the algebra St generated by the cylinder sets (6.1) for every t ≥ 0 if and only if S is dominated. In this case, the modulus semigroup |S| of S exists and (|S|(t)u, v) = |(Mt u, v)|(Ω) for every t > 0, u, v ∈ L2 (μ) with u ≥ 0 μ-a.e. and v ≥ 0 μ-a.e. Theorem 6.2 ([64]). Suppose that (6.2) is a σ-additive Markov evolution process on the Banach space X such that X is progressively measurable, that is, (ω, s) −→ Xs (ω), 0 ≤ s ≤ t, is jointly (St ⊗ B([0, t]))-measurable for every t > 0.

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For any uniformly bounded E-measurable function V : Σ → C, the operator valued map $ t %  exp V ◦ Xs ds dMt , t ≥ 0 SV : t −→ Ω

0

is a semigroup of operators on X . If S is a C0 -semigroup on X with S(t) = etA , t ≥ 0, then SV (t) = et(A+Q(V )) , t ≥ 0. Proof. If x ∈ X and we let u(0) = x and   t u(t) = e 0 V ◦Xr dr d(Mt x),

t > 0,

Ω

then an application of the Fubini-Tonelli Theorem gives   t   t s s V ◦Xs .e 0 V ◦Xr dr d(Mt x) ds = V ◦ Xs e 0 V ◦Xr dr ds d(Mt x). Ω

0

Ω

0

Applying formula (6.3) to the left-hand side and the Fundamental Theorem of Calculus to the right-hand side of this equation gives  t  /  0 t e 0 V ◦Xr dr − 1 d(Mt x) = u(t) − S(t)x. S(t − s)Q(V )u(s) ds = 0

Ω

t The expression u(t) = S(t)x + 0 S(t − s)Q(V )u(s) ds is a form of Duhamel equation and by iterating it, we obtain the Dyson series expansion  s2 ∞  t  sk  ··· et(A+Q(V )) x = etA x + n=1

0

0

0

3 4 e(t−sk )A Be(sk −sk−1 )A · · · Bes1 A x ds1 · · · dsk for the infinitesimal generator A of S [76, Theorem IX.2.1], that is, we obtain the Feynman-Kac formula   t e 0 V ◦Xr dr dMt , t > 0. (6.4) et(A+Q(V )) = Ω

Here Q(V ) = Σ V dQ is a bounded linear operator on X . The identity (6.4) first appeared in [81] and an exposition in greater detail appears in [64]. We shall always assume that the dominated semigroup S is associated with a progressively measurable process (6.2) with X = L2 (μ)—the case in all examples of practical interest. In effect, this is a mild regularity assumption on S weaker than that of [119] and [97]. On the other hand, considerable effort is devoted to establishing this fact in the theory of Dirichlet forms [48]. We look more closely at this regularity assumption in the next section.

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Measurable functions

A necessary and sufficient condition for the progressive measurability for metric space valued random processes was given by K.-L. Chung and J. Doob [27, Proposition 33] using the notion of separable Borel measurability. As pointed out in the proof of a similar result in [34, Th´eor`eme IV.30] for real valued processes, separable Borel measurability is the same notion as strong measurability familiar from vector integration [38, Definition II.1]. In the context of Banach spaces, Pettis’s Measurability Theorem [38, Theorem II.2] yields the equivalence of strong measurability and separable Borel measurability. We first clarify the measurability situation for metric space valued functions. A function f : Ω → Σ from a measurable space (Ω, S) with values in a metric space (Σ, d) is called S-measurable if f −1 (B) ∈ S for every Borel subset B of Σ. An S-measurable function f : Ω → Σ with finitely many values is called S-simple. A function f : Ω → Σ is S-simple if and only if there exists a finite partition {Ω1 , . . . , Ωn }, n ∈ N, into S-measurable sets such that f (Ωj ) = {σj } for σj ∈ Σ, j = 1, . . . , n. Let T be a Hausdorff topological space and A ⊂ T . Then [A] denotes the set of all elements in T which are the limit of some sequence of elements of A. A set A ⊂ T is called sequentially closed if A = [A]. The sequential s closure A , of a set A ⊂ T , is the smallest sequentially closed subset of T which contains A, that is, the intersection of all sequentially closed subsets of T containing A. For a metric space (Σ, d) and a nonempty set Ω, the product topology of T = ΣΩ = {f |f : Ω → Σ} is called the topology of pointwise convergence. Proposition 6.1. Let B be the Borel σ-algebra of a metric space (Σ, d) and let f : Ω → Σ be a function from a measurable space (Ω, S) into Σ. Then the following conditions are equivalent. a) f is the uniform limit on Ω of a sequence of countably valued Smeasurable Σ-valued functions. b) f is Borel measurable with separable range. c) f belongs to the pointwise sequential closure of S-simple Σ-valued functions on Ω. d) f is the pointwise limit on Ω of a sequence of S-simple Σ-valued functions. Proof. The argument is omitted in the proof of [34, Th´eor`eme IV.30], so

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we give it here. Condition a) implies condition b), because countably valued functions fn , n = 1, 2, . . . , are necessarily (S-B)-measurable and if fn → f pointwise as n → ∞, then f (Ω) ⊂ ∪n fn (Ω) and ∪n fn (Ω) is separable. To see that b) implies a), let {Sn,k }k be a partition of f (Ω) into disjoint Borel sets with diameter less than 1/n: if {xk }k∈N is a dense subset of f (Ω) and Bn,k = {x ∈ Σ : d(x, xk ) < 1/(2n)},

k = 1, 2, . . . ,

then Sn,k = (Bn,k ∩ f (Ω)) \ (∪j 0, formula (6.3) defines a σ-additive measure Mt : σ(St ) → L(X ) with respect to the random process X and sample space Ω. ˜ s : Ω → Σ, s ≥ 0, Then there exists a progressively measurable process X ˜ such that {Xs = Xs } is Mt -null for all 0 ≤ s ≤ t and t > 0. The progressive measurability is with respect to the given filtration ˜ s , 0 ≤ s ≤ t. σ(St ), t ≥ 0. It turns out that σ(St ) is also generated by X In [100], an elementary proof of the existence of progressively measurable ˜ of a measurable adapted process X is given. There it is modification X ˜ is assumed that X is adapted to a given filtration F = Ft t≥0 and X proved to be progressively measurable with respect to the same filtration F . A different proof appears in [34, Th´eor`eme IV.30 (b)] and the result may be viewed as complementary to Theorem 6.3. The proof of Theorem 6.3 above appeals to the separability of L2 (Mt ) and the assumption that S is strongly continuous. Under the assumptions of Theorem 6.3, we shall construct a modifica˜ s : Ω → Σ, s ≥ 0, of X such that the mapping tion X ˜ s (ω), (s, ω) −→ X

0 ≤ s ≤ t, ω ∈ Ω,

is (B([0, t]) ⊗ σ(St ))-measurable into (Σ, B) for each t > 0. Proof. Because B is the Borel σ-algebra of a Polish space Σ, there exists a countable subfamily A of B such that σ(A) = B [46, 424B (a)]. The algebra generated by A is also countable, so we may assume that A is already a countable algebra of subsets of Σ. The algebra S 0 of subsets of Ω generated by {Xr ∈ A} for r ≥ 0 rational and A ∈ A is also countable. The σ-algebra generated by the random variables Xs , s ≥ 0, is denoted by S. For each t > 0, the operator valued measure Mt : σ(St ) → L(X ) measures the σ-algebra σ(St ) generated by the collection of all cylinder sets (6.1). Step 1. For each t ≥ 0, the linear subspace span{[χE ]Mt : E ∈ S 0 ∩ St } is dense in L2 (Mt ). Because span{[χE ]Mt : E ∈ St } is dense in L1 (Mt ) by [80, Theorem of Extension] and the topology of L2 (Mt ) is weaker than the topology of

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L1 (Mt ) on {[f ]Mt : f ∞ ≤ 1}, it is enough to show that span{[χE ]Mt : E ∈ S 0 ∩ St } is L1 (Mt )-dense in span{[χE ]Mt : E ∈ St }. Let t > 0 and E = {Xs ∈ A} ∈ St for 0 ≤ s ≤ t and A ∈ B. Suppose that En = {Xsn ∈ An } ∈ S 0 ∩ St for n = 1, 2, . . . , where sn → s and supξX  ≤1 | Qx, ξ |(AΔAn ) → 0 as n → ∞ for each x ∈ X . Because A is a countable algebra of subsets of Σ for which σ(A) = B, such sets An ∈ A, n = 1, 2, . . . , exist by [116, Proposition 2]. This observation also handles the case t = 0. Appealing to the assumption that the semigroup S in formula (6.3) defining the operator valued measure Mt is a C0 -semigroup and Q is a spectral measure, it follows that Mt (En ∩ F )x → Mt (E ∩ F )x

(6.5)

in the norm of X as n → ∞ for each x ∈ X and each F ∈ St . Because {[χF ]Mt : F ∈ St } is L1 (Mt )-dense in {[χF ]Mt : F ∈ σ(St )} [80, Theorem of Extension], the convergence (6.5) is also valid for F ∈ σ(St ). Appealing to the Hahn decomposition of a real valued measure, we also have | Mt x, ξ |(En ∩ F ) → | Mt x, ξ |(E ∩ F ) as n → ∞ for each x ∈ X , ξ ∈ X  and F ∈ σ(St ). The locally convex Hausdorff topology of L1 (m) for an L(X )-valued measure m is defined by the family of seminorms  px [m] : [f ]m −→ sup |f | d| mx, ξ |, [f ]m ∈ L1 (m), (6.6) ξX  ≤1

Ω

for x ∈ X [86, p. 24]. A result of Bartle-Dunford-Schwartz [86, Theorem II.1.1] gives finite positive measures μx,t such that μx,t (E) ≤ supξX  ≤1 | Mt x, ξ |(E) for each x ∈ X and E ∈ σ(St ) and limμx,t (E)→0 supξX  ≤1 | Mt x, ξ |(E) = 0. A glance at the construction of the measure μx,t in [86, Theorem II.1.1] and dominated convergence ensures that μx,t (En ∩ F ) → μx,t (E ∩ F ) as n → ∞ for each x ∈ X and F ∈ σ(St ), that is, χEn → χE weakly in L1 (μx,t ) as n → ∞, for each x ∈ X . By the Hahn-Banach Theorem, χE belongs to the norm closure in L1 (μx,t ) of the balanced convex hull of {[χF ]μx,t : F ∈ S 0 ∩ St } for each x ∈ X and so, by [86, Theorem III.2.2], also to the closure in L1 (Mt ) of the balanced convex hull of {[χF ]Mt : F ∈ S 0 ∩ St }. Because the same argument applies to the intersection of finitely many cylinder sets like E, it follows that span{[χF ]Mt : F ∈ S 0 ∩ St } is L1 (Mt )-dense in span{[χF ]Mt : F ∈ St }. The measurable space (Σ, B) is isomorphic to either (C, PC) for some countable set C or (R, B(R)) [46, 424C]. It suffices to prove the result for

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Σ = R. The countable case is handled in a similar manner. Under the measure space isomorphism, Borel measurability for an R-valued function guarantees Borel measurability for the corresponding Σ-valued function f and Proposition 6.1 ensures that f : Ω → Σ is strongly measurable. The space L0 (Mt ) = {[f ]Mt |f : Ω → R is Mt -measurable} is given the weakest topology for which the map % $ |f | , [f ]Mt ∈ L0 (Mt ), [f ]Mt −→ 1 + |f | Mt is continuous in L1 (Mt ). Because X is separable, for some countable dense subset D of the unit ball of X , the topology of L1 (Mt ) is determined by the countable family of seminorms px [Mt ], x ∈ D, defined by equation (6.6). It follows that both L1 (Mt ) and L0 (Mt ) are metrisable and complete. Step 2. The mapping s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in L0 (Mt ) for each t ≥ 0. As mentioned above, the limit of a sequence of strongly measurable (n) functions is strongly measurable. Let Xs = χ{|Xs |≤n} Xs for s ≥ 0 and n = 1, 2, . . . . Because L1 (Mt ) embeds in L0 (Mt ) and for each 0 ≤ s ≤ t, (n) [Xs ]Mt → [Xs ]Mt Mt -a.e. and in L0 (Mt ) as n → ∞, it suffices to prove (n) that mapping s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in 2 L (Mt ) for each t ≥ 0. By Step 1, L2 (Mt ) is separable because we may take the span over the rationals to obtain our countable dense set. The separability of X ensures that L2 (Mt ) is metrisable. The uniform boundedness principle ensures that L2 (Mt ) is sequentially complete, so L2 (Mt ) is a Polish space as well. For each n = 1, 2, . . . , let Yn denote the closure in L2 (Mt ) of the linear (n) span of the collection Ξn = {[Xs ]Mt : 0 ≤ s ≤ t} of random variables. For each bounded σ(St )-measurable function ϕ : Ω → C, u ∈ X and v ∈ X  , the continuous linear functional ξ[ϕ, u, v] : L2 (Mt ) → C is defined by 

f, ξ[ϕ, u, v] = f ϕ d| Mt u, v |, f ∈ L2 (Mt ). Ω

Then the collection {ξ[ϕ, u, v] : ϕ ∈ Ξn , u ∈ X , v ∈ X  } of continuous linear functionals on L2 (Mt ) clearly separates the closed subspace Yn of L2 (Mt ). The closed linear subspace Yn is itself a Polish space, so according to [46, 423B (a), 423F (b)], the Borel σ-algebra of Yn is generated by the continuous linear functionals y −→ y, ξ[ϕ, u, v] , y ∈ Yn , as ϕ ranges over

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Ξn and u ∈ X and v ∈ X  . Because S is a C0 -semigroup, inspection of formula (6.3) shows that the scalar valued function  Xs(n) Xr(n) d Mt u, v , 0 ≤ s ≤ t, (6.7) s −→ E

is continuous for each 0 ≤ r ≤ t, u ∈ X , v ∈ X  and E ∈ St . The collection of sets E ∈ σ(St ) for which (6.7) is Borel measurable is a monotone class, so it is the whole σ-algebra σ(St ). An appeal to the Hahn Decomposition Theorem for real valued measures establishes the Borel measurability of the (n) scalar valued function s −→ Xs , ξ[ϕ, u, v] , 0 ≤ s ≤ t, for each ϕ ∈ Ξn , (n) u ∈ X , v ∈ X  , so that the Yn -valued function s −→ [Xs ]Mt , 0 ≤ s ≤ t, is 2 Borel measurable. From the separability of L (Mt ) and Proposition 6.1, we (n) see that s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in L2 (Mt ) for each t ≥ 0. Taking the limit as n → ∞, it follows that s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in L0 (Mt ). ˆ s : Ω → Σ, 0 ≤ s ≤ T , such Step 3. Let T > 0. There exists a process X that i) ii) iii)

ˆ s } is Mt -null for all 0 ≤ s ≤ T , {Xs = X ˆ Xs is σ(Ss )-measurable for each 0 ≤ s ≤ T , and ˆ s (ω), 0 ≤ s ≤ t, ω ∈ Ω, is strongly (B([0, t]) ⊗ the mapping (s, ω) −→ X σ(St ))-measurable for each 0 < t ≤ T .

Let T > 0. The proof of [27, Proposition 33] applies verbatim in the proof of Step 3 with convergence in probability replaced by convergence in the space L0 (MT ), see also [34, Th´eor`eme IV.30]. For convenience, the elementary construction is reproduced here. By Proposition 6.1 and Step 2, the closure C of the set {[Xs ]MT : 0 ≤ s ≤ T } in L0 (MT ) is separable. Let {Sn,k }k be a partition of C into disjoint Borel sets with diameter less than 1/n2 with the (n+1)-th partition a refinement of the n-th. Define An,k,j ⊂ [0, T ] by An,k,j = {t ∈ [0, T ] : [Xt ]MT ∈ Sn,k , j2−n < t ≤ (j + 1)2−n }. Then {An,k,j }k,j is a partition of [0, T ] for each n = 1, 2, . . . . For nonempty An,k,j , choose tn,k,j ∈ An,k,j recursively so that tn,k,j ∈ {tn+1, ,m } and define φn : [0, T ] → [0, T ] by  φn = tn,k,j χAn,k,j . k,j

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Then φn is Borel measurable and if t = tm,k,j and n ≥ m,

Xφn (t) = Xt , |φn (t) − t| < 2

−n

,

if t ∈ [0, T ].

For each t ∈ [0, T ], we have Xφn (t) (ω) → Xt (ω) as n → ∞ for MT -almost all ω ∈ Ω. ˆ defined for each Let σ be any element of Σ. Then the function X t ∈ [0, T ] by  ˆ t (ω) = limn→∞ Xφn (t) (ω), for all ω ∈ Ω for which the limit exists, X σ, otherwise, has the required properties. If t > s, then the formula Mt (E) = S(t − s)Ms (E) holds for every E ∈ σ(St ). Inspection of equation (6.3) shows that the equality is valid for any cylinder set (6.1). Equality on σ(St ) follows from the σ-additivity of Mt and Ms in the strong operator topology and the continuity of the linear operator S(t − s). Because the bounded linear operator S(r) : X → X is assumed to be injective for each r > 0, the operator valued measures Mt  σ(Ss ) and Ms are mutually absolutely continuous on the σ-algebras σ(Ss ). Hence, choosing an unbounded sequence of times 0 < Tn < Tn+1 , n = 1, 2, . . . , the process above may be applied on each time interval [Tn , Tn+1 ), n = 1, 2, . . . , ˜ s(n) } is MTn+1 -null for all Tn ≤ s < Tn+1 . Then for so that the set {Xs = X ˜ s(n) } is Mt -null for all Tn ≤ s ≤ t. The Tn < t ≤ TN +1 , the set {Xs = X required progressively measurable process is obtained by setting T0 = 0 and ˜s = X

∞ 

˜ (n) χ[T ,T ) (s), X s n n+1

s ≥ 0,

n=0

˜ under the Borel isomorphism in the in the case Σ = R and the image of X case of a general Polish space Σ. Remark 6.2. a) If (Ω, F , P ) is a general probability measure space and the process Xt , t ≥ 0, is adapted to some filtration {Ft }t≥0 , then it requires more work to show that there exists a progressively measurable modification ˜ t , t ≥ 0, adapted to the same filtration {Ft }t≥0 , see [34, Th´eor`eme IV.30], X [74] and [100] on this point. Step 2 above is a crucial feature leading to the conclusion of Theorem 6.3 in the present setting, where we have appealed to the Markov property of Mt , t ≥ 0, and the assumption that S is a C0 semigroup of operators acting on the separable Banach space X . As pointed

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out in [27, Proposition 32] for the case of a general metric space (Σ, d) in the probabilistic setting, a strongly progressively measurable modification exists only if the mapping s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in the space L0 (Mt , Σ) of Mt -measurable Σ-valued functions for each t ≥ 0. b) In the probabilistic setting, a Feller semigroup is a sub-Markov C0 semigroup on the space C0 (Σ) of continuous functions vanishing at infinity. The topological space Σ is assumed to be a locally compact Hausdorff space with a countable base, so it is σ-compact and Polish. A slight modification of the proof above also applies to the weak*-continuous semigroup dual to a Feller semigroup. In this setting, a much stronger property than Theorem ˜ of the 6.3 obtains using martingale theory: there exists a modification X process which is right continuous with left limits [28, Theorem 2.6]. c) For a general metric space (Σ, d) rather than a Polish space and t > 0, the mapping s −→ [Xs ]Mt , 0 ≤ s ≤ t, is strongly Borel measurable in the space L0 (Mt , Σ) of Mt -measurable Σ-valued functions provided that lim

s→r 0≤s≤t

sup | Mt x, ξ |({d(Xs , Xr ) > }) = 0

ξX  ≤1

for each x ∈ X , and 0 ≤ r ≤ t (stochastic continuity), because then the mapping s −→ [Xs ]Mt , 0 ≤ s ≤ t, is actually a continuous map from the compact set [0, t] into the metrisable space L0 (Mt , Σ) and so it is strongly Borel measurable in L0 (Mt , Σ). Moreover, a function from [0, t] into L0 (Mt , Σ) is strongly λ-measurable in L0 (Mt , Σ) with respect to Lebesgue measure λ if and only if it is Lusin λ-measurable [46, 411M, 418E-J]. If we only require that the subset ˜ s (ω)} {(ω, s) : Xs (ω) = X of Ω × [0, t] be (Mt ⊗ λ)-null, as is sufficient for the validity of the FeynmanKac formula (6.4), then stochastic quasi-continuity is a necessary and sufficient condition for the existence of the strongly progressively measurable ˜ see [37] for the probabilistic case. As Step 2 of the proof above process X, demonstrates, the process X of Theorem 6.3 is automatically stochastically quasi-continuous when (Σ, d) is separable and complete. d) Theorem 1 also applies to the case in which (Σ, B) is a standard Borel space [46, 424A], that is, B is the Borel σ-algebra of some Polish topology on Σ. The common spaces of distributions are standard Borel spaces [125, pp. 112-117]. When Σ is a Souslin space [46, 423A], the spectral measure Q is supported by the countable union ΣQ = ∪n Σn of compact metrisable subsets Σn , n = 1, 2, . . . , of Σ [46, 432B, 423D(c)], which is a standard

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Borel space by [125, Corollary 2, p. 102]. Then a version of Theorem 1 applies to ΣQ -valued processes. e) If we omit the assumption that Σ is a Polish space (or even a Souslin space), then for any bounded measurable function V : Σ → R, a real valued progressively measurable process V ◦ X s , s ≥ 0, may be obtained as above, so that the Feynman-Kac formula becomes   t  t(A+Q(V )) = e 0 V ◦X s ds dMt . (6.8) e Ω

Equation (6.8) suffices as an ingredient in the proof of the general CLRinequality for dominated semigroups in the next chapter. The real valued t V ◦X s ds 0 random process t −→ e , t ≥ 0, is called a multiplicative functional in the probability literature [28, p. 358]. The process V ◦ X ought to be considered as a junior grade renormalisation if the process X is itself essentially wild. 6.4

