Multiple Operator Integrals: Development and Applications

Anna Tomskova

A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales.

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Surname or Family name: Tomskova

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School: School of Mathematics and Statistics Faculty: Faculty of Science

Title: Multiple Operator Integrals: Development and Applications

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Double operator integrals, originally introduced by Y.L. Daletskii and S.G. Krein in 1956, have become an indispensable tool in perturbation and scattering theory. Such an operator integral is a special mapping defined on the space of all bounded linear operators on a Hilbert space or, when it makes sense, on some operator ideal. Throughout the last 60 years the double and multiple operator integration theory has been greatly expanded in different directions and several definitions of operator integrals have been introduced reflecting the nature of a particular problem under investigation.

The present thesis develops multiple operator integration theory and demonstrates how this theory applies to solving of several deep problems in Noncommutative Analysis.

The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several important properties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this part we also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimate problem in the space of all bounded linear operators on classical Lp-spaces of scalar sequences.

The second part of the thesis develops important aspects of multiple operator integration theory. Here, we demonstrate how this theory applies to a solution of the problem on a Koplienko-Neidhardt trace formulae for a Taylor remainder of order two, which was raised by V. Peller in 2005, and also extend the solution for a Taylor remainder of an arbitrary order. Finally, using the tools from multiple operator integration theory, we present an affirmative solution of a question concerning Frechet differentiability of the norm of Lp-spaces, which has been of interest to experts in Banach space geometry for the last 50 years. We resolve this question in the most general setting, namely for the non-commutative Lp-spaces associated with an arbitrary von Neumann algebra, thus answering the open question suggested by G. Pisier and Q. Xu in their influential survey on the geometry of such spaces.

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Date ……………………………………………...... Acknowledgements

My deepest gratitude goes to my supervisor Fedor Sukochev ﬁrst of all for accepting me to his extraordinarily strong mathematical team. I always feel that he believes in my talent more than I do and he inspired me to work on long- standing interesting problems. In addition, he opened for me the mathematical world introducing me to many highly qualiﬁed mathematicians and giving me a chance to participate in a series of mathematical conferences and workshops all over the world. My deepest thanks extend to my co-supervisor Denis Potapov who has been very generous with his help and time over these three years and has displayed an astounding level of patience. I would also like to thank my colleague and co-author Dima Zanin for his patience, for sharing his thoughts and allowing me to pester him with questions so often over the years. In addition, my thanks extends to all my co-authors Jan Rozendaal, Christian Le Merdy, Clement Coine and Anna Skripka for a pleasant time of working together. Especially, I would like to thank Christian Le Merdy for the organization of visits to University de Franche-comte, Besanson, France during my study and his great hospitality there.

i Contents

1 Overview 2 1.1 Introduction ...... 2 1.2 Structure of the thesis ...... 15

2 Double operator integration 18 2.1 Double operator integral on S2 ...... 18 2.2 Double operator integral on B(H) and Schatten classes ...... 26 2.3 Double operator integral on B(X , Y) and Banach ideals ...... 37

3 Arazy conjecture concerning Schur multipliers on Schatten ideals 50

4 Commutator and Lipschitz estimates in B(X , Y) 60 4.1 General estimates ...... 60 4.2 Spaces with unconditional basis ...... 63 4.3 Absolute value function on B(ℓp, ℓq) ...... 69 4.3.1 Case p < q ...... 69 4.3.2 Case p ≥ q ...... 72 4.4 Lipschitz estimates on the ideal of p-summing operators ...... 74

5 Multiple operator integration 77 5.1 Multiple operator integrals in the ﬁnite dimensional setting . . . . 78 5.1.1 Multilinear Schur multipliers via linear ones...... 80 5.1.2 Schur multipliers via multiple operator integrals...... 83 5.1.3 Properties of multiple operator integrals ...... 84

ii 5.2 Multiple operator integrals in the setting of non-commutative Lp- spaces ...... 96 5.2.1 Classical non-commutative Lp-spaces and weak Lp-spaces . 99 5.2.2 Haagerup’s Lp-spaces ...... 102 5.2.3 Multiple operator integrals ...... 105 5.2.4 H¨older-type estimate ...... 116

6 Applications to operator Taylor remainders 128 6.1 Aﬃrmative results for Taylor remainders ...... 129 6.2 Counterexamples for Taylor remainders...... 132 6.2.1 Self-adjoint case...... 133 6.2.2 Unitary case...... 147

7 Diﬀerentiation of the norm of Haagerup Lp-space 154 7.1 Traces on L1,∞(N , τ) ...... 155 7.2 First Fr´echet derivative ...... 158 7.3 Main estimate ...... 160

p p 7.4 Taylor expansion for A 7→ ∥A∥Lp ,A ∈ L ...... 166

Bibliography 172

1 Chapter 1

Overview

1.1 Introduction

Let H be a separable complex Hilbert space. Let B(H) be the spaces of all bounded linear operators on H equipped with the uniform norm ∥ · ∥∞. Let us first consider the finite dimensional situation. Throughout the thesis N = {1, 2,...} is the set of natural numbers, R and C ∈ N H 2 are the sets of real and complex numbers respectively. Let d and = ℓd, 2 where ℓd denote the d-dimensional Hilbert space equipped with a scalar product · · 2 ( , ). Let two self-adjoint linear operators A and B on ℓd with respective complete { }d { }d { }d systems of orthonormal eigenvectors ξj j=1, ηk k=1 and eigenvalues λj j=1, { }d µk k=1 be given. Let Pξ denote the orthogonal projection on the unit vector ∈ 2 · · ξ ℓd, that is, Pξ( ) = ( , ξ)ξ. R × R → C ∈ 2 Let φ : be a complex valued function and X B(ℓd). We define A,B 2 → 2 Tφ : B(ℓd) B(ℓd) by

∑d ∑d A,B Tφ (X) := φ(λj, µk)Pξj XPηk . (1.1) j=1 k=1

A,B The operator Tφ is a ﬁnite dimensional version of so called double operator integral, which is of the main study of this work. The function φ is usually called

A,B the symbol of the operator Tφ . ∈ 2 × Obviously, every linear operator X B(ℓd) can be identiﬁed with an d d { }d complex matrix X = (X(ηk), ξj) j,k=1. For brevity, we will use the notation xjk

2 for (X(ηk), ξj). { }d { }d Note that, since the systems ξj j=1, ηk k=1 are orthonormal, the (k0, j0)- ≤ ≤ A,B entry for 1 j0, k0 d of the matrix corresponding to the operator Tφ (X) can be calculated as follows:

∑d A,B (Tφ (X)(ηk0 ), ξj0 ) = φ(λj, µk0 )((X(ηk0 ), ξj)ξj, ξj0 ) j=1

= φ(λj0 , µk0 )(X(ηk0 ), ξj0 ) = φ(λj0 , µk0 )xj0k0 .

A,B { }d Thus, the matrix Tφ (X) is simply the matrix φ(λj, µk)xjk j,k=1, that is, a { }d pointwise product (usually called a Schur product) of matrices Φ = φ(λj, µk) j,k=1 { }d ∗ and X = xjk j,k=1 and denoted Φ X. The Schur product of matrices is discussed in [18,54]. One of the central problems in perturbation theory is the study of the difference f(A) − f(B) for some function f on the spectra of the operators A and B. In particular, it is often extremely useful to know how the difference f(A) − f(B) depends on the difference A − B. Rather surprisingly, this can be done using the language of operator integrals. Let f be a real-valued function defined on a segment [a, b] ⊇ σ(A) ∪ σ(B), where σ(X) is the spectrum of the matrix X (since A and B are selfadjoint, their spectra are subsets of R). Consider the symbol f [1] given by   f(λj )−f(µk) , λj ≠ µk [1] λj −µk f (λj, µk) := , 1 ≤ j, k ≤ d.  ′ f (λj), λj = µk

The discrete symbol f [1] and the corresponding double operator integral was first studied by Löwnerin [71], when he observed the following important connection − A,B − f(A) f(B) = Tf [1] (A B) (1.2) between the difference f(A) − f(B) and the corresponding double operator integral. In other words, he showed that the difference f(A) − f(B) equals to the { [1] }d − Schur product of two matrices f (λj, µk) j,k=1 and A B.

3 The formula (1.2) is a powerful tool in studying Lipschitz operator estimates, that is the estimates of type

∥f(A) − f(B)∥ ≤ const∥A − B∥, (1.3)

∥·∥ 2 for some natural norm on B(ℓd), where the constant is independent of A, B, d. This problem has a particular interest for the operator norm, trace class norm and

2 A,B other Schatten norms on B(ℓd). If we can estimate the norm of the operator Tφ , then this estimate combined with formula (1.2) gives us the Lipschitz operator estimates (1.3) for the corresponding norm. Therefore, the main objective in the

A,B study of the operator Tφ is to obtain ”good” estimates for the norm of this 2 operator, considered on B(ℓd). ∥ · ∥ 2 For example, the Hilbert-Schmidt norm S2 on B(ℓd) is given by

∥ ∥ ∗ 1/2 ∈ 2 X S2 := (Tr(X X)) ,X B(ℓd),

2 where Tr is the standard trace on B(ℓd). A,B The following estimate for the Hilbert-Schmidt norm of Tφ (X) is pretty easy to obtain ∥ A,B ∥ ≤ | | ∥ ∥ Tφ (X) S2 max φ(λj, µk) X S2 . (1.4) j,k Combining (1.4) and (1.2) yields

∥f(A) − f(B)∥S2 ≤ ∥f∥Lip∥A − B∥S2 , (1.5)

∗ ∗ ∈ 2 for all A = A ,B = B B(ℓd), where |f(t) − f(s)| ∥f∥Lip := sup . t,s∈[a,b] |t − s| t≠ s The estimate (1.5) is the most desirable Lipschitz estimate. Estimates in other Schatten norms and in the operator norm require much more care since a straightforward application of (1.4) and (1.2) gives the following dimension- dependent estimate

2 ∥f(A) − f(B)∥ ≤ d ∥f∥Lip∥A − B∥.

The study of the theory of double operator integral in the inﬁnite dimensional case began with the problem of diﬀerentiation of an operator valued function.

4 This problem seems fundamental to the working of functional calculus, however, an immediate application of spectral theory does not make clear how we should work with perturbations such as

df(g(t)) f(g(t)) − f(g(0)) = lim , (1.6) dt t→0 t for a compact operators A and B, g(t) = A + tB, for a real parameter t and for a complex-valued function f on the spectra of the above operators. This problem was studied and resolved under rather restrictive conditions on f in 1956 by Y.L. Daletskii and S.G. Krein (see [33]). To handle (1.6), they introduced the double operator integral as an iterated Riemann-Stieltjes integral, formally denoted as ∫ A,B ∈ H Tφ (X) = φ(λ, µ)dE(λ)XdF (µ),X B( ), (1.7) C2 where E,F are spectral families of A and B respectively, and φ is a bounded complex valued function which is called the symbol of the integral. However, at the time, neither Daletskii nor Krien regarded the double operator integrals as a subject worthy of isolated study. In 1964, Krein noticed a connection of this work to that of a colleague, B.S. Birman. Motivated by Kreins remarks, Birman and his collaborator M.Z. Solomyak turned their attention to this integral. It soon became apparent, not only that the double operator integral was the appropriate tool for many related questions in perturbation theory, but further, that these questions necessitated a comprehensive theory of such integrals. A series of papers [19–21] (see also [22]) from Birman and Solomyak followed this realization, in which the construction was further developed. There are several directions of development of the double operator integration theory (theory of Schur multipliers) in the inﬁnite dimensional setting.

Arazy conjecture

The ﬁrst direction is to consider the double operator integral on Schatten ideals in B(H). Let S∞ be the spaces of all compact operators on H. By Sp we denote the Banach (if 1 ≤ p < ∞) or quasi-Banach (if 0 < p < 1) space of all operators

5 X ∈ S∞ such that ( ) 1/p ∗ p/2 ∥X∥Sp := Tr(X X) < ∞, where Tr is the standard trace on B(H). Throughout the text we ﬁx an orthonor- { }∞ H ∈ H mal basis ej j=1 in . With respect to this basis, every operator X B( ) has { }∞ a matrix representation X = xjk j,k=1, where xjk = (X(ek), ej)H, j, k = 1, 2,... ,

(·, ·)H is the scalar product in H.

{ }∞ | | Deﬁnition 1.1. A matrix M = mjk j,k=1 with complex entries such that supjk mjk < ∞ is said to be a Schur multiplier (or, brieﬂy, a multiplier) from Sq into Sp if { }∞ ∈ Sq ∗ { }∞ for every X = xjk j,k=1 the Schur product matrix M X = mjkxjk j,k=1 belongs to Sp. By M(Sq, Sp) we denote the space of all multipliers from Sq into Sp with the (quasi-)norm

∥M∥Sq→Sp := sup{∥M ∗ X∥Sp : ∥X∥Sq ≤ 1}.

Let C1([−1, 1]) be the class of all real-valued continuously differentiable functions f on [−1, 1]. The first divided difference of a function f ∈ C1([−1, 1]) is understood as follows   − f(x) f(y) , x ≠ y [1] x−y ∈ − f (x, y) :=  , x, y [ 1, 1]. f ′(x), x = y { }∞ ∈ ∞ ∥ ∥ ≤ ∞ Let also λ = λj j=1 ℓ with λ ∞ 1, where by ℓ we denote the Banach space of all bounded real valued sequences with the usual norm ∥ · ∥∞. The main object of our study in this part of the thesis is the matrix

{ [1] }∞ Ψf,λ = f (λj, λk) j,k=1, which naturally appears in the perturbation theory of linear operators.

Observe that the Schur multiplier Ψf,λ is the double operator integral (1.7) for [1] { }∞ a particular case φ = f , A = B and σ(A) consists only of points from λj j=1. The general question may be stated as follows:

Question A: Under which conditions on p, q, on the function f and on the

q p sequence λ, does the matrix Ψf,λ belong to M(S , S )? Observe that the boundedness of the derivative f ′ is clearly a necessary con-

q p dition for Ψf,λ ∈ M(S , S ). However, if p ≠ 2 or q ≠ 2 this condition is not

6 sufficient and one has to impose more restrictive smoothness conditions on f and/or some restrictions on λ. Observe also that the case p = q = 2 is triv- S2 { }∞ ial and the multipliers on are precisely the matrices M = mjk j,k=1 with | | ∞ supjk mjk < (see e.g. [7, Proposition 2.1]). As we have already mentioned above in the finite dimensional setting the multiplier Ψf,λ was first studied by K. Löwnerin [71]. The results from [71] H { }d immediately imply that if is d-dimensional and λ = λj j=1 is a finite sequence ∥ ∥ ≤ | ′ | of real numbers, then Ψf,λ S2→S2 supt∈[−1,1] f (t) (see details in [48, Section 4]). Question A was firstly raised by M. Birman and M. Solomyak in [19]. In

p p particular, [19, Theorem 10] says that Ψf,λ belongs to M(S , S ) (1 ≤ p ≤ ∞) and to M(B(H),B(H)) for any λ, provided that the function f is such that f ′ is from the class Lipα for some α > 0. By Lipα we denote the class of all continuous functions f satisfying

|f(t) − f(s)| ≤ C|t − s|α, t, s ∈ [−1, 1],

′ 1+α for some positive constant C. If f ∈ Lipα we shall also use the notation f ∈ C . An alternative approach to the study of the class of functions f such that

p p Ψf,λ ∈ M(S , S ) (1 ≤ p ≤ ∞) was developed by V. Peller in [78], where he strengthened the above result of Birman and Solomyak by showing that Ψf,λ ∈ M Sp Sp ≤ ≤ ∞ ∈ 1 ( , ) for any 1 p and f B∞1 (for deﬁnition of the Besov class 1 B∞1 see Deﬁnition 2.9 below, see also [78] and references therein). Furthermore, p p it was shown in [91, Theorem 2] that the matrix Ψf,λ belongs to M(S , S ) for 1 < p < ∞, provided that f ∈ C1([−1, 1]). Let us also mention that there are counterexamples, constructed by Yu. Farforovskaya in [45–47] (see also [78]), of

1 1 1 functions f ∈ C ([−1, 1]) such that the matrix Ψf,λ does not belong to M(S , S ), M(S∞, S∞) nor M(B(H),B(H)). Now let us consider the case when p ≠ q. The result [91, Theorem 2] presented

q p above obviously implies that the matrix Ψf,λ belongs to M(S , S ) for any 1 ≤ q p. The special case when 1 ≤ p ≤ 2 ≤ q ≤ ∞ was ﬁrst studied in 1982 by J. Arazy [7], who found suﬃcient

7 conditions on the function f and the sequence λ = {λj} that guarantee that Ψf,λ is in M(Sq, Sp). He proved the following result. By ℓr we denote the classical sequence space of all r-summing real valued sequences with the usual norm ∥ · ∥r for 1 ≤ r ≤ ∞ (quasi-norm for 0 < r < 1).

Theorem 1.2. [7, Theorem 2.5] Let 1 ≤ p ≤ 2 ≤ q ≤ ∞ be given, and let 0 < α, r < ∞ be such that 1 α 1 = + . p r q Let f be a continuously diﬀerentiable function on [−1, 1] satisfying

|f ′(t)| ≤ C|t|α, t ∈ [−1, 1] (1.8)

{ }∞ ∈ r ∥ ∥ ≤ for some constant C > 0. Let λ = λj j=1 ℓ be such that λ ∞ 1. Then Ψf,λ is a Schur multiplier from Sq into Sp. Moreover, ∥ ∥ ≤ ∥ ∥α Ψf,λ Sq→Sp C(α, f) λ r , where C(α, f) is a constant dependant on α and f only.

In addition, it was shown in [7, Proposition 3.1] that the conditions on f and λ given in Theorem 1.2 are in fact sharp. In the same paper, J. Arazy conjectured that Theorem 1.2 holds also in the cases 2 ≤ p ≤ q ≤ ∞ and 1 ≤ p ≤ q ≤ 2. In 1992, J. Arazy and Y. Friedman ( [8, Theorem 3.6]) conﬁrmed Arazy’s conjecture for the special case f(x) = x1+α, α > 0, x ∈ [0, 1]. In addition, they proved that for these special functions Arazy’s conjecture holds also in the case 0 < p < ∞, 1 < q < ∞. In the recent paper [92, Theorem 13], Arazy’s conjecture was also conﬁrmed ∈ 1+α ∩ 1 ∞ in the case where f C B∞1, 0 < α < 1 and 1 < p, q < . It is also worth noticing that the same problem was considered in [111, Corol- lary 7.6] by Q. Xu. Here the author proved an analogy of Theorem 1.2 for a general function of two variables, instead of the function f [1] and with the same conditions on p and q. In [93] jointly with D. Potapov and F. Sukochev, we have managed to extend the result of [8, Theorem 3.6] to the class of functions suggested in Theorem 1.2 and completely resolve the Arazy’s conjecture for all 0 < p < ∞ and 1 < q < ∞.

8 The main technical instruments employed to prove this result are as follows:

(i) A new approach to the double operator integration theory initiated in [84]. This approach is better suited for studying Question A in the setting when p ≠ q.

(ii) A technique from [92], introduced there to study Fr´echet diﬀerentiability

of the norm ∥ · ∥Sp , p > 1. Essentially, we apply this technique to the function of type f(t) = |t|z, 0 ≤ ℜz ≤ 1. Note that in the case ℜz = 0 this

1 function does not belong to the Besov class B∞1 and, hence, the double operator integration theory developed by Birman, Solomyak and Peller is not applicable in this case.

(iii) Complex interpolation for quasi- Banach ideals Sp for 0 < p < 1.

The components (i), (ii) and (iii) are presented in a detailed manner in this thesis (see Chapter 3).

Double operator integrals in the setting of Banach spaces

Let X and Y be Banach spaces and let B(X , Y) be the space of all bounded linear operators from X into Y equipped with the operator norm ∥·∥B(X ,Y). If X = Y we use B(X ) instead of B(X , X ). One more interesting direction of development of double operator integration theory is to consider the double operator integral on the space B(X , Y). Such an attempt was firstly made in [75], where the theory was revised and extended in various directions, including the Banach space setting. However, the results in the general setting were much weaker than in the Hilbert space setting. In this section of the thesis we show that for scalar type operators on Banach spaces one can obtain results matching those on Hilbert spaces. Note that our approach differs from [75] and allows us to resolve a series of particular problems, which we were interested in. This part of the thesis is based on [97] (a joint work with J. Rozendaal and F. Sukochev). Let A, B ∈ B(X ) be scalar type operators on X (see Definition 2.23). Let f : σ(A)∪σ(B) → C be a bounded Borel function. We are interested in Lipschitz

9 type estimates

∥f(B) − f(A)∥B(X ) ≤ const ∥B − A∥B(X ), (1.9) and more generally in commutator estimates

∥f(B)X − Xf(A)∥B(X ,Y) ≤ const ∥BX − XA∥B(X ,Y) (1.10) for scalar type operators A ∈ B(X ) and B ∈ B(Y), and X ∈ B(X , Y). As it was already mentioned above, this problem is well-known in the special case where X = Y is a separable Hilbert space, such as ℓ2, and A and B are normal operators on X . In this part of the thesis we study such estimates in the Banach space setting, and speciﬁcally for X = ℓp and Y = ℓq with p, q ∈ [1, ∞]. In the special case where A, B are self-adjoint bounded operators on a Hilbert space H, the estimate

∥f(B) − f(A)∥∞ ≤ const ∥B − A∥∞ (1.11) was established by Peller [78, 81] (see also [74]) for f : R → R in the Besov class

1 B∞,1. In this section we prove that (1.9) holds when A, B ∈ B(X ) are scalar type ∈ 1 operators with real spectrum and f B∞,1. It is immediate from the deﬁnition of a scalar type operator that every normal operator on H is of scalar type. Thus, here we extend (1.11) to the Banach space setting. More generally, (1.10) holds ∈ 1 ∈ X Y for f B∞,1 and for all S B( , ). ∈ 1 If f is the absolute value function then f / B∞,1 and the results mentioned above do not apply. Moreover, the techniques which we used to obtain (1.9) ∈ 1 for f B∞,1 cannot be applied to the absolute value function. However, the absolute value function is very important in the theory of matrix analysis and perturbation theory (see [18, Sections VII.5 and X.2]). In the case where H is an inﬁnite-dimensional Hilbert space, it was proved by Kato [60] that the function t 7→ |t|, t ∈ R does not satisfy (1.11). An earlier example of McIntosh [73] showed the failure of the commutator estimate (1.10) for this function in the case X = Y = H. Later, it was proved by Davies [34] that for 1 ≤ p ≤ ∞ and the

10 p Schatten von-Neumann ideal S with the corresponding norm ∥·∥Sp , the estimate

∥|B| − |A|∥Sp ≤ const ∥B − A∥Sp holds for all A, B ∈ Sp if and only if 1 < p < ∞ (see also [1]). Commutator estimates for the absolute value function and different Banach ideals in B(H) have also been studied in [39]. The proofs in [34, 39, 75] are based on Macaev’s celebrated theorem (see [50]) or on the UMD-property of the reflexive Schatten von-Neumann ideals. However, the spaces B(X , Y) are not UMD-spaces, and therefore the techniques used in [34, 39, 75] do not apply to them. To study (1.10) for X = ℓp and Y = ℓq, we use completely different methods from those of [34,39,75]. Instead, we use the theory of Schur multipliers on the space B(ℓp, ℓq) developed by Bennett [11], [12]. X { }∞ ⊆ Let now be a Banach space with an unconditional Schauder basis ej j=1

X . For j ∈ N, let Pj ∈ B(X ) be the projection given by Pj(x) := xjej for all ∑ ∞ ∈ X x = k=1 xkek .