Operator bilinear integration

In order to treat random evolutions in the next section and to obtain a generalisation of the Feynman-Kac formula, Theorem 6.2, relevant to random evolutions, we need to integrate operator valued functions with respect to operator valued measures. Let Y be a Banach space, (Ω, S) a measurable space, and M : S → L(Y ) an operator valued measure, by which we mean that M is σ-additive for the strong operator topology. Suppose that X is another Banach space and τ is a completely separated norm tensor product topology on X ⊗ Y . Let IX be the identity operator on X. The tensor product IX ⊗ T : X ⊗ Y → X ⊗ Y of the identity map and a continuous linear operator T : Y → Y need not be continuous for the topology τ , so this is encompassed in the conditions below. For each A ∈ S, we denote the linear map IX ⊗[M (A)] : X ⊗Y → X ⊗Y by MX (A). An operator valued measure whose range is an equicontinuous family of linear operators is called an equicontinuous operator valued measure. If there exists C > 0 such that MX (A)φτ ≤ Cφτ for every φ ∈ X ⊗ Y and A ∈ S, then we say that the Banach space X is (M, τ )admissible. Lemma 6.1. Let X be an (M, τ )-admissible Banach space. Then MX extends uniquely to an equicontinuous operator valued measure acting on τY . X⊗

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Proof. Because X is (M, τ )-admissible, {MX (A) : A ∈ S} is an equicontinuous family of operators acting on the normed space X ⊗τ Y . For each τY A ∈ S, the unique continuous linear extension of MX (A) to all of X ⊗ ˆ X (A) : A ∈ S} is an equicontinuous family ˆ X (A). Then {M is denoted by M τ Y ). of operators acting in L(X ⊗ By property (T1) of the norm tensor product topology τ (see Section 1.4), the (X ⊗ Y )-valued set function MX a is σ-additive for all elements a τ Y ; equicontinuity ensures that belonging to the dense subset X ⊗ Y of X ⊗ ˆ MX a is σ-additive for all a ∈ X ⊗τ Y . The uniquely defined operator valued measure of the above statement τ Y ). is also denoted by MX : S → L(X ⊗ Definition 6.1. Let (Ω, S) be a measurable space and X, Y Banach spaces. Suppose that τ is a completely separated norm tensor product topology on X ⊗ Y . Let M : S → L(Y ) be an operator valued measure. An operator valued function Φ : Ω → L(X) is said to be M -integrable τ Y ), if for each A ∈ T , there exists an operator [Φ ⊗ M ](A) ∈ in L(X ⊗ L(X ⊗τ Y ) such that for every x ∈ X and y ∈ Y , the X-valued function τ Y with respect to the Y Φx : ω → Φ(ω)x, ω ∈ Ω, is integrable in X ⊗ valued measure M y : A → M (A)y in the sense of Definition 2.2 and the equality  & ' Φ ⊗ M (A)(x ⊗ y) = [Φx] ⊗ d[M y] A

holds for every A ∈ S. The operator [Φ⊗M ](A) is also written as A Φ⊗dM. Sometimes, we write M (Φ) for the definite integral [Φ ⊗ M ](Ω). If the space X is not (M, τ )-admissible, then it may happen that the only operator valued function integrable with respect to M is the function equal to zero almost everywhere. Let (Ω, S) be a measurable space and X, Y Banach spaces. Suppose that τ is a completely separated norm tensor product topology on X ⊗ Y . We check that Φ ⊗ M is σ-additive. Lemma 6.2. Let M : S → L(Y ) be an operator valued measure and let the operator & valued ' function Φ : Ω → L(X) be M -integrable. The set function τ Y ). A → Φ ⊗ M (A), A ∈ S, is σ-additive in L(X ⊗

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τ Y , there exists an ∈ X ⊗ Y , n = 1, 2, . . . , such Proof. For each a ∈ X ⊗ that a − an τ → 0. Then [Φ ⊗ M ](A)an → [Φ ⊗ M ](A)a for each A ∈ S τ Y ). The Vitali-Hahn-Saks theorem ensures because [Φ ⊗ M ](A) ∈ L(X ⊗ τY . that [Φ ⊗ M ]a is σ-additive in X ⊗ It is useful to have a condition interpreting Definition 6.1 in terms of approximation by simple functions. Proposition 6.2. Let M : S → L(Y ) be an operator valued measure and Φ : Ω → L(X), an operator valued function. Suppose that for every nonzero n element η = j=1 xj ⊗ yj of X ⊗ Y , n = 1, 2, . . . , there exist X-valued Ssimple functions sj,k : Ω → X, k = 1, 2, . . . , and j = 1, . . . , n, for which there exists C > 0 independent of η and {sj,k } such that, (i) for each j = 1, . . . , n, sj,k → Φxj (M yj )-a.e. as k → ∞,       sj,l (ω) ⊗ d[M yj ](ω) lim  sj,k (ω) ⊗ d[M yj ](ω) −  =0 k,l→∞

A

A

τ

for  each A ∈ S, and   n (ii)  j=1 A sj,k (ω) ⊗ d[M yj ](ω) ≤ ητ for all A ∈ S and k = τ 1, 2, . . . . τ Y ) and Then the operator valued function Φ is M -integrable in L(X ⊗ n   [Φ ⊗ M ](A)η = lim sj,k (ω) ⊗ d[M yj ](ω). (6.9) k→∞

j=1

A

If for dense sets of x ∈ X and y ∈ Y , the Y -valued measure M y has σfinite X-semivariation in X ⊗τ Y on the set {Φx = 0}, then conditions (i) and (ii) are also necessary. Moreover, the convergence in (6.9) is uniform for all A ∈ S. Proof. If condition (i) holds, then by Definition 2.2, Φxj : Ω → X is (M yj )-integrable for each j = 1, 2, . . . , n. Condition (ii) ensures that  n p j=1 ([Φxj ] ⊗ [M yj ])(A) ≤ q(η) for all A ∈ S. For every x ∈ X and τ Y , and the map y ∈ Y , the X-valued function Φx is M y-integrable in X ⊗ (x, y) → [Φx] ⊗ [M y], x ∈ X, y ∈ Y , is bilinear. Hence, the linear map, τ Y defined by (Φ ⊗ M )(A) : X ⊗ Y → X ⊗ [Φ ⊗ M ](A)η =

n  j=1

([Φxj ] ⊗ [M yj ])(A),

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for each η as above, satisfies p([Φ ⊗ M ](A)η) ≤ q(η), for each A ∈ S and η ∈ X ⊗Y . Therefore, [Φ⊗m](A) is the restriction to X ⊗Y of a continuous τ Y . The τ Y to X ⊗ linear operator, also denoted by [Φ ⊗ m](A), from X ⊗ collection of linear maps [Φ ⊗ M ](A), A ∈ S, is equicontinuous. According τ Y ). to Definition 6.1, the function Φ : Ω → L(X) is M -integrable in L(X ⊗ Suppose now that Φ is M -integrable, X, Y and (X ⊗τ Y, p) are Banach spaces, and for dense sets X0 ⊆ X and Y0 ⊆ Y , M y has σ-finite X τ Y on {Φx = 0} for each x ∈ X0 , y ∈ Y0 . By Lemma semivariation in X ⊗ 6.2 and the condition that τ is normed, Φ ⊗ M is σ-additive in the strong τ Y ), so the range of the measure [Φ ⊗ M ]ξ is operator topology of L(X ⊗ τ Y . By the uniform bound τ Y for each ξ ∈ X ⊗ necessarily bounded in X ⊗ edness principle, the family of operators [Φ ⊗ m](A), A ∈ S, is bounded in τ Y ). the uniform operator norm of L(X ⊗ Let C > supA∈S [Φ ⊗ M ](A)L(X ⊗ ˆ τ Y ) . Let η ∈ X0 ⊗ Y0 be a vecn x ⊗ y with n = 1, 2, . . . . According to Defitor of the form j j j=1 nition 2.2 and Theorem 2.1, there exist simple functions sj,k such that τ Y , (i) is satisfied, uniformly for all A ∈ S. We for the norm p of X ⊗ can chop the simple functions sj,k off on the set {Φxj = 0}, if necessary, in order to apply Theorem 2.1 as stated. Hence, [Φ ⊗ M ](A)η = n limk→∞ j=1 A sj,k ⊗ d[M yj ], uniformly for A ∈ S. We can extract a sub ∞ ∞ sequence {s j,k }k=1 of {sj,k }k=1 , so that for every k = 1, 2, . . . and A ∈ S,  n  we have p j=1 A sj,k ⊗ d[M yj ] ≤ Cp(η). Because X0 ⊗ Y0 is dense in X ⊗τ Y by condition (T3) of a tensor product topology, it follows that τ Y and for q = Cp, conditions (i) and (ii) are satisfied for the norm p of X ⊗ for every η ∈ X ⊗ Y . Proposition 6.3. Let Y be a Banach, M : S → L(Y ) an operator valued measure and suppose that X is an (M, τ )-admissible Banach space for the completely separated norm tensor product topology τ on X ⊗ Y . Let φ : Ω → C be a scalar valued function and define the function Φ : Ω → L(X) by Φ = φI. The following conditions are equivalent. (i) φ is M -integrable in L(Y ) and IX ⊗ ([φ.M ](A)) ∈ L(X ⊗τ Y ) for each A ∈ S. (ii) φ is MX -integrable in L(X ⊗τ Y ). τ Y ). (iii) Φ is M -integrable in L(X ⊗ If any of the conditions holds, then on X ⊗ Y , the equalities [Φ ⊗ M ](A) = [φ.MX ](A) = IX ⊗ ([φ.M ](A))

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hold for every A ∈ S. If φ : Ω → C is bounded and S-measurable, then Φ τ Y ). is M -integrable in L(X ⊗ Proof. To show that (i) implies (ii), we can find simple functions φk ∈ sim(S), k = 1, 2, . . . , converging everywhere to φ, so that as k → ∞, [φk .(M y)](A) → [φ.(M y)](A) in  Y , for each A ∈ S and y ∈ Y . Then for each x ∈ X, x⊗ [φk .(M y)](A) → x⊗ [φ.(M y)](A) in X ⊗τ Y as k → ∞, because property (T3) of a tensor product topology ensures that x ⊗ Y is isomorphic to Y in the case that x = 0; if x = 0, we get the zero vector. From the definition of MX , for each x ∈ X, we have      lim φk d[MX (x ⊗ y)] = x ⊗ φ d[M y] = IX ⊗ φ dM (x ⊗ y). k→∞

A

A

A

By assumption, IX ⊗ ([φ.M ](A)) ∈ L(X ⊗τ Y ) for each A ∈ S, so φ is MX -integrable in L(X ⊗τ Y ) and [φ.MX ](A) = IX ⊗ ([φ.M ](A)) for every A ∈ S. Now suppose that (ii) is true, and find simple functions φk ∈ sim(S), k = 1, 2, . . . , converging everywhere to φ, so that for each A ∈ S, x ∈ X and y ∈ Y , we have [φk .MX ](A)(x ⊗ y) → [φ.MX ](A)(x ⊗ y) in X ⊗τ Y , as k → ∞. Then,     lim [φk x] ⊗ d[M y] = lim φk d[MX (x ⊗ y)] = φ dMX (x ⊗ y). k→∞

k→∞

A

A

A

A glance at Definition 2.2 verifies that Φx is M y-integrable and    [Φx] ⊗ d[M y]) = φ dMX (x ⊗ y). A

A

Now [φ.MX ](A) ∈ L(X ⊗τ Y ), so it has a unique continuous linear extension τ Y , denoted by the same symbol. From Definition 6.1, the to all of X ⊗ function Φ is M -integrable and Φ ⊗ M = φ.MX . Suppose that (iii) holds, y is an element of Y and x ∈ X is a nonzero vector. By the Hahn-Banach theorem, there exists x ∈ X  such that

x, x = 1. The assumption that Φ is M  -integrable means that Φx is M yintegrable and [(Φ ⊗ M )(A)](x ⊗ y) = [Φx] ⊗ [M y] (A), for each A ∈ S. Proposition 2.2 ensures that Φx, x = φx, x = φ is M y-integrable in Y " and # " # " # φ d[M y], y 

Φx, x d[M y], y 

=

A

A

[Φx] ⊗ d[M y], x ⊗ y  .

= A

The vector x ∈ X is nonzero, so Jx : y → x ⊗ y, y ∈ Y , is an isomorphism of Y onto x ⊗ Y by property (T3) of the tensor product topology τ . A calculation shows  that for each y ∈ Y , we have φ d[M y] = (x ⊗ IY ) ◦ (Φ ⊗ M )(A) ◦ Jx y.

A

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The right-hand side of this equation is an element of Y because Y is se τ Y of the quentially complete. Here, the continuous linear extension to X ⊗   map x ⊗ IY : u ⊗ y → u, x y, for u ∈ X and y ∈ Y , has been denoted by the same symbol. Consequently, φ is M -integrable in L(Y ) and  φ dM = (x ⊗ IY ) ◦ (Φ ⊗ M )(A) ◦ Jx ∈ L(Y ), for all A ∈ S. A

 Moreover, it is readily verified that [Φ ⊗ M ](A)(x ⊗ y) = IX ⊗ ([φ.M ](A)) (x ⊗ y) for all x ∈ X and y ∈ Y , so IX ⊗ ([φ.M ](A)) necessarily belongs to L(X ⊗τ Y ). If φ is bounded and S-measurable, then it is the uniform limit of Ssimple functions φk , k = 1, 2, . . . . Let Φk = φk IX for every k = 1, 2, . . . . The assumption that X is (M, τ )-admissible means that the range of MX is contained in an equicontinuous set in L(X ⊗τ Y ), so there exists an equicontinuous subset of L(X ⊗τ Y ) containing the ranges of each of the measures φk .MX = Φk ⊗M , k = 1, 2, . . . . The limit measure φ.MX = Φ⊗M τ Y ). therefore takes its values in L(X ⊗ τ Y be Banach spaces. Let M : S → Definition 6.2. Let X, Y and X ⊗ L(Y ) be an additive set function. We say that M has finite L(X)semivariation in L(X ⊗τ Y ) if (i) A ⊗ M (E) ∈ L(X ⊗τ Y ) for each A ∈ L(X) and E ∈ S, and n (ii) there exists C > 0 such that  j=1 Aj ⊗ M (Ej )L(X⊗τ Y ) ≤ C, for all Aj ∈ L(X) with Aj  ≤ 1 and pairwise disjoint Ej ∈ S, j = 1, . . . , n and n = 1, 2, . . . . Let βL(X) (M )(E) be the smallest such number C as the sets Ej above range over subsets of E ∈ S. Then the set function βL(X) (M ) is called the L(X)semivariation of M in L(X ⊗τ Y ). It follows from the property (T1) of the norm tensor product topology τ in Section 1.4, that if X = 0, then there exists K > 0, such that for every additive set function M : S → L(Y ), the semivariation M  of M in the operator norm is bounded by KβL(X) (M ). If the L(X)-semivariation of M in L(X ⊗τ Y ) is continuous and X = 0, then the fact mentioned above shows that M is σ-additive for the uniform operator topology of L(Y )—a condition which is rarely satisfied for operator valued measures arising in applications. It is therefore a useful observation that the following result does not require the L(X)-semivariation in L(X ⊗τ Y ) to be continuous.

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τ Y be Banach spaces. Let M : S → L(Y ) Theorem 6.4. Let X, Y and X ⊗ be an operator valued measure such that M has finite L(X)-semivariation βL(X) (M ) in L(X ⊗τ Y ), and for each y ∈ Y , M y has the continuous X-semivariation in X ⊗τ Y . Let Φ : Ω → L(X) be a function such that, (a) for each x ∈ X and y ∈ Y , Φx : Ω → X is strongly M y-measurable; (b) there exists C > 0 such that for each x ∈ X and y ∈ Y , the bound Φ(ω)xX ≤ CxX holds for M y-almost all ω ∈ Ω. τ Y ). If Φ∞ denotes the smallest Then Φ is M -integrable in L(X ⊗ number C satisfying (b), then (Φ ⊗ M )(Ω) ≤ Φ∞ βL(X) (M )(Ω). Proof. We verify the conditions of Proposition 6.2 to show that Φ is M integrable. Any element φ of X ⊗ Y may be written in the form φ = n j=1 xj ⊗ yj , with {xj } a linearly independent set. Each function Φxj is M yj -integrable by Theorem 2.2, so there exist X-valued S-simple functions {sj,k } such that condition (i) of Proposition 6.2 holds. We need to modify the approximating sequence to ensure condition (ii) of that proposition is also valid. Let X0 be the linear span of {xj } and equip it with the norm of X. For ˜ k (ω)xj = ˜ k (ω) : X0 → X by setting Φ each ω ∈ Ω, define the linear map Φ sj,k (ω). Because we have assumed that {xj } is a linearly independent set, ˜ k is well-defined for each k = 1, 2, . . . . The space X0 is finite dimensional, Φ so for each j = 1, . . . , n, the operators Φk (ω) converge to Φ(ω)  X0 in the uniform operator topology of L(X0 , X) as k → ∞, for M yj -almost all ω ∈ Ω. Let > 0. The set Ak =



˜ l (ω)xX ≤ (Φ∞ + )xX for all x ∈ X0 } {ω ∈ Ω : Φ

l=k

˜ l is an L(X0 , X)-valued S-simple function belongs to the σ-algebra S, for Φ ˜ k } in the uniform for every l = 1, 2, . . . . The pointwise convergence of {Φ ∞ operator topology of L(X0 , X) guarantees that ∪k=1 Ak is a set of full M yj measure for each j = 1, . . . , n . Let P : X → X0 be any norm one projection and define Φk (ω) = ˜ k (ω) ◦ P , for all k = 1, 2, . . . and ω ∈ Ω. Then Φk (ω)L(X) ≤ χAk (ω)Φ Φ∞ + for all k = 1, 2, . . . and ω ∈ Ω. Moreover, Φk is a family of L(X)-valued S-simple functions such that the X-valued S-simple functions Φk xj , for k = 1, 2, . . . and j = 1, . . . , n satisfy condition (i) of Proposition 6.2. The setwise convergence of the indefinite integrals follows from the

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argument of Vitali’s convergence theorem; see the proof of Lemma 2.1. By the finiteness of the L(X)-semivariation βL(X) (M ) of M , for every A ∈ S we have      n  [Φk xj ] ⊗ d[M yj ]  ≤ (Φ∞ + )βL(X) (M )(Ω)φτ ,  j=1

A

τ

for all k = 1, 2, . . . . Hence, condition (ii) of Proposition 6.2 also holds, so Φ is M -integrable and (6.9) holds. Because is any positive number and φ is any element of X ⊗ Y , the bound (Φ ⊗ M )(Ω) ≤ Φ∞ βL(X) (M )(Ω) is valid. The following result is a slight modification of the bounded convergence theorem of [11, Theorem 7]. Again, it is significant that we only assume that the operator valued measure M has pointwise continuous X-semivariation in X ⊗τ Y . Theorem 6.5 (Bounded Convergence Theorem). Let X, Y and τ Y be Banach spaces. Let M : S → L(Y ) be an operator valued meaX⊗ sure such that M has finite L(X)-semivariation βL(X) (M ) in L(X ⊗τ Y ), and for each y ∈ Y , (M y) has continuous X-semivariation in X ⊗τ Y . Let Φk : Ω → L(X), k = 1, 2, . . . , be functions such that (a) for every x ∈ X and y ∈ Y , the function Φk x : Ω → X is (M y)measurable; (b) there exists a positive number C with the property that for every x ∈ X and y ∈ Y , and k = 1, 2, . . . , the bound Φk (ω)xX ≤ CxX holds, for (M y)-almost all ω ∈ Ω. If for each x ∈ X and y ∈ Y , Φk (ω)x → Φ(ω)x as k → ∞, for ((M y))almost every ω ∈ Ω, then Φ is M -integrable and   Φk ⊗ dM → Φ ⊗ dM A

A

τ Y ) as k → ∞, uniformly for in the strong operator topology of L(X ⊗ A ∈ S. Proof. We already know by Theorem 6.4 that Φ is M -integrable in τ Y ) and that the bound  [Φk ⊗ M ](A)L(X ⊗ L(X ⊗ ˆ τ Y ) ≤ CβL(X) (M )(Ω) holds, for all k = 1, 2, . . . and A ∈ S. The vector valued bounded convergence theorem, Theorem 2.2, tells us that for all φ ∈ X ⊗ Y , the vec τ Y , as tors [Φk ⊗ M ](A)φ, k = 1, 2, . . . , converge to [Φ ⊗ M ](A)φ in X ⊗

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k → ∞, uniformly for A ∈ S. The equicontinuity for the values of the indefinite integrals Φk ⊗ M , k = 1, 2, . . . , guarantees the convergence for τY . every φ ∈ X ⊗ We have the following analogues of Propositions 2.2 and 2.3, and Corollaries 2.1 and 2.2. The proofs are similar to the earlier ones, so they are omitted. Proposition 6.4. Let M : S → L(Y ) be an L(Y )-valued measure. If τ Y ), then for all x ∈ X, y ∈ Y , Φ : Ω → L(X) is M -integrable in L(X ⊗ x ∈ X  and y  ∈ Y  , the L(X)-valued function Φ is integrable with respect to the scalar measure M y, y  , the scalar valued function Φx, x is integrable with respect to the L(Y )-valued measure M and the following equalities hold for all A ∈ S: # " # " Φ d M y, y  , x ⊗ x =

Φx, x dM , y ⊗ y  A  A = Φx, x d M y, y  . A

Corollary 6.1. Let M : S → L(Y ) be an L(Y )-valued measure. If Φ : τ Y ), then Φ ⊗ M  M . Ω → L(X) is M -integrable in L(X ⊗ Corollary 6.2. Suppose that M : S → L(Y ) is an L(Y )-valued measure. τ Y ), and f : Ω → C is a bounded If Φ : Ω → L(X) is M -integrable in L(X ⊗ S-measurable function, then f Φ is M Φ is f.M -integrable and  -integrable,  the equalities (f Φ) ⊗ M = Φ ⊗ f.M = f. Φ ⊗ M hold. Proposition 6.5. Suppose that Xj , Yj , j = 1, 2 are Banach spaces and τ1 is a completely separated norm tensor product topology on X1 ⊗ Y1 , and τ2 is a completely separated norm tensor product topology on X2 ⊗ Y2 . Let M : S → L(Y1 ) be a measure and suppose that S : X1 → X2 and T : Y1 → Y2 are continuous linear maps whose tensor product S ⊗ T : X1 ⊗τ1 Y1 → X2 ⊗τ2 Y2 is continuous. If Φ : Ω → L(X1 ) is M -integrable τ1 Y1 ), then SΦ is T M -integrable in L(X2 ⊗ τ2 Y2 ) and in L(X1 ⊗   Φ ⊗ dM = [SΦ] ⊗ d[T M ], (S ⊗ T ) A

A

for every A ∈ S. Similarly, if M : S → L(Y2 ) is a measure and Φ : Ω → L(X2 ) is M τ2 Y2 ), then the function ΦS is M T -integrable in the integrable in L(X2 ⊗ &

'

space L(X1 ⊗τ1 Y1 ) and the equality A Φ ⊗ dM (S ⊗ T ) = A [ΦS] ⊗ d[M T ] holds for every A ∈ S.