∈ X { }∞ Deﬁnition 1.3. An operator A B( ) is diagonalizable (with respect to ej j=1) ∈ X { }∞ ∈ ∞ if there exists U B( ) invertible and a sequence λj j=1 ℓ of complex numbers such that ∑∞ −1 UAU x = λjPjx, x ∈ X , j=1 { }∞ where the series converges since ek k=1 is unconditional.

Let p, q ∈ [1, ∞] with p < q. We show that, for diagonalizable operators A ∈ B(ℓp) and B ∈ B(ℓq) with real spectrum, and for the absolute value function f,

∥f(B)X − Xf(A)∥B(ℓp,ℓq) ≤ const ∥BX − XA∥B(ℓp,ℓq) (1.12)

p q ∞ holds for all X ∈ B(ℓ , ℓ ) (where ℓ should be replaced by c0, that is the space of all sequences converging to zero). Moreover, if p, q ∈ [1, ∞] with p = 1 or q = ∞ then (1.12) holds for any Lipschitz function f. In particular,

∥f(B) − f(A)∥B(ℓ1) ≤ const ∥B − A∥B(ℓ1)

11 and

∥ − ∥ ≤ ∥ − ∥ f(B) f(A) B(c0) const B A B(c0)

1 for diagonalizable operators on ℓ respectively c0. Therefore we show that, even though (1.12) fails for p = q = 2 and f the absolute value function, and in particular (1.9) fails for X = ℓ2, (1.12) does hold for p < q and f the absolute

1 value function, and (1.9) holds for X = ℓ or X = c0 and each Lipschitz function f. We also study the commutator estimate in (1.10) in the more general form

∥f(B)X − Xf(A)∥I ≤ const ∥BX − XA∥I , (1.13) where I is a Banach ideal in B(X , Y). We also present an example of a Banach I ∥ · ∥ p∗ p ∈ ∞ 1 1 ∞ ideal ( , I ) in B(l , ℓ ), for p [1, ) and p + p∗ = 1 (with ℓ replaced by c0), namely the ideal of p-summing operators, such that any Lipschitz function f (in particular, the absolute value function) satisﬁes (1.13). These results are presented in Chapter 4.

Operator Taylor remainders

Another important direction in the double operator integration theory consists of considering multiple operator integrals. The study of multiple operator integration theory is applied here to the problem associated with operator Taylor remainders. This part of the thesis is based on [87] (a joint work with D. Potapov, A. Skripka and F. Sukochev). Let A and B be bounded self-adjoint operators and let U be a unitary operator on H. Let n ∈ N, let f and g be n times continuously diﬀerentiable functions on the real line R (denotes f ∈ Cn(R)) and on the unit circle T (denotes g ∈ Cn(T)), respectively. We investigate the question whether the nth Taylor remainder − ∑n 1 k 1 d Rn,f,A(B) := f(A + B) − f(A + tB) (1.14) k! dtk k=0 t=0 in the self-adjoint case or − ∑n 1 k iB 1 d itB Qn,g,U (B) := g(e U) − g(e U) (1.15) k! dtk k=0 t=0

12 in the unitary case belongs to a Schatten class under suitable assumptions on B and, if so, what best estimates it satisﬁes. The ﬁrst main result here (Theorem 6.1) shows that when the perturbation B is in the Schatten class Sp with p > n and f is in Cn(R), then the remainder

p/n Rn,f,A(B) belongs to S and has an estimate analogous to the one in the scalar case. The second main result (Theorem 6.7) shows that the above assumption on B to be in Sp with p > n is sharp, that is, there exist a Cn-function f and self-

n 1 adjoint bounded operators A and B ∈ S such that Rn,f,A(B) ∈/ S . In particular, when n = 2, we solve Peller’s problem [80, Problem 2] (see also [29]).

Completely analogous results are also established for Qn,g,U (B) (Theorems 6.2 and 6.15). In particular, when n = 2, we solve Peller’s problem [80, Problem 1] (see also [30]). The results above substantially relax up to date conditions on f and g (see below) guaranteeing

p p/n Rn,f,A,Qn,g,U : S → S , p > n. (1.16)

It followed from [81] and [86] that (1.16) holds for f in the intersection of Besov n ∩ 1 n T ∈ n R classes B∞1 B∞1 and g in B∞1( ), while we prove (1.16) for f C ( ) and g ∈ Cn(T). The estimates for the remainders obtained in this part of the thesis extend the respective estimates in [10, 61, 84, 85, 91] to arbitrary f ∈ Cn(R) and g ∈ Cn(T), n ∈ N. The key step in our solution is to present the remainders (1.14) and (1.15) in terms of multiple operator integrals and then to establish dimension dependent estimates from below for trace class norms of the values of these operator integrals. These dimension dependent estimates are applied in construction of our counterexamples. The results are presented in Chapter 5.

Fr´echet diﬀerentiability of the norm of Lp-spaces

One more direction in the development of the double and multiple operator integration theory consists of considering the operator integrals not only on

13 the Schatten von Neumann ideals Sp but also on the non-commutative Lp-space associated with arbitrary von Neumann algebras. In this part of the thesis, we will demonstrate how the tools of the theory of multiple operator integrals can be applied to important questions from the geometry of Banach spaces. To demonstrate this we consider the question concerning the diﬀerentiability of the norms of non-commutative Lp-spaces associated with an arbitrary von Neumann algebra M suggested by G. Pisier and Q. Xu in their survey [83]. In the special case, when the algebra M is of type I this question is fully resolved in [26, 92, 105, 108], however, the general case (even when M is semiﬁnite) required new ideas. The results presented in this part of the thesis are announced in [94] and will be published in [95] (a joint work with D. Potapov, F. Sukochev and D. Zanin). The main result here is stated in Theorem 1.4 below. Let 1 ≤ p < ∞ and let M ⊆ B(H) be a von Neumann algebra. By Lp(M) we denote the Haagerup

p L -space on the algebra M with the norm ∥ · ∥Lp (the definition of Haagerup Lp-space is given in Section 5.2.2 and for the notion of Fréchet differentiability we refer to Definition 7.1).

p p Theorem 1.4. The function A 7→ ∥A∥Lp ,A ∈ L (M) is

(i) infinitely many times Fréchetdifferentiable, whenever p is an even integer;

(ii) (p − 1)-times Fr´echetdiﬀerentiable, whenever p is an odd integer;

(iii) ⌊p⌋-times Fr´echetdiﬀerentiable, whenever p is not an integer, where ⌊·⌋ denotes the integer part of a real number

Moreover, the result of Theorem 1.4 is sharp. In the special case when M is a semiﬁnite von Neumann algebra on a Hilbert space H and τ is a normal faithful semiﬁnite trace on M we denote by (Lp(M, τ), ∥·

p ∥Lp ) the classical noncommutative L -space with respect to (M, τ). It is well- p p known that in this case the spaces (L (M), ∥ · ∥Lp ) and (L (M, τ), ∥ · ∥Lp ) are isometric for all 1 ≤ p < ∞. Therefore, the result of Theorem 1.4 also describes the diﬀerentiability properties in the classical noncommutative Lp-spaces

14 p (L (M, τ), ∥ · ∥Lp ), 1 ≤ p < ∞ and thus substantially strengthens results from [92,108]. It should be pointed out that our strategy of the proof of Theorem 1.4 is completely different from that of used in [92], where the result is proved for a von Neumann algebra of type I. In particular, we do not use the complex interpolation method, which is a principal tool used in [92]. Two additional important points concerning our approach are worthy to em- phasize. Firstly, we define Fréchet differentials in Theorem 1.4 via singular traces on weak non-commutative L1-spaces associated with semifinite (non-finite) von Neumann algebras. Our definition of such traces is new and not covered by earlier treatment of singular traces on semifinite von Neumann algebras (see [51, 70]). Nontrivial applications of singular traces are manifold in noncommutative geometry (see e.g. [70] and references therein), however, their usage in Banach space geometry is very rare. The fact that we resort to this rather unusual instrument

p is intimately linked with Haagerup’s definition of (L (M), ∥ · ∥Lp ). The second novelty of our approach is the extension of the theory of multiple operator integrals developed recently in [84] in the setting of semifinite von Neu- mann algebras (see, earlier versions in [9,21,75,76,81]) to the realm of Haagerup Lp-spaces. We believe that this extension is potentially useful for further study of geometrical and analytical properties of Haagerup Lp-spaces. Acting in this vein, we have significantly simplified the proof of the Höldertype estimates given in [92, Theorem 14] and clarified [92, Remark 15], proving the Lipschitz type estimate as well (see Theorem 5.36). These results are presented in Chapter 6.

1.2 Structure of the thesis

This thesis may be split into two main parts. The ﬁrst part consists of Chap- ters 2, 3 and 4, where only double operator integrals are considered. Chapter 2 contains chronologically all main steps in the development of double operator integration theory. We also include here some important properties of double operator integrals, which we use later, in Chapters 3 and 4. In Chapter 3 the

15 solution of the Arazy conjecture is presented. Next in Chapter 4 we study the double operator integrals in the setting of Banach spaces. The second part of this thesis studies multiple operator integration theory, which is contained in Chapters 5, 6 and 7. The main ideas of multiple operator integration are studied in Chapter 5. Chapter 6 solves the problem of Taylor operator reminders using tools from Chapter 5. In Chapter 7 we establish Fr´echet diﬀerentiability properties of the norm of Lp-spaces again using results from multiple operator integration theory established in Chapter 5. Most of the results of this thesis are published in the following papers and presented at the talks on various international and local conferences listed below.

1. A. Skripka, D. Potapov, F. Sukochev, A. Tomskova, Schatten properties of multilinear Schur multipliers and applications to operator Taylor remainders, preprint.

2. D. Potapov, F. Sukochev, A. Tomskova, D. Zanin, Fr´echetdiﬀerentiability

of the norm of Lp-spaces associated with arbitrary von Neumann algebras, Trans. Amer. Math. Soc., to appear.

3. C. Coine, C. Le Merdy, D. Potapov, F. Sukochev, A. Tomskova, Peller’s problem concerning Koplienko-Neidhardt trace formula: the unitary case, J. Funct. Anal. 271 (2016), no. 7, 1747–1763.

4. J. C. Coine, C. Le Merdy, D. Potapov, F. Sukochev, A. Tomskova, Reso- lution of Peller’s problem concerning Koplienko-Neidhardt trace formulae, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 113–139.

5. J. Rozendaal, F. Sukochev, A. Tomskova, Operator Lipschitz functions on Banach spaces, Studia Math. 232 (2016), no. 1, 57–92.

6. D. Potapov, F. Sukochev, A. Tomskova, On the Arazy conjecture concerning Schur multipliers on Schatten ideals, Adv. Math. 268 (2015) 404–422.

7. D. Potapov, F. Sukochev, A. Tomskova, D. Zanin, Fr´echetdiﬀerentiability

of the norm of Lp-spaces associated with arbitrary von Neumann algebras, C. R. Acad. Sci. Paris, Ser. I 352 (2014) 923–927 (the short version).

16 8. A. Tomskova, Frchet diﬀerentiability of the norm of non-commutative Lp- spaces, Analysis and Partial Diﬀerential Equations workshop, 9-10 Febru- ary, 2016, Australian National University, Canberra, Australia.

9. A. Tomskova, Resolution of Peller’s problem concerning Koplienko-Neidhardt trace formulae, International Conference ”Algebra, Analysis and Quantum Probability”, Tashkent, September 10-12, 2015.

10. A. Tomskova, On the Arazy conjecture concerning Schur multipliers on Schatten ideals, International Workshop on Operator Theory and Applica- tions, IWOTA 2015 Tbilisi, Georgia, July 6-10, 2015.

11. A. Tomskova, On the Arazy conjecture concerning Schur multipliers on Schatten ideals, Annual School Postgraduate Conference, UNSW, Aus- tralia, June 2015.

12. A. Tomskova, Fréchet differentiability of the norm of Lp-spaces, part II, Conference on Operator Spaces and Quantum Probability, December 15- 19, 2014, University de Franche-Comté,Besancon, France.

13. A. Tomskova, Schur multipliers and applications, plenary talk at Annual School Postgraduate Conference, UNSW, Australia, Oct. 2014.

17 Chapter 2

Double operator integration

In this chapter we present different approaches to double operator integration theory. We start with a detailed exposition of definition of double operator integral on S2 given by Birman and Solomyak in [19]. We explain the basics of this theory here since the paper [19] is written in somewhat telegraphic style and since this definition of double operator integral is essential for the exposition. Next we introduce the double operator integral on B(H) and Sp (1 ≤ p ≤ ∞) using definition suggested in [9,81] which later will become a basis for a definition of double operator integral on the space B(X ) of all bounded linear operators on a Banach space X . Here we also introduce a notion of double operator integral in the sense of [84], which plays a fundamental role in this thesis. Finally, double operator integration theory on B(X ) for an arbitrary Banach space X is developed. This approach exercised here is new and is linked to the ideas explored in [9,81]. Although the results we obtain match the corresponding results in the Hilbert space setting, the Banach space setting requires rather different technical tools. The results presented here are taken from [97] (a joint work with J. Rozendaal and F. Sukochev).

2.1 Double operator integral on S2

Let an inﬁnite dimensional separable Hilbert space H be ﬁxed and let B(H) be the C∗-algebra of all linear bounded operators on H.

18 Recall that by S2 we denote the Hilbert-Schmidt class, that is, the ideal in B(H) of all compact operators X on a Hilbert space H, such that

∗ 1/2 ∥X∥S2 := (Tr(X X)) < ∞, where Tr is the standard trace on B(H). The ideal S2 with the scalar product

(X,Y ) := Tr(Y ∗X),X,Y ∈ S2 becomes a Hilbert space. Naturally, the next step in studying of operator integrals should be defining a continuous variant of the operator integral (1.1) on the algebra B(H). In this section we define the double operator integral on S2 only, since the Hilbert space structure of S2 makes such a definition much easier. We will freely apply facts from general spectral theory of self-adjoint operators (for details see, e.g., [23, Chapters 5 and 6]).

Spectral measure product

Let (Ωj, Σj), j = 1, 2 be measurable space. Let also two spectral measures E and F be given on Σ1 and Σ2 respectively with respect to the same Hilbert space H. In this subsection we deﬁne the continuous version of the double operator integral as an integral with respect to the spectral measure constructed as a product of the spectral measures E and F . Such an approach was ﬁrstly suggested by Birman and Solomyak in [19] (see also [20,21]), where the authors have built the foundation of the theory of double operator integrals. Let us consider

E 7→ (σ1): X E(σ1)X, 2 σ1 ∈ Σ1, σ2 ∈ Σ2,X ∈ S . F(σ2): X 7→ XF (σ2),

It is clear that E and F are commuting spectral measures on (Ω1, Σ1) and (Ω2, Σ2), respectively, with respect to the Hilbert space S2. Deﬁne the product of the measures E and F

G(σ1 × σ2) := E(σ1)F(σ2), σ1 ∈ Σ1, σ2 ∈ Σ2, (2.1)

19 so that

2 G(σ1 × σ2)(X) = E(σ1)XF (σ2),X ∈ S . (2.2)

To verify that G is indeed a spectral measure we need to show that G is σ-additive. It is interesting to observe that in order to deﬁne a double operator integral on S2, in [19] the authors used the fact that G is a σ-additive measure without proving it. Moreover, it turns out that the product of two spectral measures can in general (for an arbitrary Hilbert space) fail to be σ-additive, as it is shown in [25]. In [24, Theorem 2] it is established that G constructed above is in fact σ-additive. The proof uses the speciﬁcity of the space S2. More general result for the product of n spectral measures, n ≥ 2 is proved in [76, Theorem 1]. Since this result is important in the exposition, let us present the proof of [24, Theorem 2] here in details.

Proposition 2.1. Let (Ω1, Σ1) and (Ω2, Σ2) be measure spaces, H be a separable

Hilbert space and let E and F be spectral measures with respect to H on (Ω1, Σ1) 2 and (Ω2, Σ2), respectively. Then, the mapping G :Σ1 × Σ2 → B(S ) deﬁned by 2 (2.2) extends to a spectral measure on (Ω, Σ) with respect to S , where Ω = Ω1×Ω2 and Σ = Σ1 ⊗ Σ2 is the minimal σ-algebra generated by the algebra Σ1 × Σ2 of ”measurable rectangles”. ∪ ( ) n (k) (k) (i) (i) Proof. If σ ∈ Σ1 × Σ2 is such that σ = σ × σ , where σ × σ ∩ ( ) k=1 1 2 1 2 (j) × (j) ∅ ̸ σ1 σ2 = , for i = j, then we deﬁne ∑n (k) × (k) G(σ)X := G(σ1 σ2 )X. (2.3) k=1

(k) The fact that the deﬁnition of G(σ) does not depend on the choice of σ1 and (k) σ2 can be checked using standard arguments. Let us prove that G takes values in orthogonal projections on S2. Indeed, for

2 2 X,Y ∈ S and δ = σ1 × σ2, σ1 ∈ Σ1, σ2 ∈ Σ2, we have that G(δ) = G(δ) and

∗ (G(δ)X,Y ) = (E(σ1)XF (σ2),Y ) = Tr(Y E(σ1)XF (σ2))

∗ = Tr((E(σ1)YF (σ2)) X) = (X,E(σ1)YF (σ2)) = (X,G(δ)Y ),

20 ∗ that is, G(δ) = G(δ). For an arbitrary set δ ∈ Σ1 × Σ2 the proof is similar using (2.3). Since E and F are spectral measures, it follows that

G(Ω)X = E(Ω1)XF (Ω2) = X, that is, G(Ω) is the identity operator on S2.

2 We show that G is σ-additive on Σ1 × Σ2. For X,Y ∈ S consider the scalar measure

νX,Y : σ1 × σ2 7→ (G(σ1 × σ2)X,Y ), σ1 ∈ Σ1, σ2 ∈ Σ2. ∪ ∈ × ∞ ∈ × ∩ ∅ ̸ Let δ Σ1 Σ2 be such that δ = n=1 δn, δn Σ1 Σ2 and δn δm = , n = m, n, m = 1, 2,.... Recalling the inequality (see, e.g., [70, Lemma 12.4.7]) ∑∞ 2 2 2 ∥G(δn)X∥S2 ≤ ∥X∥S2 ,X ∈ S , n=1 and evaluating ∑∞ ∑∞ |(G(δn)X,Y )| = |(G(δn)X,G(δn)Y )| n=1 n=1 ∑∞ ≤ ∥G(δn)X∥S2 ∥G(δn)Y ∥S2 n=1 (2.4) ( ∑∞ ) 1 ( ∑∞ ) 1 2 2 2 2 ≤ ∥G(δn)X∥S2 ∥G(δn)Y ∥S2 n=1 n=1

≤ ∥X∥S2 ∥Y ∥S2 , ∑ ∞ ∈ S2 we obtain that the series n=1(G(δn)X,Y ) converges for any X,Y .

Next we prove that νX,Y is a σ-additive scalar measure, i.e., ( ( ∪∞ ) ) ∑∞ G δn X,Y = (G(δn)X,Y ). (2.5) n=1 n=1

If X = (·, ξ2)ξ1 and Y = (·, η2)η1 for some ξj, ηj ∈ H, and σj ∈ Σj, j = 1, 2, then we have that

νX,Y (σ1 × σ2) = Tr(YE(σ1)XF (σ2)) = (E(σ1)ξ1, η2)(F (σ2)η1, ξ2), which is a product of two scalar σ-additive measures. Thus, νX,Y is σ-additive for any one-dimensional operators X and Y. Therefore, any one-dimensional operators X and Y satisfy (2.5).

21 If each of X and Y is represented as a ﬁnite sum of one-dimensional operators, then by linearity of the inner product, we obtain that X and Y satisfy (2.5).

2 Let now X,Y ∈ S and {Xk}k≥1 and {Yk}k≥1 be sequences of ﬁnite-dimensional ∪ → → S2 → ∞ ∞ ∈ operators such that Xk X and Yk Y in as k . Since δ = n=1 δn

Σ1 × Σ2, it follows from (2.3) that

(G(δ)Xk,Yk) → (G(δ)X,Y ) as k → ∞. (2.6)

Applying (2.4), we have that

∞ ∞ ∞ ∑ ∑ ∑ (G(δn)Xk,Yk) − (G(δn)X,Y ) ≤ |(G(δn)Xk,Yk) − (G(δn)X,Y )| n=1 n=1 n=1 ∑∞ ∑∞ ≤ |(G(δn)Xk,Yk) − (G(δn)X,Yk)| + |(G(δn)X,Yk) − (G(δn)X,Y )| n=1 n=1 ∑∞ ∑∞ = |(G(δn)(Xk − X),Yk)| + |(G(δn)X,Yk − Y )| n=1 n=1

≤ ∥Xk − X∥S2 ∥Yk∥S2 + ∥X∥S2 ∥Yk − Y ∥S2 .

Hence, ∑∞ ∑∞ (G(δn)Xk,Yk) → (G(δn)X,Y ) as k → ∞, n=1 n=1 which along with (2.6) proves (2.5) for all X,Y ∈ S2. The observation that σ- additivity of νX,Y implies σ-additivity of G completes the proof. D

Double operator integrals on S2

∞ ∞ Let Ω = Ω1 ×Ω2 and φ ∈ L (Ω,G), where G is deﬁned in (2.1) and L (Ω,G) consists of all measurable functions φ on Ω for which

∥φ∥∞ := G-sup φ = inf{α ∈ R+ : |φ(·)| ≤ α, G-a.e.}

G S2 → S2 The Birman-Solomyak double operator integral Tφ : is deﬁned as the integral of the symbol φ with respect to the spectral measure G, i.e., ∫ G ∈ S2 Tφ (X) := φ(ω) dG(ω)(X),X . (2.7) Ω The notation ∫ G ∈ S2 Tφ (X) =: φ(ω1, ω2)dE(ω1)XdF (ω2),X , Ω1×Ω2

22 is frequently used. ∫ ∥ ∥ ∥ ∥ The property Ω φ(ω) dG(ω) S2→S2 = φ ∞ of the functional calculus implies the following important result.

G S2 ∈ Proposition 2.2. The operator integral Tφ is bounded on if and only if φ L∞(Ω,G).

We also highlight the following fact. Assume that the function φ ∈ L∞(Ω,G) has the representation φ(ω1, ω2) = a1(ω1)a2(ω2) for some measurable functions a1 and a2. It is not diﬃcult to see that the following formula ∫ ∫ G · · Tφ (X) = a1(ω1)dE(ω1) X a2(ω2)dF (ω2) Ω1 Ω2 holds for all X ∈ S2. Indeed, observing that ∫ ∫ G E E · Ta1 (X) = Ta1 (X) = a1(ω1)d (ω1)X = a1(ω1)dE(ω1) X, Ω1 Ω1 ∫ G F · and similarly, Ta2 (X) = Ta2 (X) = X a2(ω2)dF (ω2), (2.8) Ω2 and using multiplicativity of the spectral integral, we obtain ( ∫ ) G G G G G · Tφ (X) = Ta1a2 (X) = (Ta1 Ta2 )(X) = Ta1 X a2(ω2)dF (ω2) ∫ Ω2 ∫

= a1(ω1)dE(ω1) · X · a2(ω2)dF (ω2). (2.9) Ω1 Ω2 Let A, B be self-adjoint operators on H and E and F be spectral measures of

A and B, respectively. Let Ω1 = Ω2 = R and let Σ1 = Σ2 be the σ-algebra of all Borel sets on R. In this case, we use the notation

A,B G Tφ := Tφ , (2.10) for φ ∈ L∞(R × R,G). Assume again that the function φ ∈ L∞(R × R,G) has the representation

φ(λ, µ) = a1(λ)a2(µ). By (2.9), using spectral resolutions of a1(A) and a2(B), we obtain

A,B · · Ta1·a2 (X) = a1(A) X a2(B). (2.11)

The resolution (2.11) plays a crucial role in the further presentation. In particular, the deﬁnitions of the operator integral on B(H) and Schatten ideals are based on this formula (see, e.g., Section 2.2).

23 Commutator and Lipschitz estimates

Let f be a Lipschitz function defined on a segment [a, b] ⊇ σ(A)∪σ(B). Recall that f [1] denotes the first divided difference of the function f. Now we discuss the A,B following nice property of the integral Tf [1] .

Theorem 2.3. Let A = A∗,B = B∗ ∈ B(H) and let f be a Lipschitz function on a segment [a, b] ⊇ σ(A) ∪ σ(B). Then

− A,B − ∈ S2 f(A)X Xf(B) = Tf [1] (AX XB),X .

Proof. Let p1(λ, µ) = λ and p2(λ, µ) = µ, for λ, µ ∈ R. Since f is Lipschitz, it ∞ follows that f is continuous on [a, b] and, therefore, f ◦ p1, f ◦ p2 ∈ L (R × R,G), where G is deﬁned in (2.1). Moreover, by (2.8), we have that

A,B A,B T ◦ (X) = f(A)X,T ◦ (X) = Xf(B), f p1 f p2 X ∈ S2. A,B A,B and Tp1 (X) = AX, Tp2 (X) = XB Thus, using properties of the functional calculus, we obtain that

f(A)X − Xf(B) = T A,B (X) − T A,B (X) = T A,B (X) f◦p1 f◦p2 f◦p1−f◦p2 A,B A,B − A,B = T [1] (X) = T [1] (X) T [1] (X) (p1−p2)f p1f p2f = T A,B(T A,B(X)) − T A,B(T A,B(X)) = T A,B(AX − XB). f [1] p1 f [1] p2 f [1]

The following corollary is an immediate consequence of Theorem 2.3 and the fact that

A,B [1] ∥ ∥S2→S2 ∥ ∥∞ ∥ ∥ Tf [1] = f = f Lip (see (2.7) and (2.10)).

Corollary 2.4. Let A = A∗,B = B∗ ∈ B(H) and let f be a Lipschitz function on a segment [a, b] ⊇ σ(A) ∪ σ(B). Then

2 ∥f(A)X − Xf(B)∥S2 ≤ ∥f∥Lip∥AX − XB∥S2 ,X ∈ S .

Although Theorem 2.3 is not applicable to the identity operator X = I, the following result remains valid; its proof is rather technical and so is omitted here.

24 Theorem 2.5. [21, Theorem 4.5] Let A = A∗,B = B∗ ∈ B(H) be such that A − B ∈ S2 and let f be a Lipschitz function on a segment [a, b] ⊇ σ(A) ∪ σ(B). Then f(A) − f(B) ∈ S2 and

− A,B − f(A) f(B) = Tf [1] (A B).

The following result follows immediately from the theorem above. A diﬀerent proof of it, which uses a similar result for ﬁnite matrices, is presented in [48, Theorem 4.1].

Theorem 2.6. For every A = A∗,B = B∗ ∈ B(H) such that A − B ∈ S2 and every Lipschitz function f on a segment [a, b] ⊇ σ(A) ∪ σ(B) the estimate

∥f(A) − f(B)∥S2 ≤ ∥f∥Lip∥A − B∥S2 holds, that is f is operator Lipschitz on S2.

Double operator integrals as Schur multipliers on S2

Next we study the notion of Schur multipliers on S2 and its connection with double operator integrals on S2 deﬁned in (2.10). { }∞ { }∞ H Fix two orthonormal bases ξj j=1 and ηk k=1 in . Then, every operator ∈ S2 { }∞ X can be represented as inﬁnite matrix X = xjk j,k=1, where xjk := { }∞ (X(ηk), ξj), j, k = 1, 2,.... Recall that for a matrix M = mjk j,k=1 the product

M ∗ X := {mjkxjk} is said to be the Schur product of the matrices M and X. Recall also that the matrix M is called the Schur multiplier on S2 if the mapping M ∗ X 7→ X is a bounded operator on S2. Suppose that each of self-adjoint operators A and B has a discrete spectrum. { }∞ { }∞ Consider sequences λ = λj j=1 and µ = µk k=1 consisting of the points of { }∞ the spectrum of A and B, respectively, counted with multiplicities. Let ξj j=1, { }∞ ηk k=1 be corresponding orthonormal bases of eigenvectors of the operators A and B, respectively, and let φ ∈ L∞(R × R,G). Then, we have the formulas ∑ A,B Tφ (X) = φ(λj, µk)Pξj XPηk , j,k

A,B (Tφ (X)ηk, ξj) = φ(λj, µk)xjk,

25 where by Pξ, ξ ∈ H, we denote the orthogonal projection on the vector ξ. In A,B { } ∗ other words, Tφ (X) is represented as a Schur product φ(λj, µk) j,k X of the matrices {φ(λj, µk)}j,k and X = {xjk}. Thus, the double operator integral can be considered as a continuous version

A,B of the Schur multiplier and, therefore, sometimes the operator integral Tφ is called a Schur multiplier.

2.2 Double operator integral on B(H) and Schat- ten classes

Let 1 ≤ p < ∞. Recall that Sp is the Schatten (von Neumann) class, that is the ideal in B(H) of all compact operators X on a Hilbert space H, such that

p 1/p ∥X∥Sp := (Tr(|X| )) < ∞, where Tr is the standard trace on B(H). By S∞ we denote the Banach space of all compact operators on H equipped with the uniform norm ∥ · ∥∞ (we also use

∥ · ∥S∞ or ∥ · ∥B(H)).

Observe that the norm ∥ · ∥Sp satisﬁes the H¨olderinequality, which is stated in the following lemma.

≤ ≤ ∞ 1 1 1 ∈ Sp ∈ Sq Lemma 2.7. Let 1 p, q, r be such that r = p + q . For X ,Y , we have that

∥XY ∥Sr ≤ ∥X∥Sp ∥Y ∥Sq .

The simplest approach to the deﬁnition of double operator integral on Sp is usage of the property (2.11). Let A, B ∈ B(H) be self-adjoint operators and let φ ∈ L∞(R × R) be given by the formula

∑n φ(λ, µ) = a1(λ, j)a2(µ, j), λ, µ ∈ R, j=1

∞ ∞ ∞ where a1(·, j), a2(·, j) ∈ L (R) for all j = 1, . . . , n, L (R × R), L (R) are usual Lebesgue spaces of bounded functions on R × R and R, respectively. Deﬁne the

26 A,B Sp → Sp operator Tφ : as follows

∑n A,B Tφ (X) := a1(A, j)Xa2(B, j). j=1

A,B From the H¨olderinequality it follows that Tφ is a bounded linear operator on Sp for 1 ≤ p ≤ ∞ and

∑n ∥ A,B∥ ≤ ∥ · ∥ ∥ · ∥ Tφ Sp→Sp a1( , j) ∞ a2( , j) ∞. j=1

A,B H Similarly, the operator Tφ is deﬁned on B( ). This approach was ﬁrst suggested in [19, Section 6] and developed widely in many papers on operator integration theory, including [9,81]. Next we present the largest class of functions, for which this approach works. Let A be the class of functions φ : R × R → C admitting the representation ∫

φ(λ, µ) = a1(λ, ω) a2(µ, ω)dν(ω), (2.12) Ω for some ﬁnite measure space (Ω, ν) and bounded Borel functions

aj (·, ω): R → C, j = 1, 2.

The class A is an algebra with respect to the operations of pointwise addition and multiplication [9, Proposition 4.10]. The formula ∫ ∥ ∥ ∥ · ∥ ∥ · ∥ | | φ A = inf a1( , ω) ∞ a2( , ω) ∞ d ν (ω), Ω where the infimum is taken over all possible representations (2.12) defines a norm on A (see [74, Lemma 4.6]). The class A can also be defined (see, e.g., [74,90]) as the class of functions φ : R × R → C admitting the representation ∫

φ(λ, µ) = b1(λ, ω) b2(µ, ω)dν1(ω), (2.13) Ω for some (not necessarily ﬁnite) measure space (Ω, ν1) and bounded Borel functions

bj (·, ω): R → C, j = 1, 2

27 such that ∫

∥b1(·, ω)∥∞ ∥b2(·, ω)∥∞ d |ν1| (ω) < ∞. Ω These deﬁnitions coincide since the representation (2.12) of the function φ can be obtained from (2.13) with

b1(λ, ω) b2(λ, ω) a1(λ, ω) = , a2(λ, ω) = ∥b1(·, ω)∥∞ ∥b2(·, ω)∥∞ and the ﬁnite measure ν deﬁned by

ν = ∥b1(·, ω)∥∞ ∥b2(·, ω)∥∞ ν1.

Let 1 ≤ p ≤ ∞. For every φ ∈ A, and a fixed couple of self-adjoint opera- A,B Sp → Sp tors A, B, the double operator integral Tφ : , is defined by ∫ A,B ∈ Sp Tφ (X) := a1(A, ω) X a2(B, ω)dν(ω),X , (2.14) Ω where aj’s and (Ω, ν) are taken from the representation (2.12) and the integral is understood in the sense of the Bochner integral (see, e.g., [9, Definition 4.1]). A,B H → H The operator Tφ : B( ) B( ) is defined similarly. One of important A,B results of this theory is that the value Tφ (X) does not depend on the particular representation on the right-hand side of (2.12) (see [81, Lemma 3.1], [9, Lemma 4.3]). A,B Sp ≤ ≤ ∞ Observe that Tφ is a bounded linear operator on , 1 p , and on B(H). Moreover, due to the ideal property, we have

T A,B , T A,B ≤ ∥φ∥ . (2.15) φ B(H)→B(H) φ Sp→Sp A

Similarly to the proof of (2.11), one can show that if φ ∈ A, then the deﬁnition A,B S2 of Tφ on given in this section in (2.14) coincides with that of Birman and Solomyak given in (2.7). Now we discuss the class of functions f : R → C such that the divided diﬀerence f [1] belongs to the class A. In Lemma 2.8 below we provide examples of such functions. Let Ff and F −1f be the Fourier transform and the inverse Fourier transform,

28 respectively, of the function f : R → C, i.e.,1 ∫ ∫ Ff(t) = f(s) e−ist ds, F −1f(s) = f(t) eist dt. R R Let L1(R) be the usual Lebesgue space of all Lebesgue integrable functions on

′ 1 R. By W1(R) we denote the class of functions f : R → C such that Ff ∈ L (R).

[1] Lemma 2.8. If f ∈ W1(R), then f ∈ A.

Proof. Recalling that F −1Ff = f, we have ∫ f(λ) − f(µ) 1 ( ) f [1](λ, µ) = = eiλt − eiµt Ff(t)dt λ − µ λ − µ R ∫ ∫ ( t ) = i ei(λs+µ(t−s))ds Ff(t)dt. R 0 Making the substitution u = s, v = t − s in the latter integral, we obtain ∫ f [1](λ, µ) = i eiλueiµvFf(u + v)dudv. R×R

iλu Taking Ω = R × R and dν(u, v) = Ff(u + v)dudv, a1((u, v), λ) = e and iµv a2((u, v), µ) = e , u, v ∈ R, we obtain the representation (2.12) for the function f [1]. We also have f [1] ∈ A because ∫ | F | ∥F ′∥ ∞ t f(t) dt = f L1 < R D

1 Another condition involves the Besov space B∞,1. Denote the set of all integers by Z. Following [79], let {ψk}k∈Z be a sequence of Schwartz functions on R such k−1 k+1 that, for each k ∈ Z, the Fourier transform Fψk of ψk is supported on [2 , 2 ] ∑ F F ∞ F and ψk+1(x) = ψk(2x) for all x > 0, and such that k=−∞ ψk(x) = 1 ∗ F ∗ F − ∈ Z ∈ for all x > 0. Let ψk be deﬁned by ψk(x) = ψk( x) for k and x k R. If f is a distribution on R such that {2 ∥f ∗ ψk∥L∞(R)}k∈Z ∈ ℓ1(Z) and ∗ { k∥ ∗ ∥ ∞ } ∈ Z 2 f ψk L (R) k∈Z ℓ1( ), then f admits a representation ∑ ∑ ∗ ∗ ∗ f = f ψk + f ψk + P, k∈Z k∈Z where P is a polynomial.

1For brevity, we omit the coeﬃcients in the deﬁnition of the Fourier transform, since they do not play any role in the exposition.

29 1 Deﬁnition 2.9. We let the homogeneous Besov space B∞,1 consist of all distri- 1 butions as above for which P = 0. Then B∞,1 is a Banach space when equipped with the norm ∑∞ ∑∞ k k ∗ 1 ∥f∥ 1 := 2 ∥f ∗ ψ ∥ ∞ R + 2 ∥f ∗ ψ ∥ ∞ R , f ∈ B∞ . B∞,1 k L ( ) k L ( ) ,1 k=−∞ k=−∞

The following result is established in [78, Theorem 2] (see also [79, Theorem 2]).

Lemma 2.10. There exists a constant C ≥ 0 such that f [1] ∈ A for each f ∈

1 [1] B∞ , with ∥f ∥ ≤ C∥f∥ 1 . ,1 A B∞,1

Below we present another interesting subclass of A. By L2(R) we denote the usual Lebesgue spaces of square integrable functions on R. Let W1,2(R) be the space of all g ∈ L2(R) with weak derivative g′ ∈ L2(R), equipped with the norm

′ 1,2 ∥g∥W1,2(R) := ∥g∥L2(R) + ∥g ∥L2(R) for g ∈ W (R).

Lemma 2.11. [90, Theorem 9] Let g ∈ W1,2(R) and let   λ g(log( µ )) if λ, µ > 0 ψg(λ, µ) :=  . (2.16) 0 otherwise √

Then ψ ∈ with ψ ≤ 2∥g∥ 1,2 R . g A g A W ( )

Since the functions in A are bounded, the class A does not contain polynomials. Moreover, it is known that f [1] does not belongs to this class if f is the absolute value function. Therefore, there have been many attempts to find an- A,B Sp other appropriate definition of the operator Tφ on . In [19], the following A,B Sp ∈ ∞ R × R definition of Tφ on for an arbitrary φ L ( ) is suggested. ∈ ∞ R×R A,B ∈ S2 S1 ⊂ S2 Let φ L ( ). In this case, we know that Tφ B( ). Since , A,B ∈ S1 A,B we may consider Tφ (X) for an arbitrary X . Now we say that Tφ is bounded on S1 if the mapping

7→ A,B X Tφ (X) deﬁnes a bounded operator on S1. In this case, by duality, this map extends to a bounded linear transformation on the space B(H). If this extension restricted on

30 Sp Sp ⊂ H Sp A,B (using that B( )) is bounded on , then we say that Tφ is bounded Sp A,B ∈ Sp on and denote this fact as Tφ B( ). Let 1 ≤ p ≤ ∞ and deﬁne

{ ∈ ∞ R × R A,B ∈ Sp } Mp := φ L ( ): Tφ B( ).

We have the following important characterization of M1 and M∞.

Theorem 2.12. [78, Theorem 1] M1 = M∞ = A.

We also know from Proposition 2.2 that

∞ M2 = L (R × R).

The class Mp, 1 < p ≠ 2 < ∞, has not been described yet. However, the following inclusions are known (see, e.g., [22, (5.7)]):

∞ A = M1 = M∞ ⊂ Mp ⊂ M2 = L (R × R). (2.17)

The left inclusion in (2.17) is strict. For instance, it is proved in [91] that for

[1] the absolute value function f, its divided diﬀerence f belongs to Mp whenever 1 < p < ∞ (recall that f [1] ∈/ A). There is no counterexample, which shows that the right inclusion in (2.17) is strict. Let 1 ≤ p ≤ ∞ and deﬁne

[1] Fp := {f : R → C : f ∈ Mp}.

≤ ≤ ∞ 1 1 The class Fp, 1 p , has the following description, where B11 and B∞1 are Besov classes deﬁned in [81] and Deﬁnition 2.9.

1 ⊇ ⊇ 1 Theorem 2.13. (i) B11 F1 = F∞ B∞1;

(ii) F2 = Lip(R);

(iii) Fp = Lip(R), 1 < p < ∞.

Proof. The assertion (i) is proved in [78, Theorems 2 and 4]. Proposition 2.2 implies (ii). The assertion (iii) has been recently proved in [91]. D

Observe that (i) in Theorem 2.13 above is improved in [8].

31 Commutator and Lipschitz estimates

The proof of the following theorem goes along the lines of the proof of Theo- rem 2.3 and applies properties of the double operator integral

Tφ1+φ2 = Tφ1 + Tφ2 ,Tφ1φ2 = Tφ1 Tφ2 established in [9].

Theorem 2.14. Let 1 ≤ p ≤ ∞ and let A = A∗,B = B∗ ∈ B(H). If f is a Lipschitz function on the segment [a, b] ⊇ σ(A) ∪ σ(B) such that f [1] ∈ A (in particular, f ∈ W1(R)), then

− A,B − ∈ H f(A)X Xf(B) = Tf [1] (AX XB),X B( ).

The following result extends Theorems 2.5 and 2.6 to the case of Sp, 1 ≤ p ≤ ∞ and B(H).

Corollary 2.15. Let A = A∗,B = B∗ ∈ B(H). If f is a Lipschitz function on

[1] the segment [a, b] ⊇ σ(A) ∪ σ(B) such that f ∈ A (in particular, f ∈ W1(R)), then

(i) − A,B − f(A) f(B) = Tf [1] (A B);

(ii) f(A) − f(B) ∈ Sp whenever A − B ∈ Sp, 1 ≤ p ≤ ∞. Moreover,

[1] ∥f(A) − f(B)∥Sp ≤ ∥f ∥A∥A − B∥Sp . (2.18)

Proof. Taking X = I in Theorem 2.14, we arrive at (i). From (i) and (2.15), it follows that

∥ − ∥ A,B − ≤ ∥ [1]∥ ∥ − ∥ f(A) f(B) Sp = T [1] (A B) f A A B Sp . f Sp

Observe that if f is as in Corollary 2.15, then

[1] ∥f∥Lip ≤ ∥f ∥A

32 by the straightforward estimate ∫ [1] |f (λ, µ)| ≤ |a1(λ, ω)| · |a2(λ, ω)| d|ν|(ω) ∫Ω

≤ ∥a1(·, ω)∥∞∥a2(·, ω)∥∞ d|ν|(ω). Ω Therefore, the constant in the estimate (2.18) is worse than in the case p = 2 (Theorem 2.6). However, recently the result for 1 < p < ∞ has been highly improved in [91], where the following important result is established.

Theorem 2.16. [91, Theorem 1] Let A = A∗,B = B∗ ∈ B(H) and 1 < p < ∞. If f is a Lipschitz function on the segment [a, b] ⊇ σ(A) ∪ σ(B), then

∥f(A) − f(B)∥Sp ≤ ∥f∥Lip∥A − B∥Sp .

New deﬁnition of double operator integral

The paper [91] is a breakthrough in the operator integration theory. In this paper another concept of the notion of a double operator integral is introduced. This concept is developed in detail in the paper [84], where a new deﬁnition of double operator integral is presented and studied. This discovery is fundamental in this thesis. It makes it possible to study double operator integrals on Schatten ideals for arbitrary bounded functions φ. Moreover, this new approach allows to consider double operator integrals on quasi-Banach Schatten ideals Sp (0 < p < 1).

Let A and B be self-adjoint operators and let {Et}t∈R, {Fs}s∈R be the families of spectral projections corresponding to A and B respectively. We set El,m := l l+1 l l+1 ∈ N ∈ Z E[ m , m ), Fl,m := F [ m , m ), for every m , l .

Deﬁnition 2.17. Let m ∈ N and let 0 < p, q ≤ ∞. Fix a bounded Borel function φ : R × R → C. Suppose that for every m ∈ N the series ( ) ∑ l l S (X) := φ 0 , 1 E XF (2.19) φ,m m m l0,m l1,m l0,l1∈Z converges in the norm of Sp and

X 7→ Sφ,m(X), m ∈ N

33 is a sequence of bounded linear operators from Sq into Sp. If the sequence { } A,B Sφ,m m≥1 converges in the strong operator topology to some linear operator Tφ , then, according to Banach-Steinhaus theorem, {Sφ,m}m≥1 is uniformly bounded A,B A,B and the operator Tφ is also bounded. If Tφ exists as above and is bounded, A,B Sq Sp we simply say that Tφ is bounded from into and, as before, the operator A,B Tφ will be called a double operator integral.

The definition above is different from the original approach to double operator integration introduced in [19–21] and in [9, 75, 81]. However, for a large class of [1] ∈ 1 ≤ ≤ ∞ functions φ, such as φ = f , f B∞1, and 1 p = q , these definitions coincide (see e.g. [84, Lemma 3.5] or [92], where a detailed comparison of these definitions is presented).

A,B From now on the double operator integral Tφ in the setting of Hilbert spaces A A,A is always assumed to be the operator deﬁned in Deﬁnition 2.17. Set Tφ := Tφ . A The operator Tφ has the following simple algebraic properties.

∥ ∥ ≤ A Sp A Lemma 2.18. Let be A be such that A ∞ 1. If Tφ is bounded on , then Tψ

p s0 s1 is also bounded on S , where ψ(x, y) = x y φ(x, y), s0, s1 are non-negative integers. Moreover,

A A s0 s1 A s0 A s1 Tψ (X) = Tφ (A XA ) and Tψ (X) = A Tφ (X)A . (2.20)

s s Proof. Since A El,m = El,mA , for any l ∈ Z, m ∈ N, s ≥ 0, we have that ∑ ( ) l0 l1 S (As0 XAs1 ) = φ , E As0 X As1 E φ,m m m l0,m l1,m l0,l1∈Z ( ∑ ( ) ) l0 l1 = As0 φ , E X E As1 = As0 S (X)As1 . (2.21) m m l0,m l1,m φ,m l0,l1∈Z

Taking the limit in the strong operator topology in (2.21), we arrive at

A s0 s1 s0 A s1 Tφ (A XA ) = A Tφ (X)A .

To complete the proof we need to verify that

A s0 A s1 Tψ (X) = A Tφ (X)A . (2.22)

34 Using the fact that for a ﬁxed m ∈ N all projections {El,m}l∈Z are pairwise orthogonal, we have that ∑ ( ) ( ) ( ) l s0 l s1 l l S (X) = 0 1 φ 0 , 1 E XE ψ,m m m m m l0,m l1,m l0,l1∈Z ( ∑ ( ) )( ∑ ( ) )( ∑ ( ) ) k s0 l l k s0 = 0 E φ 0 , 1 E XE 1 E m k0,m m m l0,m l1,m m k1,m k0∈Z l0,l1∈Z k1∈Z ( ∑ ( ) ) ( ∑ ( ) ) k s0 k s0 = 0 E S (X) 1 E . (2.23) m k0,m φ,m m k1,m k0∈Z k1∈Z Taking the limit in the strong operator topology in (2.23), using the fact that

{El,m} are spectral projections of A, we obtain (2.22). D

H Remark 2.19. Note also that if the function φ is such that the operator Tφ is bounded for any self-adjoint operator H, then by choosing the matrix   A 0 H˜ =   , 0 B and by considering operator H˜ S˜q → S˜p Tφ : , given by     X X T A(X ) T A,B(X ) H˜  11 12   φ 11 φ 12  ∈ Sq Tφ = ,Xjk , j, k = 1, 2, B,A B X21 X22 Tφ (X21) Tφ (X22) where by S˜p we denote the Schatten class on the Hilbert space H ⊗ H. The

A,B H˜ boundedness of Tφ follows from that of Tφ . Let us now introduce a class of symbols φ, for which the double operator integral is well defined. Let P be the class of polynomials on R with real coefficients. Given a bounded function h : R → C, and Q ∈ P, we set ∫ 1 φh,Q (x, y) := Q (t) h ((1 − t)x + ty) dt. (2.24) 0 Definition 2.20. Following the terminology from [84] we shall call the function φh,Q polynomial integral momentum. In case Q(t) ≡ 1, we will abbreviate

φh := φh,Q.