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As shown in Section 1.5.2, operator valued measures taking their values in the Banach lattice of positive operators on an Lp -space have bounded L(X)-semivariation for any Banach space X and the same holds true for operator valued measures dominated by a positive measure. The bounded (S, Q)-processes on Lp -spaces considered in Section 6.5 below have this property, so Theorem 1.10 and Theorem 6.6 below see use in the next section on random evolutions. Let (Γ, E, μ) be a σ-finite measure space and 1 ≤ p ≤ ∞. Theorem 6.6. Suppose that M is positive and that N is dominated by M . Let F : Ω → L(X) be a function such that for each x ∈ X and y ∈ Lp (μ), the function F x is strongly (M y)-measurable and F L(X) is M -integrable in L(Lp (μ)). Then the function F is N -integrable in L(Lp (μ; X)) and the estimate           F ⊗ dN  ≤  F L(X) dM     L(Lp (μ;X))

A

holds for all A ∈ S.

L(Lp (μ))

A

Proof. We establish the result for the special case N = M ≥ 0 first. For  n ∈ N, let g = nj=1 xj χG be an X-valued E-simple function with xj ∈ X j and Gj pairwise disjoint for j = 1, . . . , n, such that gLp(μ;X) ≤ 1. Then, making  use of Theorem2.3,pwe have    F (ω) ⊗ dM (ω) g    p A L (μ;X)  p       = [F x ](ω) ⊗ d[M χ ](ω) j   Gj  j A  p L (μ;X) p ⎛ ⎞         ⎝ = [F xj ](ω) ⊗ d[M χGj ](ω)⎠ (γ)  dμ(γ)  Γ A  j X ⎛ ⎞p        ⎝ ⎠ dμ(γ) ≤ [F xj ](ω) ⊗ d[M χGj ](ω) (γ)   Γ

 ≤

Γ

⎛ ⎝

A

j

  j

A

X

⎞p

 F (ω)xj X d[M χG ](ω) (γ)⎠ dμ(γ) j

 p       = F (ω)x  d[M χ ](ω) j X   Gj  j A  p

L (μ)

.

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167

By assumption, the nonnegative function F L(X) is M -integrable in L(Lp (μ)), so the right-hand side is bounded by  p        F (ω) x  d[M χ ](ω) j X L(X)   Gj  j A  p L (μ)  ⎞p ⎛        ⎠ ⎝ = F (ω)L(X) dM (ω) xj X χG   j  A  p j L (μ)  p    p        ≤  F (ω)L(X) dM (ω) xj X χGj    A  p L(Lp (μ))  j L (μ)  p    = gpLp(μ;X)  F (ω)L(X) dM (ω) L(Lp (μ))

A

holding for all A ∈ S. Since X-valued E-simple functions are dense in Lp (μ; X) this establishes the required inequality and completes the proof for the positive case. The inequality for the case where N is dominated by M follows analogously to the positive case taking note that, when x ∈ X and E ∈ E,          [F x](ω) ⊗ d[N χE ](ω) (γ) ≤ F (ω)xX d[M χE ](ω) (γ)  A

X

A

holds true for all A ∈ S and μ-almost every γ ∈ Γ by Theorem 2.3. 6.5

Random evolutions

A random evolution Ft : Ω → L(X ), t ≥ 0, is a family of operator valued random variables acting on some Banach space X of states, whose evolution is influenced by some random changes in the environment. Random events are measured by a family of operator valued measures Mt : σ(St ) → L(Lp (μ)), t ≥ 0. For example, suppose that a particle moves in a straight line with constant speed, until it suffers a random collision, after which it changes velocity, and again moves in a straight line with a new constant speed. The situation may be described, more abstractly, as an evolving system whose mode of evolution changes due to random changes in the environment. The notion of a ‘random evolution’ was introduced by R. Griego and R. Hersh [56] to provide a mathematical formulation of such a randomly influenced dynamical system (see [61] for a later survey, [77, Chapter 10]

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and [62] for elementary and historical accounts). A random evolution or more prosaically, a multiplicative operator functional is an operator valued function M satisfying a linear differential equation of the form dM (s, t) = −V (Xs )M (s, t), 0 ≤ s < t. (6.10) ds The coefficient V is an operator valued function and Xs is a random variable for each s ≥ 0. In the case that Xs is Markovian with respect to a family of probability measures P x , the expected value u(x, t) = P x [M (0, t)] satisfies the equation du (x, t) = Gu(x, t) + V (x)u(x, t). (6.11) dt Here G denotes the generator of the Markov process Xs s≥0 . This is a generalisation of the Feynman-Kac formula considered in Theorem 6.2 where V is now allowed to be an operator valued function. For example, a finite state Markov chain is defined on the state space Σ = {1, . . . , n} by its generator, an (n × n) matrix Q = {qij } with the n property that qij ≥ 0 for i = j and j=1 qij = 0 for each i = 1, . . . , n. If we define qi = −qii and Πij = qij /qi if qi = 0 and zero otherwise, then there exists a Markov process Xt , t ≥ 0, with probability measures P i , i = 1, . . . , n, such that P i (X0 = j) = δij . The sample path t → Xt is piecewise constant with jumps at the instants τ1 < τ2 < . . . , where P i (τ1 > t) = e−qi t and P i (Xτ1 = j) = Πij for 1 ≤ i, j ≤ n, i = j. Given real numbers v1 , . . . , vn , there is a continuous group Ti of linear operators on the space C0 (R) of continuous functions vanishing at infinity, defined by the formula (Ti (t)f )(x) = f (x + vi t),

x∈R

for each f ∈ C0 (R) and t ∈ R. Then M (0, t] = TX0 (τ1 )TXτ1 (τ2 − τ1 ) · · · TXτN (t) (t − τN (t) )

(6.12)

is a random evolution. The random integer N (t) is defined by the formula τN (t) ≤ t < τN (t)+1 . For smooth fi ∈ C0 (R), i = 1, . . . , n, the expectations ui (t) = Ei (M (0, t]fi ), t ≥ 0, is the solution of the system of equations ∂ui  ∂ui = vi + qij uj , ∂t ∂x j=1 n

ui (0) = fi , i = 1, . . . , n.

The point of departure in this chapter is to re-interpret the term ‘random’ so as to obtain the representation of solutions to a more general class of partial differential equations. The representation we are aiming for is

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one in which the expectation value above is replaced by the integral with respect to the operator valued measures Mt of an (S, Q)-process, associated with a semigroup S of bounded linear operators acting on an Lp -space and the spectral measure Q of multiplication by characteristic functions. We want to obtain a representation of solutions ut , t ≥ 0, to initial value problems for certain partial differential equations in the form   Ft ⊗ dMt u0 . ut = Ω

Here Ft : Ω → L(X) is an operator valued ‘random variable’. It turns out that we need to define Ft in terms of the adjoint of a functional satisfying equation (6.10), but we retain the term multiplicative operator functional. In the example described by formula (6.12), instead we consider the multiplicative operator functional Ft = TXτN (t) (t − τN (t) ) · · · TXτ1 (τ2 − τ1 )TX0 (τ1 ) in the natural ordering between increasing times and operator actions. For the case of a scalar perturbation V , the' multiplicative operator & t functional Ft is given by ω → exp 0 V (ω(s)) ds , ω ∈ Ω. Because of our more general setting, equation (6.11) can be solved in cases where G is not the generator of a probabilistic Markov process. In this section, we apply the theory of bilinear integration developed in Section 6.4 to integration with respect to (S, Q, t)-set functions Mt in order to establish conditions for which the integral of a multiplicative operator functional Ft , t ≥ 0, defines a C0 -semigroup t → Ω Ft ⊗ dMt , t ≥ 0, acting on Lp (μ; X). The construction of the multiplicative operator function

Ft t≥0 itself follows work of J. Hagood [58]. Now suppose that the system (Ω, St t≥0 , Mt t≥0 ; Xt t≥0 ) is a σadditive time homogeneous Markov evolution process as in Section 6.1, that is, there exist a L(Y)-valued spectral measure Q on E and a semigroup S of operators acting on Y such that for each t ≥ 0, the operator Mt (E) ∈ L(Y) is given by Mt (E) = S(t − tn )Q(Bn )S(tn − tn−1 ) · · · Q(B1 )S(t1 )

(6.13)

for every cylinder set E ∈ St of the form (6.1) and the process is called an (S, Q)-process. The basic ingredients are the semigroup S describing the evolution of states and the spectral measure Q describing observation of states represented by vectors in Y. For our purposes, attention is restricted to the Banach space Y = Lp (μ) with 1 ≤ p < ∞ and μ a σ-finite measure. Now let X be a Banach space. A multiplicative operator functional is a measurable mapping Ft : Ω → L(X ), t ≥ 0, such that a.e.,

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(i) t → Ft (ω) is continuous for the weak operator topology (ii) Fs+t (ω) = Fs (θt ω)Ft (ω), F0 (ω) = IdX . Here θt : Ω → Ω is a shift map: Xs+t (ω) = Xs (θt ω). If X = R and t V : Σ → R is a suitable measurable function, Ft = e− 0 V ◦Xs ds is an example. For a random evolution in the sense of probability theory, the operators in (ii) are usually written in the opposite order. With the right idea of bilinear integration, it is straightforward that for an integrable multiplicative operator functional Ft , t ≥ 0, the formula

R(t) = Ω Ft (ω) ⊗ dMt (ω), t ≥ 0, gives a semigroup R of continuous linear operators acting on Lp (μ, X ). Proving that R is a C0 -semigroup requires additional assumptions [64,  Chapter 5]. For example, j Aj ⊗ Mt (Ej ) should form a bounded subset of L(Lp (μ, X )) if Aj L(X ) ≤ 1 and (Ej ) are pairwise disjoint, that is, Mt has finite L(X )-semivariation in L(Lp (μ, X )). The boundedness properties are satisfied if Mt comes from a Markov process or according to Theorem 1.10, is dominated by a positive L(Lp (μ))valued measure. In the case that Q is the spectral measure of multiplication by characteristic functions acting on Lp (μ), Theorem 6.1 shows that Mt is dominated by a positive L(Lp (μ))-valued measure if and only if S is a dominated semigroup of operators on Lp (μ). The following example is from [64, Theorem 5.3.3]. Example 6.3. Let cj : R → (0, ∞), j = 1, 2, . . . , be continuous functions such that supx,j cj (x) < ∞ and inf x cj (x) > 0. Suppose that  A = {ajk } is an infinite matrix such that supj∈N ∞ k=1 |ajk | < ∞ and ∞ supk∈N j=1 |ajk | < ∞. Then the solution to the equations ∞  ∂uj ∂uj (t, x) = cj (x) (t, x) + ajk uk (t, x) ∂t ∂x k=1

uj (0, x) = fj (x), j = 1, 2, . . . ∞ 2 with j=1 fj 2 + fj 22 < ∞ can be written as %  $ uj (t, x) = Ft ⊗ dMt f (x). Ω

j

Here Σ = N, X = L2 (R), the multiplicative operator functional Ft is constructed from the operators Vj : f → cj f  and Mt is the (S, Q, t)measure constructed from S(t) = eAt on 2 and Q multiplication by characteristic functions. The sample space Ω consists of piecewise constant paths. Now {ajk } need not be the generator of a Markov process.

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Chapter 7

The Cwikel-Lieb-Rozenbljum inequality for dominated semigroups

The subject of scattering theory touched upon in Chapter 5 dealt with the long time asymptotics of quantum systems. The asymptotics of quantum systems as  → 0 is termed the semiclassical approximation, where the leading term in the asymptotic approximation is usually given by a quantity obtained from Hamiltonian mechanics on classical phase space, as would be consistent with physical intuition. For example, the number of bound states in a quantum system should correlate with the volume of classical states with negative energy in the corresponding classical phase space. Volume in classical phase space is determined by Liouville measure. An inequality establishes that this is the case in wide generality. Moreover, its proof appeals to the theory that has been outlined in the preceding chapters. The essential idea of the proof given in this chapter is a tour de force of classical functional analysis due to E. Lieb [90]. We can implement Lieb’s proof under the rather weak assumption on the free Hamiltonian H0 that e−tH0 , t > 0, is a dominated semigroup of operators on L2 (μ), such as in the case that the Schr¨ odinger operator is coupled to a magnetic field in 3n R , n = 1, 2, . . . . An appeal to our Feynman-Kac formula, Theorem 6.2, and our treatment of operator traces in Chapter 3 via bilinear integration serve to push Lieb’s proof into the setting of general measure spaces and quantum field theory.

7.1

Asymptotic estimates for bound states

The Cwikel-Lieb-Rozenbljum or CLR inequality refers to an upper estimate for the number of negative eigenvalues (bound state energies) for a Schr¨odinger operator, proved by very different methods in [30, 90, 118]. In the case of the Laplacian operator Δ = ∂ 2 /∂x21 + · · ·+ ∂ 2 /∂x2d defined 171

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in L2 (Rd ), the spectrum σ(H0 ) of selfadjoint operator H0 = −Δ is [0, ∞) and the semigroup S(t) = e−tH0 , t ≥ 0, is defined by the functional calculus for the selfadjoint operator H0 . Then for each f ∈ L2 (Rd ) and t > 0,  |x−y|2 1 (S(t)f )(x) = e− 4t f (y) dy, f ∈ L2 (Rd ), x ∈ Rd . d/2 (4πt) Rd Suppose that V : Σ → R is a measurable function with positive and negative parts V+ = V ∨ 0 and V− = (−V ) ∨ 0. Let N (H0 + Q(V )) denote the number of eigenvalues of H0 + Q(V ) belonging to the half-line (−∞, 0]. The selfadjoint operator of multiplication by V is denoted by Q(V ) and H0 + Q(V ) is defined as a form sum. [115, Section VII.6] The CLR inequality for the Schr¨ odinger operator H0 is  d N (H0 + Q(V )) ≤ cd V− (x) 2 dx, d = 3, 4, . . . , (7.1) Rd

where the constant cd depends on the dimension d but not the interaction potential V . The significance of the bound (7.1) is that the asymptotic limit  N (H0 + λV ) d −d = (2π) σ |V (x)| 2 dx (7.2) lim d d λ→∞ λ2 Rd holds if V ≤ 0 belongs to Ld/2 (Rd ), where σd is the volume of the unit ball in Rd [128, Theorem 10.7]. In classical phase space Rd × Rd with the Hamiltonian H(p, x) = p2 + V (x), the volume of {H ≤ 0 } ⊆ Rd × {V ≤ 0} is   d 1 dpdx = σd V−2 (x) dx, {H≤0}

Rd

2

so in the classical limit with λ = 1/ and Planck’s constant h → 0 ( = h/(2π)), the number N (−2 Δ + V ) = N (−Δ + −2 V ) of bound states of a quantum system is given asymptotically by the corresponding volume of phase space with H ≤ 0 divided by hd = (2π)d d . The bound (7.1) replaces the asymptotic limit (7.2) by an inequality. Moreover, such a bound is needed to prove (7.2) in the stated generality. Similar asymptotic estimates are useful in considering the problem ‘Can you hear the shape of a drum?’, see [115, Section XIII.15]. The bound (7.1) admits the following generalisation which encompasses many quantum systems, see [97]. Let (Σ, E, μ) be a σ-finite standard Borel space [46, 424A] with a given Lusin μ-filtration F, so that L2 (μ) is separable as described in Subsection 3.4.1. Suppose that H0 is a selfadjoint operator defined in L2 (μ) and with spectrum contained in [0, ∞).

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CLR inequality

The semigroup S(t) = e−tH0 , t ≥ 0, is assumed to consist of absolute integral operators and have the property that there exists a (smallest) semigroup |S|(t), t ≥ 0, of (pointwise) positive operators such that |S(t)f | ≤ |S|(t)f μ-a.e. for every t ≥ 0 and nonnegative f ∈ L2 (μ)—we take the term positive operator to mean positive in the partial order of operators acting on the Banach lattice L2 (μ). To avoid confusion, we say that a selfadjoint operator T is hermitian positive if its spectrum σ(T ) is a nonnegative set of real numbers. Also, suppose that supt>0 |S|(t)L(L2 (μ)) < ∞ and suppose that ( I + H0 )−1 and Q(V )( I + H0 )−1 are compact linear operators for each (some) > 0. Theorem 7.1. Let G : [0, ∞) → [0, ∞) be a convex function with  ∞ dz e−z G(z) = 1. z 0

Let m : E → χE , E ∈ E, and suppose that E |S|(t), dm < ∞ for every t > 0 and every set E with finite μ-measure. Then   ∞ dt

|S|(t)G(tV− ), dm . (7.3) N (H0 + Q(V )) ≤ t Σ 0 Some explanation of the bound (7.3) is in order. The precise definition of the bilinear integral E T, dm used in this chapter is given in Definition 7.1 below. If the right-hand side of (7.3) is infinite, then there is nothing to prove.

In applications, the bilinear integral Σ |S|(t)G(tV− ), dm is the trace of 2 the trace class operator |S|(t)G(tV − ) on L (μ), see Chapter 3, but we don’t

preclude the possibility that Σ |S|(t)G(tV− ), dm = ∞ for some t > 0. For the operator H0 = −Δ in L2 (Rd ), d = 3, 4, . . . , S = |S| and for Lebesgue measure λ on Rd , the integral 

S(t), dm = (4πt)−d/2 λ(E) E

is finite for each set E with finite Lebesgue measure and each t > 0. Moreover, we can choose G(z) = ∞ a

(z − a)+ (z − a)e−z dz z

to obtain the CLR inequality N (H0 + Q(V )) ≤ C(G)



d

Rd

V− (x) 2 dx

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where



C(G) =

(t − a)t−d/2−1 dt

∞ . (2π)d/2 a (z − a)e−z dz z a

For d = 3, the optimal value for C(G) is 0.1156 obtained when a = 1/4 [90]. The idea that the CLR inequality (7.1) applies to quantum systems dominated by a Markov process first appears in [119], where the dominating process is assumed to be determined by an ultracontractive semigroup |S|. The proof of a bound similar to (7.3) is achieved in [119] by a version of the Trotter product formula rather than the Feynman-Kac formula that is a feature of Lieb’s proof [90]. The proof in [119] also applies in the situation when |S| is not a contraction L1 (μ). Despite the title of paper [119], the proof of the bound (7.3) in the general setting is achieved with path integration in the operator setting espoused by I. Kluv´anek, see Theorem 6.2 above. The CLR inequality in the form (7.1) is equivalent to the global Sobolev inequality on the underlying space with dimension d = 3, 4, . . . . The argument of Li and Yau [89] gives the first intimation and the connection is more fully explored in [91,122]. However, our main result, Theorem 7.1, belongs to the domain of measure theory. Additional geometric information may then be extracted from Lp -related bounds on the semigroup e−tH0 , t ≥ 0. The work of [119] suggests that the bound (7.3) may be related to a weak form of logarithmic Sobolev inequality for an abstract measure space, independent of dimension, see [57]. The operator version of the Feynman-Kac formula Theorem 6.2 is used to prove the bound (7.3) with few assumptions about the regularity of the dominating semigroup. In particular, |S|(t), t > 0, need not be a semigroup associated with the transition functions pt (x, dy), x ∈ Σ, of a Markov process in Σ. For example, if H0 = −Δ + cδ is the Laplacian with a point interaction defined in L2 (R3 ), then e−tH0 , t ≥ 0, is a semigroup of (pointwise) positive operators bounded on Lp (R3 ) only for 3/2 < p < 3 [2, equation (3.4)] and such a semigroup cannot have a family of integral kernels which are the transition functions of a Markov process. Besides the operator version of the Feynman-Kac formula, a critical ingredient in the approach of this chapter

to the proof of the CLR bound (7.3) is the use of the bilinear integral Σ T, dm ∈ [0, ∞] for a positive operator T on L2 (μ) as developed in Chapter 3. A similar notion of bilinear integration features in the connection between stationary state and timedependent scattering theory in Chapter 5.