35 The divided diﬀerence f [1] is the simplest example of the polynomial integral momentum. Indeed, the function f [1] admits the following integral representation ∫ 1 f [1](x, y) = f ′ ((1 − t)x + ty) dt. (2.25) 0

[1] In other words, the function f is the polynomial integral momentum φf ′ associated with the polynomial Q ≡ 1 and the function h = f ′. The notion of polynomial integral momentum plays a crucial role in our present approach. The following theorem describes the class of polynomial inte-

A gral momenta φ = φh,Q for which the double operator integral Tφ is well-deﬁned.

By Cb we denote the class of all continuous bounded complex-valued functions on R.

Theorem 2.21. [84, Theorem 5.3] Let 1 < p < ∞,Q ∈ P and h ∈ Cb. Then the operator T A is bounded on Sp and φh,Q

A ∥T ∥Sp→Sp ≤ c(p, Q)∥h∥∞, φh,Q where the constant c(p, Q) > 0 above depends on p and the polynomial Q only. A Sp ∞ In particular, the operator Tf [1] is bounded on (1 < p < ) provided that ′ f ∈ Cb.

A Remark 2.22. Let φ be such that the operator Tφ is bounded and let A be such that ∥A∥∞ ≤ 1. From deﬁnition of double operator integral, we have that

A A Tφ = Tφ˜ , where

φ˜(x, y) := χ[−1,1](x)φ(x, y)χ[−1,1](y), and by χ[−1,1] we denote the indicator function of the segment [−1, 1]. Let φ be the polynomial integral momentum φh for a given h. Since −1 ≤ (1−t)x+ty ≤ 1 if −1 ≤ x, y ≤ 1, we also have that

A A Tφ = Tφ . h χ[−1,1]h

A In other words, if ∥A∥∞ ≤ 1, then T depends only on values of h on [−1, 1]. In φh general, T A depends only on the values of h on σ(A). φh

36 2.3 Double operator integral on B(X , Y) and Ba- nach ideals

Throughout this section, X and Y denote Banach spaces over the complex ﬁeld, the space of all bounded linear operators from X to Y is B(X , Y), and B(X ) := B(X , X ). We identify the algebraic tensor product X ∗ ⊗ Y with the space of ﬁnite rank operators in B(X , Y) via (x∗ ⊗ y)(x) := ⟨x∗, x⟩y for x ∈ X , ∗ ∗ ∗ x ∈ X and y ∈ Y, where X is the dual space of X . By IX ∈ B(X ) we denote the identity operator on X . The Borel σ-algebra on a Borel measurable subset ς ⊆ C will be denoted by Bς , and B := BC. For measurable spaces (Ω1, Σ1) and (Ω2, Σ2) as before we denote by Σ1 ⊗ Σ2 the σ-algebra on Ω1 × Ω2 generated by all measurable rectangles ς1 × ς2 with ς1 ∈ Σ1 and ς2 ∈ Σ2. If (Ω, Σ) is a measurable space then L∞(Ω) is the space of all bounded Σ- measurable complex-valued functions (equivalence classes) on Ω, a Banach algebra with the supremum norm

∞ ∥f∥∞ := sup |f(ω)|, f ∈ L (Ω). ω∈Ω

If µ is a complex Borel measure on a measurable space (Ω, Σ) and X is a Banach space, then a function f :Ω → X is µ-measurable if there exists a sequence of X -valued simple functions converging to f µ-almost everywhere. For Banach spaces X and Y, we say that a function f :Ω → B(X , Y) is strongly measurable if ω 7→ f(ω)x is a µ-measurable mapping Ω → Y for each x ∈ X . If µ is a positive measure on a measurable space (Ω, Σ) and f :Ω → [0, ∞] is a function, we let ∫ ∫ f(ω) dµ(ω) := inf g(ω) dµ(ω) ∈ [0, ∞], Ω Ω where the infimum is taken over all measurable g :Ω → [0, ∞] such that g(ω) ≥ f(ω) for ω ∈ Ω. ∈ ∞ ∗ 1 1 The Hölderconjugate of p [1, ] is denoted by p and is defined by p + p∗ =

1. The indicator function of a subset ς of a set Ω is denoted by χς . We will often

37 identify functions deﬁned on ς with their extensions to Ω by setting them equal to zero oﬀ ς.

Scalar type operators

In this part we summarize some of the basics of scalar type operators, as taken from [41]. Let X be a Banach space. A spectral measure on X is a map E : B → B(X ) such that the following hold:

• E(∅) = 0 and E(C) = IX ;

• E(ς1 ∩ ς2) = E(ς1)E(ς2) for all ς1, ς2 ∈ B;

• E(ς1 ∪ ς2) = E(ς1) + E(ς2) − E(ς1)E(ς2) for all ς1, ς2 ∈ B;

• E is σ-additive in the strong operator topology.

Note that these conditions imply that E is projection-valued. Moreover, by [41, Corollary XV.2.4] there exists a constant K such that

∥E(ς)∥B(X ) ≤ K, ς ∈ B. (2.26)

An operator A ∈ B(X ) is a spectral operator if there exists a spectral measure E on X such that AE(ς) = E(ς)A and σ(A, E(ς)X ) ⊆ -ς for all ς ∈ B, where σ(A, E(ς)X ) denotes the spectrum of A in the space E(ς)X . For a spectral operator A, we let spec(A) denote the minimal constant K occurring in (2.26) and call spec(A) the spectral constant of A. This is well-defined since the spectral measure E associated with A is unique, cf. [41, Corollary XV.3.8]. Moreover, E is supported on σ(A) in the sense that E(σ(A)) = IX [41, Corollary XV.3.5]. Hence we can define an integral with respect to E of bounded Borel measurable ∑ n functions on σ(A), as follows. For f = j=1 αjχςj a finite simple function with

αj ∈ C and ςj ⊆ σ(A) mutually disjoint Borel sets for 1 ≤ j ≤ n, we let

∫ ∑n f dE := αjE(ςj). (2.27) σ(A) j=1

38 This deﬁnition is independent of the representation of f, and

∫ ∑n ∗ f dE = sup αjx E(ςj)x B(X ) ∥ ∥ ∥ ∗∥ ∗ σ(A) x X = x X =1 j=1 ∗ ≤ sup |αj| sup ∥x E(·)x∥var ∗ j ∥x∥X =∥x ∥X ∗ =1 ∗ ≤ 4∥f∥∞ sup sup |x E(ς)x| ∗ ∥x∥X =∥x ∥X ∗ =1 ς⊆σ(A)

≤ 4 spec(A)∥f∥∞,

∗ ∗ where ∥x E(·)x∥var is the variation norm of the measure x E(·)x. Since the simple functions lie dense in L∞(σ(A)), for general f ∈ L∞(σ(A)) we can deﬁne ∫ ∫ f dE := lim f dE ∈ B(X ) →∞ n σ(A) n σ(A) { }∞ ⊆ ∞ ∥ − ∥ → for fn n=1 L (σ(A)) a sequence of simple functions with fn f ∞ 0 as n → ∞. This deﬁnition is independent of the choice of approximating sequence and ∫

f dE ≤ 4 spec(A)∥f∥∞. (2.28) X σ(A) B( ) It is straightforward to check that ∫ ∫ ∫ (αf + g) dE = α f dE + g dE, σ(A) ∫ (∫σ(A) )(∫σ(A) ) fg dE = f dE g dE σ(A) σ(A) σ(A) for all α ∈ C and simple f, g ∈ L∞(σ(A)), and approximation then extends these ∫ ∞ identities to general f, g ∈ L (σ(A)). Moreover, χσ(A) dE = E(σ(A)) = IX . ∫ σ(A) 7→ ∞ → X Hence the map f σ(A) f dE is a continuous morphism L (σ(A)) B( ) of unital Banach algebras. Since the spectrum of A is compact, the identity function ∫ 7→ ∈ X λ λ is bounded on σ(A) and σ(A) λ dE(λ) B( ) is well deﬁned.

Deﬁnition 2.23. A spectral operator A ∈ B(X ) with spectral measure E is a scalar type operator if ∫ A = λ dE(λ). σ(A)

The class of scalar type operators on X is denoted by Bs(X ).

39 ∞ For A ∈ Bs(X ) with spectral measure E and f ∈ L (σ(A)) we deﬁne ∫ f(A) := f dE. (2.29) σ(A) As remarked above, f 7→ f(A) is a continuous morphism L∞(σ(A)) → B(X ) of unital Banach algebras with norm bounded by 4 spec(A). Note also that ∫ ⟨x∗, f(A)x⟩ = f(λ) d⟨x∗,E(λ)x⟩ (2.30) σ(A) for all f ∈ L∞(σ(A)), x ∈ X and x∗ ∈ X ∗. Indeed, for simple functions this follows from (2.27), and by taking limits one obtains (2.30) for general f ∈ L∞(σ(A)). Finally, we note that a normal operator A on a Hilbert space H is a scalar type operator with spec(A) = 1, and in this case (2.28) improves to ∫

f dE ≤ ∥f∥∞, (2.31) H σ(A) B( ) as is known from the Borel functional calculus for normal operators.

Spaces of operators

In this section we discuss some properties of spaces of operators that we will need later in this thesis. First we provide a lemma about approximation by ﬁnite rank operators. Re- call that a Banach space (X , ∥ · ∥X ) has the bounded approximation property if there exists M ≥ 1 such that, for each K ⊆ X compact and ϵ > 0, there exists ∈ X ∗ ⊗X ∥ ∥ ≤ ∥ − ∥ X with X B(X ) M and supx∈K Xx x X < ϵ.

Lemma 2.24. Let (X , ∥ · ∥X ) and (Y, ∥ · ∥Y ) be Banach spaces such that X is separable and either X or Y has the bounded approximation property. Then each T ∈ B(X , Y) is the limit in the strong operator topology (SOT-limit) of a norm bounded sequence of ﬁnite rank operators.

Proof. Fix T ∈ B(X , Y). By [68, Proposition 1.e.14] there exists a norm bounded { } ⊆ X ∗⊗ Y net Tj j∈J having T as its SOT-limit. It is straightforward to see that the strong operator topology is metrizable on bounded subsets of B(X , Y) by ∑∞ −k d(X1,X2) := 2 ∥X1xk − X2xk∥Y ,X1,X2 ∈ B(X , Y), k=1

40 { } ⊆ X X where xk k∈N is a countable subset that is dense in the unit ball of . { } Hence there exists a subsequence of Tj j∈J with T as its SOT-limit. D

Let now (Z, ∥ · ∥Z ) be a Banach space which is continuously embedded in B(X , Y). Following [110] (in the case where Z is a subspace of B(X , Y)), we say that Z has the strong convex compactness property if the following holds. For any finite measure space (Ω, Σ, µ) and any strongly measurable bounded f :Ω → Z, the operator T ∈ B(X , Y) defined by ∫ T x := f(ω)x dµ(ω), x ∈ X , (2.32) Ω ∫ Z ∥ ∥ ≤ ∥ ∥ belongs to with T Z Ω f(ω) Z dµ(ω). By the Pettis Measurability Theo- rem, any separable Z has this property. Indeed, if Z is separable then combining Propositions 1.9 and 1.10 in [109] shows that any strongly measurable f :Ω → Z is µ-measurable as a map to Z. If f is bounded as well, then (2.32) defines an element of Z with ∫

∥T ∥Z ≤ ∥f(ω)∥Z dµ(ω). Ω It is shown in [110] and [98] that the subspaces of compact and weakly compact operators in B(X , Y) have the strong convex compactness property, but not all subspaces of B(X , Y) do. Moreover, if N is a semiﬁnite von Neumann algebra on a separable Hilbert space H, with faithful normal semiﬁnite trace τ, and F is a rearrangement invariant Banach function space with the Fatou property, then E = N ∩ F(N , τ) has the strong convex compactness property (see [9, Lemma 3.5]).

Lemma 2.25. Let X and Y be separable Banach spaces and let Z be a Banach space continuously embedded in B(X , Y). If BZ := {z ∈ Z : ∥z∥Z ≤ 1} is SOT- closed in B(X , Y), then Z has the strong convex compactness property.

Proof. The proof follows that of [9, Lemma 3.5]. First we show that BZ is a Polish space in the strong operator topology. As in the proof of Lemma 2.24, bounded subsets of B(X , Y) are SOT-metrizable. The ﬁnite rank operators are

SOT-dense in B(X , Y), hence B(X , Y) is SOT-separable. Therefore BZ is SOT- separable and metrizable. By assumption, BZ is complete.

41 Now let (Ω, µ) be a ﬁnite measure space and let f :Ω → Z be bounded and strongly measurable. Without loss of generality, we may assume that f(Ω) ⊆ BZ and that µ is a probability measure. For each y∗ ∈ Y∗ and x ∈ X , the mapping ∗ BZ → [0, ∞), T 7→ |⟨y , T x⟩|, is continuous. The collection of all these mappings, ∗ ∗ ∗ for y ∈ Y and x ∈ X , separates the points of BZ . Moreover, ω 7→ |⟨y , f(ω)x⟩| is a measurable mapping Ω → [0, ∞) for each y∗ ∈ Y∗ and x ∈ X . By [109, Propositions 1.9 and 1.10], f is the µ-almost everywhere SOT-limit of a sequence ∫ ∞ of BZ -valued simple functions {fk} . Let Tk := fk dµ ∈ BZ for k ∈ N. k=1 Ω ∫ → By the dominated convergence theorem, Tk(x) T (x) := Ω f(ω)x dµ(ω) as k → ∞, for all x ∈ X . Since BZ is SOT-closed by the assumption, we conclude that T ∈ BZ .

Now let g :Ω → [0, ∞) be measurable such that 1 ≥ g(ω) ≥ ∥f(ω)∥Z for ∈ f(ω) ∫ g(ω) ∈ ω Ω, and deﬁne h(ω) := g(ω) and dν(ω) := g(η) dµ(η) dµ(ω) for ω Ω. By ∫ Ω 7→ what we have shown above, x Ω h(ω)x dν(ω) deﬁnes an element of BZ . Since ∫ ∫ ∫ T x = f(ω)x dµ(ω) = g(ω) dµ(ω) h(ω)x dν(ω), Ω Ω Ω ∫ ∥ ∥ ≤ we obtain T Z Ω g(ω) dµ(ω), as remained to be shown. D

Remark 2.26. Note that the converse implication does not hold. Indeed, if X is a Hilbert space (or more generally, a Banach space with the metric approximation property) then the ﬁnite rank operators of norm less than or equal to 1 are SOT- dense in the unit ball of B(X ). Therefore the compact operators of norm less than or equal to 1 are not SOT-closed in B(X ) if X is inﬁnite-dimensional. However, by [110, Theorem 1.3], the space of compact operators on X has the strong convex compactness property.

Let X and Y be Banach spaces and I a Banach space which is continuously embedded in B(X , Y). We say that (I, ∥ · ∥I ) is a Banach ideal in B(X , Y) if

• For all R ∈ B(Y), X ∈ I and T ∈ B(X ), RXT ∈ I with ∥RXT ∥I ≤

∥R∥B(Y)∥X∥I ∥T ∥B(X );

∗ ∗ ∗ ∗ ∗ •X ⊗ Y ⊆ I with ∥x ⊗ y∥I = ∥x ∥X ∗∥y∥Y for all x ∈ X and y ∈ Y.

42 By Lemma 2.25 and [35, Proposition 17.21] for separable X and Y, any maximal Banach ideal (for the deﬁnition see e.g. [82]) in B(X , Y) has the strong convex compactness property. This includes a large class of operator ideals, such as the ideal of absolutely p-summing operators, the ideal of integral operators, etc (see [35, p. 203]).

Double operator integral

Fix Banach spaces X and Y, scalar type operators A ∈ Bs(X ) and B ∈ Bs(Y) with spectral measures E respectively F , and φ ∈ A. Let a representation as in (2.12) for φ be given, with corresponding (Ω, ν) and a1, a2. For ω ∈ Ω, let a1(A, ω) := a1(·, ω)(A) ∈ B(X ) and a2(B, ω) := a2(·, ω)(B) ∈ B(Y) be deﬁned by the functional calculus for A respectively B.

Lemma 2.27. Let X ∈ B(X , Y) have separable range. Then, for each x ∈ X ,

ω 7→ a2(B, ω)Xa1(A, ω)x is a weakly measurable map Ω → Y.

Proof. Fix x ∈ X . If a1 = χς for some measurable ς ⊆ σ(A) × Ω then it ∗ ∗ ∗ is straightforward to show that ⟨x , a1(A, ·)x⟩ is measurable for each x ∈ X .

As X has separable range, Xa1(A, ·)x is ν-measurable by the Pettis Measur- ability Theorem. If a2 is an indicator function as well, the same argument shows that a2(B, ·)y is weakly measurable for each y ∈ Y. General arguments, approximating Xa1(A, ·)x by simple functions, show that a2(B, ·)Xa1(A, ·)x is weakly measurable. By linearity this extends to simple a1 and a2, and for general a1 and a2 let {fk}k∈N, {gk}k∈N be sequences of simple functions such that a1 = limk→∞ fk and a2 = limk→∞ gk uniformly. Then a1(A, ω) = limk→∞ fk(A) and a2(B, ω) = limk→∞ gk(B) in the operator norm, for each ω ∈ Ω. The desired measurability now follows. D

Now suppose that Y is separable, that I is a Banach ideal in B(X , Y) and let X ∈ B(X , Y). By (2.28),

∥a2(B, ω)Xa1(A, ω)∥I ≤16 spec(A) spec(B)∥X∥I ∥a1(·, ω)∥∞∥a2(·, ω)∥∞ (2.33) for w ∈ Ω. Since I is continuously embedded in B(X , Y), by the Pettis Mea- surability Theorem, Lemma 2.27 and (2.33) we can deﬁne the double operator

43 integral ∫ A,B ∈ Y ∈ X Tφ (X)x := a2(B, ω)Xa1(A, ω)x dν(ω) , x . (2.34) Ω Proposition 2.28. Let X and Y be separable Banach spaces such that X or Y has the bounded approximation property, and let A ∈ Bs(X ), B ∈ Bs(Y), and φ ∈ A. Let I be a Banach ideal in B(X , Y) with the strong convex compactness A,B ∈ I property. Then (2.34) deﬁnes an operator Tφ B( ) which is independent of the choice of representation of φ in (2.12), with

∥ A,B∥ ≤ ∥ ∥ Tφ I→I 16 spec(A) spec(B) φ A. (2.35)

Proof. By (2.33) and the strong convex compactness property, it holds that A,B ∈ I ∈ I Tφ (X) B( ) for all X , and ∫ ∥ A,B ∥ ≤ ∥ ∥ ∥ · ∥ ∥ · ∥ Tφ (X) I 16 spec(A) spec(B) X I a1( , ω) ∞ a2( , ω) ∞dν(ω). Ω A,B A,B Clearly Tφ is linear, hence the result follows if we establish that Tφ is independent of the representation of φ. For this it suﬃces to let φ ≡ 0. Now, ﬁrst consider X = x∗ ⊗ y for x∗ ∈ X ∗ and y ∈ Y, and let x ∈ X , y∗ ∈ Y∗ and w ∈ Ω. Recall that E and F are the spectral measures of A and B, respectively. Then ∫ ∗ ∗ ⟨y , a2(B, ω)Xa1(A, ω)x⟩ = a2(µ, ω) d⟨y ,F (µ)Xa1(A, ω)x⟩ ∫ σ(B) ∗ ∗ = a2(µ, ω)⟨x , a1(A, ω)x⟩d⟨y ,F (µ)y⟩ ∫σ(B) ∫ ∗ ∗ = a1(λ, ω)a2(µ, ω)d⟨x ,E(λ)x⟩d⟨y ,F (µ)y⟩ σ(B) σ(A) by (2.30). Now Fubini’s Theorem and the assumption on φ yield ∫ ⟨ ∗ A,B ⟩ ⟨ ∗ ⟩ y ,Tφ (X)x = y , a2(B, ω)Xa1(A, ω)x dν(ω) ∫ ∫ ∫ Ω ∗ ∗ = a1(λ, ω)a2(µ, ω)d⟨x ,E(λ)x⟩d⟨y ,F (µ)y⟩dν(ω) ∫Ω σ(∫B) σ(∫A) ∗ ∗ = a1(λ, ω)a2(µ, ω)dν(ω)d⟨x ,E(λ)x⟩d⟨y ,F (µ)y⟩ ∫σ(B) ∫σ(A) Ω = φ(λ, µ)d⟨x∗,E(λ)x⟩d⟨y∗,F (µ)y⟩ = 0. σ(B) σ(A) A,B ∈ X ∗ ⊗Y By linearity, Tφ (X) = 0 for all X . By Lemma 2.24, a general element X ∈ I is the SOT-limit of a bounded (in the operator norm on B(X , Y)) sequence

44 ∗ {Xn}n∈N ⊆ X ⊗ Y. Hence for each x ∈ X there exists a constant C ≥ 0 such that, for all n ∈ N, ∫

∥a2(B, ω)Xna1(A, ω)x∥Y dν(ω) Ω ∫

≤ 16 spec(A) spec(B)∥Xn∥B(X ,Y)∥x∥ ∥a1(·, ω)∥∞∥a2(·, ω)∥∞dν(ω) ∫ Ω

≤ C ∥a1(·, ω)∥∞∥a2(·, ω)∥∞dν(ω) < ∞, Ω where we have used (2.28). Now the dominated convergence theorem shows that A,B A,B ∈ X A,B Tφ (X)x = limn→∞ Tφ (Xn)x = 0 for all x , which implies that Tφ is independent of the representation of φ and concludes the proof. D

If A and B are self-adjoint operators on separable Hilbert spaces X and Y, then (2.35) improves to

∥ A,B∥ ≤ ∥ ∥ Tφ I→I φ A (2.36) by appealing to (2.31) instead of (2.28) in (2.33).

Remark 2.29. Let X = Y = H be an infinite-dimensional separable Hilbert space and I = S2 the ideal of Hilbert-Schmidt operators on H. Then the definition of a double operator integral on S2 given in Section 2.1 coincides with the definition above for all φ ∈ A and self-adjoint operators A, B ∈ B(H). One could wonder whether Proposition 2.28 can be extended to a larger class of functions than A. A,B I As we have seen above, Tφ is a bounded operator on = B(H) if and only if φ ∈ A (see Theorem 2.12). Hence, Proposition 2.28 cannot be extended to a larger function class than A in general. However, for specific Banach ideals, e.g. ideals with the UMD property, results have been obtained for larger classes of functions [75,91] (see Theorem 2.13 (iii)).

2 Let p1, p2 : R → R be as before the coordinate projections given by p1(λ, µ) := 2 ∞ λ, p2(λ, µ) := µ for (λ, µ) ∈ R . Note that f ◦ p1, f ◦ p2 ∈ A for all f ∈ L (R). For selfadjoint operators A and B on a Hilbert space and for a Schatten von Neumann ideal I, the following lemma is [90, Lemma 3].

Lemma 2.30. Under the assumptions of Proposition 2.28, the following hold:

45 7→ A,B → I (i) The map φ Tφ is a morphism A B( ) of unital Banach algebras.

(ii) Let f ∈ L∞(R) and X ∈ B(X , Y). Then T A,B (X) = Xf(A) and T A,B (X) = f◦p1 f◦p2 A,B A,B f(B)X. In particular, Tp1 (X) = XA and Tp2 (X) = BX.