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If (T u, u) ≥ 0 for all u ∈ L2 (μ) and Σ T, dm < ∞, then according to Theorem 3.3 above, T is a trace class operators on L2 (μ) and 

T, dm = tr(T ). Σ

2

The space C1 (E, L (μ)) of absolute integral operators T for which 

|T |, dm < ∞ Σ

considered in Chapter 3 forms a lattice ideal in the space of regular operators on L2 (μ), whereas the space C1 (L2 (μ)) of trace class operators is an operator ideal in the space of continuous linear operators on L2 (μ), that is, if T ∈ C1 (E, L2 (μ)) and U : L2 (μ) → L2 (μ) is a continuous linear operator with |U f | ≤ |T f | μ-a.e. for all f ∈ L2 (μ) with f ≥ 0 μ-a.e., then U ∈ C1 (E, L2 (μ)), but on the other hand, AT, T A ∈ C1 (L2 (μ)) for all A ∈ L(L2 (μ)) and T ∈ C1 (L2 (μ)). As mentioned above, the intersections C1 (E, L2 (μ)) ∩ P+ and C1 (L2 (μ)) ∩ P+ with the collection P+ of bounded selfadjoint operators with nonnegative spectra coincide. The proof of Theorem 7.1 is achieved in two steps. First a generalisation of Lieb’s inequality [90] to dominated semigroups is given. The FeynmanKac formula with respect to the evolution process associated with the semigroup S and the interchange of the order of integration with respect to the bilinear integral with respect to m and Lebesgue measure are essential ingredients. The last step is a standard application of the Birman-Schwinger principle. 7.2

Lattice traces for positive operators



˜ x) dμ(x) for an absolute integral The trace integral Σ T, dm = Σ k(x, operator with integral kernel k was defined in Chapter 3 for operators T ∈ C1 (E, L2 (μ)), with E a given Lusin filtration and k˜ = lim sup E(k|En ⊗ En ) n→∞

˜ x) = k(x, x) for μ-almost on Σ × Σ. As shown in Proposition 3.7, k(x, all x ∈ Σ in the case that μ is a σ-finite radon measure on the Hausdorff topological space Σ, the kernel k : Σ×Σ → C is continuous and E is suitably associated with k. In particular, k is necessarily (∨E)2 -measurable. For a general positive operator T : L2+ (μ) → L2+ (μ), we find that the requisite analysis is facilitated by approximating T from below by elements

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of C1 (E, L2 (μ)) in a similar fashion to measure theory: for a nonnegative measurable function f and a semifinite measure μ, the equality    f dμ = sup s dμ : 0 ≤ s ≤ f ∈ [0, ∞] Σ

Σ

holds with respect to the supremum over μ-integrable simple functions s [45, 213B]. Let (Σ, B, μ) be a σ-finite measure space equipped with a Lusin μfiltration E = En n∈N . The closure in C1 (E, L2 (μ)) of the collection of all finite rank operators T : L2 (μ) → L2 (μ) with integral kernels of the form k=

n 

fj ⊗ gj ,

j=1

for fj ∈ L2 (μ), gj ∈ L2 (μ) for j = 1, . . . , n and n = 1, 2, . . . , is denoted by E L2 (μ). L2 (μ)⊗ The map J : Σ → Σ × Σ defined by J(x) = (x, x), x ∈ Σ, maps Σ bijectively onto the diagonal of Σ × Σ. The following statement is what we observed in Proposition 3.5 above. Proposition 7.1. If k is the integral kernel of an operator belonging to E L2 (μ), then limn→∞ E(f |En ⊗ En ) ◦ J converges μ-almost everyL2 (μ)⊗ where in Σ and in L1 (μ). Definition 7.1. Let (Σ, B, μ) be a σ-finite measure space equipped with a Lusin μ-filtration E = En n∈N . For a (lattice) positive operator T on L2 (μ) and B ∈ B,    ˜ x) dμ(x) : 0 ≤ Tk ≤ T, Tk ∈ L2 (μ)⊗ E L2 (μ) .

T, dm = sup k(x, B

B

(7.4) E L2 (μ). The Here k is the integral kernel of the operator Tk ∈ L2 (μ)⊗ supremum may be infinite. According to Theorem 3.1, the space C1 (E, L2 (μ)) is a lattice ideal in the space Lr (L2 (μ)) of regular operators on L2 (μ) and there is a one-toone correspondence between absolute integral operators Tk ∈ Lr (L2 (μ)) and their integral kernels k [96, Theorem 3.3.5]. Consequently, the linear E L2 (μ), is a Banach lattice homomorphism map Tk −→ k˜ ◦ J, Tk ∈ L2 (μ)⊗ 1 into L (μ) and the supremum (7.4) is the limit of an upwards directed set

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of finite integrals. Hence, if Σ T, dm < ∞, then B −→ B T, dm , B ∈ B, is a finite measure whose density with respect to μ is E L2 (μ)} sup{k˜ ◦ J : 0 ≤ Tk ≤ T, Tk ∈ L2 (μ)⊗ in the Dedekind complete Banach lattice L1 (μ). In particular, for Tk ∈ E L2 (μ), we have L2 (μ)⊗   ˜ x) dμ(x), B ∈ B.

Tk , dm = k(x, B

B

It is clear that for 0 ≤ T1 ≤ T2 we have  

T1 , dm ≤ T2 , dm . Σ

(7.5)

Σ

As noted above, if T is trace class or the integral kernel kT of T is con E L2 (μ) and it follows that for a positive tinuous, then T belongs to L2 (μ)⊗ 2 2 kernel operator T : L (μ) → L (μ), we have 

T, dm = tr(T ) Σ

E L2 (μ), then if T is a trace class operator. In the case that T ∈ L2 (μ)⊗  

T, dm = kT (x, x) dμ(x) Σ

Σ

if μ is a σ-finite Radon measure on the Hausdorff space Σ and kT is continuous, provided that the partitions P E that determine our Lusin μ-filtration E consist of relatively compact subsets of Σ. E L2 ([0, 1]) such that Tk ≤ Note that the only operator Tk ∈ L2 ([0, 1])⊗

1 Id, the identity operator, is the zero operator, so 0 Id, dm = 0. On the other hand, if Σ = {1, . . . , n} and μ is counting measure, then  1

Id, dm = n. 0

Because we are dealing with a σ-finite measure μ it is also worthwhile to look at local approximations, leading to Theorem 7.2 below, which is a refinement of Theorem 3.3. For a nonempty subset Γ of Σ, the σ-algebra {B ∩ Γ : B ∈ B} is denoted by B ∩ Γ. For a set B ∈ B with μ(B) > 0, μB : E −→ μ(E ∩ B), E ∈ B, is the relative measure on B. The completion of L2 (μB ) ⊗ L2 (μB ) with respect to the norm  f −→ f L1 (μB ⊗μB )) + ME 2 (f χB×B ) ◦ J dμ B

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5 E L2 (μB ). The tilde is used here to emphasise that is denoted by L2 (μB )⊗ we are now dealing with the integral kernels of operators rather than the absolute integral operators themselves. As in Proposition 3.5, the functions E(f χB×B |En ⊗ En ) ◦ J, n = 1, 2, . . . , 5 E L2 (μB ). converge μB -a.e. and in L1 (μB ) for each f ∈ L2 (μB )⊗ The following observations are worth noting. a) By the argument of Proposition 3.4, the projective tensor product π L2 (μB ) embeds onto a dense subspace of L2 (μB )⊗ 5 E L2 (μB ) and L2 (μB )⊗ lim E(k χB×B |En ⊗ En )(x, x) =

n→∞

∞ 

φj (x)ψj (x)

j=1

∞ for μ-almost all x ∈ B, if k χB×B = j=1 φj ⊗ ψj (μ ⊗ μ)-a.e. with ∞ φ  ψ  < ∞. j 2 j 2 j=1 b) According to Proposition 3.7, if μ is a Radon measure on Σ, K is a compact subset of Σ and k : Σ × Σ → C is continuous, then k χK×K ∈ 5 E L2 (μK ) and L2 (μK )⊗ lim E(k χK×K |En ⊗ En )(x, x) = k(x, x) n→∞

for μ-almost all x ∈ K. There is an underlying technical assumption here that Σ possesses a Lusin filtration, so the Borel σ-algebra of Σ is countably generated. 5 E,σ L2 (μ) consists of all (μ ⊗ μ)-equivalence classes [f ] The space L2 (μ)⊗ of functions f : Σ × Σ → C for which there exist a partition U of Σ by sets 5 E L2 (μU ) for every U ∈ U, that is, in ∪P E such that [f χU ×U ] ∈ L2 (μU )⊗ 2 2 5 functions belonging to L (μ)⊗E,σ L (μ) are locally traceable with respect to the Lusin μ-filtration E. 5 E,σ L2 (μ), then limn→∞ E(f |En ⊗ En ) ◦ J Proposition 7.2. If f ∈ L2 (μ)⊗ converges μ-almost everywhere in Σ. 5 E,σ L2 (μ) and let U be a partition of Σ by sets in Proof. Let f ∈ L2 (μ)⊗ E 5 E L2 (μU ) for every U ∈ U. ∪P such that [f χU ×U ] ∈ L2 (μU )⊗ Then for each U ∈ U, there exists nU = 1, 2, . . . , such that U ∈ PnEU and E(f |En ⊗ En )(x, y) = E(f χU×U |En ⊗ En )(x, y), n ≥ nU , for every x, y ∈ U because Un (x) ⊂ U and Un (y) ⊂ U for all n ≥ nU , so that lim E(f |En ⊗ En ) ◦ J

n→∞

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179

converges μ-a.e. in U . Because U is a partition of Σ, limn→∞ E(f |En ⊗En )◦J converges μ-almost everywhere in Σ. In the following result, we obtain a necessary and sufficient condition for a hermitian positive operator with kernel k to be trace class directly in ˜ x) dμ(x) along the diagonal. terms of the finiteness of the integral Σ k(x, Theorem 7.2. Let Tk : L2 (μ) → L2 (μ) be a hermitian positive integral operator with kernel k and let E be a Lusin μ-filtration for which E(|k||En ⊗ En ) has finite values for each n = 1, 2, . . . . The operator Tk is trace class ˜ x) dμ(x) < ∞. If Tk is trace 5 E,σ L2 (μ) and k(x, if and only if k ∈ L2 (μ)⊗ Σ class, then the formula  ˜ x) dμ(x) tr(Tk ) = k(x, (7.6) Σ

holds with respect to the integral kernel k˜ of the operator Tk defined by k˜ = lim E(k|En ⊗ En ), n→∞

wherever the limit exists. Moreover,  ME 2 (k)(x, x) dμ(x) ≤ 4tr(Tk ). Σ

5 E,σ L2 (μ) then for disjoint Proof. As in Proposition 7.1, if k ∈ L2 (μ)⊗ E Σj ∈ ∪P , j = 1, 2, . . . , such that ∪j Σj has full measure, the function 5 E L2 (μΣj ) and f χΣj ×Σj belongs to L2 (μΣj )⊗ ˜ x) = lim k χ k(x, Σj ×Σj (x, x) j→∞

for μ-almost all x ∈ Σj and every j = 1, 2, . . . . Moreover, k χΣj ×Σj is the integral kernel of the hermitian positive operator Q(Σj )Tk Q(Σj ) for each j = 1, 2, . . . . Applying Theorem 3.3, the operator Q(Σj )Tk Q(Σj ) is trace class and  Σj

ME 2 (k χΣj ×Σj ) ◦ J dμ ≤ 4tr(Q(Σj )Tk Q(Σj )).

For any finite subset I of N, let nI = max{nΣj : j ∈ I }, where Σj ∈ PnΣj and E(k|En ⊗ En )(x, y) = E(k χ(∪i∈I Σi )×(∪i∈I Σi ) |En ⊗ En )(x, y), n ≥ nI , for every x, y ∈ ∪i∈I Σi because Un (x) ⊂ ∪i∈I Σi and Un (y) ⊂ ∪i∈I Σi for all n ≥ nI . In particular, for j ∈ I and x ∈ Σj , Un (x) ⊂ Σj for all n ≥ nI and

k d(μ ⊗ μ) U (x)×Un (x) . E(k|En ⊗ En )(x, x) = n μ(Un (x))2

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Consequently, E(k|En ⊗ En )(x, x) = E(k χB×B |En ⊗ En )(x, x) for x ∈ B =

χB×B (x, x) dμ(x) by equation (7.6) ∪i∈I Σi and tr(Q(B)Tk Q(B)) = B k and the bound  ˜ x) dμ(x) tr(Q(B)Tk Q(B)) ≤ k(x, Σ

follows. The noncommutative Fatou lemma shows that Tk is trace class. The rest of the proof follows that of Theorem 3.3. Proposition 7.3. Let T : L2 (μ) → L2 (μ) be a positive operator. For any uniformly bounded nonnegative μ-measurable functions V1 , V2 , the equalities   

Q(V2 )T Q(V1 ), dm = Q(V1 V2 )T, dm = T Q(V1 V2 ), dm Σ

Σ

Σ

of extended real numbers hold. For any essentially bounded μ-measurable function V ≥ 0,  

Q(V )T, dm ≤ V ∞ T, dm ∈ [0, ∞].

(7.7)

Proof. We will first prove the inequality  

Q(V1 V2 )T, dm ≤ Q(V2 )T Q(V1 ), dm

(7.8)

Σ

Σ

Σ

Σ

of extended real numbers. The equalities are established by a similar argument. () Let > 0, Vj = Vj ∨ , for j = 1, 2, and let λ be a positive number such

() () E L2 (μ) that λ < Σ Q(V1 V2 )T, dm . Then there exists Tk ∈ L2 (μ)⊗ () () such that Tk ≤ Q(V1 V2 )T and   () () ˜ λ< k ◦ J dμ ≤ Q(V1 V2 )T, dm . Σ

Then

() () Q((V1 )−1 )Tk Q(V1 )

Σ

E L2 (μ), the inequality ∈ L (μ)⊗ 2

()

()

()

()

Q((V1 )−1 )Tk Q(V1 ) ≤ Q(V2 )T Q(V1 )



() () holds and Σ Q((V1 )−1 )Tk Q(V1 ), dm = Σ k˜ ◦ J dμ. Consequently, λ <

() () Σ Q(V2 )T Q(V1 , dm , so that   () () () ()

Q(V1 V2 )T, dm ≤ Q(V2 )T Q(V1 ), dm .

Σ

() () sup>0 Σ Q(V2 )T Q(V1 ), dm

Σ

< ∞, then the inequality (7.8) obtains If in the limit by monotone convergence. The other inequalities are proved in a similar fashion.

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() A similar argument shows that for λ < V −1 )T, dm there ∞ Σ Q(V 2 2 −1 () E L (μ) such that Tk ≤ V ∞ Q(V )T and exists Tk ∈ L (μ)⊗   λ
λ. Consequently, λ < Σ T, dm , so that   () V −1

Q(V )T, dm ≤

T, dm ∞ Σ



Σ

for all > 0. If Σ T, dm < ∞, monotone convergence ensures that sup>0 Σ Q(V () )T, dm = Σ Q(V )T, dm , which establishes the bound (7.7). It is well known that if T is a trace class operator on a Hilbert space H and B is any bounded linear operator on H then BT and T B are also trace class operators (C1 (H) is an operator ideal ) and tr(BT ) = tr(T B). By contrast, the space C1 (E, L2 (μ)) is a lattice ideal in Lr (L2 (μ)). For T ∈ C1 (E, L2 (μ)) and B ∈ L(L2 (μ)), the operator BT may not even be a kernel operator, see [20, Section 4.6.2], but we have the following trace property. Proposition 7.4. Let Tj : L2 (μ) → L2 (μ), j = 1, 2, be positive kernel operators. Then the equalities  

T1 T2 , dm = T2 T1 , dm Σ

Σ

of extend real numbers hold. Proof. We can find a partition Σn , n = 1, 2, . . . , of Σ into sets with finite μ-measure. Let μn = μ  (∪ ≤n Σn ) ∩ B. Let f ∗ (x, y) = f (y, x) for a function of two variables x, y ∈ Σ. An appeal to [60, Theorem 10.7] shows that the kernel of T1 T2 is  k1 ∗ k2 : (x, y) −→ k1 (x, z)k2 (z, y) dμ(z) Σ

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for (μ ⊗ μ)-almost all (x, y) ∈ Σ × Σ and    u(x)k1 (x, z)k2 (z, y)v(y) dμ(y)dμ(z)dμ(x) < ∞ Σ

Σ

Σ

for every u, v ∈ L2+ (μ). Suppose that f, g ∈ L∞ + (μ ⊗ μ) and f ≤ k1 and g ≤ k2 . Then Tf and Tg are Hilbert-Schmidt operators on L2 (Σn ) so Tf ∗g is a trace class operator such that 

Tf Tg , dm = f g ∗ L1 (μn ⊗μn ) ≤ k1 .k2∗ L1 (μn ⊗μn ) = k2 .k1∗ L1 (μn ⊗μn ) , Σn



 Σn

Tf Tg , dm ≤

Σ

T1 T2 , dm

k1 .k2∗ L1 (μ⊗μ) = sup{f g ∗ L1 (μ⊗μ) }. It follows that for μ(Σn ) < ∞ and

k1 .k2∗ L1 (μ⊗μ) ≤ Σ T1 T2 , dm .

E L2 (μ) and 0 ≤ f ≤ k1 ∗k2 , then Tf , dm ≤ Moreover, if Tf ∈ L2 (μ)⊗ Σ k1 .k2∗ L1 (μ⊗μ) . The same argument applies to T2 T1 , so  

T1 T2 , dm = T2 T1 , dm = k1 .k2∗ L1 (μ⊗μ) ∈ [0, ∞]. Σ

Σ

We also note that a bilinear version of the Fubini-Tonelli Theorem holds. Let (Ξ, B, ν) be a σ-finite measure space. For any function f : Ξ →

2 L+ (L (μ)) such that Ξ Σ f (ξ), dm dν(ξ) < ∞, we say that f is (m ⊗ ν)integrable if for each u, v ∈ L2 (μ), the scalar function (f u, v) : ξ −→ (f (ξ)u, v) is ν-integrable and there exists T ∈ C1 (E, L2 (μ)) such that   

f (ξ), dm dν(ξ) = T, dm (7.9) Ξ Σ Σ  (f (ξ)u, v) dν = (T u, v) (7.10) Ξ

for all u, v ∈ L2 (μ). Then we set  

f, d(m ⊗ ν) = T, dm . Σ×Ξ

Σ

Because C 1 (E, L2 (μ)) is a lattice ideal, for each A ∈ B there exists a positive operator A f dν ∈ C1 (E, L2 (μ)) such that     f dν u, v = (f (ξ)u, v) dν ≤ (T u, v) A

for all u, v ∈ L2 (μ)+ .

A

Remark 7.1. For each u, v ∈ L2 (μ), the tensor product u ⊗ v, and T −→

T, dm are continuous functionals on C1 (E, L2 (μ)), so it is natural to Σ assume that both equalities (7.9) and (7.10) hold.

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The following statement is a consequence of the definitions. Proposition 7.5. Let f : Ξ → L+ (L2 (μ)) be a positive operator valued function such that f is (m ⊗ ν)-integrable. Suppose that f (ξ) ∈ E L2 (μ) for ν-almost all ξ ∈ Ξ. L2 (μ)⊗

Then the scalar valued function ξ −→ Σ f (ξ), dm is ν-integrable and the equalities #  " 

f, d(m ⊗ ν) = f dν, dm (7.11) Σ×Ξ Σ  Ξ =

f (ξ), dm dν(ξ) (7.12) Ξ Σ



!



hold. Moreover, Σ A f dν, dm = A Σ f (ξ), dm dν(ξ) for every A ∈ B.

Proof. Equation (7.11) is the definition of Σ×Ξ f, d(m ⊗ ν) and (7.12) is a reformulation of assumption (7.9). By assumption, for ν-almost all ξ ∈ Ξ, we can find a martingale Fξ with respect to the filtration {En } and a regularisation kξ (x, y), x, y ∈ Σ, of the kernel associated with f (ξ) such that       f dν u, v

=

A

A

Σ

Σ

kξ (x, y)u(x)v(y) dμ(x)dμ(y)dν(ξ)

for all A ∈ B and u, v ∈ L2 (μ). Then for each A ∈ B, we have #    " f dν, dm = kξ (x, x) dμ(x)dν(ξ) Σ A A Σ   =

f (ξ), dm dν(ξ) A

Σ

by the scalar Fubini-Tonelli Theorem. The following result follows from the observation in Proposition 3.1 that C1 (E, L2 (μ)) is a lattice ideal and an application of monotone convergence. Proposition 7.6. Let M : B → L+ (L2 (μ))

be a positive operator valued

M (Ξ), dm < ∞, then the set measure on a measurable space (Ξ, B). If Σ

function M, m : A → Σ M (A), dm , A ∈ B, is a finite measure such that  

M (A), dm ≤ M (Ξ), dm , A ∈ B, Σ

and



Σ



 Σ

M (f ), dm =

Ξ

f d M, m ≤ f ∞

for all B-measurable functions f : Ξ → [0, ∞].

Σ

M (Ξ), dm

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The CLR inequality for dominated semigroups

Proof of Theorem 7.1. Step 1. (Lieb’s inequality) If X denotes the (S, Q)-process (6.2), then according to Theorem 6.1, the dominating (|S|, Q)-process |X | = (Ω, St t≥0 , Pt t≥0 ; Xt t≥0 ) with Pt (Ω) = |S|(t), t ≥ 0, is σ-additive. According to Theorem 6.3 and the assumption that (Σ, B) is a standard Borel space, we may assume from the outset that X is a progressively measurable process. Also, by an appeal to Theorem 6.1 and the assumption that S(t) is an absolute integral operator for each t > 0, the bounded linear operator |S|(t) is the supremum in Lr (L2 (μ)) of an increasing family of absolute integral operators for each t > 0, directed by the cylindrical σ-algebras σ( Xs s∈J ) as the finite subsets J ⊂ [0, t] increase by inclusion. Because the absolute integral operators form a band in the Banach lattice Lr (L2 (μ)) of regular linear operators on L2 (μ) [96, Theorem 3.3.6], it follows that the positive linear operator |S|(t) is a kernel operator on L2 (μ) for each t > 0. For a bounded E-measurable function U ≥ 0, suppose that the limit 1

1

KU = lim U 2 ( I + H0 )−1 U 2 →0+

exists in the strong operator topology. Then KU is selfadjoint and ϕ(KU ) is defined by the functional calculus for selfadjoint operators for any bounded measurable function ϕ. Let f : [0, ∞) → [0, ∞) be a lower semicontinuous function with f (0) = 0 and set  ∞ dt (7.13) e−t f (xt) , x ≥ 0. F (x) = t 0

We now show that Σ F (KU ), dm ∈ [0, ∞] and  t  #  "   ∞ dt

F (KU ), dm ≤ f U (Xs ) ds dPt , dm . (7.14) t Σ Σ Ω 0 0 In the case that F is a uniformly bounded function on [0, ∞) and F (KU ) is a trace class operator, then an application of Proposition 3.3 ensures that the left-hand side of the equation is finite and 

F (KU ), dm = tr(F (KU )). Σ

To prove the bound (7.14), suppose first that U is a bounded measurable function and let 1

1

KU ( ) = U 2 ( I + H0 )−1 U 2 ,

> 0.

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Then KU ( ) is a compact selfadjoint linear operator because under our assumptions, ( I + H0 )−1 is a compact operator for each > 0. By the spectral theorem for selfadjoint operators, KU ( ) has a countable set of real eigenvalues whose only limit point is zero. For each λ > 0, the operator 1

1

AU (λ, ) = U 2 ( I + λU + H0 )−1 U 2 has the property that AU (λ, ) = KU ( )(I + λKU ( ))−1  ∞  1 1 −t −t(H +λU) 0 = Q(U 2 ) e e dt Q(U 2 )    0 ∞ t 1 1 −t −λ U◦X ds s 0 = Q(U 2 ) e e dMt dt Q(U 2 ), Ω

0

by the Feynman-Kac formula (6.2). Hence, the expression   t    ∞  1 1 −t 2 e g U ◦ Xs ds dMt dt Q(U 2 ) F (KU ( )) = Q(U ) Ω

0

0

(7.15) , x ≥ 0, and F : x → x/(1 + λx), x ≥ 0, so we take is valid for g : x → e  ∞  ∞ dt −t F (x) = x e g(xt) dt = e−t f (xt) t 0 0 −λx

for the function f : x → xg(x), x ≥ 0. By the Stone-Weierstrass Theorem, the collection K of all linear combinations of functions x → e−λx , x ≥ 0, for λ > 0, is dense in the space C0 ([0, ∞)) of continuous functions on [0, ∞) vanishing at infinity. The equality (7.15) is valid in the case that U is a bounded μ-measurable function and f (x) = xg(x), x ≥ 0, for g ∈ C0 ([0, ∞)) because the left-hand side is continuous in the uniform operator topology and the uniform norm on g by the functional calculus for selfadjoint operators. The right-hand side is bounded by Cg∞ U ∞ / with C = supt>0 Pt (Ω)L(L2 (μ)) , which we have assumed to be finite. Next we show that for the functions g : x → e−λx , x ≥ 0, and f : x → xg(x), x ≥ 0, the equality  t  #  " 1 f W ◦ Xs ds dPt , dm t Σ Ω 0    t  #  " 1 1 2 2 Q(W ) g W ◦ Xs ds dPt Q(W ), dm (7.16) = Σ

Ω

0

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holds for every t > 0 and every nonnegative bounded measurable function W vanishing off a set of finite μ-measure. The right-hand side is finite because    t  #  " 1 1 2 2 Q(W ) g W ◦ Xs ds dPt Q(W ), dm Σ Ω 0  6 7 1 1 Q(W 2 )Pt (Ω)Q(W 2 ), dm ≤ Σ

Q(W )Pt (Ω), dm < ∞ = Σ

by appealing to the monotone property (7.5) and Proposition 7.3. Also, for

any t > 0 and bounded St -measurable function ϕ : Ω → C, the operator Ω ϕ dPt is necessarily an absolute integral operator because      Q(E)  ϕ dPt  ≤ ϕ∞ Q(E)Pt (Ω) Ω

for every set E ∈ E with finite μ-measure. To prove the equality (7.16) for g : x → e−λx , x ≥ 0, we first look at the inner integral on the left-hand side of the equation. By an application of Fubini’s Theorem and the Markov property of the process |X |, we have  t   f W ◦ Xs ds dPt Ω 0   t    t = W ◦ Xs ds g W ◦ Xr dr dPt Ω 0 0  t   t = W ◦ Xs .g W ◦ Xr dr dPt ds Ω 0 0     t t−s Pt−s g W ◦ Xr dr = 0 0    s W ◦ Xs .g W ◦ Xr dr dPs ds × Ω 0   t−s   t Pt−s g W ◦ Xr dr = 0    0  s g W ◦ Xr dr dPs ds. ×Q(W ) Ω

0

Applying the bilinear integral with respect to m to the integrand, for each

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0 ≤ s ≤ t, we have   t−s   #   s  " Pt−s g W ◦ Xr dr Q(W ) g W ◦ Xr dr dPs , dm Σ Ω 0 0   s   " Q(W )Ps g W ◦ Xr dr = Σ 0   t−s  # W ◦ Xr dr , dm ×Pt−s g 0  #   t  " Q(W ) g W ◦ Xs ds dPt , dm = Σ

Ω

0

by Proposition 7.4 and the semigroup property of the map    t g W ◦ Xs ds dPt , t ≥ 0, t −→ Ω

0

mentioned in Theorem 6.2, so an appeal to the Fubini-Tonelli Theorem, Proposition 7.5, with respect to dm ⊗ ds and Proposition 7.3 gives  #  t   " 1 f W ◦ Xs ds dPt , dm t Σ Ω 0 "   t−s    1 t = Pt−s g W ◦ Xr dr t 0 Σ 0   #  s g W ◦ Xr dr dPs ds, dm ×Q(W ) Ω 0  #   "   t 1 t Q(W ) g W ◦ Xr dr dPt , dm ds = t 0 Σ Ω 0  #   t  " Q(W ) g W ◦ Xs ds dPt , dm = Σ Ω 0    t  #  " 1 1 2 2 Q(W ) g W ◦ Xs ds dPt Q(W ), dm . = Σ

Ω

0

Appealing to the Fubini-Tonelli Theorem again for dm ⊗ dt, it follows that 

F (KW ( )), dm Σ "   t    ∞  # 1 1 −t 2 2 Q(W ) e g W ◦ Xs ds dMt dt Q(W ), dm = Ω 0 t   0 ∞  # Σ " 1 1 Q(W 2 ) e−t g W ◦ Xs ds dPt dt Q(W 2 ), dm ≤   Ω  t0  #  " 0 Σ∞ 1 1 −t 2 2 Q(W ) e g W ◦ Xs ds dPt Q(W ), dm dt = 0

Σ

Ω

0

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= 0

e−t t

 "

 f

Σ

Ω

0



t

W ◦ Xs ds

 dPt , dm

# dt.

(7.17)

The inequality (7.17) is also valid if W is replaced by a bounded and μ-measurable function

tU and f (x) = xg(x), x ≥ 0, for g ∈ C0 ([0, ∞)). To prove this, let Φ t = 0 W ◦ Xs ds. According to Proposition 7.6 and the assumption that Σ W Pt (Ω), dm < ∞, the measures W (Pt ◦Φ−1 t ), m and ), m are finite Borel measures on [0, ∞). The equality (7.16) is

id.(Pt ◦Φ−1 t valid for all g ∈ K and so for all g ∈ C0 ([0, ∞)) by uniform convergence. By monotone convergence, equality (7.16) is valid for all uniformly bounded and continuous functions g : [0, ∞) → C. Replacing W by any bounded and measurable function U ≥ 0, there is nothing to prove if the right-hand side of (7.17) is infinite, so suppose that f : [0, ∞) → [0, ∞) is a lower semicontinuous function with f (0) = 0 and  #  t   ∞ −t  " e f U ◦ Xs ds dPt , dm dt < ∞. t Σ Ω 0 0 Then for any g ∈ C0 ([0, ∞)) such that 0 ≤ f˜(x) ≤ f (x), with f˜(x) = xg(x) for x ∈ [0, ∞), Proposition 7.6 ensures that  #   t  ∞ −t  " e U ◦ Xs ds dPt , dm dt < ∞. f˜ t Σ Ω 0 0 Let F˜ be the function corresponding to f˜ according to formula (7.13). Now let Σn , n = 1, 2 . . . , be increasing sets with finite μ-measure such χ that Σ = ∪∞ n=1 Σn and put Un = U Σn for n = 1, 2, . . . . According to for∞ −t ˜ ˜ , is increasing. The equality mula (7.13), the function7F (x) =6 0 e x f (t) dt t7

6

˜ ˜ F (KUn ( )), dm = F (KU ( )), dm follows from Theorem 3.3 supn Σ

Σ

and the monotonicity of the spectrum. The integrals  #   t  ∞ −t  " e Un ◦ Xs ds dPt , dm dt f˜ t Σ Ω 0 0 0 7 0 /

∞ e−t 6/

t converge to 0 f˜ 0 U ◦ Xs ds dPt , dm dt as n → ∞ by t Σ Ω Proposition 7.6 and dominated convergence. Then  6 7 F˜ (KU ( )), dm Σ   #   t ∞ −t  " e ˜ U ◦ Xs ds dPt , dm dt. f ≤ t Σ Ω 0 0

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t An appeal to Proposition 7.6 ensures that for Φt = 0 U ◦ Xs ds, t > 0, the set function   # ∞ −t  " e f d(Pt ◦ Φ−1 ) , dm dt A −→ t t Σ A 0 is a finite Borel measure on [0, ∞), so finally, taking the supremum over all 0 ≤ f˜ ≤ f gives the inequality (7.17) with W replaced by U . Taking → 0+, monotone convergence now gives the inequality (7.14). Step 2. (Birman-Schwinger Principle) Let F : [0, ∞) → [0, ∞) be a strictly increasing function and let {μk } denote the set of eigenvalues of F (KU ). Because KU is selfadjoint and F is nonnegative, it follows that μk ≥ 0. According to the Birman-Schwinger Principle [128, Theorem 8.1], for U = V− , we have  1 N (H0 + Q(V )) ≤ {k:μk ≥F (1)}



1 F (1)



μk

{k:μk ≥F (1)}

1 tr(F (KU )) F (1)  1 =

F (KU ), dm F (1) Σ ≤

by Theorem 3.3. Then N (H0 + Q(V )) ≤ F (1)−1



F (KU ), dm  t  #  " dt −1 f U (Xs ) ds dPt , dm ≤ F (1) t Σ Ω 0 0 by Lieb’s inequality (7.14) when F is given by formula (7.13), f is lower semicontinuous and F (1) is finite. If f is convex, then by Jensen’s inequality   t  t U (Xs ) ds ≤ t−1 f (tU (Xs )) ds, f 0

so that

Σ  ∞

0

 t  #  " dt f U (Xs ) ds dPt , dm t Σ Ω 0 0  #  "  t  ∞ dt f (tU (X )) ds dP , dm ≤ s t t2 Σ Ω 0 0 #   " t   ∞ dt ds , dm . f (tU (Xs )) dPt = t Σ t Ω 0 0





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According to the Markov property (6.3) for the (|S|, Q)-process |X |, the right-hand side is equal to #  " t  ∞ dt ds Pt−s (Ω)Q(f (tU ))Ps (Ω) , dm t Σ t 0 0  t  ∞ dt ds =

Pt (Ω)Q(f (tU ), dm t 0 Σ t 0   ∞ dt

Pt (Ω)Q(f (tU ), dm . = t Σ 0 The convex function f : [0, ∞) → [0, ∞) is necessarily continuous, so if F (1) is finite, then f (0) = 0. Consequently, the bound (7.3) is valid if we take G = F (1)−1 f . The bound is valid for any V satisfying the stated conditions, for then H0 + Q(V ) is defined by a form sum, so by replacing V by V χ{|V |≤n} + I and H0 by H0 + I, if necessary, and taking the limit as n → ∞ and → 0+, the bound (7.3) is obtained. Example 7.1. Let P be the momentum operator −id/dx acting in L2 (R). With H(A) = 12 (P − A)2 and S(t) = e−tH(A) , t ≥ 0, the σ-additive evolution process (Ω, St t≥0 , MtA t≥0 ; Xt t≥0 ) corresponds to a quantum particle on the line subject to a magnetic vector potential A. Then (Ω, St t≥0 , Mt t≥0 ; Xt t≥0 ) is the dominating process 2 1 with Mt (Ω) = e− 2 tP , t ≥ 0. According to Theorem 7.1, the bound   ∞ 2 1 dt

e− 2 tP G(tV− ), dm N (H(A) + Q(V )) ≤ t Σ 0 holds for any A : R → R such that A and A belong to L2loc (R) and V ∈ L1 (R) + Lp (R) with 1 < p < ∞ [128, Theorem 8.2, Theorem 15.5]. Other examples of dominated semigroups for which the CLR inequality is valid may be found in [119], [97].

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Linear operator equations

The analysis of the equation AX − XB = Y for linear operators A, B, X and Y acting in a Hilbert space H has many applications in operator theory, differential equations and quantum physics, see [14] for a relaxed discussion with numerous examples. Starting with the case of scalars, the equation ax − xb = y has a unique solution provided that a = b. For the case of diagonal matrices A = diag(λ1 , . . . , λn ) and B = diag(μ1 , . . . , μn ), for any matrix Y = {yij }ni,j=1 there exists a unique solution X of the equation AX − XB = Y if and only if λi − μj = 0 for i, j = 1, . . . , n and then the solution X = {xij }ni,j=1 is given by yij , i, j = 1, . . . , n. xij = λi − μj The operator version is called the Sylvester-Rosenblum Theorem in [14], although earlier versions are due to M. Krein and Yu. Daletskii [14, p. 1]. For a continuous linear operator A on a Banach space X , the spectrum σ(A) of A is the set of all λ ∈ C for which λI − A is not invertible. Theorem 8.1 (Sylvester-Rosenblum Theorem). Let X be a Banach space and let A and B be continuous linear operators on X for which σ(A)∩ σ(B) = ∅. Then for each operator Y ∈ L(X ), the equation AX − XB = Y has a unique solution X ∈ L(X ). As a taster for applications of the Sylvester-Rosenblum Theorem, suppose that A and B bounded normal operators on a Hilbert space H with spectral measures PA and PB , respectively. Then there exists c > 0, such that for any two Borel subsets S1 and S2 of C separated by a distance δ = inf{|x − y| : x ∈ S1 , y ∈ S2 }, 191

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the projections E = PA (S1 ), F = PB (S2 ), satisfy the norm estimate c EF  ≤ A − B. δ The norm EF  represents the angle between the subspaces ran(E) and ran(F ). Such estimates are useful in numerical computations. Even in finite dimensional Hilbert spaces, the Sylvester-Rosenblum Theorem leads to eigenvalue estimates for matrix norms independent of dimension. Theorem 8.2 ([13, Theorem 5.1a]). Let A and B be two normal (n×n) matrices with eigenvalues α1 , . . . , αn and β1 , . . . , βn respectively, counting multiplicity. With the same constant c mentioned above, if A − B ≤ /c, then there exists a permutation π of the index set {1, . . . , n} such that |αi − βπi | < for i = 1, . . . , n. The Sylvester-Rosenblum Theorem also comes with a representation of the solution X of the equation AX − XB = Y if A and B are bounded linear operators for which σ(A) ∩ σ(B) = ∅. Suppose that the contour Γ is the union of closed contours in the plane, with total winding numbers 1 around σ(A) and 0 around σ(B). Then  1 (ζI − A)−1 Y (ζI − B)−1 . (8.1) X= 2πi Γ Other representations of the solution are possible by utilising the spectral properties of the operators A and B, see [14, Section 9]. In this chapter, we are concerned with solutions X of the operator equation AX −XB = Y when A is an unbounded selfadjoint or normal operator acting in a Hilbert space H and B is a closed unbounded operator. If the spectra σ(A) and σ(B) are a positive distance apart, then we hope to construct the solution X of AX − XB = Y by the formula  dPA (ζ)Y (ζI − B)−1 (8.2) X= σ(A)

in place of (8.1) with respect to the spectral measure PA of A. The operator valued measure PA acts on the values of the operator valued function ζ −→ Y (ζI − B)−1 . As in the case of scattering theory considered in Chapter 5, for h ∈ H, the vector Xh ∈ H often has the decoupled representation  PA (dζ) ⊗ (Y (ζI − B)−1 h), Xh = J σ(A)

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where the H-valued function ζ −→ Y (ζI − B)−1 h, σ ∈ σ(A), is PA τ H → H is the continuous τ H and J : L(H)⊗ integrable in the space L(H)⊗ linear extension of the composition map T ⊗ h → T h, T ∈ L(H), h ∈ H. If the operator B is itself a bounded linear operator, then the simpler representation (8.1) may be employed with the contour Γ winding once around σ(B) and zero times around σ(A). Because we shall be dealing with unbounded operators A and B, we have to be careful about domains when interpreting the equation AX −XB = Y . We follow the treatment in [3, Section 2]. Applications of equation (8.2) to perturbation theory and the spectral shift function may also be found in [3]. 8.1

Operator equations

Definition 8.1. Let H and K be Hilbert spaces. Suppose that A : D(A) → K and B : D(B) → H are closed and densely defined linear operators with domains D(A) ⊂ K and D(B) ⊂ H. Given Y ∈ L(H, K), a continuous linear operator X ∈ L(H, K) is said to be a weak solution of the equation AX − XB = Y

(8.3)



if for every h ∈ D(B) and k ∈ D(A ), the equality (Xh, A∗ k) − (XBh, k) = (Y h, k) holds with respect to the inner product (·, ·) of K. The domain D(A∗ ) of the adjoint A∗ of A is the set of all elements k of K such that the linear map h −→ (Ah, k), h ∈ D(A), is the restriction to D(A) of h −→ (h, y), h ∈ H, for an element y ∈ K and then y = A∗ k. A strong solution X ∈ L(H, K) of (8.3) has the property that ran(X  D(B)) ⊂ D(A) and AXh − XBh = Y h,

h ∈ D(B).

The existence of strong solutions of the operator equation (8.3) is discussed in [106] under the assumption that A and −B are the generators of C0 -semigroups, a situation that arises in delay or partial differential equations and control theory. Strong solutions of (8.3) may not exist in this setting, even when the spectra σ(A) and σ(B) are separated by a vertical strip [106, Example 9].

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In the case that A and B are both selfadjoint operators, the following result is a consequence of [13, Theorem 4.1], see [3, Theorem 2.7]. Theorem 8.3. Let H and K be Hilbert spaces. Suppose that A : D(A) → K and B : D(B) → H are selfadjoint operators whose spectra σ(A) and σ(B) are a distance δ > 0 apart. Then equation (8.3) has a unique weak solution  e−itA Y eitB fδ (t) dt X= R

for any function fδ ∈ L1 (R), continuous on R \ {0}, such that  1 1 for |x| > . e−isx fδ (s) ds = x δ R π Moreover X ≤ 2δ Y . The integral representing the solution X is a Bochner integral for the strong operator topology. For a selfadjoint operator A in a Hilbert space K and a closed, densely defined operator B in a Hilbert space H, the domains D(B) and D(A) are endowed with the respective graph norms associated with the closed operators B and A. Suppose also, that τ is the topology on the tensor product L(X ) ⊗ X defined by formula (5.13) with X = K and Y = H τ K be the completion of the tensor product with the and let E = L(K)⊗ norm topology τ . According to Lemma 5.3, the Banach space E is bilinear admissible for the Hilbert space K in the sense of Section 2.3 and the composition map T ⊗ k −→ T k,

T ∈ L(K),

k ∈ K,

has a continuous linear extension JE : E → K. We may adopt the definition analogous to Definition 2.3. Definition 8.2. Let K be a Hilbert space. A function f : Ω → K is τ K for an operator valued measure said to be m-integrable in E = L(K)⊗   m : S → L(K), if for each x, x , y ∈ K, the scalar function (f, x ) is integrable with respect to the scalar measure (mx, y  ) and for each S ∈ S, there exists an element (m ⊗ f )(S) of E such that  ((m ⊗ f )(S), x ⊗ y  ⊗ x ) = (f, x ) d(mx, y  ) S

for every x, x , y  ∈ K. If f is m-integrable in E, then mf (S) ∈ K is defined for each S ∈ S by   mf (S) = JE (m ⊗ f )(S) .



We also denote mf (S) by S dm f or S dm(ω) f (ω).