Proof. The structure of the proof is the same as that of [90, Lemma 3]. Linear- ity in (i) is straightforward. Fix φ1, φ2 ∈ A with representations as in (2.12), with corresponding measure spaces (Ωj, νj) and bounded Borel functions a1,j ∈ ∞ ∞ L (σ(A)×Ωj) and a2,j ∈ L (σ(B)×Ωj) for j ∈ {1, 2}. Then φ := φ1φ2 also has a representation as in (2.12), with Ω = Ω1 × Ω2, ν = ν1 × ν2 the product measure and a1 = a1,1a1,2, a2 = a2,1a2,2. By multiplicativity of the functional calculus for A, ( ) a1(A, (ω1, ω2)) = a1,1(·, ω1)a1,2(·, ω2) (A) = a1,1(A, ω1)a1,2(A, ω2)

for all (ω1, ω2) ∈ Ω, and similarly for a2(B, (ω1, ω2)). Applying this to (2.34) yields ∫ A,B Tφ (X)x = a2(B, ω)Xa1(A, ω)x dµ(ω) ∫Ω A,B = a2,1(B, ω1)Tφ2 (X)a1,1(A, ω1)x dµ1(ω1) Ω1 A,B A,B = Tφ1 (Tφ2 (X))x for all X ∈ I and x ∈ X, which proves (i). Part (ii) follows from (2.34) and the

A,B fact that Tφ is independent of the representation of φ. D

Schur multipliers on B(ℓp, ℓq)

∈ ∞ { }∞ ⊆ p p For p [1, ) let ej j=1 ℓ be the standard Schauder basis of ℓ , with ∑ P ∞ ∈ N the corresponding projections j(x) := xjej for x = k=1 xkek and j . For q ∈ [1, ∞], any operator X ∈ B(ℓp, ℓq) can be identified with an infinite matrix { }∞ ∈ N X = xjk j,k=1, where xjk := (X(ek), ej) for j, k . Recall that for an infinite { }∞ ∗ { } matrix M = mjk j,k=1 the product M X := mjksjk is the Schur product of the matrices M and X. The matrix M is said to be (p, q)-multipliers if the mapping X 7→ M ∗ X is a bounded operator on B(ℓp, ℓq). Note that a (2, 2)- multiplier is the same as a Schur multiplier on S2 defined in Section 2.1. We

46 denote by M(p, q) the Banach space of (p, q)-multipliers with the norm { } ∥M∥(p,q) := sup ∥M ∗ X∥B(ℓp,ℓq) : ∥X∥B(ℓp,ℓq) ≤ 1 .

Remark 2.31. We also consider (p, q)-multipliers M for p = ∞ and q ∈ [1, ∞]. ∈ q { }∞ Any operator X B(c0, ℓ ) corresponds to an inﬁnite matrix X = xjk j,k=1, and M is said to be a (∞, q)-multiplier if the mapping X 7→ M ∗ X is a bounded

q operator on B(c0, ℓ ). We deﬁne the Banach space M(∞, q) in the obvious way. Often we do not explicitly distinguish the case p = ∞ from 1 ≤ p < ∞, but the

p reader should keep in mind that for p = ∞ the space ℓ should be replaced by c0. ∥ ∥ ≥ | | Remark 2.32. It is straightforward to see that M (p,q) supj,k∈N mj,k for all p, q ∈ [1, ∞] and M ∈ M(p, q).

For p, q ∈ [1, ∞] and X ∈ B(ℓp, ℓq), deﬁne ∑ T∆(X) := PkXPj, (2.37) k≤j which is a well-deﬁned element of B(ℓr, ℓs) for suitable r, s ∈ [1, ∞] by Proposition

2.33 below. The operator T∆ is the (standard) triangular truncation (see [67]). ′ This operator can be identiﬁed with the following Schur multiplier. Let T∆ = { ′ }∞ ′ ≤ ′ tjk j,k=1 be a matrix given by tjk = 1 for k j and tjk = 0 otherwise. It is p q ′ clear that T∆ extends to a bounded linear operator on B(ℓ , ℓ ) if and only if T∆ is a (p, q)-multiplier. For n ∈ N and r, s ∈ [1, ∞] we will consider the operators

p q r s T∆,n ∈ B(B(ℓ , ℓ ),B(ℓ , ℓ )), given by ∑ p q T∆,n(X) := PkXPj,X ∈ B(ℓ , ℓ ). 1≤k≤j≤n

The dependence of the (p, q)-norm of T∆ on the indices p and q was determined in [11] and [67] (see also [104]), and is as follows.

Proposition 2.33. Let p, q ∈ [1, ∞]. Then the following statements hold.

(i) [11, Theorem 5.1] If p < q, 1 = p = q or p = q = ∞, then T∆ ∈ M(p, q).

(ii) [67, Proposition 1.2] If 1 ≠ p ≥ q ≠ ∞, then there is a constant C > 0 such that

∥T∆,n∥B(ℓp,ℓq)→B(ℓp,ℓq) ≥ C ln n

for all n ∈ N.

47 (iii) [11, Theorem 5.2] If 1 ≠ p ≥ q ≠ ∞, then for each s > q and r < p,

p q p s p q r q T∆ : B(ℓ , ℓ ) → B(ℓ , ℓ ) and T∆ : B(ℓ , ℓ ) → B(ℓ , ℓ )

are bounded.

Remark 2.34. In Proposition 2.33 (i), a stronger statement holds if p = 1 or ∞ { }∞ ∈ M q = . Then, for M = mjk j,k=1 a matrix, M (p, q) if and only if | | ∞ ∥ ∥ | | supj,k∈N mjk < , in which case M (p,q) = supj,k∈N mjk . This follows immediately from the well-known identities (see [12, p. 605, (2) and (3)])

( ∞ ) ∑ 1/q q ∥X∥B(ℓ1,ℓq) = sup |xjk| k∈N j=1

∈ ∞ { }∞ ∈ 1 q for q [1, ) and X = xjk j,k=1 B(ℓ , ℓ ), and

( ∞ ) ∗ ∑ 1/p p∗ ∥X∥B(ℓp,ℓ∞) = sup |xjk| j∈N k=1

∈ ∞ { }∞ ∈ p ∞ for p [1, ] and X = xjk j,k=1 B(ℓ , ℓ ) (with the obvious modiﬁcation for p = 1).

We will also need the following result, a generalization of [12, Theorem 4.1]. { }∞ f { e }∞ For a matrix M = mjk j,k=1, let M = mjk j,k=1 be obtained from M by repeating the ﬁrst column, i.e. me j1 = mj1 and me jk = mj(k−1) for j ∈ N and k ≥ 2.

∈ ∞ ≤ { }∞ Proposition 2.35. Let p, q, r, s [1, ] with r p. Let M = mjk j,k=1 be such that X 7→ M ∗ X is a bounded mapping B(ℓp, ℓq) → B(ℓr, ℓs). Then X 7→ Mf ∗ X is also a bounded mapping B(ℓp, ℓq) → B(ℓr, ℓs), with

f ∥M∥B(ℓp,ℓq)→B(ℓr,ℓs) = ∥M∥B(ℓp,ℓq)→B(ℓr,ℓs).

f f In particular, if M ∈ M(p, q) then M ∈ M(p, q) with ∥M∥(p,q) = ∥M∥(p,q).

Proof. The proof is almost identical to that of [12, Theorem 4.1], and the condi-

p p r r p/r tion r ≤ p is used to ensure that |x1| + |x2| ≤ (|x1| + |x2| ) for all x1, x2 ∈ C (with the obvious modiﬁcation for p = ∞ or r = ∞). D

48 Remark 2.36. By considering the transpose M ′ of a matrix M, and using that

∗ ∗ ∗ ∗ M ′ : B(ℓq , ℓp ) → B(ℓs , ℓr ) with

′ ∥ ∥ ∗ ∗ ∗ ∗ ∥ ∥ M B(ℓq ,ℓp )→B(ℓs ,ℓr ) = M B(ℓp,ℓq)→B(ℓr,ℓs),

Proposition 2.35 implies that the B(B(ℓp, ℓq),B(ℓr, ℓs))-norm of a matrix is invariant under row repetitions if s ≤ q. Moreover, since ∥X∥B(ℓp,ℓq) is invariant under permutations of the columns and rows of X ∈ B(ℓp, ℓq), rearrangements

p q r s of the rows and columns of M ∈ B(B(ℓ , ℓ ),B(ℓ , ℓ )) leave ∥M∥B(ℓp,ℓq)→B(ℓr,ℓs) invariant.

The following lemma is technical but is crucial to our main results in Chap- { }∞ { }∞ ∈ N ter 4. Let λ = λj j=1 and µ = µk k=1 be sequences of real numbers. For n λ,µ deﬁne T∆,n by ∑n ∑ λ,µ P P ∈ p q T∆,n(X) := kX j,X B(ℓ , ℓ ). j,k=1 µk≤λj

λ,µ We call T∆,n the triangular truncation associated to λ and µ.

∈ ∞ ≤ ≤ { }∞ Lemma 2.37. Let p, q, r, s [1, ] with r p and s q. Let λ = λj j=1 and { }∞ µ = µk k=1 be sequences of real numbers. Then

∥ λ,µ∥ ≤ ∥ ∥ T∆,n B(ℓp,ℓq)→B(ℓr,ℓs) T∆,n B(ℓp,ℓq)→B(ℓr,ℓs) for all n ∈ N.

λ,µ ∗ ∈ p q { }∞ Proof. Note that T∆,n(X) = M X for all X B(ℓ , ℓ ), where M = mjk j,k=1 is such that mjk = 1 if 1 ≤ j, k ≤ n and µk ≤ λj, and mjk = 0 otherwise. We show that ∥M∥B(ℓp,ℓq)→B(ℓr,ℓs) ≤ ∥T∆,n∥B(ℓp,ℓq)→B(ℓr,ℓs). Assume that M is non-zero, otherwise the statement is trivial. By Remark 2.36, rearrangement of the rows { }n and columns of M does not change its norm. Hence we may assume that λj j=1 { }n and µk k=1 are decreasing. Now M has the property that if mjk = 1 then mil = 1 for all i ≤ j and k ≤ l ≤ m2. By Proposition 2.35 and Remark 2.36, we may omit repeated rows and columns of M, and doing this repeatedly reduces M to T∆,N for some 1 ≤ N ≤ n. Noting that ∥T∆,N ∥B(ℓp,ℓq)→B(ℓr,ℓs) ≤ ∥T∆,n∥B(ℓp,ℓq)→B(ℓr,ℓs) concludes the proof. D

49 Chapter 3

Arazy conjecture concerning Schur multipliers on Schatten ideals

In this chapter we prove the following theorem, thus, we conﬁrm the Arazy conjecture stated in [7]. Observe that the special case of the Arazy conjecture, when q = ∞, 2 ≤ p < ∞ and 0 < α < 2 remains open, we will present more details about this case later in this chapter. This part of the thesis is based on the results published in [93] (a joint work with D. Potapov and F. Sukochev). ∈ 1 − { }∞ ∈ ∞ ∥ ∥ ≤ Recall that for f C ([ 1, 1]) and for λ = λj j=1 ℓ with λ ∞ 1 we denote { [1] }∞ Ψf,λ = f (λj, λk) j,k=1.

Theorem 3.1. Let 0 < p < ∞, 1 < q < ∞ and 0 < α, r < ∞ be such that 1 α 1 = + , p r q r 1 and let λ ∈ ℓ be such that ∥λ∥∞ ≤ 1. Then, for function f ∈ C ([−1, 1]) satisfying |f ′(t)| ≤ C|t|α, t ∈ [−1, 1] (3.1) for some constant C > 0, we have the following estimate

∥ ∥ ≤ ∥ ∥α Ψf,λ Sq→Sp const λ r , where the constant const is independent of λ.

50 Statement of the problem in terms of double operator integrals.

Let 0 < α < ∞. Throughout this chapter we assume that f is a continuously diﬀerentiable function on [−1, 1] satisfying (3.1). Let   f ′(t) ∈ − \{ } |t|α , t [ 1, 1] 0 , ω(t) :=  0, t = 0

Observe that the function ω(t) is bounded by C. Throughout this chapter the notation fα stands for a continuous bounded function on R such that fα is supported on [−2, 2] and coincides with f ′(t) = ω(t)|t|α on [−1, 1].

Remark. We shall consider the double operator integral T A , with A = A∗ and φfα

∥A∥∞ ≤ 1 (see Deﬁnition 2.17). However, according to Remark 2.22, we can

A A A identify T with T ′ . The latter operator coincides with T [1] , as it was already φfα φf f mentioned above (see (2.25)).

Our main result in this chapter is the following theorem, taken from [93] (a joint work with D. Potapov and F. Sukochev).

Theorem 3.2. Let 0 < p < ∞, 1 < q < ∞ and 0 < α, r < ∞ be such that 1 α 1 = + . p r q

∗ r Let A = A ∈ S be such that ∥A∥∞ ≤ 1. Then, for function f satisfying (3.1), we have the following estimate

A ≤ ∥ ∥α T [1] const A Sr , f Sq→Sp where the constant is independent of A.

The proof of Theorem 3.2 requires some preparation presented in the next subsection. We complete this subsection with the proof of Theorem 3.1. { }∞ Let λ = λj j=1 be a bounded sequence of real numbers. By Aλ we denote { }∞ the self-adjoint diagonal operator given by Aλ(ej) = λjej, where ej j=1 is a ﬁxed orthonormal basis in H. Before we prove Theorem 3.1, let us present the following lemma, which shows that in the special case A = Aλ the double operator integral given by Deﬁnition 2.17 is, in fact, a Schur multiplier.

51 { }∞ Lemma 3.3. Let λ = λj j=1 be a bounded sequence of real numbers. Suppose

Aλ Sq that φ is such that the operator integral Tφ is well-deﬁned and bounded from Sp ≤ ∞ { }∞ ∈ Sq into (0 < p, q ). Let Φλ = φ(λj, λk) j,k=1. Then for every X , we have

Aλ ∗ Tφ (X) = Φλ X.

In particular,

∥ Aλ ∥ ∥ ∥ Tφ Sq→Sp = Φλ Sq→Sp .

Proof. Let Pj be the one-dimensional self-adjoint projection on the element ej, j = 1, 2,... If A = Aλ, then the spectral projections of A are as follows ∑ E[a, b) = Pj, a < b ∈ R.

λj ∈[a,b)

q For ﬁxed j0, j1 ∈ N and X ∈ S we have the convergence ( ) ∑ l l P S (X)P = φ 0 , 1 E XE → φ(λ , λ )P XP as m → ∞ j0 φ,m j1 m m l0,m l1,m j0 j1 j0 j1 ∈ ls ls+1 λjs [ m , m ) ls∈Z, s=0,1 in the strong operator topology, where Sφ,m(X) is deﬁned in (2.19). Therefore, we have

Aλ Pj0 Tφ (X)Pj1 = φ(λj0 , λj1 )Pj0 XPj1 .

So ∑∞ Aλ ∈ Sq Tφ (X) = φ(λj, λk)PjXPk,X . j,k=1

The observation that xjk = PjXPk, j, k = 1, 2, ... completes the proof of the lemma. D

r Proof of Theorem 3.1. Consider the diagonal operator Aλ. Since λ ∈ ℓ , we have r Aλ ∈ S and ∥Aλ∥Sr = ∥λ∥r. Applying Theorem 3.2 we obtain that

Aλ ≤ ∥ ∥α ∥ ∥α T [1] const Aλ Sr = const λ r . f Sq→Sp

Applying Lemma 3.3, we conclude that

Aλ ∗ ∈ Sq Tf [1] (X) = Ψf,λ X, for any X , which completes the proof of the theorem. D

52 Proof of Theorem 3.2

The case α is an even integer

The following lemma is an analogue of [92, Theorem 17] and a crucial technical step in the proof of Theorem 3.2. Recall that by fα, (0 < α < ∞) we denote a continuous bounded function on R supported on [−2, 2] and coincides with f ′(t) = ω(t)|t|α on [−1, 1].

z Lemma 3.4. Let fz(x) = ω(x)|x| , z ∈ C be the analytic continuation of the mapping α 7→ fα and let m > 0 be an integer such that ℜz ≥ 2m, where ℜz denotes the real part of the complex number z. Let p < q satisfy one of the following conditions

(i) 0 < p < ∞, 1 < q < ∞;

(ii) 1 < p < ∞, q = ∞ and be such that 1 2m 1 = + , p r q r for some 0 < r < ∞. Let also A ∈ S be such that ∥A∥∞ ≤ 1. Suppose also that f satisﬁes (3.1). Then, the following estimate

A ≤ ∥ ∥2m ∥ ∥ ∈ Sq Tφ (X) const A Sr X Sq ,X (3.2) fz Sp holds, with a constant const independent of z, A and X.

Proof. Using the polynomial expansion ∑ | − |2m − m0 · m1 (1 t)x + ty = Cm0,m1 ((1 t)x) (ty) , m0+m1=2m (2m)! where Cm0,m1 = , m0! · m1!

53 we represent the momentum φfz via (2.24) as follows ∫ 1 − φfz (x, y) = fz((1 t)x + ty) dt ∫0 1 = ω((1 − t)x + ty) |(1 − t)x + ty|z dt ∫ 0 1 − = ω((1 − t)x + ty) |(1 − t)x + ty|z 2m+2m dt ∫0 1 2m = hz((1 − t)x + ty) |(1 − t)x + ty| dt ∫0 1 ∑ − − m0 · m1 = hz((1 t)x + ty) Cm0,m1 ((1 t)x) (ty) dt 0 m0+m∫1=2m ∑ 1 m0 m1 − m0 · m1 − = Cm0,m1 x y (1 t) t hz((1 t)x + ty) dt m +m =2m 0 0 ∑1 m0 m1 = Cm0,m1 x y φhz,Q (x, y) m +m =2m 0 ∑1

= Cm0,m1 ψ(x, y), m0+m1=2m where the function ψ is given by the equality

m0 m1 ∈ − ψ(x, y) := x y φhz,Q (x, y) , x, y [ 2, 2], the polynomial Q is Q(t) = (1 − t)m0 tm1 and the function hz is z−2m hz(x) = ω(x) |x| , for x ∈ [−1, 1], which is continuous and which vanishes outside the interval [−2, 2].

Observe that, since ℜz ≥ 2m, the function hz is uniformly bounded. Let us estimate an individual summand in the operator integral ∑ A A T = Cm ,m T , φfz 0 1 ψ m0+m1=2m

whose decomposition is provided by the representation φfz above and Deﬁni- tion 2.17. Fix m0, m1 ∈ N such that m0 + m1 = 2m.

1 m0 m1 1 (i). Applying subsequently (2.20), the H¨olderinequality with p = r + r + q ,

54 and then Theorem 2.21, we obtain

A m0 A m1 Tψ (X) Sp = A Tφ (X) A hz,Q Sp

≤ ∥ ∥m0 A ∥ ∥m1 ≤ ∥ ∥2m ∥ ∥ A Sr Tφ (X) A Sr const A Sr X Sq . hz,Q Sq

(ii). Applying subsequently (2.20), Theorem 2.21, and then the H¨olderin-

1 m0 m1 equality with p = r + r , we arrive at

A A m0 m1 m0 m1 ≤ ∥ ∥ p ≤ Tψ (X) p = Tφ (A XA ) const A XA S S hz,Q Sp

m0 m1 2m ≤ const ∥A∥Sr ∥X∥∞ ∥A∥Sr ≤ const ∥A∥Sr ∥X∥∞ .

Thus, the estimate (3.2) is established. The proof of the lemma is completed. D

The following corollary is an immediate consequence of Lemma 3.4 with z = α. It establishes the assertion of Theorem 3.2 for the special case, when α is an even number.

Corollary 3.5. Let α be an even integer and let p < q satisfy one of the following

(i) 0 < p < ∞, 1 < q < ∞;

(ii) 1 < p < ∞, q = ∞ and be such that 1 α 1 = + , p r q r for some 0 < r < ∞. Let A ∈ S be such that ∥A∥∞ ≤ 1. Suppose also that the function f below satisﬁes (3.1). Then, we have the following estimate

A ≤ ∥ ∥α T [1] const A Sr , f Sq→Sp where the constant is independent of A.

To prove the result for an arbitrary α, we shall use a complex interpolation method for quasi-Banach spaces Sp.

55 Complex method of interpolation for quasi-Banach spaces

We shall consider a complex interpolation method for the spaces ℓp and Sp (0 < p ≤ ∞). In fact, the construction of complex interpolation for arbitrary quasi-Banach spaces has many problems (see e.g. [17]), that occur due to the fact that general quasi-Banach spaces do not satisfy the Maximum Modulus Principle. However, in [57] it was proved that for the spaces ℓp and Sp (0 < p < 1) the Max- imum Modulus Principle holds (for details see [57, Theorem 4.10] and references therein). Therefore, the complex interpolation theory can be developed for these spaces, similarly to the Banach setting. We shall now brieﬂy recall the complex

p method of interpolation. From now on we suppose that X0 and X1 are both ℓ or Sp spaces, 0 < p ≤ ∞.

For a compatible pair of (quasi-)Banach spaces (X0, X1), and 0 < θ < 1, the complex interpolation (quasi-)Banach space (X0, X1)[θ] is deﬁned as follows (see e.g. [66, Section 4.1]):

(X0, X1)[θ] := {x ∈ X0 + X1 : ∃f ∈ F(X0, X1) such that x = f(θ)} .

Here the class F(X0, X1) consists of all bounded and continuous functions f : ¯ S 7→ X0 + X1 deﬁned on the closed strip

S¯ := {z ∈ C : 0 ≤ ℜz ≤ 1} such that f is analytic on the open strip S := {z ∈ C : 0 < ℜz < 1} and such that t → f(j + it) ∈ Xj, j = 0, 1 are continuous and bounded functions on the real line. We provide F(X0, X1) with the (quasi-)norm

∥f∥F(X ,X ) := max{c0(f), c1(f)}, 0 1 j=0,1 where c (f) := sup ∈R ∥f(j + it)∥ , j = 0, 1. j t Xj By setting

∥ ∥ { ∥ ∥ ∈ F X X } x [θ] := inf f F(X0,X1) : f(θ) = x, f ( 0, 1)

X X ∥ · ∥ ∥ ∥ ≤ we obtain a (quasi-)Banach space (( 0, 1)[θ], [θ]). The fact that x [θ] 1−θ θ ∈ F X X const c0 (f) c1(f), where f(θ) = x, f ( 0, 1) can be proved similarly to the case when X0, X1 are Banach spaces.

56 The proof of the following lemma is a verbatim repetition of [16, Theorem 5.1.2] or [17, Theorem 3].

Lemma 3.6. Let 0 < θ < 1. If 0 < p0, p1 < ∞, then

1 1 − θ θ p0 p1 p (ℓ , ℓ )[θ] = ℓ , where = + . p p0 p1

In the case of Schatten classes Sp we also present a similar result, which follows from Lemma 3.6 and by verbatim repetition of the proof of [6, Theorem 2.4]. We omit the proof.

Lemma 3.7. Let 0 < θ < 1. If 0 < p0, p1 < ∞, then

1 1 − θ θ p0 p1 p (S , S )[θ] = S , where = + . p p0 p1

Now we are fully equipped to prove Theorem 3.2. Due to the result of Corol- lary 3.1, we omit the case when α is even.

Proof of Theorem 3.2. Let n ∈ N ∪ {0} be such that 2n < α < 2n + 2. Let us select p1 < p0 satisfying the equalities

1 2n 1 1 2n + 2 1 = + , and = + . (3.3) p0 α q p1 α q

α−2n Setting 0 < θ := 2 < 1, we obtain that

1 (1 − θ) + 1 θ = 1 and p0 p1 p 2n(1 − θ) + (2n + 2)θ = α.