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In the present context, the representation of solutions of equation (8.3) via bilinear integration is analogous to Example 5.1 in scattering theory. Example 8.1. Suppose that A is a bounded selfadjoint operator defined on a Hilbert space K such that σ(A) ⊂ (−∞, −δ) for some δ > 0. Let −B be the generator of a uniformly bounded C0 -semigroup e−tB , t ≥ 0, on the Hilbert space H. We can employ (8.1) in this situation to represent the weak solution of equation (8.3), but it is instructive to see how the integral (8.2) converges with the assumptions above. π K be the projective tensor product of the Hilbert space Let E = L(K)⊗ K with the space L(K) of bounded linear operators on K with the uniform norm. Then etA ⊗ (Y e−tB h) belongs to the tensor product L(K) ⊗ K for each t ≥ 0 and h ∈ H and the function t −→ etA ⊗ (Y e−tB h), t ≥ 0, is π K, because A is assumed to be bounded so continuous in L(K)⊗ ∞ n  t (An ⊗ (Y e−tB h)) etA ⊗ (Y e−tB h) = I ⊗ (Y e−tB h) + n! n=1 π K uniformly for t in any bounded interval. The inconverges in L(K)⊗ equalities  ∞  ∞  tA  e ⊗ (Y e−tB h) etA .(Y e−tB h) dt  πK ≤ L(K)⊗ 0 0  ∞  −δt −tB ≤ e e  dt .Y L(H,K) .h 0

∞ tA ensure that 0 e ⊗ (Y e−tB h) dt converges as a Bochner integral in the π K and projective tensor productin L(K)⊗   ∞      −1 ζt −tB PA (dζ) ⊗ Y (ζI − B) h = PA (dζ) ⊗ Y e e h dt σ(A)

0

σ(A)







=

e PA (dζ)

0

 =

0

  ⊗ Y e−tB h dt

σ(A) ∞

etA ⊗ (Y e−tB h) dt

π K too. Then belongs to L(K)⊗     −1 PA (dζ) Y (ζI − B) h = JE σ(A)

  PA (dζ) ⊗ Y (ζI − B)−1 h

σ(A)

defines a continuous linear operator   PA (dζ)Y (ζI − B)−1 : h −→ σ(A)



ζt

σ(A)

  PA (dζ) Y (ζI − B)−1 h , h ∈ H,

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belonging to L(H, K) with norm bounded by supt≥0 e−tB  Y L(H,K) . δ In order to deal with unbounded operators, we replace the projective tensor product topology π by the topology τ defined by formula (5.13). Lemma 8.1. Let H and K be Hilbert spaces. Suppose that A : D(A) → K is a selfadjoint operator with spectral measure PA and B : D(B) → H is a densely defined, closed linear operator such that σ(A) ∩ σ(B) = ∅. Let Y ∈ L(H, K). For each h ∈ H, the K-valued function Φh : ζ −→ Y (ζI − B)−1 h,

ζ ∈ σ(A),

(8.4)

π K on every compact subset of σ(A). is PA -integrable in L(K)⊗ Furthermore, there exist L(H, K)-valued B(σ(A))-simple functions sn : σ(A) → L(H, K),

n = 1, 2, . . . ,

such that for each h ∈ H, sn (ω)h → Φh (ω) in K as n → ∞ for PA -almost all ω ∈ σ(A) and for each compact subset of K of σ(A), sup (PA ⊗ Φh )(S) − (PA ⊗ (sn h))(S)L(K)⊗  πK → 0

S∈B(K)

as n → ∞. Proof. For a closed and densely defined operator T , the resolvent (λI − T )−1 is defined for all complex numbers λ belonging to the resolvent set ρ(T ) = C \ σ(T ). Suppose that ρ(T ) is nonempty. Then the resolvent equation (λI − T )−1 − (μI − T )−1 = (μ − λ)(λI − T )−1 (μI − T )−1 for λ, μ ∈ ρ(T ) ensures that λ −→ (λI − T )−1 , λ ∈ ρ(T ), is a holomorphic operator valued function for the uniform operator topology [76, Equation I (5.6)]. It follows that for each h ∈ H, the function λ −→ (λI − A)−1 ⊗ (Y (λI − B)−1 h) π K for the uniform is continuous in the projective tensor product L(K)⊗ norm on L(K). For a compact subset K of σ(A), let AK = PA (K)A be the part of A on K. Then for a contour ΓK with winding number 1 around K and zero around the closed set σ(B), the integral  (λI − AK )−1 .Y (λI − B)−1 hK |dλ| ΓK

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is bounded by (|ΓK |. sup (λI − AK )−1 .(λI − B)−1 ).Y L(H,K) .hH , λ∈ΓK

so the function ΓK (λI −AK )−1 ⊗(Y (λI −B)−1 h) dλ converges as a Bochner π K. For every Borel subset S of the set K and x, x , y  ∈ integral in L(K)⊗ K, an application of Cauchy’s integral formula yields  (PA x, x )(dζ)(Y (ζI − B)−1 h, y  ) S   (Y (λI − B)−1 h, y  ) 1 = dλ (PA x, x )(dζ) 2πi S λ−ζ ΓK  1 = ((λI − PA (S)AK )−1 x, x )(Y (λI − B)−1 h, y  ) dλ, 2πi ΓK so according to Definition 8.2 (replacing the topology τ by the stronger projective topology π), the function ζ −→ Y (ζI − B)−1 h, ζ ∈ σ(A), is π K on the set K and PA -integrable in L(K)⊗  dPA (ζ) ⊗ (Y (ζI − B)−1 h) S  1 = (λI − P (S)AK )−1 ⊗ (Y (λI − B)−1 h) dλ (8.5) 2πi ΓK π K for each Borel as an element of the projective tensor product L(K)⊗ subset S of K. Because the operator valued function λ −→ (λI − B)−1 , λ ∈ σ(A), is uniformly continuous on the compact set K, for each > 0, there exists an L(H)-valued B(σ(A))-simple function ϕ such that sup (λI − B)−1 − ϕ (λ)L(H) < ,

λ∈K

so that



sup S∈B(K)

ΓK

(λI − P (S)AK )−1 .Y (λI − B)−1 h − Y ϕ (λ)hK |dλ| → 0

as → 0+ for each h ∈ H. According to the identity (8.5), it follows that sup (PA ⊗ Φh )(S) − (PA ⊗ (Y ϕ h))(S)L(K)⊗ πK → 0

S∈B(K)

as → 0+. Because the spectral measure PA is inner regular on compact sets, the simple functions sn , n = 1, 2, . . . , can be pieced together from the simple functions ϕ1/n , n = 1, 2, . . . , on each compact set K.

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If both operators A and B are selfadjoint, then Theorem 8.3 ensures that a weak solution X of equation (8.3) exists and gives a norm estimate for X. If just one operator is selfadjoint, the following result is applicable. Theorem 8.4. Let H and K be Hilbert spaces. Suppose that A : D(A) → K is a selfadjoint operator with spectral measure PA and B : D(B) → H is a densely defined, closed linear operator such that σ(A) ∩ σ(B) = ∅. Let Y ∈ L(H, K). (i) Equation (8.3) has a strong solution if and only if there exists an operator valued measure M : B(σ(A)) → L(H, K) such that  dPA (ζ)(Y (ζI − B)−1 h), h ∈ H, (8.6) M (K)h = K

for each compact subset K of σ(A). The operator valued measure M exists if and only if     −1  dP (ζ)(Y (ζI − B) h) 0 and Y ∈ L(H), then for each h ∈ H the function ζ → Y (ζI − B)−1 h, ζ ∈ σ(A), is PA -integrable in

−1 τ H and X = is the unique strong solution L(H)⊗ σ(A) dPA (ζ)Y (ζI − B) of equation (8.3). Because B is selfadjoint, we can rewrite the solution X as an iterated integral    dPB (μ) dPA (ζ)Y X= σ(A) σ(B) ζ − μ with respect to the spectral measures PA , PB associated with the A and B. An application of the Fubini strategy sees the expression  dPA (ζ)Y dPB (μ) (8.9) X= ζ −μ σ(A)×σ(B) as a representation of the strong solution of the operator equation AX − XB = Y in the case that both A and B are selfadjoint operators. Integrals like (8.9) have been studied extensively in the case that Y ∈ L(H) is a Hilbert-Schmidt operator [15] and, more generally, when Y belongs to the Schatten ideal Cp (H) in L(H) for some 1 ≤ p < ∞, where they are called double operator integrals. In this section, the operator ideal Cp (H) consists of compact linear operators on H whose singular values belong to p . For a bounded linear operator T on a Hilbert space H, the expression  Iϕ (T ) = ϕ(λ, μ) F (dλ)T E(dμ) Λ×M

is a double operator integral if F is an L(H)-valued spectral measure on the measurable space (M, F ) and E is an L(H)-valued spectral measure on

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the measurable space (Λ, E). The function ϕ : Λ × M → C is taken to be uniformly bounded on Λ × M . In formula (8.9), ϕ(λ, μ) = (λ − μ)−1 , so that |ϕ(λ, μ)| is bounded by 1/δ for (λ, μ) ∈ σ(A) × σ(B) when the spectra σ(A) and σ(B) are a distance δ apart. The map T −→ Iϕ (T ), T ∈ C2 (H), is continuous into the space C2 (H) of Hilbert-Schmidt operators and Iϕ C2 (H) = ϕL∞ (Λ×M) , so that the map (E ⊗ F )C2 (H) : U −→ IχU , U ∈ E ⊗ F, is actually a countably additive spectral measure acting on C2 (H) and the equality  Iϕ = ϕ d(E ⊗ F )C2 (H) Λ×M

holds for all bounded measurable functions ϕ : Λ × M → C [15, Section 3.1]. The situation is more complicated if the space C2 (H) of Hilbert-Schmidt operators (with the Hilbert-Schmidt norm) is replaced by the Schatten ideal S = Cp (H) in L(H) for some 1 ≤ p < ∞ not equal to 2, or as in the case of formula (8.9), by S = L(H) itself, because the map U × V → IχU ×V , U ∈ E, V ∈ F , only defines a finitely additive set function (E ⊗ F )S acting on elements T ∈ S so that (E ⊗ F )S (U × V )T = E(U )T F (V ),

U ∈ E, V ∈ F .

For a bounded function ϕ : Λ × M → C, the double operator integral Iϕ may be viewed as a continuous generalisation of a classical Schur multiplier   μij αij eij , x = αij eij , (8.10) Tμ : x −→ i,j

i,j

for an infinite matrix μ = {μij } ∈ M, with respect to the matrix units eij corresponding to an orthonormal basis {hj } of H. If Pj denotes the orthogonal projection onto the linear space span{hj } for each j = 1, 2, . . . , then  μij (Pi ⊗ Pj )M Tμ = i,j

for the operators (Pi ⊗ Pj )M : x −→ Pi xPj acting on the infinite matrix x ∈ M for i, j = 1, 2 . . . . To be more precise, let S be a symmetrically normed ideal in L(H). The linear map JS : L(H) ⊗ L(H) → L(S) is defined by JS (A ⊗ B)T = AT B for T ∈ S and A, B ∈ L(H). In the language of [15, Section 4], the element JS (A ⊗ B) of L(S) is the transformer on S associated with A ⊗ B.

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Definition 8.3. Let (Λ, E) and (M, F ) be measurable spaces and H a separable Hilbert space. Let m : E → Ls (H) be an operator valued measure for the strong operator topology and n : F → H be a H-valued measure. Suppose that τ is the topology on the tensor product L(H) ⊗ H defined by formula (5.13). An (E ⊗ F)-measurable function ϕ : Λ × M → C is said to be (m ⊗ n) τ H if for every x, x , y  ∈ H, the function ϕ is integrable in E = L(H)⊗ integrable with respect to the scalar measure (mx, x ) ⊗ (n, y  ) and for τ H such that every A ∈ E ⊗ F , there exists ϕ.(m ⊗ n)(A) ∈ L(H)⊗  ϕ d((mx, x ) ⊗ (n, y  ))

ϕ.(m ⊗ n)(A), x ⊗ x ⊗ y  = A

for every x, x , y  ∈ H. τ H → H is the τ H and JE : L(H)⊗ If ϕ is (m ⊗ n)-integrable in L(H)⊗ multiplication map, then  ϕ d(mn) = JE (ϕ.(m ⊗ n)(A)), A ∈ E ⊗ F . A

The following observation is useful for treating double operator integrals. τ H defined by L(H, H⊗ π H) as Let τw be the relative topology on L(H)⊗ in Lemma 5.1 above. Proposition 8.1. Let (Λ, S) and (M, T ) be measurable spaces and let H, m : S → Ls (H) and n : T → H be as in Definition 8.3. If U ∈ C1 (H), then there exists a unique vector measure τH m ⊗ (U n) : S ⊗ T → L(H)⊗ σ-additive for the topology τw such that (m ⊗ (U n))(S × T ) = m(S) ⊗ (U n(T )) ∈ L(H) ⊗ H for each S ∈ S and T ∈ T . Consequently, every bounded (S⊗T )-measurable τ H and function ϕ : Λ × M → C is (m ⊗ (U n))-integrable in E = L(H)⊗  ϕ d(m(U n)) = JE (ϕ.(m ⊗ (U n))(A)), A ∈ S ⊗ T . A

Proof. If U is a trace class operator on H, then there exist orthonormal sets {φj }j , {ψj }j and a summable sequence {λj }j of scalars such that ∞ U h = j=1 λj φj (h, ψj ) for every h ∈ H. For each j = 1, 2, . . . , the total variation of the product measure (mh, k) ⊗ (n, ψj ) : S × T −→ (m(S)h, k) ⊗ (n(T ), ψj ),

S ∈ S, T ∈ T ,

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is bounded by m(Λ).n(M ).h.k for every h, k ∈ H. Here m and n denote the semivariation of m and n, respectively. Appealing to formula (5.13), the norm |(n, ψj )|(M )(m(S) ⊗ φj )τ of the finitely additive set function |(n, ψj )|(M )(m ⊗ φj ) evaluated at the set S ∈ S is |(n, ψj )|(M )(m(S) ⊗ φj )τ = |(n, ψj )|(M ) sup φj ⊗ (m(S)∗ k)π k≤1

≤ n(M )m(Λ). It follows that the finitely additive set function S × T −→ (n(S), ψj )(m(T ) ⊗ φj ),

S ∈ S, T ∈ T ,

admits a unique τw -countably additive extension Mj : S ⊗ T → Ls (H) ⊗ H whose semivariation with respect to the norm (5.13) is bounded by  m(Λ).n(M ) and m ⊗ (U n) = j λj Mj converges in L(H)⊗τ H uniformly on S ⊗ T . Corollary 8.1. Let (Λ, S) and (M, T ) be measurable spaces and H a separable Hilbert space. Let m : S → Ls (H) and n : T → Ls (H) be operator valued measures for the strong operator topology. Then there exists a unique operator valued measure JC1 (H) (m ⊗ n) : S ⊗ T → Ls (C1 (H), Ls (H)) such that JC1 (H) (m ⊗ n)(S × T ) = JC1 (H) (m(S) ⊗ n(T )),

S ∈ S, T ∈ T .

Proof. It is easy to check that for A ∈ S ⊗ T and T ∈ C1 (H), the formula '  & JC1 (H) (m ⊗ n)(A) T h = JE ((m ⊗ (T (nh)))(A)), h ∈ H, ' & defines a linear operator JC1 (H) (m ⊗ n)(A) T on H& whose operator norm ' is bounded by m(Λ).n(M )T C1(H) and A −→ JC1 (H) (m ⊗ n)(A) T , A ∈ S ⊗ T , is countably additive in L(H) for the strong operator topology for each T ∈ C1 (H). The following notation gives an interpretation of formula (8.9) in the case that Y belongs to the ideal S = Cp (H), 1 ≤ p < ∞ or S = L(H). The collection Cp (H) of trace class operators is a linear subspace of S in each case. Let (m ⊗ n)S be the finitely additive set function defined by (m ⊗ n)S (E × F ) = JS (m(E) ⊗ n(F )),

E ∈ E, F ∈ F ,

that is, (m ⊗ n)S : A → L(S) is finitely additive on the algebra A of all finite unions of product sets E × F for E ∈ E, F ∈ F .

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Suppose that the function ϕ : Λ × M → C is integrable with respect to the Ls (C1 (H), Ls (H))-valued measure JC1 (H) (m ⊗ n). If for E ∈ E and F ∈ F , the linear map   u −→ E×F

ϕ d[JC1 (H) (m ⊗ n)] u,

u ∈ C1 (H),

is the restriction to C1 (H) of a continuous linear map Tϕ ∈ L(S), then we ϕ d(m ⊗ n)S for Tϕ and we say that ϕ is (m ⊗ n)S -integrable if write E×F

ϕ d(m ⊗ n)S ∈ L(S) for every E ∈ E and F ∈ F . E×F

To check that the operator E×F ϕ d(m ⊗ n)S ∈ L(S) is uniquely defined, observe that C1 (H) is norm dense in Cp (H) for 1 < p < ∞. In the case S = L(H), the closure in the ultraweak topology σ(L(H), C1 (H)) can be taken, so that    ϕ d(m ⊗ n)L(H) = ϕ d(m ⊗ n)C1 (H) . E×F

E×F

Although (m ⊗ n)S is only a finitely additive set function, the L(S)valued set function  ϕ d(m ⊗ n)S , E ∈ E, F ∈ F , E × F −→ E×F

of an (m⊗ n)S -integrable function ϕ defines a finitely additive L(S)-valued set function on the algebra generated by all product sets E × F for E ∈ E and F ∈ F . Corollary 8.1 tells us that for an (m ⊗ n)C1 (H) -integrable function ϕ : Λ × M → C, the L(H)-valued set function  A −→ A

ϕ d(m ⊗ n)C1 (H) T,

A ∈ E ⊗ F,

is countably additive in the strong operator topology for each T ∈ C1 (H). The following result describes the situation for other operator ideals S. Proposition 8.2. Suppose that ϕ : Λ × M → C is an (m ⊗ n)S -integrable function. For each T ∈ S, the set function ϕ d(m ⊗ n)S T, E ∈ E, F ∈ F , E × F −→ E×F

isseparately σ-additive in the operator topology of L(H),  strong   that is,  ∞  ϕ d(m ⊗ n)S T = ϕ d(m ⊗ n)S T, F ∈ F , (∪∞ j=1 Ej )×F



E×(∪∞ j=1 Fj )

 ϕ d(m ⊗ n)S

T =

j=1

Ej ×F

j=1

E×Fj

 ∞ 

 ϕ d(m ⊗ n)S

T,

E ∈ E,

for all pairwise disjoint Ej ∈ E, j = 1, 2, . . . , and all pairwise disjoint Fj ∈ F , j = 1, 2, . . . .

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The following result was proved by M. Birman and M. Solomyak [15, Section 3.1]. Theorem 8.5. Let (Λ, E) and (M, F ) be measurable spaces and H a separable Hilbert space. Let P : E → Ls (H) and Q : F → Ls (H) be spectral measures. Then there exists a unique spectral measure (P ⊗Q)C2 (H) : E ⊗ F → L(C2 (H)) such that (P ⊗Q)C2 (H) (A) = (P ⊗ Q)C2 (H) (A) for every set A belonging to the algebra A of all finite unions of product setsE × F for E ∈ E, F ∈ F , and A

ϕ d(P ⊗ Q)C2 (H) =

A

ϕ d(P ⊗Q)C2 (H) ∈ L(C2 (H)),

A ∈ E ⊗ F,

for every bounded (E ⊗ F)-measurable function ϕ : Λ × M → C. Moreover, (P ⊗Q)C2 (H) (ϕ)L(C2 (H)) = ϕ∞ . For spectral measures P and   Q, the  formula  ϕ d(P ⊗ Q)S T = ϕ d(P ⊗ Q)S P (E)T Q(F ) Λ×M

E×F

holds for each E ∈ E, F ∈ F and T ∈ S, so it is only necessary to verify that

ϕ d(P ⊗ Q)S ∈ L(S) in order to show that ϕ is (P ⊗ Q)S -integrable. Λ×M The following observation gives an interpretation of formula (8.9) as a double operator integral. The Fourier transform of f ∈ L1 (R) is the

−iξx f (x) dx for ξ ∈ R. function fˆ : R → C defined by fˆ(ξ) = R e Theorem 8.6. Let H be a separable Hilbert space. Let P : B(R) → Ls (H) and Q : B(R) → Ls (H) be spectral measures on R. Let S = Cp (H) for some 1 ˆ 1 ≤ p < ∞ or S = L(H).

Suppose that f ∈ L (R) and ϕ(λ, μ) = f (λ − μ) for all λ, μ ∈ R. Then R×R ϕ d(P ⊗ Q)S ∈ L(S) and      ϕ d(P ⊗ Q)S  ≤ f 1 .   R×R

L(S)

Proof. For each T ∈ C1 (H), the set function E × F −→ P (E)T Q(F ), E, F ∈ B(R), is the restriction to all measurable rectangles of an L(H)valued the strong operatortopology and the integral  for  measure σ-additive e−it(λ−μ)t f (t) dt

ϕ d(P T Q) = R×R

R×R

 

R

= R = R

d(P T Q)(λ, μ)  −it(λ−μ)t e d(P T Q)(λ, μ) f (t) dt

R×R

e−itA T eitB f (t) dt

(8.11)

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converges as a Bochner integral in the strong operator topology to an element of the operator ideal C1 (H) of trace class operators. The interchange of integrals is verified scalarly. It follows that ϕ is a (P ⊗ Q)C1 (H) -integrable function and       ϕ d(P ⊗ Q) ≤ f 1. C1 (H)   R×R

L(C1 (H))

The corresponding bound for S = Cp (H) for 1 ≤ p < ∞ and S = L(H) follows by duality and interpolation, or directly from formula (8.11). The following corollary is a consequence of Theorem 8.3. Corollary 8.2. Let H be a separable Hilbert space and let A, B be selfadjoint operators with spectral measures PA : B(σ(A)) → Ls (H) and PB : B(σ(B)) → Ls (H), respectively. Let S = Cp (H) for some 1 ≤ p < ∞ or S = L(H). If the spectra of A and B are separated by a distance d(σ(A), σ(B)) = δ > 0, then σ(A)×σ(B) (λ − μ)−1 (PA ⊗ PB )S (dλ, dμ) ∈ L(S) and    (PA ⊗ PB )S (dλ, dμ)  π   . ≤     σ(A)×σ(B) λ−μ 2δ L(S)

In particular, equation (8.3) has a unique strong solution for Y ∈ S given by the double operator integral    dPA (λ)Y dPB (μ) (PA ⊗ PB )S (dλ, dμ) := Y, X= λ−μ λ−μ σ(A)×σ(B) σ(A)×σ(B) so that XS ≤

π 2δ Y

S .

Although the Heaviside function χ(0,∞) is not the Fourier transform of an L1 -function, the following result of I. Gohberg and M. Krein [52, Section III.6] holds, in case P = Q. Theorem 8.7. Let H be a separable Hilbert space. Let P : B(R) → Ls (H) and Q : B(R) → Ls (H) be spectral measures on R. Then  χ{λ>μ} d(P ⊗ Q)Cp (H) ∈ L(Cp (H)) R×R

for every 1 < p < ∞.

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The following recent result of F. Sukochev and D. Potapov [113] settled a long outstanding conjecture of M. Krein for the index p in the range 1 < p < ∞. Theorem 8.8. Let H be a separable Hilbert space. Let P : B(R) → Ls (H) and Q : B(R) → Ls (H) be spectral measures on R. Suppose that f : R → R is a continuous function for which the difference quotient + f (λ)−f (μ) , λ = μ, λ−μ ϕf (λ, μ) = 0 , λ = μ, is uniformly bounded. Then for every 1 < p < ∞,  ϕf d(P ⊗ Q)Cp (H) ∈ L(Cp (H)) R×R

and there exists Cp > 0 such that      ϕf d(P ⊗ Q)Cp (H)    R×R

Cp (H)

≤ Cp ϕf ∞ .

Such a function f is said to be uniformly Lipschitz on R and f Lip1 := ϕf ∞ . Corollary 8.3. Suppose that f : R → R is a uniformly Lipschitz function. Then for every 1 < p < ∞, there exists Cp > 0 such that f (A) − f (B)Cp (H) ≤ Cp f Lip1 A − BCp (H) for any selfadjoint operators A and B on a separable Hilbert space H. Proof. Let PA and PB be the spectral meaures of A and B, respectively, and suppose that A − BCp (H) < ∞. Then according to [15, Theorem 8.1] (see also [101, Corollary 7.2]), the equality   f (A) − f (B) = ϕf d(PA ⊗ PB )Cp (H) (A − B) R×R

holds and the norm estimate follows from Theorem 8.8. 8.3

Traces of double operator integrals

In this section, let (Λ, E) and (M, F ) be given measurable spaces, H a separable Hilbert space and P : E → Ls (H), Q : F → Ls (H) spectral measures. Let S = Cp (H) for some 1 ≤ p < ∞ or S = L(H). The

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Banach space L1 (P ) of P -integrable functions is isomorphic to the C*algebra L∞ (P ) of P -essentially bounded functions. The analagous result for (P ⊗ Q)S -integrable functions follows. Proposition 8.3. For an (E ⊗ F)-measurable function ϕ : Λ × M → C, let [ϕ] be the equivalence class of all functions equal to ϕ (P ⊗ Q)-almost everywhere. Let L1 ((P ⊗ Q)S ) = {[ϕ] : ϕ is (P ⊗ Q)S -integrable} with the pointwise operations of addition and scalar multiplication with the norm      ϕ d(P ⊗ Q)S  . [ϕ]S =   Λ×Λ

L(S)

1

Then [ϕ]∞ ≤ [ϕ]S and L ((P ⊗ Q)S ) is a commutative Banach *algebra under pointwise multiplication. If S = C2 (H), then L1 ((P ⊗ Q)S ) = L∞ (P ⊗ Q) is a commutative ∗ C -algebra. Furthermore, the Banach *-algebras L1 ((P ⊗ Q)C1 (H) ) = L1 ((P ⊗ Q)C∞ (H) ) = L1 ((P ⊗ Q)L(H) ), are isometric, where C∞ (H) is the uniformly closed subspace of L(H) consisting of compact operators on H. Remark 8.2. The analogy of double operator integrals with multiplier theory in harmonic analysis is fleshed out in [101, Example 2.13], as follows. If Λ is a locally compact abelian group with Fourier transform F , the spectral measure Q is defined by multiplication by characteristic functions on L2 (Λ) and P = F −1 QF is the spectral measure of the “momentum operator” on Λ, then for 1 < p < ∞, the space Mp (Λ) of Fourier multipliers on Lp (Λ) coincides with the commutative Banach *-algebra L1 (Pp ) for the finitely additive set function Pp : A → L(Lp (Λ)) defined as in [101, Example 2.13] by the spectral measure P acting on L2 (Λ). For example, when Λ = R,

the operator R sgn dPp ∈ L(Lp (R)) is the Hilbert transform for 1 < p < ∞. It is only in the case p = 2, that L1 (P2 ) = L∞ (P ). One might argue that multiplier theory in commutative harmonic analysis is devoted to the study of the commutative Banach *-algebra L1 (Pp ) for 1 < p < ∞. The analysis of the commutative Banach *-algebra L1 ((E ⊗ F )S ) for a general spectral measures E and F and symmetric operator ideal S has many applications to scattering theory and quantum physics [15]. The commutative Banach *-algebra L1 ((P ⊗ Q)L(H) ) is characterised by a result of V. Peller [104]. Theorem 8.9. Let ϕ : Λ × M → C be a uniformly bounded function. Then [ϕ] ∈ L1 ((P ⊗ Q)L(H) ) if and only if there exists a finite measure space

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(T, S, ν) and measurable functions α : Λ × T → C and β : M × T → C such that T α(·, t)L∞ (P ) β(·, t)L∞ (Q) dν(t) < ∞ and  ϕ(λ, μ) = α(λ, t)β(μ, t) dν(t), λ ∈ Λ, μ ∈ M. (8.12) T

Then norm of [ϕ] ∈ L1 ((P ⊗Q)L(H) ) with the representation (8.12) satisfies  −1 KG α(·, t)L∞ (P ) β(·, t)L∞ (Q) dν(t) ≤ [ϕ]L1 ((P ⊗Q)L(H) ) T    12   12          2 2 ≤ |α(·, t)| dν(t)  |β(·, t)| dν(t)  (8.13)   ∞  T  ∞  T L

(P )

L

(Q)

for Grothendieck’s constant KG . Moreover, [ϕ]L1 ((P ⊗Q)L(H) ) is the infimum of all numbers on the right hand side of the inequality (8.13) for which there exists a finite measure ν such that the representation (8.12) holds for ϕ. Formula (8.12) is to be interpreted in the sense that ϕ is a special representative of the equivalence class [ϕ] ∈ L1 ((P ⊗ Q)L(H) ). It is worthwhile to make a few remarks on the significance of formula (8.12) in order to motivate its proof below. If the functions α and β in the representation (8.12) have the property that t −→ α(·, t), t ∈ T , and t −→ β(·, t), t ∈ T , are strongly νmeasurable in L∞ (P ) and L∞ (Q), respectively, then the function t −→ α(·, t) ⊗ β(·, t), t ∈ T , is strongly measurable in the projective tensor prod π L∞ (Q), and uct L∞ (P )⊗  α(·, t)L∞ (P ) β(·, t)L∞ (Q) dν(t) < ∞, T

hence the function t −→ α(·, t) ⊗ β(·, t), t ∈ T , is Bochner integrable in π L∞ (Q), that is, [ϕ] ∈ L∞ (P )⊗ π L∞ (Q). However, it is only L∞ (P )⊗ assumed α is (E ⊗ S)-measurable and β is (F ⊗ S)-measurable, so this conclusion is unavailable. Let νP : E → [0, ∞) be a finite measure such that νP (E) ≤ P (E) for E ∈ E and limνP (E)→0 P h(E) = 0 for all h ∈ H with h ≤ 1. Such a measure exists by the Bartle-Dunford-Schwartz Theorem 1.2, or  −n (P en , en ) for some orthonormal basis {en }n more simply, νP = ∞ n=1 2 of H. Let νQ : F → [0, ∞) be a finite measure corresponding to Q. Then L∞ (P ) = L∞ (νP ) and L∞ (Q) = L∞ (νQ ).

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There is a bijective correspondence between elements [k] of the projec π L∞ (νQ ) ⊂ L∞ (νP ⊗ νQ ) and nuclear operative tensor product L∞ (νP )⊗ 1 ∞ tors Tk : L (νQ ) → L (νP ) such that for each f ∈ L1 (νQ ),  k(λ, μ)f (μ) dνQ (μ) (Tk f )(λ) = M

for νP -almost all λ ∈ Λ, in the sense that for functions with ∞ 

the kernel [k] =

∞

φj L∞ (νP ) ψj L∞ (νQ ) < ∞

j=1

φj ⊗ ψj correspond to the nuclear operator  ∞  (Tk f ) = φj f ψj dνQ , f ∈ L1 (νQ ). j=1

j=1

M

Nuclear operators between Banach space are discussed in [107]. In the case that H = 2 and P = Q are projections onto the standard basis vectors, then N×N ϕ d(P ⊗ Q)L( 2 ) is the classical Schur multiplier operator (8.10) and Grothendieck’s inequality ensures that L1 ((P ⊗Q)L( 2 ) ) = π ∞ , see Proposition 1.3 below and [112, Theorem 3.2]. In this case, ∞ ⊗ the measure ν in formula (8.12) is the counting measure on N and there is no difficulty with strong ν-measurability in an L∞ -space. The passage from the discrete case to the case of general spectral measures P and Q sees the nuclear operators from L1 (νQ ) to L∞ (νP ) replaced by 1-integral operators from L1 (νQ ) to L∞ (νP ), which leads to the Peller representation (8.12). 8.3.1

Schur multipliers and Grothendieck’s inequality

Grothendieck’s inequality (1.7) has already been employed to prove that for any measure μ, an L2 (μ)-valued measure has bounded K-semivariation in L2 (μ, K) for any Hilbert space K. If E is any L(H)-valued spectral measure and h ∈ H, the identity    ∞ ∞   2 2 E(fn )hH = E |fn | h, h n=1

n=1

ensures that the H-valued measure Eh has bounded 2 -semivariation in 2 = ⊕∞ 2 (H)—the Hilbert space tensor product H⊗ j=1 H with norm ∞ 2 2 u  . It follows from Theorem 2.2 that for any esu 2 (H) = j H j=1 2 2 sentially bounded functions f : Λ →  and g : M →  and h ∈ H, the

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2 -valued function f is (P h)-integrable in 2 (H) and the 2 -valued function g is (Qh)-integrable in 2 (H). Then there exist operator valued measures f ⊗ P : E → L(H, 2 (H)) and g ⊗ Q : F → L(H, 2 (H)) such that (f ⊗ P )(E)h = (f ⊗ (P h))(E),

E ∈ E, h ∈ H and

(g ⊗ Q)(F )h = (g ⊗ (Qh))(F ),

F ∈ F , h ∈ H.

There is a simple sufficient condition for ϕ ∈ L1 ((P ⊗ Q)L(H) ). Observe π H defined by first that the linear map J : 2 (H) ⊗ 2 (H) → H⊗ J(({φn }n ) ⊗ ({ψm }m )) =

∞ 

φj ⊗ ψj

j=1

has a continuous linear extension to a contraction J : C1 (2 (H)) → C1 (H) corresponding to taking the trace in the discrete index. The formula (((f ⊗P )⊗(g⊗Q))C1 (H) (E ×F ))(h⊗k) := (((f ⊗(P h))(E))⊗(g⊗(Qk))(F )) for h, k ∈ H, E ∈ E and F ∈ F , defines a finitely additive set function ((f ⊗ P ) ⊗ (g ⊗ Q))C1 (H) with values in L(C1 (H), C1 (2 (H))), because C1 (K) can be identified with π K for any Hilbert space K. Moreover, K⊗ ((f ⊗ P ) ⊗ (g ⊗ Q))C1 (H) (Λ × M )L(C1 (H),C1 ( 2 (H))) ≤ f ∞.g∞ . Then the operator  (f, g) d(P ⊗ Q)C1 (H) = J[((f ⊗ P ) ⊗ (g ⊗ Q))C1 (H) (Λ × M )] Λ×M

is an element of L(C1 (H)), that is, ϕ = (f, g) belongs to L1 ((P ⊗Q)C1 (H) ) = L1 ((P ⊗ Q)L(H) ) and we have the representation ϕ(λ, μ) =

∞ 

fn (λ)gn (μ),

(8.14)

n=1

∞ ∞ where P -ess.sup{ n=1 |fn |2 } < ∞ and Q-ess.sup{ n=1 |gn |2 } < ∞. Moreover, the bound       ∞ ∞            ϕ d(P ⊗ Q)L(H)  ≤ |fn |2  . |gn |2        ∞ Λ×M L(L(H)) ∞ n=1

L

(P )

n=1

L

(Q)

(8.15) holds. Alternatively, for each T ∈ L(H), the linear operator   ϕ d(P ⊗ Q)L(H) T ∈ L(H) Λ×M

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can be realised as the operator associated with the bounded sesquilinear form ∞  (T Q(gn )h, P (fn )k), (h, k) −→ n=1

see [15, Theorem 4.1]. A remarkable consequence of Grothendick’s inequality is that for ϕ ∈ L1 ((P ⊗Q)L(H) ), Peller’s representation (8.12) is necessary (νP ⊗νQ )-almost everywhere. The analysis of G. Pisier [112] leads the way. 8.3.2

Schur multipliers on measure spaces

We first note that for any choice of finite measures νP , νQ equivalent to P and Q respectively, the Banach algebra L1 ((P ⊗ Q)L(H) ) is isometrically isomorphic to the set of multipliers of the projective tensor product π L2 (νQ ), that is, [ϕ] ∈ L1 ((P ⊗ Q)L(H) ) if and only if for every L2 (νP )⊗ π L2 (νQ ), the function ϕ.h is equal (νP ⊗ νQ )-a.e. to an el[h] ∈ L2 (νP )⊗ 2 π L2 (νQ ) and [ϕ]L1 ((P ⊗Q)L(H) is equal to the norm of ement of L (νP )⊗ π L2 (νQ ), on L2 (νP )⊗ π L2 (νQ ). the linear map [h] −→ [ϕ.h], [h] ∈ L2 (νP )⊗  are another pair of such equivalent measures, then the If νP and νQ 8 operator of multiplication by dνP /dνP is a unitary map 8 from L2 (νP ) to L2 (νP ) and similarly for νQ , so that multiplication by dνP /dνP ⊗ . 2  /dν π L2 (νQ ) onto dνQ Q is an isometric isomorphism from L (νP )⊗  π L2 (νQ L2 (νP )⊗ ).

Proposition 8.4. Let νP , νQ be finite measures equivalent to the spectral measures P , Q respectively. Then L1 ((P ⊗ Q)L(H) ) is isometrically isomorphic to the set of multipliers of the projective tensor product π L2 (νQ ) and the identity L2 (νP )⊗ [ϕ]L1 ((P ⊗Q)L(H) ) =

sup hH ≤1,gH ≤1

[ϕ]L2 ((P h,h))⊗  π L2 ((Qg,g))

(8.16)

holds.  Proof. Let {hn }n be a sequence of vectors in H with n hn 2 < ∞ such that {P (E)hn : n = 1, 2, . . . } is an orthogonal set of vectors in H for each E ∈ E. Such a sequence of vectors can always be manufactured by taking  any vectors ξn ∈ H with n ξn 2 < ∞ and for a measure νP equivalent to * P , the sets Λn where d(P ξn , ξn )/dνP > 0. Then hn = P (Λn \ m 0 such that ϕn  ∞ ⊗  π ∞ ≤ M for all n = 1, 2, . . . . Then tr(Pn∗ Tϕn Qn U ) = tr(T(En ⊗Fn )ϕ U ) = tr(Tϕ En U Fn ), and taking n → ∞, the martingale convergence theorem shows that the bound |tr(Tϕ U )| ≤ M U L(L∞ ,L1 ) holds for every finite rank operator U : L∞ (νP ) → L1 (νQ ). It follows from [36, Theorem 6.16] that Tϕ belongs to the Banach ideal I1 (L1 (νQ ), L∞ (νP )) of 1-integral operators from L1 (νQ ) to L∞ (νP )—see Example 1.4 above. Because L∞ (νP ) is a dual space, [36, Corollary 5.4] ensures that Tϕ enjoys the factorisation L1 (νQ ) T1 ↓ L∞ (ν)

Tϕ −→

−→ j

L∞ (νP ) ↑ T2 L1 (ν)

for some bounded linear operators T1 and T2 and finite measure space (T, S, ν). The given factorisation also follows by the original 1954 Grothendieck argument with the choice E = L1 (νQ ), F = L1 (νP ) in [123, Section IV.9.2]. Every bounded linear operator u from L1 (η1 ) to L∞ (η2 ) is an integral operator with a bounded kernel because f ⊗g → uf, g defines a continuous π L1 (η2 ) ≡ L1 (η1 ⊗ η2 ) (see [63, Lemma 2.2] linear functional on L1 (η1 )⊗ for a compactness argument), so there exist bounded measurable functions α : Λ × T → C and β : M × T → C such that  β(μ, t)f (λ)dνQ (λ), f ∈ L1 (νQ ), (T1 f )(t) = M (T2 g)(λ) = α(λ, t)g(t)dν(t), g ∈ L1 (ν). T

The representation (8.12) and the associated bounds follow if we can take M = KG [ϕ]L1 ((P ⊗Q)L(H) ) . We know from the bounds (1.3) that ϕn  ∞ ⊗  π ∞ ≤ KG Mϕn L(L( 2 )) = KG ϕn  ∞ ⊗ γ

2

∞ .

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The norm γ2 defined on ∞ ⊗ ∞ is the norm of factorisation through a Hilbert space. For any bounded linear operator u : X → Y between Banach spaces X and Y , γ2 (u) = inf{u1 , u2} where the infimum runs over all Hilbert spaces H and all possible factorisations u2

u1

u : X −→ H −→ Y of u through H with u = u1 ◦ u2 . Taking X = L1 (νQ ) and Y = L∞ (νP ), the bound (8.15) says that ϕL1 ((P ⊗Q)L(H) ) ≤ ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

γ2 L∞ (νQ ) of L∞ (νP ) ⊗ L∞ (νQ ) with respect to the completion L∞ (νP )⊗ ∞ in the norm φ → γ2 (Tφ ), φ ∈ L (νP ) ⊗ L∞ (νQ ). The norm estimates ϕn  ∞ ⊗ γ

2



= (En ⊗ Fn )ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

≤ ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

follow from the definition of γ2 and the contractivity of the conditional expectation operators En , Fn . According to Proposition 8.4, the norm of the linear operator Mϕ : C1 (L2 (νQ ), L2 (νP )) → C1 (L2 (νQ ), L2 (νP )) π L2 (νQ ) is equal to associated with multiplication by ϕ on L2 (νP )⊗ [ϕ]L1 ((P ⊗Q)L(H) ) = [ϕ]L1 ((P ⊗Q)C1 (H) ) . The equality ϕL∞ (νP )⊗  γ L∞ (νQ ) = Mϕ L(L(L2 (νQ ),L2 (νP ))) is proved 2 in [129, Theorem 3.3] using complete boundedness techniques, but this can be established in a more elementary way by noting that if [ϕ] ∈ L1 ((P ⊗ Q)C1 (H) ), then the Martingale Convergence Theorem ensures that M(En ⊗Fn )ϕ → Mϕ in the strong operator topology of   L C1 (L2 (νQ ), L2 (νP )), C1 (L2 (νQ ), L2 (νP )) as n → ∞ and also (En ⊗ Fn )ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

−→ ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

as n → ∞. Then Mϕ  = supn M(En ⊗Fn )ϕ)  by duality. The equality (En ⊗ Fn )ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

= M(En ⊗Fn )ϕ) L(L(L2 (νQ ),L2 (νP )))

follows for each n = 1, 2, . . . from Proposition 1.2 by replacing ei ⊗ej in (iii) by χAi ×Bj for i, j = 1, 2, . . . . The final assertion of Theorem 8.9 follows

from the equality ϕL∞ (νP )⊗ γ

2

L∞ (νQ )

= Mϕ L(L(L2 (νQ ),L2 (νP ))) .

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Remark 8.3. a) The original proof of V. Peller [104], [63, Theorem 2.2] factorises the finite rank operator U : L∞ (νP ) → L1 (νQ ) instead, so the 2 appears in place of KG in the bound associated with (8.12). constant KG ˜ 1 (νQ ) be the closure of the linear space of all k ∈ b) Let L1 (νP )⊗L L1 (νP ) ⊗ L1 (νQ ) in the uniform norm of the space of operators Tk ∈ L(L∞ (νQ ), L1 (νP )) corresponding to the compact linear operators from L∞ (νQ ) to L1 (νP ). By [123, Section IV.9.2], the function α ⊗ β in formula (8.12) is ν-integrable in the space of 1-integral operators ˜ 1 (νQ )) I1 (L1 (νQ ), L∞ (νP )) ≡ (L1 (νP )⊗L

and ϕ = T α ⊗ β dν. c) The proof above shows that operator Tϕ : L1 (νQ ) → L∞ (νP ) is (strictly) 1-integral in the sense of [36] and [123, Section IV.9.2] if and only if [ϕ] ∈ L1 ((P ⊗ Q)L(H) ). The reason that [ϕ]L∞ (νP )⊗  π L∞ (νQ ) = ∞ for 1 some [ϕ] ∈ L ((P ⊗ Q)L(H) ), that is, the function α ⊗ β associated with the π L∞ (νQ ) so that representation (8.12) fails to be ν-integrable in L∞ (νP )⊗ 1 ∞ Tϕ : L (νQ ) → L (νP ) thereby fails to be a nuclear operator, is that the vector measure E −→ uχE associated with a continuous linear map u from L1 to L∞ has a weak*-density, but not necessarily a strongly measurable density in L∞ . For any u ∈ C1 (H) and ϕ ∈ L1 ((P ⊗ P )L(H) ), the operator   Mϕ u = ϕ d(P ⊗ P )C1 (H) u Λ×Λ

is trace class. Moreover, the expression E −→ tr(uP (E)), E ∈ E, is a complex measure μu on the σ-algebra E such that |μu |  νP . As indicated in [15, Section 9.1], the identity  ϕ(λ, λ) dμu (λ) (8.18) tr(Mϕ u) = Λ

holds. In the case that u : H → H is is a finite rank operator, together with the polarisation, the bound (8.16) shows that the operator Tϕ : L2 (μu ) → L2 (μu ) with integral kernel ϕ is trace class and TϕC1 (L2 (μu )) ≤ 16[ϕ]L1(P ⊗Q)L(H) uC1 (H) . The same bound holds for all u ∈ C1 (H). The identity |ψ|2 .νP = (P (UP ψ), (UP ψ)),

ψ ∈ L2 (νP ),

ensures that tr(Mφ1 ⊗φ2 u) = tr(Tφ1 ⊗φ2 ) for Tφ1 ⊗φ2 ∈ C1 (L2 (μu )) with u ∈ C1 (H) and φ1 , φ2 bounded on Λ. Then the equality tr(Mϕ u) = tr(Tϕ )

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γ2 L∞ (νP ). holds because both sides are continuous for ϕ ∈ L∞ (νP )⊗ ∞ γ2 L∞ (νP ) and there The representation (8.14) converges in L (νP )⊗ exists a set Λ0 of full νP -measure such that ∞ 

ϕ(λ, μ) =

fn (λ)gn (μ)

n=1

for all λ, μ ∈ Λ0 where the right-hand sum converges absolutely. The expression above constitutes a distinguished element of the equivalence class [ϕ]. Consequently, formula (8.18) is valid because tr(Tϕ ) = =

∞ 

tr(Tfn ⊗gn )

n=1 ∞   n=1

 =

Λ

Λ

fn (λ)gn (λ) dμu (λ)

ϕ(λ, λ) dμu (λ).