Fix X ∈ Sq and set F (z) := T A (X), φf2z+2n where f2z+2n is deﬁned in Lemma 3.4 with the substitution z 7→ 2z + 2n. Suppose ﬁrstly that 2 < α. Then, by Lemma 3.4 (i), for any X ∈ Sq we have the following two estimates

2n ∥ ∥ ≤ ∥ ∥ ∥ ∥ q F (it) Sp0 const A Sr X S ,

2n+2 ∥ ∥ ≤ ∥ ∥ ∥ ∥ q ∈ R F (it + 1) Sp1 const A Sr X S , t . (3.4)

57 Now assume that 0 < α < 2. Then n = 0, and so the ﬁrst equality in (3.3) implies that p0 = q. In this case the ﬁrst estimate in (3.4) is replaced with

∥F (it)∥Sq ≤ const ∥X∥Sq , t ∈ R (3.5)

which holds due to Theorem 2.21 with h = fit, appealing to the fact that 1 < q < ∞. The second estimate in (3.4) continues to hold in this case. Thus, (3.4) holds for all 0 < α < ∞ and we are now in a position to use complex interpolation. By Lemma 3.7, using (3.4), we conclude that

A ∥ ∥ Tφ (X) = F (θ) Sp fα Sp 2n(1−θ) (2n+2)θ α ≤ const ∥A∥Sr ∥A∥Sr ∥X∥Sq = const ∥A∥Sr ∥X∥Sq , that completes the proof of Theorem 3.2. D

Remarks in conclusion

Arazy-Friedman inequality

The Arazy conjecture in the case 0 < p < ∞, 1 < q < ∞ and f(x) = x1+α, x ∈ [0, 1], 0 < α < ∞ is answered in the aﬃrmative in [8, Theorem 3.6]. In A,B α α ∥ ∥Sq→Sp ∥ ∥ ∥ ∥ particular, the fact that the norm Tf [1] is majorized by A Sr + B Sr , is established. In the following theorem we prove [8, Theorem 3.6] for all functions satisfying (3.1).

Theorem 3.8. Let 0 < p < ∞, 1 < q < ∞ and 0 < α, r < ∞ be such that

1 α 1 = + , p r q

∗ ∗ r and let A = A , B = B ∈ S , with ∥A∥∞, ∥B∥∞ ≤ 1. Let also f satisfy (3.1). Then, we have the following estimate

A,B ≤ ∥ ∥r ∥ ∥r α/r T [1] const ( A Sr + B Sr ) , (3.6) f Sq→Sp where the constant is independent of A, B.

58 Proof. Using the 2 × 2 matrix approach from Remark 2.19, by Theorem 3.2, we have that

A,B H˜ ˜ T [1] (X) = T [1] (X) f Sp f S˜p ≤ ∥ ˜ ∥α ∥ ˜∥ ∥ ∥r ∥ ∥r α/r ∥ ∥ const H S˜r X S˜q = const ( A Sr + B Sr ) X Sq , where   0 X X˜ =   ,X ∈ Sq. 0 0

Thus, we obtain (3.6). D

The case q = ∞

Similarly, we can prove Theorem 3.2 and Theorem 3.8 in the case when 1 < p < ∞, q = ∞, with using Lemma 3.4 (ii) in place of Lemma 3.4 (i). However, our method allows us to treat the case α ≥ 2 only, since 0 < α < 2, the ﬁrst equality in (3.3) implies that p0 = q = ∞. Therefore, Theorem 2.21 is not applicable anymore and the estimate (3.5) is not clear.

59 Chapter 4

Commutator and Lipschitz estimates in B(X , Y)

In this chapter, we study commutator and Lipschitz estimates in B(X , Y) and Banach ideals I. We also consider a particular case X = ℓp, Y = ℓq, and study such estimates for the absolute value function. The results are based on the preliminary material given in Section 2.3. In particular, we use the double operator integral deﬁned by (2.34). All main results presented in this chapter are taken from [97] (a joint work with J. Rozendaal and F. Sukochev).

4.1 General estimates

Theorem 4.1. Let X and Y be separable Banach spaces such that X or Y has the bounded approximation property, and let I be a Banach ideal in B(X , Y) with the strong convex compactness property. Let A ∈ Bs(X ) and B ∈ Bs(Y) be such that σ(A) ∪ σ(B) ⊆ R, and let f ∈ L∞(R) be such that f [1] ∈ A. Then

[1] ∥f(B)X − Xf(A)∥I ≤ 16 spec(A) spec(B) f ∥BX − XA∥I (4.1) A for all X ∈ B(X , Y) such that BX − XA ∈ I. In particular, if X = Y and B − A ∈ I,

[1] ∥f(B) − f(A)∥I ≤ 16 spec(A) spec(B) f ∥B − A∥I . A

60 [1] Proof. Note that (p2 − p1)f = f ◦ p2 − f ◦ p1. By Lemma 2.30,

− A,B − A,B A,B f(B)X Xf(A) = T ◦ (X) T ◦ (X) = T [1] (X) f p2 f p1 (p2−p1)f A,B − A,B A,B A,B − A,B = T [1] (X) T [1] (X) = T [1] (T (X) T (X)) p2f p1f f p2 p1 A,B − = Tf [1] (BX XA) for all X ∈ B(X , Y) such that BX − XA ∈ I. Proposition 2.28 now concludes the proof. D

In the special case when X and Y are Hilbert spaces and A and B are self-adjoint operators, the result of Theorem 4.1 extends a number of results from [21, 90] to all Banach ideals with the strong convex compactness property. As mentioned in Section 2.3, the latter class includes all separable ideals and all so-called maximal operator ideals, which in turn is a large class of ideals containing the absolutely (p, q)-summing operators, the integral operators, and more [35, p. 203]. Note that, for self-adjoint operators, we can improve the estimate in (4.1) (see (2.36)).

Corollary 4.2. Let A ∈ B(X ) and B ∈ B(Y) be self-adjoint operators on separable Hilbert spaces X and Y. Let I be a Banach ideal in B(X , Y) with the strong convex compactness property, and let f ∈ L∞(R) be such that f [1] ∈ A. Then

[1] ∥f(B)X − Xf(A)∥I ≤ f ∥BX − XA∥I A for all X ∈ B(X , Y) such that BX − XA ∈ I. In particular, if X = Y and B − A ∈ I, then we have

[1] ∥f(B) − f(A)∥I ≤ f ∥B − A∥I . A

Combining Theorem 4.1 with Lemma 2.10 yields the following generalization of [79, Theorem 4].

Corollary 4.3. Let X and Y be separable Banach spaces such that X or Y has the bounded approximation property, and let I be a Banach ideal in B(X , Y) with ∈ 1 ∈ X the strong convex compactness property. Let f B∞,1, and let A Bs( ) and

61 B ∈ Bs(Y) be such that σ(A) ∪ σ(B) ⊆ R. Then there exists a universal constant C ≥ 0 such that

∥f(B)X − Xf(A)∥I ≤ C spec(A) spec(B)∥f∥ 1 ∥BX − XA∥I (4.2) B∞,1 holds for all X ∈ B(X , Y) such that BX − XA ∈ I. In particular, if X = Y and B − A ∈ I, then we have

∥f(B) − f(A)∥I ≤ C spec(A) spec(B)∥f∥ 1 ∥B − A∥I . B∞,1

In the case where the Banach ideal I is the space B(X , Y) of bounded operators from X to Y, we obtain the following corollary.

Corollary 4.4. There exists a universal constant C ≥ 0 such that the following holds. Let X and Y be separable Banach spaces such that either X or Y has the ∈ 1 ∈ X ∈ Y bounded approximation property. Let f B∞,1, and let A Bs( ),B Bs( ) be such that σ(A) ∪ σ(B) ⊆ R. Then

∥f(B)X − Xf(A)∥ X Y ≤ C spec(A) spec(B)∥f∥ 1 ∥BX − XA∥ X Y (4.3) B( , ) B∞,1 B( , ) for all X ∈ B(X , Y). In particular, if X = Y then

∥f(B) − f(A)∥ X ≤ C spec(A) spec(B)∥f∥ 1 ∥B − A∥ X . B( ) B∞,1 B( )

Remark 4.5. Corollaries 4.3 and 4.4 yield global estimates, in the sense that (4.2) and (4.3) hold for all scalar type operators A and B with real spectra, and the constant in the estimate depends on A and B only through their spectral constants spec(A) and spec(B). Local estimates follow if f ∈ L∞(R) is contained in the Besov class locally. More precisely, given scalar type operators A ∈ Bs(X ) ∈ Y ∈ 1 and B Bs( ) with real spectra, suppose there exists g B∞,1 with g(s) = f(s) for all s ∈ σ(A) ∪ σ(B). Then (with notation as in Corollary 4.3), Theorem 4.1 yields

∥f(B)X − Xf(A)∥I ≤ C spec(A) spec(B)∥g∥ 1 ∥BX − XA∥I B∞,1 for all X ∈ B(X , Y) such that BX − XA ∈ I.

62 4.2 Spaces with unconditional basis

In this section we prove some results for speciﬁc scalar type operators, namely operators which are diagonalizable with respect to an unconditional Schauder basis. These results will be used in later sections.

Diagonalizable operators

X { }∞ ⊆ X Let be a Banach space with an unconditional Schauder basis ej j=1 (the deﬁnition of an unconditional Schauder basis may be found e.g. in [68]).

Recall that for j ∈ N, let Pj ∈ B(X ) be the projection given by Pj(x) := xjej ∑ ∞ ∈ X for all x = k=1 xkek . ∑ ∥ P ∥ Assumption 4.6. For simplicity, assume in this section that j∈N j B(X ) = 1 for all non-empty N ⊆ N. This condition is satisﬁed in the examples we consider in the later sections, and simpliﬁes the presentation. For general unconditional bases there are additional constants in the estimates.

Recall Deﬁnition 1.3 that an operator A ∈ B(X ) is diagonalizable (with re- { }∞ ∈ X { }∞ ∈ ∞ spect to ej j=1) if there exists U B( ) invertible and a sequence λj j=1 ℓ of complex numbers such that ∑∞ −1 UAU x = λjPjx, x ∈ X , j=1 { }∞ where the series converges since ek k=1 is unconditional (see [99, Lemma 16.1]). { }∞ In this case A is a scalar type operator, with point spectrum λj j=1, σ(A) = { }∞ λj j=1 and the spectral measure E is given by ∑ −1 E(ς) = U PjU (4.4)

λj ∈ς for ς ⊆ C being Borel. The set of all diagonalizable operators on X is denoted X { }∞ by Bd( ). We do not explicitly specify the basis ej j=1 with respect to which an operator is diagonalizable, since this basis will be ﬁxed throughout. Often we ∈ X { }∞ write A Bd( , λ, U) to identify the operator U and the sequence λ = λj j=1 ∞ from above. For A ∈ Bd(X , λ, U) and f ∈ L (C), it follows from (2.29) that ( ∑∞ ) −1 f(A) = U f(λj)Pj U. (4.5) j=1

63 Since any Banach space with a Schauder basis is separable and has the bounded approximation property, we can apply the results from the previous section to diagonalizable operators, and obtain, for instance, the following corollary, which presents a signiﬁcant extension of classical Peller’s [78] and Birman-Solomyak’s [19] results which are only available for the special case when the spaces X and Y coincide with a Hilbert space.

Corollary 4.7. There exists a universal constant C ≥ 0 such that the following holds. Let X and Y be Banach spaces with unconditional Schauder bases, and let I be a Banach ideal in B(X , Y) with the strong convex compactness property. ∈ 1 ∈ X ∈ Y ∪ ⊆ R Let f B∞,1, and let A Bd( ) and B Bd( ) be such that σ(A) σ(B) . Then

∥f(B)X − Xf(A)∥I ≤ C spec(A) spec(B)∥f∥ 1 ∥BX − XA∥I B∞,1 for all X ∈ B(X , Y) such that BX − XA ∈ I. In particular, if X = Y and B − A ∈ I,

∥f(B) − f(A)∥I ≤ C spec(A) spec(B)∥f∥ 1 ∥B − A∥I . B∞,1

Since this result does not apply to the absolute value function (and neither does the more general Theorem 4.1), and because of the importance of the absolute value function, we will now study Lipschitz estimates for more general functions. Y { }∞ Let be a Banach space with an unconditional Schauder basis fk k=1, and ∑ Q ∈ Y Q ∞ ∈ Y let the projections k B( ) be given by k(y) := ykfk for all y = l=1 ylfl ∈ N { }∞ { }∞ and k . Let λ = λj j=1 and µ = µk k=1 be sequences of real numbers, and R2 → C ∈ N λ,µ ∈ X Y let φ : . For n , deﬁne Tφ,n B(B( , )) by

∑n λ,µ Q P ∈ X Y Tφ,n (X) := φ(λj, µk) kX j,X B( , ). j,k=1

λ,µ ∈ I I X Y Note that Tφ,n B( ) for each Banach ideal in B( , ).

Lemma 4.8. Let X and Y be Banach spaces with unconditional Schauder bases, I X Y { }∞ { }∞ and let be a Banach ideal in B( , ). Let λ = λj j=1 and µ = µk k=1 be

64 ∞ sequences of real numbers, and let A ∈ Bd(X , λ, U), B ∈ Bd(Y, µ, V ), f ∈ L (R) and n ∈ N. Then

−1 λ,µ −1 ∥ − ∥I ≤ ∥ ∥ X ∥ ∥ Y ∥ − ∥I f(B)Xn Xnf(A) U B( ) V B( ) Tf [1],n(V (BX XA)U ) ∑ ∈ X Y − ∈ I n −1Q −1P for all X B( , ) with BX XA , where Xn := j,k=1V kVXU jU.

Proof. Let X ∈ B(X , Y) be such that BX − XA ∈ I. For brevity we write

−1 −1 Pj := U PjU ∈ B(X ) and Qk := V QkV ∈ B(Y) for j, k ∈ N. By (4.5), and using that PjPk = 0 and QjQk = 0 for j ≠ k, we have ∑∞ ( ∑n ) ∑∞ ( ∑n ) f(B)Xn − Xnf(A) = f(µk)Qk QlXPi − f(λj) QlXPi Pj k=1 i,l=1 j=1 i,l=1 ∑n = (f(µk) − f(λj))QkXPj j,k=1 ∑n ∑ f(µk) − f(λj) = (µkQkXPj − λjQkXPj) µk − λj j,k=1 µk≠ λj ∑n (( ∑∞ ) ( ∑∞ )) [1] = f (λj, µk)Qk µlQl X − X λiPi Pj j,k=1 l=1 i=1 ∑n [1] = f (λj, µk)Qk(BX − XA)Pj j,k=1 −1 A,B − −1 = V Tf [1] (V (BX XA)U )U.

The proof is concluded by appealing to the ideal property of I. D

For a sequence λ of complex numbers and A ∈ Bd(X , λ, U), deﬁne { } −1 KA := inf ∥U∥B(X )∥U ∥B(X ) : A ∈ Bd(X , λ, U) .

We will call KA the diagonalizability constant of A. Using the unconditionality of the Schauder basis of X and Assumption 4.6, one can show that KA does not depend on the speciﬁc ordering of the sequence λ. Since the sequence λ is, up to ordering, uniquely determined by A (it is the point spectrum of A),

KA only depends on A. Moreover, by Assumption 4.6 and (4.4), ∥E(ς)∥B(X ) ≤ −1 ∥U ∥B(X )∥U∥B(X ) for all ς ⊆ C Borel and U ∈ B(X ) such that A ∈ Bd(X , λ, U), where E is the spectral measure of A. Hence

spec(A) ≤ KA, (4.6)

65 where spec(A) is the spectral constant of A from Section 2.3. We now derive commutator estimates for A and B in the operator norm, under a boundedness assumption which will be veriﬁed for speciﬁc X and Y in later sections.

Proposition 4.9. Let X and Y be Banach spaces with unconditional Schauder { }∞ { }∞ bases, let λ = λj j=1 and µ = µk k=1 be sequences of real numbers and let ∞ A ∈ Bd(X , λ, U), B ∈ Bd(Y, µ, V ) and f ∈ L (R). Suppose that

λ,µ ∥ ∥ X Y → X Y ∞ C := sup Tf [1],n B( , ) B( , ) < . (4.7) n∈N Then

∥f(B)X − Xf(A)∥B(X ,Y) ≤ CKAKB∥BX − XA∥B(X ,Y) for all X ∈ B(X , Y).

Proof. Let X ∈ B(X , Y) and for n ∈ N let Xn ∈ B(X , Y) be as in Lemma 4.8.

It is straightforward to show that, for each x ∈ X , Xnx → Xx as n → ∞. Hence f(B)Xnx − Xnf(A)x → f(B)Xx − Xf(A)x as n → ∞, for each x ∈ X . Lemma 4.8 and (4.7) now yield

∥f(B)X − Xf(A)∥B(X ,Y) ≤ lim sup ∥f(B)Xn − Xnf(A)∥B(X ,Y) n→∞ −1 −1 ≤ C∥U∥∥V ∥∥V (BX − XA)U ∥B(X ,Y)

−1 −1 ≤ C∥U∥∥U ∥∥V ∥∥V ∥ ∥BX − XA∥B(X ,Y).

Taking the inﬁmum over U and V concludes the proof. D

Remark 4.10. Proposition 4.9 also holds for more general Banach ideals in B(X , Y). I X Y { }∞ ⊆ Indeed, let be a Banach ideal in B( , ) with the property that, if Xm m=1 I is an I-bounded sequence which SOT-converges to some X ∈ B(X , Y) as → ∞ ∈ I ∥ ∥ ≤ ∥ ∥ m , then X with X I lim supm→∞ Xm I . If

λ,µ ∥ ∥I→I ∞ C := sup Tf [1],n < n∈N then the proof of Proposition 4.9 shows that

∥f(B)X − Xf(A)∥I ≤ CKAKB∥BX − XA∥I for all X ∈ B(X , Y) such that BX − XA ∈ I.

66 Estimates for the absolute value function

It is known that Lipschitz estimates for the absolute value function are related to estimates for so-called triangular truncation operators. For example, in [62] and [39] it is shown that the boundedness of the standard triangular truncation on many operator spaces is equivalent to Lipschitz estimates for the absolute value function. We prove that triangular truncation operator is related to Lipschitz estimates for the absolute value function in our setting as well. { }∞ { }∞ Let λ = λj j=1 and µ = µk k=1 be sequences of real numbers, and let X Y {P }∞ {Q }∞ ∈ N , , j j=1 and k k=1 be as before. For n recall that the operator λ,µ ∈ X Y T∆,n B(B( , )) is deﬁned by

∑n ∑ λ,µ Q P ∈ X Y T∆,n(X) := kX j,X B( , ). (4.8) j,k=1 µk≤λj

λ,µ We call T∆,n the triangular truncation associated to λ and µ. For f(t) := |t| for t ∈ R, deﬁne f [1] : R2 → C by   |t|−|s| ̸ [1] ---t−s if t = s f (t, s) :=  . 1 otherwise

λ,µ λ,µ The following result relates the norm of Tf [1],n to that of T∆,n.

Proposition 4.11. Let X and Y be Banach spaces with unconditional Schauder bases and let I be a Banach ideal in B(X , Y) with the strong convex compactness property. Let λ and µ be bounded sequences of real numbers. Let f(t) := |t| for t ∈ R. Then there exists a universal constant c ≥ 0 such that ( ) λ,µ λ,µ ∥ ∥I ≤ ∥ ∥I ∥ ∥I Tf [1],n(X) c X + T∆,n(X)

∈ N ∈ I ∥ λ,µ ∥ holds for all n and X . In particular, if supn∈N T∆,n(X) B(X ,Y)→B(X ,Y) < ∞ then (4.7) holds.

∈ N ∈ I { }∞ { }∞ Proof. Let n and X , and let λ = λj j=1 and µ = µk k=1. Throughout the proof we will only consider λj and µk for 1 ≤ j, k ≤ n, but to simplify the

67 λ,µ presentation we will not mention this explicitly. We can decompose Tf [1],n(X) as ∑ ∑ λ,µ Q P − µk + λj Q P Tf [1],n(X) = kX j kX j+ µk − λj λk,µk≥0 µk<0<λj ∑ ∑ ∑ µk + λj QkXPj − QkXPj + QkXPj. µk − λj λj <0<µk λk,µk≤0 λk,µk=0 Note that some of these terms may be zero. By the ideal property of I and Assumption 4.6,

∑ ∑ ∑ QkXPj ≤ Qk ∥X∥I Pj ≤ ∥X∥I . (4.9) I B(Y) B(X ) λj ,µk≥0 µk≥0 λj ≥0 ∑ ∑ Similarly, ∥ Q XP ∥I and ∥ Q XP ∥I are each bounded by λk,µk≤0 k j λk,µk=0 k j

∥X∥I . To bound the other terms it is suﬃcient to show that ( ) ∑ µ − λ k j Q P ≤ ′ ∥ ∥ ∥ λ,µ ∥ kX j C X I + T∆,n(X) I µk + λj I λj ,µk>0 for some universal constant C′ ≥ 0. Indeed, replacing λ by −λ and µ by −µ then yields the desired conclusion. Let ∑ µk − λj Φ(X) := QkXPj, µk + λj λj ,µk>0

∈ 1,2 R 2 ∈ R and deﬁne g W ( ) by setting g(t) := e|t|+1 for t . Note that Φ(X) is equal to ∑ ( ( ) ) ∑ ( ( )) λj λj g log − 1 QkXPj + 1 − g log QkXPj. µk µk 0<µk≤λj 0<λj <µk ∑ R2 → C ∞ P ∈ X Now let ψg : be as in (2.16), and let A := j=1 λj j B( ) and ∑∞ B := µ Q ∈ B(Y). Let T A,B be as in (2.34). One can check that k=1 k k ψg ∑ Φ(X) =T A,B(T λ,µ(X)) − Q T λ,µ(X)P + ψg ∆,n k ∆,n j λj ,µ >0 ∑ k Q (X − T λ,µ(X))P − T A,B(X − T λ,µ(X)). k ∆,n j ψg ∆,n λj ,µk>0 Since each Banach space with a Schauder basis is separable and has the bounded approximation property, Lemma 2.11 and Proposition 2.28 yield √ A,B λ,µ λ,µ ∥T (T (X))∥I ≤ 16 2 spec(A) spec(B)∥g∥ 1,2 R ∥T (X))∥I . ψg ∆,n W ( ) ∆,n

68 By (4.6), spec(A) = spec(B) = 1. Similarly, √ ( ) A,B λ,µ λ,µ ∥T (S − T (S))∥I ≤ 16 2∥g∥W1,2(R) ∥S∥I + T (S)) . ψg ∆,n ∆,n I

By the same arguments as in (4.9), ∑ ∑ λ,µ λ,µ QkT (X)Pj + Qk(X − T (X))Pj ∆,n I ∆,n λj ,µk>0 λj ,µk>0 λ,µ ≤2∥X∥I + T (X) . ∆,n I

Combining all these estimates yields ( √ )( ) λ,µ ∥Φ(X)∥I ≤ 2 + 32 2∥g∥W1,2(R) ∥X∥I + T (X) , ∆,n I as desired. D

4.3 Absolute value function on B(ℓp, ℓq)

In this section we study the absolute value function on B(ℓp, ℓq). We obtain the commutator estimate (1.10) for the absolute value function and X = ℓp and Y = ℓq with p < q, and we obtain (1.9) for each Lipschitz function and X = B(ℓ1) or X = B(c0). We also obtain results for p ≥ q. The key idea of the proof is entirely different from the techniques used in [34, 39, 62, 75], which are based on a special geometric property of the reflexive Schatten von Neumann ideals (the UMD-property), a property which B(ℓp, ℓq) does not have. Instead, we prove our results by relating estimates for the operators from (4.8) to the standard triangular truncation operator, defined in (2.37) above. For this we use the theory of Schur multipliers on B(ℓp, ℓq) developed in [12]. We then appeal to results from [11] about the boundedness of the standard triangular truncation on B(ℓp, ℓq).

4.3.1 Case p < q

We now combine the theory from the previous sections to deduce our main results.