From the point of view of our study of traces in Chapter 3, for any Lusin μP filtration F = Ek k of Λ, for each k = 1, 2, . . . , the conditional expectation operators Ek : f −→ E(f |Ek ) with respect to the σ-algebra Ek and the finite measure νP have the property that ∞ 

|fn .gn − Ek (fn ).Ek (gn )|

n=1



∞ 

|(fn − Ek (fn )).gn | +

n=1





 12  |fn − Ek (fn )|

2

n=1



∞ 

.

 12  |Ek (fn )|2

n=1

→0

|Ek (fn ).(gn − Ek (gn ))|

n=1

∞ 

+

∞ 

.

∞ 

 12 |gn |

2

n=1 ∞ 

 12

|gn − Ek (gn )|2

n=1

νP -almost everywhere as k → ∞,

by the Martingale Convergence Theorem. Consequently, setting ϕ˜ = lim (Ek ⊗ Ek )ϕ k→∞

wherever the limit exists, the equality ϕ(λ, ˜ λ) = ϕ(λ, λ) holds for νP -almost all λ ∈ Λ, so that formula (8.18) is also a consequence of [19, Theorem 3.1].

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Remark 8.4. There is a representative function ϕ of the equivalence class [ϕ] that is continuous for the so-called ω-topology of [78, Proposition 9.1], so formula (8.18) is also a consequence of Theorem 8.9. In fact, Peller’s representation (8.12) can be deduced directly from Proposition 1.3 by employing the ω-continuity of ϕ rather than the Martingale Convergence Theorem, see [78, Remark on p. 139]. 8.4

The spectral shift function

The following perturbation formula of Birman and Solomyak [15, theorem 8.1] was mentioned in the proof of Corollary 8.3. The operator ideal S is taken to be Cp (H) for 1 ≤ p < ∞ or L(H) for a given Hilbert space H. Theorem 8.10. Let H be a separable Hilbert space and let A and B be selfadjoint operators with the same domain such that A − B ∈ S. Let PA : B(R) → Ls (H) and PB : B(R) → Ls (H) be the spectral measures on R associated with A and B, respectively. Suppose that f : R → R is a continuous function for which the difference quotient + f (λ)−f (μ) , λ = μ, λ−μ ϕf (λ, μ) = 0, λ = μ, is uniformly bounded and ϕf ∈ L1 ((PA ⊗ QB )S ). Then  ϕf d(PA ⊗ PB )S ∈ L(S) R×R

and



 f (A) − f (B) =

R×R

ϕf d(PA ⊗ PB )S (A − B).

If S = C1 (H), then we would like to calculate the trace of f (A) − f (B). The method of the preceding section is unavailable with different spectral measures PA , PB so we can try to invoke the Daletskii-Krein formula [15, Equation (9.10)]. For a sufficiently smooth function f , this takes the form   1  ϕf (λ, μ) d(PA(t) ⊗ PA(t) )C1 (H) (A − B) dt f (A) − f (B) = 0

R×R

with A(t) = B + t(A − B), 0 ≤ t ≤ 1, and ϕf (λ, λ) = f  (λ), λ ∈ R. At each point 0 ≤ t ≤ 1, the same spectral measure PA(t) is involved, so from formula (8.18), we can expect that  tr(f (A) − f (B)) = f  (λ) dΞ(λ) R

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1 for the complex measure Ξ : E −→ 0 tr(V PA(t) (E)) dt, E ∈ B(R), with V = (A − B) ∈ C1 (H). It turns out that Ξ is absolutely continuous with respect to Lebesgue measure on R from which the formula  tr(f (A) − f (B)) = f  (λ)ξ(λ) dλ (8.19) R

is obtained. The function ξ : R → C is Krein’s spectral shift function. We now turn to establishing the validity of formula (8.19) for a restricted class of functions f . Better results are known, for example from [113], but our purpose is to describe applications of singular bilinear integrals such as double operator integrals to problems in the perturbation theory of linear operators. The approach of K. Boyadzhiev [17] best suits our purpose. Setting V = A − B ∈ C1 (H), we first note that eisA(t) − eisB ∈ C1 (H) for each s ∈ R and 0 ≤ t ≤ 1 because the perturbation series eisA(t) = eisB  t  ∞  + (is)n ··· n=1

0

0

s2

eisB(s−sn ) V · · · eisB(s2 −s1 ) V eisBs1 ds1 · · · dsn

converges in the norm of C1 (H) and t −→ eisA(t) −eisB is norm differentiable in C1 (H). Moreover, eisA(t) − eisB C1 (H) ≤ (e|s|V C1 (H) − 1).

(8.20)

The following result is straightforward but it depends on some measure theoretic facts. It establishes that Ξ is a complex measure. Lemma 8.2. The function t −→ PA(t) (E)h, t ∈ [0, 1], is strongly measurable in H for each h ∈ H and E ∈ B(R). There exists an operator valued measure M : B([0, 1]) ⊗ B(R) → Ls (H), σ-additive for the strong operator topology, such that the equality  (M (X × Y )h, h) = (PA(t) (Y )h, h) dt, X ∈ B([0, 1]), Y ∈ B(R), X

holds for each h ∈ H. For each V ∈ C1 (H), the set function E −→ tr(V M (E)), E ∈ B([0, 1]) ⊗ B(R), is a complex measure and we have  1 tr(V M ([0, 1] × Y )) = tr(V PA(t) (Y )) dt. (8.21) 0

Proof. If f = μ ˆ is the Fourier transform of a finite measure μ, then  (PA(t) h)(f ) = e−iξA(t) h dμ(ξ) R

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as a Bochner integral and by dominated convergence t −→ (PA(t) h)(f ), 0 ≤ t ≤ 1, is continuous in H for each h ∈ H. By a monotone class argument, t −→ (PA(t) h)(f ), 0 ≤ t ≤ 1, is strongly measurable for all bounded Borel measurable functions f . For each h ∈ H, the set function (M h, h) is nonnegative and finitely additive and the algebra A generated by product sets X × Y for X ∈ B([0, 1]) and Y ∈ B(R), so |(M (A)h, h)| ≤ h21 , A ∈ A. The set function (M h, h) : A → [0, h2 ] is separately countably additive with respect to Borel sets, so it is inner regular with respect to compact product sets and so, countably additive (note that countable additivity may fail without inner-regularity). Denoting the extended measure by the same symbol, |(M (E)h, h)| ≤ h21 for all E ∈ B([0, 1]) ⊗ B(R). The H-valued measure M h is weakly countable additive by polarity and so norm countably additive by the Orlicz-Pettis Theorem. For each V ∈ C1 (H) and orthonormal basis {hj }j of H, the bound ∞  |(V M (E)hj , hj )| ≤ 4V C1 (H) , E ∈ B([0, 1]) ⊗ B(R) j=1

holds and tr(V M ([0, 1] × Y )) =

∞ 

(V M ([0, 1] × Y )hj , hj )

j=1

 =

∞ 1

0 j=1

(V PA(t) (Y )hj , hj ) dt

by the Beppo Levi convergence theorem, because ∞  |(V PA(t) (Y )hj , hj )| ≤ 4V C1 (H) ,

0 ≤ t ≤ 1,

j=1

so equation (8.21) holds. An application of Fubini’s Theorem for disintegrations of measures [16, Section 10.6] shows that  1    −iλs −iλs e d(M h, h)(t, λ) = e (PA(t) h, h) dt (dλ) [0,1]×R



R 1



= 0

 =

0

1

R

0

e−iλs (PA(t) h, h)(dλ) dt

(e−isA(t) h, h) dt

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for each h ∈ H. The identity  1    1 −iλs e (V PA(t) h, h) dt (dλ) = (V e−isA(t) h, h) dt R

0

0

follows for each h ∈ H by polarisation. Because  1 ∞  1  tr(V PA(t) (E)) dt = (V PA(t) (E)hj , hj ) dt, Ξ(E) = 0

0

j=1

E ∈ B(R),

for any orthonormal basis {hj }j of H, the Fourier transform of the measure Ξ is  e

−iλs



1

dΞ(λ) = 0

R



tr(V e−isA(t) ) dt 1

d tr(e−isA(t) ) dt dt 0 tr(e−isA − e−isB ) =i . s ˇ of the uniformly We need to establish that the inverse Fourier transform Φ bounded, continuous function =i

s−1

tr(e−isA − e−isB ) , s ∈ R \ {0}, Φ(0) = tr(V ), s ˇ is the spectral shift function. Clearly, the belongs to L1 (R). Then ξ = Φ value of Φ at 0 is irrelevant. It suffices to show that there exists ξ ∈ L1 (R) such that   μ(t) dt = ξ(t)ˆ μ(t) dt μ(Φ) = 2π ξ(t)ˇ Φ : s −→ i

−1



R



R

with μ ˇ (t) = (2π) dμ(s) and μ ˆ(t) = R e−ist dμ(s), t ∈ R, for every Re ˇ = ξ as elements of the space S  of finite positive measure μ, because then Φ Schwartz distributions on R [121, Definition 7.11]. So, we consider the class ˆ and f (0) = 0 and consequently of functions f : R → R for which f  = μ tr(f (A) − f (B)) = (2π)−1 μ(Φ). ist

Theorem 8.11. Let H be a separable Hilbert space and let A and B be selfadjoint operators with the same domain such that A − B ∈ C1 (H). Then there exists a function ξ ∈ L1 (R) such that  f  (λ)ξ(λ) dλ (8.22) tr(f (A) − f (B)) = R

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for every function f : R → C for which there exists a finite positive Borel measure μ on R such that  −isx e −1 dμ(s), x ∈ R. f (x) = i s R Furthermore, ξ possesses the following properties.

a) tr(A − B) = R ξ(λ) dλ. b) ξ1 ≤ A − BC1 (H) . c) If B ≤ A, then ξ ≥ 0 a.e. d) ξ is zero a.e. outside the interval (inf(σ(A) ∪ σ(B)), sup(σ(A) ∪ σ(B))). Proof. The following proof is adapted from [17]. The estimate f (A) − f (B)C1 (H) ≤ μ(R)A − BC1 (H)

(8.23)

follows from the bound (8.20) and the calculation  −isA e − e−isB i f (A) − f (B) = dμ(s) 2π R s obtained from an application of Fubini’s theorem with respect to PA ⊗ μ and PB ⊗ μ on R × [ , ∞) for > 0. Then  1 Φ dμ. tr(f (A) − f (B)) = 2π R An expression for the spectral shift function ξ may be obtained from Fatou’s Theorem [120, Theorem 11.24]. Suppose that ν is a finite measure on R  dν(λ) 1 , z ∈ C \ R, φν (z) = 2πi R λ − z is the Cauchy transform of ν. Then ν is absolutely continuous if  e−iξx (φν (x + i0+) − φν (x + i0−)) dx, ξ ∈ R. νˆ(ξ) = R

The function x −→ φν (x + i0+) − φν (x + i0−) defined for almost all x ∈ R is then the density of ν with respect to Lebesgue measure. For ν = Ξ, if the representation tr(e−isA − e−isB ) Φ(s) = i s    1 1 −isx = lim e tr(V (A + tV − x − i )−1 − →0+ 0 2πi R  −1 V (A + tV − x + i ) ) dt dx

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ˇ has the representation were valid, we would expect that ξ = Φ  tr(e−ixA − e−ixB ) 1 lim dx, s ∈ R, eisx−|x| ξ(s) = 2πi →0+ R x $    % 1 A − sI B − sI = lim tr arctan − arctan , →0+ π where the arctan function may be expressed as  ist e − 1 −|s| 1 e ds, t ∈ R. arctan t = 2i R s For the function defined by $    % A − xI B − xI 1 − arctan h(x, y) = tr arctan π y y we have the bounds       B − xI  A − xI  − arctan π|h(x, y)| ≤  arctan   y y C1 (H)

(8.24)

(8.25)

1 A − BC1 (H) , y from the bound (8.23) and the representation (8.25). Rewriting % $ isA  − eisB 1 e −ixs−y|s| ds h(x, y) = e tr 2πi R s using (8.25), it follows that h(x, y) is harmonic in the upper half-plane {(x, y) : x ∈ R, y > 0}. We first look at the case that A − B = α(·, w)w for α > 0 and w ∈ H, w = 1, so that A is a rank one perturbation of the bounded selfadjoint operator B. If we set B−x A−x , Y = 2 arctan , X = 2 arctan y y then 2πh = tr(X − Y ). The formula tr log(eiX e−iY ) = itr(X − Y ) follows from the Baker-Campbell-Hausdorff formula for large y > 0, see [17, Lemma 1.1]. Let TA = e−iX , TB = e−iY . Then for z = x + iy, spectral theory gives ≤

TA = (A − zI)(A − zI)−1 = I + 2iy(A − zI)−1 , TB = (B − zI)(B − zI)−1 = I + 2iy(B − zI)−1 . Our aim is to compute tr log(U ) for the unitary operator U = TA∗ TB . Because U − I = TA∗ TB − TB∗ TB = (TA∗ − TB∗ )TB = −i2y[(A − zI)−1 − (B − zI)−1 ]TB ,

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we obtain U = I + i2y(A − zI)−1 (A − B)(B − zI)−1 . Substituting A − B = α(·, w)w gives U = I + i2yα(·, (B − zI)−1 w)(A − zI)−1 w. The vector (A − zI)−1 w is an eigenvector for the unitary operator U with eigenvalue 1 + i2yα((A − zI)−1 w, (B − zI)−1 w) which can be expressed as ei2πθ(x,y) for a continuous function θ in the upper half plane such that 0 < θ < 1. Consequently, for large y > 0, i2πθ = tr log(U ) = itr(X − Y ) = i2πh. Then θ is harmonic for large y > 0 so it is harmonic on the upper half plane and it is equal to h there, so 0 < h < 1. By Fatou’s Theorem, the boundary values ξ(x) = limy→0+ h(x, y) are defined for almost all x ∈ R and satisfy  ξ(t) dt = ξ1 ≤ A − BC1 (H) lim πyh(x, y) = y→∞

R

for every x ∈ R, so in the case that A − B has rank one, formula (8.24) is valid. ∞ For an arbitrary selfadjoint perturbation V = j=1 αj (·, wj )wj with ∞ 1 |α | = A − B < ∞, the function ξ ∈ L (R) may be defined j n C1 (H) j=1 n in a similar fashion for An = B + j=1 αj (·, wj )wj , n = 1, 2, . . . , so that ˇ ξn → ξ in L1 (R) as n → ∞ from which it verified that ξ = Φ. ˇ obtained above may be viewed as the Fourier The representation ξ = Φ transform approach. In the case of a rank one perturbation V = α(·, w)w, the Cauchy transform approach is developed by B. Simon [126] with the formula  ξ(λ) dλ tr((A − zI)−1 − (B − zI)−1 ) = − 2 R (λ − z) for z ∈ C \ [a, ∞) for some a ∈ R, established in [126, Theorem 1.9] by computing a contour integral. Here the boundary value ξ(x) = limy→0+ h(x, y) is expressed as ξ(x) =

1 Arg(1 + αF (λ + i0+)) π

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for almost all x ∈ R with respect to the Cauchy transform  d(PB w, w)(λ) , z ∈ C \ (−∞, a). F (z) = λ−z R The Cauchy transform approach is generalised to type II von Neumann algebras in [10]. Many different proofs of Krein’s formula (8.22) are available for a wide class of functions f , especially in a form that translates into the setting of noncommutative integration [10, 105, 114]. As remarked in [15, p. 163], an ingredient additional to double operator integrals (such as complex function theory) is needed to show that the measure Ξ is absolutely continuous with respect to Lebesgue measure on R. Krein’s original argument uses perturbation determinants from which follows the representation Det(S(λ)) = e−2πiξ(λ) for the scattering matrix S(λ) for A and B [136, Chapter 8].

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[135] J. Weidman, Integraloperatoren der spurklasse, Math. Ann. 163 (1966), 340–345. [136] D. Yafaev, Mathematical Scattering Theory: General Theory, Providence, RI, Amer. Math. Soc., 1992. Scattering Theory: Some Old and New Problems Lecture Notes in [137] Mathematics 1735, Berlin, Springer, 2000. [138] A.C. Zaanen, Riesz Spaces II, North Holland, Amsterdam, New York, Oxford, 1983.

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Index

m-equivalent, 12 m-integrable, 10 m-null, 12

completely separated, 116 conditional expectation, 85, 105 conditional probability, 104 convolution, 73 cross-norm, 16 Cwikel-Lieb-Rozenbljum inequality, 94

approximation property, 44, 55, 59, 130, 133 Banach function space, 59, 75, 83 Banach lattice, 27, 32 Banach space type 2, 117 UMD, 119 Bartle-Dunford-Schwartz Theorem, 9 Bessel potential, 78 bilinear admissible, 55 bilinear form integral, 20 separately continuous, 19 bilinear integral regular, 40 bilinear integration, 3 regular, 3 singular, 3 Birman-Schwinger Principle, 189 Bochner integrable function, 14, 56 pth, 14 Brownian motion, 112–114

Dedekind complete, 27 density, 103 distribution, 102 function, 102 joint, 103 double operator integral, 5, 202 dual pair, 6 dynamical flow, 146 events, 101 independent, 104 evolution process, 144, 169 σ-additive, 145 expectation, 103 Feynman-Kac formula, 143, 147, 168 filtration, 83, 91, 106, 111 Lusin, 5, 91 Lusin μ-, 92, 216 standard, 111 strict Lusin, 91 strict Lusin μ-, 92 financial derivatives, 109

Calkin Theorem, 94 Carath´eodory-Hahn-Kluv´ anek Theorem, 10 237

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financial markets, 109 Fourier multiplier, 210

multiplicative operator functional, 168

gambling strategy, 108 Gaussian random measure, 43, 115 Grothendieck’s inequality, 5, 23, 29, 31

Nikodym Boundedness Theorem, 9 null set, 7

Hamiltonian, 123 Hardy-Littlewood maximal operator, 74 hitting time, 109 independence, 105 integral kernel, 72, 84 Krein’s spectral shift function, 5, 222 lattice ideal, 71, 79, 86 Lebesgue’s differentiation theorem, 74 Lidskii’s equality, 72 Liouville measure, 146 Lippmann-Schwinger equations, 126, 138 locally convex space, 2 complete, 10 quasicomplete, 10 Markov chain, 168 martingale, 107 Martingale Convergence Theorem, 85, 88 maximal function, 86 Hardy-Littlewood, 74 measurable space, 6 measure modulus, 28, 53 Radon, 90 regular conditional, 85, 91 scalar, 7 variation, 8 measure space, 6 complete probability, 111 probability, 101 multiplicative functional, 157

operator 1-integral, 20, 212, 217 absolute integral, 75, 84 compact, 95 conditional expectation, 96 free Hamiltonian, 131 generalised Carleman, 64 hermitian positive, 4, 95 Hilbert-Schmidt, 61, 79, 94, 95, 98, 203 Hille-Tamarkin, 60 integral, 95 Laplacian, 171 modulus, 27, 84 nuclear, 55, 59, 62, 95, 212 positive, 83 regular, 84 strictly 1-integral, 21 trace class, 72 traceable, 4 Volterra integral, 95 operator ideal, 72, 79 Marcinkiewicz, 94 operator valued measure, 7 Optional Stopping Theorem, 111 Orlicz-Pettis Theorem, 7 partition refinement of a, 84 Pettis integrable function, 13, 56 Pettis’s Measurability Theorem, 150 phase space, 146, 171 positive operator, 27 positive operator valued measure, 33 potential, 123 Coulomb, 131 short-range, 138 uniformly bounded, 138 process (S, Q)-, 144, 169

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239

Index

Feller, 145 Markov, 145 progressively measurable, 146, 150 stochastic quasi-continuous, 156 projective tensor product, 17 Radon-Nikodym Theorem, 103, 105 random variables, 102 random evolution, 32, 167 random variables adapted, 107 discrete, 104 independent, 113 normally distributed, 103, 113 uncorrelated, 105 rational central planning, 109 resolvent, 196 ring, 6 δ-, 6, 56, 116 sample space, 101 scalarly integrable, 13 scattering theory stationary, 124, 125 time-dependent, 123 Schatten ideal, 202 Schur multiplier, 22, 203 Selberg trace formula, 94 semi-algebra, 143 semiclassical approximation, 171 semigroup of operators, 144 C0 -, 144 dominated, 146 Feller, 156 modulus, 146 semimartingale, 121 semivariation, 8, 35 E-, 35 X-, 25, 36 L(E, F )-, 25 continuous X-, 26 total, 1 separable Borel measurability, 148 sequence ideal, 94 sequential closure, 148, 150 shift map, 170

singular values, 72 spectral measure, 2, 144 spectrum, 2 Stieltjes integral strong operator valued, 201 stochastic integral, 109, 115 stochastic process, 112 adapted, 112 elementary progressively measurable, 118 left continuous, 112 measurable, 112 progressively measurable, 117 right continuous, 112 stopping time, 109 strongly μ-measurable, 149 strongly measurable, 14, 148, 149 submartingale, 107 supermartingale, 107 Sylvester-Rosenblum Theorem, 191 tensor product, 15 injective, 64 norm, 15 projective, 44, 61, 64, 81 seminorm, 17 topological space Lusin, 92 Polish, 92, 150 Souslin, 92 topology completely separated, 44 injective tensor product, 19 pointwise convergence, 148 projective tensor product, 17 strong operator, 2 tensor product, 15 weak, 6 total winnings, 109 trace, 71, 72, 94 Dixmier, 94 transition functions, 145 unconditionally summable, 7 weakly, 7 uniformly countably additive, 9

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16:37

240

variation, 1, 7, 8 2-, 42 p-, 31 total, 1 vector lattice, 27

10381 - Singular Bilinear Integrals

9789813207578

Singular Bilinear Integrals

complex, 27 vector measure, 6 Vitali-Hahn-Saks Theorem, 9, 35, 45 Wiener measure, 114

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