69 Theorem 4.12. Let p, q ∈ [1, ∞] with p < q, and let f(t) := |t| for t ∈ R. Then there exists a constant c ≥ 0 such that the following holds (where ℓ∞ should be

p q replaced by c0). Let A ∈ Bd(ℓ ) and B ∈ Bd(ℓ ) have real spectra. Then

∥f(B)X − Xf(A)∥B(ℓp,ℓq) ≤ cKAKB∥BX − XA∥B(ℓp,ℓq) for all X ∈ B(ℓp, ℓq).

Proof. Simply combine Propositions 4.9 and 4.11, Lemma 2.37 and Proposition

2.33 (i), using that ∥T∆,n∥(p,q) ≤ ∥T∆∥(p,q) for all n ∈ N. D

We can deduce a stronger statement if p = 1 or q = ∞ in Theorem 4.12.

∞ Theorem 4.13. Let p, q ∈ [1, ∞] with p = 1 or q = ∞ (with ℓ replaced by c0). p q Let A ∈ Bd(ℓ ) and B ∈ Bd(ℓ ) have real spectra, and let f be Lipschitz on R. Then

∥f(B)X − Xf(A)∥B(ℓp,ℓq) ≤ KAKB∥f∥Lip∥BX − XA∥B(ℓp,ℓq) (4.10) for all X ∈ B(ℓp, ℓq). In particular, for p = q = 1,

∥f(B) − f(A)∥B(ℓ1) ≤ KAKB∥f∥Lip∥B − A∥B(ℓ1), and for p = q = ∞,

∥ − ∥ ≤ ∥ ∥ ∥ − ∥ f(B) f(A) B(c0) KAKB f Lip B A B(c0).

{ }∞ { }∞ ∈ Proof. Let λ = λj j=1 and µ = µk k=1 be real sequences such that A p q p q Bd(ℓ , λ, U) and B ∈ Bd(ℓ , µ, V ) for certain U ∈ B(ℓ ) and V ∈ B(ℓ ). By λ,µ ∥ ∥ p q → p q ≤ ∥ ∥ Proposition 4.9, it suﬃces to prove that supn∈N Tf [1],n B(ℓ ,ℓ ) B(ℓ ,ℓ ) f Lip. ∈ N λ,µ ∗ ∈ p q Fix n and note that Tf [1],n(S) = M S for all S B(ℓ , ℓ ), where { }∞ [1] ≤ ≤ M = mjk j,k=1 is the matrix given by mjk = f (λj, µk) for 1 j, k n, and mjk = 0 otherwise. Then

[1] sup |mjk| ≤ sup |f (λj, µk)| ≤ ∥f∥Lip. j,k∈N j,k∈N

Remark 2.34 now concludes the proof. D

70 Remark 4.14. Theorem 4.13 shows that each Lipschitz function f is operator

1 Lipschitz on ℓ and c0, in the following sense. For ﬁxed M ≥ 1 and f Lipschitz on R, there exists a constant C ≥ 0 such that

∥f(B) − f(A)∥B(ℓ1) ≤ C∥B − A∥B(ℓ1)

1 for all A, B ∈ Bd(ℓ ) with real spectra such that KA,KB ≤ M, and C is independent of A and B. Similarly for c0. For p < q an analogous statement holds. By considering A, f(A) ∈ B(ℓp) and B, f(B) ∈ B(ℓq) as operators from ℓp to ℓq, and by letting X be the inclusion mapping ℓp ,→ ℓq in Theorems 4.12 and 4.13, one can suggestively write

∥f(B) − f(A)∥B(ℓp,ℓq) ≤ C∥B − A∥B(ℓp,ℓq),

p q for all A ∈ Bd(ℓ ) and B ∈ Bd(ℓ ) with real spectra and KA,KB ≤ M. Here f is the absolute value function for general p < q in [1, ∞] and any Lipschitz function if p = 1 or q = ∞. This remark also applies to Corollaries 4.15 and 4.16 below.

In the case of Theorems 4.12 and 4.13 where p = 2 or q = 2, we can apply our results to compact self-adjoint operators. By the spectral theorem, any compact self-adjoint operator A ∈ B(ℓ2) has an orthonormal basis of eigenvectors, and

2 therefore A ∈ Bd(ℓ , λ, U) for some sequence λ of real numbers and an isometry U ∈ B(ℓ2). Thus Theorems 4.12 and 4.13 yield the following corollaries.

Corollary 4.15. Let p ∈ (1, 2). Then there exists a constant c ≥ 0 such that the

p 2 following holds. Let A ∈ Bd(ℓ ) have real spectrum and let B ∈ B(ℓ ) be compact and self-adjoint. Then

∥f(B)X − Xf(A)∥B(ℓp,ℓ2) ≤ cKA∥BX − XA∥B(ℓp,ℓ2) for all X ∈ B(ℓp, ℓ2), where f(t) := |t| for t ∈ R. Moreover,

∥f(B)X − Xf(A)∥B(ℓ1,ℓ2) ≤ KA∥f∥Lip∥BX − XA∥B(ℓ1,ℓ2)

1 1 2 2 for each A ∈ Bd(ℓ ) and X ∈ B(ℓ , ℓ ), each compact self-adjoint B ∈ B(ℓ ) and each Lipschitz function f.

71 Corollary 4.16. Let q ∈ (2, ∞). Then there exists a constant C ≥ 0 such that the

2 q following holds. Let A ∈ B(ℓ ) be compact and self-adjoint, and let B ∈ Bd(ℓ ) have real spectrum. Then

∥f(B)X − Xf(A)∥B(ℓ2,ℓq) ≤ CKB∥BX − XA∥B(ℓ2,ℓq) for all X ∈ B(ℓ2, ℓq), where f(t) := |t| for t ∈ R. Moreover,

∥ − ∥ 2 ≤ ∥ ∥ ∥ − ∥ 2 f(B)X Xf(A) B(ℓ ,c0) KB f Lip BX XA B(ℓ ,c0)

2 2 for each compact self-adjoint A ∈ Bd(ℓ ), each B ∈ Bd(c0) and X ∈ B(ℓ , c0), and each Lipschitz function f.

4.3.2 Case p ≥ q

We now examine the absolute value function f on B(ℓp, ℓq) for p ≥ q, and obtain the following result.

Proposition 4.17. Let p, q ∈ (1, ∞] with p ≥ q. Then for each s < q there exists a constant C ≥ 0 such that the following holds (where ℓ∞ should be replaced by

p q c0). Let A ∈ Bd(ℓ , λ, U) and B ∈ Bd(ℓ , µ, V ) have real spectra. Then

−1 −1 ∥f(B)X − Xf(A)∥B(ℓp,ℓq) ≤C∥U∥B(ℓp)∥V ∥B(ℓq)∥V (BX − XA)U ∥B(ℓp,ℓs) for all X ∈ B(ℓp, ℓq) such that V (BX − XA)U −1 ∈ B(ℓp, ℓs). In particular, if p = q and V (B − A)U −1 ∈ B(ℓp, ℓs), then

−1 −1 ∥f(B) − f(A)∥B(ℓp) ≤ C∥U∥B(ℓp)∥V ∥B(ℓp)∥V (B − A)U ∥B(ℓp,ℓs).

Proof. Let R := V (BS − SA)U −1. With notation as in Lemma 4.8,

−1 λ,µ ∥ − ∥ p q ≤ ∥ ∥ p ∥ ∥ q ∥ ∥ p q f(B)Sn Snf(A) B(ℓ ,ℓ ) U B(ℓ ) V B(ℓ ) Tf [1],n(R) B(ℓ ,ℓ ) for each n ∈ N. Proposition 4.11, Lemma 2.37 (with p = r and with q and s interchanged) and Proposition 2.33 (iii) (with q and s interchanged) yield a constant C′ ≥ 0 such that ( ) λ,µ ′ ∥ ∥ p q ≤ ∥ ∥ p q ∥ ∥ p s Tf [1],n(R) B(ℓ ,ℓ ) C R B(ℓ ,ℓ ) + R B(ℓ ,ℓ ) .

72 Since B(ℓp, ℓs) ,→ B(ℓp, ℓq) contractively,

∥f(B)Sn − Snf(A)∥B(ℓp,ℓq)

−1 −1 ≤C∥U∥B(ℓp)∥V ∥B(ℓq)∥V (BS − SA)U ∥B(ℓp,ℓs) for all n ∈ N, where C = 2C′. Finally, as in the proof of Proposition 4.9, one lets n tend to inﬁnity to conclude the proof. D

In the same way, appealing to the second part of Proposition 2.33 (iii), one deduces the following result.

Proposition 4.18. Let p, q ∈ [1, ∞) with p ≥ q. Then for each r > p there exists a constant C ≥ 0 such that the following holds (where ℓ∞ should be replaced by

p q c0). Let A ∈ Bd(ℓ , λ, U) and B ∈ Bd(ℓ , µ, V ) have real spectra. Then

−1 −1 ∥f(B)X − Xf(A)∥B(ℓp,ℓq) ≤C∥U∥B(ℓp)∥V ∥B(ℓq)∥V (BX − XA)U ∥B(ℓr,ℓq) for all X ∈ B(ℓp, ℓq) be such that V (BX − XA)U −1 ∈ B(ℓr, ℓq). In particular, if p = q and V (B − A)U −1 ∈ B(ℓr, ℓq), then

−1 −1 ∥f(B) − f(A)∥B(ℓp) ≤ C∥U∥B(ℓp)∥V ∥B(ℓp)∥V (B − A)U ∥B(ℓr,ℓq).

We single out the case where p = q = 2. Here we write f(A) = |A| for a √ normal operator A ∈ B(ℓ2), since then f(A) is equal to |A| := A∗A. Note also that the following result applies in particular to compact self-adjoint operators. For simplicity of the presentation we only consider ϵ ∈ (0, 1].

Corollary 4.19. For each ϵ ∈ (0, 1] there exists a constant C ≥ 0 such that

2 2 the following holds. Let A ∈ Bd(ℓ , λ, U) and B ∈ Bd(ℓ , µ, V ) be self-adjoint operators, with U and V unitaries, and let X ∈ B(ℓ2). If V (BX − XA)U −1 ∈ B(ℓ2, ℓ2−ϵ), then

−1 ∥|B|X − X|A|∥B(ℓ2) ≤ C∥V (BX − XA)U ∥B(ℓ2,ℓ2−ϵ) and if V (BX − XA)U −1 ∈ B(ℓ2+ϵ, ℓ2) then

−1 ∥|B|X − X|A|∥B(ℓ2) ≤ C∥V (BX − XA)U ∥B(ℓ2+ϵ,ℓ2).

73 In particular, if V (B − A)U −1 ∈ B(ℓ2, ℓ2−ϵ), then

−1 ∥|B| − |A|∥B(ℓ2) ≤ C∥V (B − A)U ∥B(ℓ2,ℓ2−ϵ) and if V (B − A)U −1 ∈ B(ℓ2+ϵ, ℓ2), then

−1 ∥|B| − |A|∥B(ℓ2) ≤ C∥V (B − A)U ∥B(ℓ2+ϵ,ℓ2).

Remark 4.20. Let J be the class of all f : R → R such that ∫ t f(t) = at + b + (t − s) dµ(s) (4.11) −∞ for all t ∈ R, where a, b ∈ R and µ is a signed measure of compact support. This class is introduced by Davies in [34, p. 156], and he states that f : R → R satisﬁes (4.11) for a positive µ if and only if f is convex and linear for large |t|. The results in this section for f the absolute value function can be extended to all f ∈ J , in the same way as in [34, Theorem 17]. We leave the details to the reader.

4.4 Lipschitz estimates on the ideal of p-summing operators

Let H be a separable inﬁnite-dimensional Hilbert space. Recall that it was { }∞ shown in [7] that a matrix M = mjk j,k=1 is a Schur multiplier on the Hilbert- S ⊂ H | | ∞ S Schmidt class 2 B( ) if and only if supj,k mjk < . By [77], 2 coincides with the Banach ideal Πp(H) of all p-summing operators (see the deﬁnition below) ∈ ∞ { }∞ H for all p [1, ). Hence a matrix M = mjk j,k=1 is a Schur multiplier on Πp( ) | | ∞ if and only if supj,k mjk < . In Corollary 4.22 below we show that the same p∗ p p∗ p statement is true for the Banach ideal Πp(ℓ , ℓ ) in B(ℓ , ℓ ), for p ∈ [1, ∞). As a p∗ p corollary we obtain operator Lipschitz estimates on Πp(ℓ , ℓ ) for each Lipschitz function f on C. Let X and Y be Banach spaces and 1 ≤ p < ∞. An operator X : X → Y is p-absolutely summing if there exists a constant C such that for each n ∈ N and { }n ⊆ X each collection xj j=1 ,

( ∑n ) 1 ( ∑n ) 1 p p ∗ p p ∥X(xj)∥Y ≤ C sup |⟨x , xj⟩| . (4.12) ∥ ∗∥ ∗ ≤ j=1 x X 1 j=1

74 The smallest such constant is denoted by πp, and Πp(X , Y) is the space of p-absolutely summing operators from X to Y. We let Πp(X ) := Πp(X , X ).

By Propositions 2.3, 2.4 and 2.6 in [37], (Πp(X , Y), πp(·)) is a Banach ideal in B(X , Y).

∗ Below we consider p-absolutely summing operators from ℓp to ℓp. We ﬁrst present the following result.

∈ ∞ { }∞ ∈ p∗ p Lemma 4.21. Let p [1, ) and X = xjk j,k=1. Then X Πp(ℓ , ℓ ) (with ∞ ℓ replaced by c0) if and only if

( ∑∞ ∑∞ ) 1 p p cp := |xjk| < ∞. j=1 k=1

In this case, πp(X) = cp.

Proof. It follows from [37, Example 2.11] that, if cp < ∞ for p ∈ (1, ∞), then p∗ p X ∈ Πp(ℓ , ℓ ) with πp(X) ≤ cp. An inspection of the proof of [37, Example 2.11] shows that this statement in fact also holds for p = 1. For the converse, let

p∗ p∗ p n ∈ N and let xj := ej ∈ ℓ for 1 ≤ j ≤ n. By (4.12) (with X = ℓ and Y = ℓ ),

( ∑n ∑∞ ) 1 p p |xjk| ≤ πp(X). k=1 j=1

Letting n tend to inﬁnity concludes the proof. D

For the following corollary of Lemma 4.21, recall that a matrix M is said to be a Schur multiplier on a subspace I ⊆ B(ℓp, ℓq) if X 7→ M ∗ X is a bounded map on I. Recall also the deﬁnition of the standard triangular truncation T∆ from (2.37).

∈ ∞ { }∞ Corollary 4.22. Let p [1, ) and let M = mjk j,k=1 be a matrix. Then p∗ p ∞ M is a Schur multiplier on Πp(ℓ , ℓ ) (with ℓ replaced by c0) if and only if | | ∞ supj,k∈N mjk < . In this case,

∥ ∥ p∗ p | | M B(Πp(ℓ ,ℓ )) = sup mjk . j,k∈N

p∗ p ∈ ∥ ∥ p∗ p In particular, T∆ B(Πp(ℓ , ℓ )) with T∆ B(Πp(ℓ ,ℓ )) = 1.

75 p∗ p ∗ Observe that T∆ ∈/ B(B(ℓ , ℓ )) if p ≥ p, by Proposition 2.33 (ii). Never- p∗ p p∗ p theless, T∆ is bounded on the ideal Πp(ℓ , ℓ ) ⊂ B(ℓ , ℓ ) for all p ∈ [1, ∞). p∗ p We now prove our main result concerning commutator estimates on Πp(ℓ , ℓ ).

p∗ ∞ Theorem 4.23. Let p ∈ [1, ∞), A ∈ Bd(ℓ ) (with ℓ replaced by c0) and B ∈ p Bd(ℓ ) be operators with real spectra. Let f be Lipschitz on R. Then

πp(f(B)X − Xf(A)) ≤ KAKB∥f∥Lipπp(BX − XA) (4.13)

p∗ p p∗ p for all X ∈ B(ℓ , ℓ ) such that BX − XA ∈ Πp(ℓ , ℓ ).

∈ p∗ ∈ p { }∞ Proof. Let A Bd(ℓ , λ, U) and B Bd(ℓ , µ, V ) for certain λ = λj j=1, { }∞ ∈ p∗ ∈ p { }∞ ⊆ p∗ p µ = µk k=1, U B(ℓ ) and V B(ℓ ). If Xm m=1 Πp(ℓ , ℓ ) is a πp- p∗ p bounded sequence which SOT-converges to X ∈ B(X , Y), then X ∈ Πp(ℓ , ℓ ) ≤ with πp(X) lim supm→∞ πp(Xm), by (4.12). Hence, by Remark 4.10, it suﬃces λ,µ to prove that sup ∥T ∥ p∗ p p∗ p ≤ ∥f∥ . This is done as in the n∈N f [1],n Πp(ℓ ,ℓ )→Πp(ℓ ,ℓ ) Lip proof of Theorem 4.13, using Corollary 4.22 instead of Remark 2.34. D

76 Chapter 5

Multiple operator integration

In this chapter we develop multiple operator integration theory considering firstly for simplicity the finite dimensional setting. An important step here is introducing the notion of a multilinear Schur multiplier and connecting it to a classical Schur multiplier defined in Chapter 2. This approach is new and is developed in [87] (a joint work with A. Skripka, D. Potapov and F. Sukochev). It is also worth mentioning that here we prove important properties of the multiple operator integrals such as the perturbation formula (Lemma 5.7), continuity (Lemma 5.8), representing derivatives of operator valued functions in terms of multiple operator integrals (Theorem 5.9 in the self-adjoint case, Theorem 5.11 in the unitary case). These properties are proved for arbitrary Cn-functions and have analogies in the infinite dimensional setting (albeit for a smaller class of functions), except for Theorem 5.11, which is a significant improvement compar- atively with its prototype given in [86]. In this part of the thesis we prove the most important property of multiple operator integrals, which connects this notion with the operator Taylor remainder (see Theorem 5.12), where the formula in the unitary case is new. Next we introduce the notion of multiple operator integral in the setting of non-commutative Lp-spaces, where we consider the most general possible case, that is non-commutative Lp-spaces for an arbitrary von Neumann algebra. The definition of a multiple operator integral here is based on that one given in [84]. One of the important results of this chapter is the Hölder-type estimate given

77 in Theorem 5.36, which may be compared with [92, Corollary 13 and Theorem 14], where a von Neumann algebra of type I is considered. However, our proof of Theorem 5.36 diﬀers from the one given in [92], and will be published in [95] (a joint work with D. Potapov, F. Sukochev and D. Zanin). ∈ N X ∥ · ∥ ≤ ≤ Let n and a family ( j, Xj ) 1 j n of Banach spaces be given. By

X1 × ... × Xn we denote the direct product of the Banach spaces X1,..., Xn. In ×n the case when X = X1 = ... = Xn for X1 ×...×Xn we use the notation X . Let

(Y, ∥·∥Y ) be a Banach space. For a multilinear operator T : X1 ×...×Xn → Y by dom(T ) we denote the domain of T. A multilinear operator T : X1 ×...×Xn → Y is said to be bounded if

∥ ∥ {∥ ∥ ∥ ∥ ≤ ≤ ≤ } ∞ T X1×...×Xn→Y := sup T (x1, ..., xn) Y : xj Xj 1, 1 j n < .

The set of all bounded multilinear operators T : X1 × ... × Xn → Y is denoted by B(X1 × ... × Xn, Y).

5.1 Multiple operator integrals in the ﬁnite dimensional setting

∈ N 2 Recall that for d , ℓd denotes the d-dimensional complex Hilbert space 2 equipped with the norm ∥ · ∥ 2 . The identity operator on ℓ is denoted by Id. Let ℓd d

Md denote the algebra of d × d matrices over the field C. As before we naturally 2 2 ∈ identify Md with the space B(ℓd) of bounded linear operators on ℓd. If X Md, then the entries of X are denoted by xij; if (X1,...,Xn) ∈ Md × ... × Md, then (k) ≤ ≤ Sp the entries of Xk are denoted by xij , 1 i, j d. The symbol d stands for the space Md equipped with the Schatten p-norm, 1 ≤ p ≤ ∞. The elementary matrix with the only nonzero entry at the position (i, j) is denoted by Eij and Ejj is denoted by Ej. In the case when the dimension d (d) needs to be specified, we will also use the notation Eij instead of Eij. The spectral projection of a normal operator X ∈ B(H) when H is finite-dimensional associated with a spectral point λ is denoted by EX (λ). Recall also that the spectrum of an operator X is denoted by σ(X).

78 Let n ∈ N and let

{ }d ⊂ C m(n) := mr0,...,rn r0,...,rn=1 .

An n-linear Schur multiplier (or a (linear) Schur multiplier in case n = 1)

× × → Mn = Mm(n) : |Md ...{z M}d Md n times associated with symbol m(n) is deﬁned via

∑d · (1) (2) (n) · Mn(X1,...,Xn) := mr0,...,rn xr0r1 xr1r2 . . . xrn−1rn Er0rn , (5.1) r0,...,rn=1 where Xj ∈ Md, 1 ≤ j ≤ d. The latter can be rewritten in the form

∑d · Mn(X1,...,Xn) = mr0,...,rn Er0 X1Er1 X2Er2 ...XnErn . (5.2) r0,...,rn=1 A Schur multiplier can also be viewed as a map on a Cartesian product of

Schatten classes. Let 1 ≤ p1, . . . , pn, p ≤ ∞. The n-linear transformation deﬁned in (5.1) can be viewed as a transformation

Sp1 × × Spn → Sp Mn : d ... d d with the norm

∥Mn∥Sp1 × ×Spn →Sp := sup ∥Mn(X1,...,Xn)∥Sp . d ... d d ∥X1∥Sp1 ,...,∥Xn∥Spn ≤1 The following simple result shows that any bounded n-dimensional matrix gen- erates n-linear Schur multiplier on the direct product of Hilbert-Schmidt classes.

{ }d ⊂ C Lemma 5.1. Let m(n) := mr0,...,rn r0,...,rn=1 . Then, the Schur multiplier deﬁned in (5.1) satisﬁes

∥Mn∥S2× ×S2→S2 = sup |mr ,...,r | d ... d d 0 n 1≤r0,...,rn≤d

Proof. For a ﬁxed n-tuple (r0, . . . , rn), by (5.1), we have that

Mn(Er0r1 ,Er1r2 ,...,Ern−1rn ) = mr0,...,rn Er0rn and thus,

∥Mn∥S2× ×S2→S2 ≥ |mr ,...,r | d ... d d 0 n

79 for all 1 ≤ r0, . . . , rn ≤ d. Hence, trivially we have

∥Mn∥S2× ×S2→S2 ≥ sup |mr ,...,r |. d ... d d 0 n 1≤r0,...,rn≤d Conversely, by (5.1), we have that d d ∑ ∑ 2 2 (1) (n) ∥ ∥ 2 · Mn(X1,...,Xn) S = mr0,...,rn xr0r1 . . . xrn−1rn (5.3) r0,rn=1 r1,...,rn−1=1 d ( d ) ∑ ∑ 2 ≤ | |2 | (1) (n) | sup mr0,...,rn xr0r1 . . . xrn−1rn . 1≤r ,...,r ≤d 0 n r0,rn=1 r1,...,rn−1=1 Using the Cauchy-Schwartz inequality n-times, we obtain that d ( d ) ∑ ∑ 2 | (1) (n) | xr0r1 . . . xrn−1rn r0,rn=1 r1,...,rn−1=1 d ( d ( d )) ∑ ∑ ∑ 2 | (1) | · | (2) (n) | = xr0r1 xr1r2 . . . xrn−1rn r0,rn=1 r1=1 r2,...,rn−1=1 d d d ( d ) ∑ ∑ ∑ ∑ 2 ≤ | (1) |2 · | (2) (n) | xr0r1 xr1r2 . . . xrn−1rn r0,rn=1 r1=1 r1=1 r2,...,rn−1=1 d ( d ) ∑ ∑ 2 2 (2) (n) 2 2 ∥ ∥ 2 | | ≤ ≤ ∥ ∥ 2 ∥ ∥ 2 = X1 S xr1r2 . . . xrn−1rn ... X1 S ... Xn S . r1,rn=1 r2,...,rn−1=1 Combining the latter with (5.3) proves the converse inequality, which completes the proof of the lemma. D

The following property of a linear Schur multiplier is a straightforward consequence of (5.2).

{ }d ⊂ C Lemma 5.2. Let m(1) := mr0,r1 r0,r1=1 . Then, the Schur multiplier deﬁned in (5.1) satisﬁes

∥ ∥ p p ∥ ∥ ∗ ∗ M1 S →S = M1 Sp →Sp , d d d d ≤ ∗ ≤ ∞ 1 1 where 1 p, p and p + p∗ = 1.

5.1.1 Multilinear Schur multipliers via linear ones.

≤ ≤ ∞ ∈ { } ˜ Let 1 p . Let k 1, . . . , d and denote k := (|k, .{z . . , k}). A particular n−1 times → case of (5.1) is the linear multiplier M1,k˜ : Md Md deﬁned by ∑d ∑d · · · M1,k˜(X) := mr,k,s˜ xrs Ers = mr,k,s˜ ErXEs (5.4) r,s=1 r,s=1

80 with the norm

∥M ˜∥S1→Sp = sup ∥M ˜(X)∥Sp . 1,k d d 1,k ∥X∥S1 61

1 1 1 Theorem 5.3. Let 1 ≤ p1, . . . , pn, p ≤ ∞ be such that + . .. + = . For Mn p1 pn p deﬁned in (5.1) and M1,k˜ deﬁned in (5.4), we have

∥ ∥ p ≥ ∥ ∥ p Mn S 1 ×...×Spn →Sp max M1,k˜ :S1→S . d d d 1≤k≤d d d

Proof. Choose and ﬁx arbitrary 1 ≤ k ≤ d and ε > 0. Let X ∈ Md be such that

∥X∥S1 ≤ 1 and ∥M ˜∥S1→Sp − ε < ∥M ˜(X)∥Sp ≤ ∥M ˜∥S1→Sp . 1,k d d 1,k 1,k d d

The operator X can be represented in the form

∑d X = xl ⊗ yl, (5.5) l=1

∈ 2 where xl, yl ℓd are such that

∑d ∥xl∥ 2 · ∥yl∥ 2 ≤ 1 (5.6) ℓd ℓd l=1

⊗ 2 ∈ 2 and x y is the rank-one operator on ℓd deﬁned for x, y ℓd by

⊗ ⟨ ⟩ ∈ 2 (x y)(h) := h, y x, h ℓd.

Indeed, represent X in the polar form X = V |X|. Since |X| is diagonalizable in { }d 2 an orthogonal basis, there is a set of vectors yl l=1 in ℓd such that

∑d ∑d |X| = yl ⊗ yl and ∥yl∥ 2 ≤ 1. ℓd l=1 l=1

Setting xl := V yl we obtain (5.5) and (5.6). { }d 2 Let el l=1 be the standard basis in ℓd and let

⊗ ⊗ Al,k˜ := xl ek,Bl,k˜ := ek yl.

A straightforward computation yields

⊗ Al,k˜Bl,k˜ = xl yl. (5.7)

81 Observe that for every 1 ≤ q ≤ ∞,

∥A ˜∥Sq = ∥xl∥ 2 and ∥B ˜∥Sq = ∥yl∥ 2 . (5.8) l,k ℓd l,k ℓd

By (5.5) and (5.7), we obtain

∑d

X = Al,k˜Bl,k˜ (5.9) l=1 and from (5.6) and (5.8),

∑d ∥ ∥ · ∥ ∥ ≤ Al,k˜ Sp1 Bl,k˜ Spn 1. (5.10) l=1 Note that   Al,k˜Bl,k˜ if k = r1 = ... = rn−1 A ˜Er EkEr ...Er − EkEr − B ˜ = (5.11) l,k 1 2 n 2 n 1 l,k  0 otherwise.

We have the following representation of M1,k˜ via n-linear Schur multipliers Mn:

∑d ∑d (5.4)&(5.9) M (X) = m · E A B E 1,k˜ r0,k,r˜ n r0 l,k˜ l,k˜ rn l=1 r0,rn=1 ∑d ∑d (5.11) · = mr0,...,rn Al,k˜Er1 EkEr2 ...Ern−2 EkErn−1 Bl,k˜ l=1 r0,...,rn=1 ∑d

= Mn(Al,k˜,Ek,...,Ek,Bl,k˜). l=1 Hence,

∑d ∥ ∥ ≤ ∥ ∥ M1,k˜(X) Sp Mn(Al,k˜,Ek,...,Ek,Bl,k˜) Sp l=1 ∑d

≤ ∥Mn∥Sp1 × ×Spn →Sp · ∥A ˜∥Sp1 · ∥B ˜∥Spn d ... d d l,k l,k l=1 (5.10) ≤ ∥Mn∥Sp1 × ×Spn →Sp . (5.12) d ... d d

Since (5.12) holds for every X ∈ Md with ∥X∥S1 ≤ 1 and every k ∈ {1, . . . , d}, the theorem is proved. D

82 5.1.2 Schur multipliers via multiple operator integrals.

∈ 2 For a unit vector x ℓd, let Px be as before the projection on the linear span of ∈ 2 ∈ 2 x, that is, Px(y) = (y, x)x, y ℓd. Let A0,...,An B(ℓd) be self-adjoint (unitary) (j) { (j)}d operators and for any j = 0, . . . , n, let g = gi i=1 be an orthonormal basis of { (j)}d eigenvectors of Aj and let λi i=1 be the corresponding d-tuple of eigenvalues. Let φ : Cn+1 → C be a bounded Borel function. Deﬁne a multilinear operator

A0,...,An 2 × × 2 → 2 Tφ : B| (ℓd) ...{z B(ℓd}) B(ℓd) by setting n times ∑d ( ) A0,...,An (0) (1) (n) (0) (1) (n) Tφ (X1,...,Xn) = φ λr0 , λr1 . . . , λrn P X1P ...XnP , gr0 gr1 grn r0,...,rn=1 (5.13)

∈ 2 A0,...,An for X0,...,Xn B(ℓd). The operator Tφ is a ﬁnite-dimensional multiple operator integral with symbol φ. Multiple operator integration theory started its development in Pavlov’s paper [76] in 1969 and was greatly expanded later in [9,81,84]. { (j)}dj Assume that λi i=1 is the set of pairwise distinct eigenvalues of the operator

Aj, where dj ∈ N, dj ≤ d. Denote

∑d (j) ≤ ≤ Ei = P (j) , 1 i dj, (5.14) gk k=1 (j) (j) λk =λi

(j) (j) that is, Ei is the spectral projection of Aj associated with the eigenvalue λi . Alternatively, using the spectral projections (5.14), we can write (5.13) as

∑d0 ∑dn A0,...,An (0) (n) (0) (n) Tφ (X1,...,Xn) = ... φ(λr0 , . . . , λrn )Er0 X1 ...XnErn , (5.15) r0=1 rn=1 ∈ 2 for X1,...,Xn B(ℓd). The following fact is derived in [29, Subsection 3.2] in the case n = 2. The proof of the case n > 2 follows along the lines of that given in [29] and is therefore omitted.

Lemma 5.4. Let n ∈ N. Let A0,...,An be all self-adjoint or all unitary elements 2 ∈ 2 (k) in B(ℓd) and X1,...,Xn B(ℓd). Let g be an orthonormal basis of eigenvectors { (k)}d of Ak with the corresponding d-tuple of eigenvalues λl l=1 and suppose that Xk

83 { (k)}d { (k) (k−1)} has the matrix representation xij i,j=1 in the bases g , g , k = 1, . . . , n. Suppose φ : Cn+1 → C is a bounded Borel function. The matrix representation of the multiple operator integral with symbol φ given by (5.13) in the bases {g(n), g(0)} coincides with the value of the Schur multiplier associated with the matrix

{ (0) (n) }d m(n) = ψ(λr0 , . . . , λrn ) r0,...,rn=1 ( ) { (1)}d { (n)}d given by (5.1) on the n-tuple xij i,j=1,..., xij i,j=1 , that is,

A0,...,An Mn(X1,...,Xn) = Tφ (X1,...,Xn).

5.1.3 Properties of multiple operator integrals

In this section we discuss properties of multiple operator integrals deﬁned in (5.15). In particular, in Theorem 5.12 we will represent Taylor remainder as a multiple operator integral, which is an important step in applications of this theory. The second interesting component is contained in Theorem 5.5 below, which reduces the estimates for multiple operator integrals to estimates for double operator integrals.

1 1 1 Theorem 5.5. Let 1 ≤ p1, . . . , pn, p ≤ ∞ be such that + . .. + = and let p1 pn p Cn+1 → C ∈ 2 ψ : be a bounded Borel function. Let A, B B(ℓd). The following assertions hold.

(i) If A and B are self-adjoint, 0 ∈ σ(A), and

φ(x0, xn) := ψ(x0, 0,..., 0, xn), for x0, xn ∈ R,

then

A+B,A,...,A A+B,A ∥T ∥Sp1 × ×Spn →Sp ≥ ∥T ∥S1→Sp . ψ d ... d d φ d d

(ii) If A and A + B are unitary, 1 ∈ σ(A), and

φ(z0, zn) := ψ(z0, 1,..., 1, zn), for z0, zn ∈ T,

then

A+B,A,...,A A+B,A ∥T ∥Sp1 × ×Spn →Sp ≥ ∥T ∥S1→Sp . ψ d ... d d φ d d

84 ≤ ∗ ≤ ∞ 1 1 (iii) If n = 1, 1 p, p , and p + p∗ = 1, then

A+B,A A+B,A ∗ ∗ Tψ Sp→Sp = Tψ Sp →Sp . d d d d

Proof. The proof is an immediate consequence of Theorem 5.3 and Lemmas 5.4 and 5.2. D

The tensor product of A ⊗ B of a d × d-matrix A and m × m-matrix B is the dm × dm-matrix given by   ···  a11B a1dB    ⊗  . .  A B =  . . 

ad1B ··· addB The following lemma extends the result of [29, Lemma 11] to a higher order case.

∈ 2 ∈ Lemma 5.6. Let A, B B(ℓd) be self-adjoint or unitary operators and X1,...,Xn 2 B(ℓd). Let ˜ ˜ H = E11 ⊗ A + E22 ⊗ B, X1 = E12 ⊗ X1 and Xj = E22 ⊗ Xj, 2 ≤ j ≤ n.

Then, for a bounded Borel function ψ : Cn+1 → C,

H,...,H ˜ ˜ ⊗ A,B,...,B Tψ (X1,..., Xn) = E12 Tψ (X1,...,Xn).

{ }m { ′ }m′ ′ ≤ Proof. Let σ(A) = λi i=1 and σ(B) = λj j=1 for m, m d. The spectral projection of the operator H associated with λ ∈ σ(A) ∪ σ(B) is given by

EH (λ) = E11 ⊗ EA(λ) + E22 ⊗ EB(λ).

By (5.15), ∑ H,...,H ˜ ˜ ˜ ˜ Tψ (X1,..., Xn) = ψ(λr0 , . . . , λrn )EH (λr0 )X1 ... XnEH (λrn ). ∈ ∪ λr0 ,...,λrn σ(A) σ(B) Moreover,

˜ ˜ EH (λr0 )X1 ... XnEH (λrn ) ( ) ⊗ ⊗ ⊗ ˜ ˜ = E11 EA(λr0 ) + E22 EB(λr0 ) (E12 X1)EH (λr1 )X2 ... XnEH (λrn ) ( ) ⊗ ⊗ ˜ ˜ = E12 EA(λr0 )X1 EH (λr1 )(E22 X2)EH (λr2 )X3 ... XnEH (λrn ) ( ) ⊗ ˜ ˜ = E12 EA(λr0 )X1EB(λr1 )X2 EH (λr2 )X3 ... XnEH (λrn ) ( ) ⊗ = ... = E12 EA(λr0 )X1EB(λr1 )X2 ...XnEB(λrn ) .

85 Therefore,

H,...,H ˜ ˜ Tψ (X1, ..., Xn) ∑ ( ) ⊗ = ψ(λr0 , . . . , λrn )E12 EA(λr0 )X1EB(λr1 )X2 ...XnEB(λrn ) ∈ ∈ λr0 σ(A),λr1 ...,λrn σ(B) ⊗ A,B,...,B = E12 Tψ (X1,...,Xn).

In this section we will work with multiple operator integrals whose symbols are divided diﬀerences of functions in Cn(R) or Cn(T). As usual, Cn(R) stands for the set of functions n times continuously diﬀerentiable on R. For φ ∈ C(T), by its derivative at z0 ∈ T, we understand the limit

′ φ(z) − φ(z0) φ (z0) := lim , (5.16) T∋ → z z0 z − z0 provided it exists. The symbol Cn(T) denotes the set of functions n times continuously diﬀerentiable on T in the sense of (5.16). We will also work with the

n+1 space of bounded continuous multivariable functions Cb(R ). We recall that the divided diﬀerence of the zeroth order f [0] is the function

n f itself. Let λ0, λ1, . . . , λn be points in R (respectively, in T) and let f ∈ C (R) (respectively, f ∈ Cn(T)). The divided difference f [n] of order n is defined recursively by   [n−1] [n−1]  f (λ0,λ1,...,λn−1)−f (λ1,λ2,...,λn) if λ0 ≠ λn [n] λ0−λn f (λ0, λ1, λ2, . . . , λn) =   ∂ [n−1] f (λ, λ1, λ2, . . . , λn−1) if λ0 = λn. ∂λ λ=λ0 Basic properties of the divided difference of a function defined on R can be found in, e.g., [36, Section 4.7]. In particular, for f ∈ Cn(R), 1 f [n](λ , . . . λ ) = f (n)(λ ), (5.17) 0 0 n! 0

[n] 1 (n) f ≤ ∥f ∥ ∞ , (5.18) L∞([a,b]n+1) n! L ([a,b]) and the divided diﬀerence is invariant with respect to any permutation of its variables. Directly from [31, Theorem 2.1 and Lemma 2.2], for f ∈ Cn(T), we have π(n+3)/2 f [n] ≤ f (n) , (5.19) L∞(Tn) 2n+1Γ((n + 1)/2) L∞(T)

86 where Γ(·) is the Gamma function. The following perturbation formulas are standard results on multiple operator integration.

n n Lemma 5.7. Let n ∈ N, f ∈ C (R)(or f ∈ C (T)), and A, B, H1,...,Hn ∈ 2 B(ℓd) be self-adjoint (or unitary) operators. Then, the following results hold.

(i) − A,B − f(A) f(B) = Tf [1] (A B) (5.20)

∈ 2 ≤ ≤ − ≤ ≤ (ii) For all X1,...,Xk B(ℓd), 0 k n 1, 1 j k + 1,

H1,...,Hj−1,A,Hj ,...,Hk − H1,...,Hj−1,B,Hj ,...,Hk Tf [k] (X1,...,Xk) Tf [k] (X1,...,Xk)

H1,...,Hj−1,A,B,Hj ,...,Hk − − = Tf [k+1] (X1,...,Xj 1,A B,Xj,...,Xk), (5.21)

H1,...,Hj−1,A,Hj ,...,Hk A,H1,...,Hk where Tf [k] for j = 1 is understood as Tf [k] and for j = k + 1

H1,...,Hk,A as Tf [k] .

Proof. (i) The proof of (5.20) can be found in, e.g., [29, (22)]. (ii) The formula (5.21) is proved in [86, Lemma 2.4(i)] for operators on inﬁnite dimensional spaces and a smaller set of functions f; it is proved in [29, Theorem 15] for matrices in the special case n = 2. Let us prove it for an arbitrary n. Fix n ≥ 2 and 0 ≤ k ≤ n − 1. For 1 ≤ j ≤ k + 1, let us denote

[k+1] ψj(x0, . . . , xk+1) := xj f (x0, . . . , xk+1);

[k] ϕj(x0, . . . , xk+1) := f (x0, . . . , xj−1, xj+1, . . . , xk+1).

From the deﬁnition of the divided diﬀerence, we see that

ψj−1 − ψj = ϕj − ϕj−1.

For brevity we set

˜ ˜ Hj−1 := (H1, ..., Hj−1), Xj−1 := (X1, ..., Xj−1), ˜ ˜ jHk := (Hj,Hj+1, ..., Hk), jXk := (Xj,Xj+1, ..., Xk). (5.22)

87 By the deﬁnition of the multiple operator integral (5.13),

˜ ˜ Hj−1,A,B, j Hk ˜ − − ˜ RHS of (5.21) = Tf [k+1] (Xj 1,A B, jXk) ˜ ˜ ˜ ˜ Hj−1,A,B, j Hk Hj−1,A,B, j Hk ˜ − ˜ − ˜ − ˜ = Tf [k+1] (Xj 1, A, jXk) Tf [k+1] (Xj 1,B, jXk) ˜ ˜ ˜ ˜ Hj−1,A,B, j Hk Hj−1,A,B, j Hk = T (X˜ − ,I , X˜ ) − T (X˜ − ,I , X˜ ) ψj−1 j 1 d j k ψj j 1 d j k ˜ ˜ Hj−1,A,B, j Hk = T (X˜ − ,I , X˜ ) ψj−1−ψj j 1 d j k ˜ ˜ Hj−1,A,B, j Hk = T (X˜ − ,I , X˜ ) ϕj −ϕj−1 j 1 d j k ˜ ˜ ˜ ˜ Hj−1,A,B, j Hk Hj−1,A,B, j Hk = T (X˜ − ,I , X˜ ) − T (X˜ − ,I , X˜ ) ϕj j 1 d j k ϕj−1 j 1 d j k ˜ ˜ ˜ ˜ Hj−1,A, j Hk ˜ − Hj−1,B, j Hk ˜ = Tf [k] (Xk) Tf [k] (Xk) = LHS of (5.21), proving the assertion. D

The following lemma establishes a simple fact that the multiple operator integral is continuous with respect to its parameters.

∈ 2 ∈ N Lemma 5.8. (i) Let Am, A, B B(ℓd), m be self-adjoint (respectively, n+1 unitary) operators such that Am → A as m → ∞. Let ψ ∈ Cb(R ) (respectively, ψ ∈ C(Tn+1)). Then,

Am,B,...,B −→ A,B,...,B → ∞ Tψ (X1,...,Xn) Tψ (X1,...,Xn), m ,

∈ 2 for all X1,...,Xn B(ℓd).

(m) ∈ 2 ∈ N (ii) Let A ,A B(ℓd), m , be self-adjoint (respectively, unitary) operators such that A(m) → A, as m → ∞. Let k ∈ N and let f ∈ Ck+1(R) (respectively, f ∈ Ck+1(T)). Then,

A(m),...,A(m) → A,...,A → ∞ Tf [k] (X1,...,Xk) Tf [k] (X1,...,Xk), m ,

∈ 2 for all X1,...,Xk B(ℓd).

Proof. (i) We prove the assertion for self-adjoint operators. The case of unitary

′ ′′ { (m)}dm { }d { }d operators is similar. Let λj j=1, λj j=1 and µk k=1 be the sequences of ′ ′′ distinct eigenvalues of Am,A, and B, respectively, for some dm, d , d ≤ d, m ∈ N. n For a ﬁxed tuple (x1, . . . , xn) ∈ R , since ψ is continuous with respect to the ﬁrst variable, it follows (using functional calculus) that

∑dm (m) (m) ψ(λj , x1, . . . , xn)Ej = ψ(Am, x1, . . . , xn) j=1

88 approaches ∑d′ ψ(A, x1, . . . , xn) = ψ(λj, x1, . . . , xn)EA(λj), j=1 as m → ∞. Therefore, by (5.15), and the preceding convergence,

Am,B,...,B Tψ (X1,...,Xn) ′′ ∑dm ∑d (m) (m) = ψ(λj , µr1 . . . , µrn )Ej X1EB(λr1 ) ...EB(λrn−1 )XnEB(λrn ) j=1 r1,...,rn=1 ′′ ∑d ( ∑dm ) (m) (m) = ψ(λj , µr1 . . . , µrn )Ej X1EB(λr1 ) ...EB(λrn−1 )XnEB(λrn ) r1,...,rn=1 j=1 converges to

∑d′′ ( ∑d′ ) A ψ(λj, µr1 . . . , µrn )Ej X1EB(λr1 ) ...EB(λrn−1 )XnEB(λrn ) r1,...,rn=1 j=1 A,B,...,B = Tψ (X1,...,Xn), as m → ∞. (ii) Applying Lemma 5.7 (ii) with A = A(m),B = A and m = j, via telescop- ing summation we obtain

(m) (m) A ,...,A − A,...,A T [k] (X1,...,Xk) T [k] (X1,...,Xk) f f S2

∑k+1 (m) (m) A˜ − , A˜ A˜ , A˜ j 1 j k+1 − j j+1 k+1 = T [k] (X1,...,Xk) T [k] (X1,...,Xk) f f S2 j=1

∑k+1 (m) A˜ − ,A(m),A, A˜ j 1 j k+1 (m) − = T [k+1] (X1,...,Xj−1,A A, Xj,...Xk) , f S2 j=1

˜ − ˜(m) (m) (m) (m) where Aj−1 := (A, ..., A) is (j 1)-tuple, and jAk+1 := (A ,A , ..., A ) is − ˜ ˜(m) (k + 1 j + 1)-tuple and Aj,j+1 Ak+1 have similar meaning. Using the triangle inequality and applying Lemmas 5.1 and 5.4 with the inequality (5.18) (or (5.19) for the unitary case), we obtain

(m) (m) A ,...,A − A,...,A T [k] (X1,...,Xk) T [k] (X1,...,Xk) f f S2 (k+1) (m) ≤ const · ∥f ∥∞ · ∥A − A∥S2 · ∥X1∥S2 · ... · ∥Xk∥S2 , which approaches 0 as m → ∞. D

89 The formula (5.23) below in the case of an inﬁnite dimensional Hilbert space and a smaller class of functions can be found in [81] (see also [9, Theorem 5.7]).

∈ n R ∈ 2 Theorem 5.9. Let f C ( ) and let A, B B(ℓd) be self-adjoint operators. Then, the function f is n − 1 times diﬀerentiable at A along the direction B in the sense of Gˆateaux and the respective derivative evaluated in the operator norm has the representation

k 1 d A+sB,...,A+sB k f(A + tB) = Tf [k] (B,...,B), (5.23) k! dt t=s for all 1 ≤ k ≤ n − 1.

∈ N ∈ n R ∈ 2 Proof. Fix n , f C ( ) and self-adjoint operators A, B B(ℓd). We will prove the lemma by induction on k. The case k = 1 follows from the formula (5.20) and Lemma 5.8 (i) as follows:

d f(A + tB) − f(A + sB) f(A + tB) = lim t→s − dt t=s t s T A+tB,A+sB((t − s)B) (5.20) f [1] A+tB,A+sB Lemma 5.8 (i) A+sB,A+sB = lim = lim T [1] (B) = T [1] (B). t→s t − s t→s f f Assume that (5.23) holds for all k = p < n − 1. Denoting H(t) = A + tB, we obtain

p+1 1 d 1 d H(t),...,H(t) p+1 f(H(t)) = Tf [p] (B,...,B) (p + 1)! dt t=s p + 1 dt t=s H(t),...,H(t) − H(s),...,H(s) 1 T [p] (B,...,B) T [p] (B,...,B) = lim f f p + 1 t→s t − s H](s) , H](t) H](s) , H](t) ∑p+1 j−1 j p+1 − j j+1 p+1 1 T [p] (B,...,B) T [p] (B,...,B) = lim f f , p + 1 t→s t − s j=1 ] where jH(t)k is deﬁned as in (5.22). Applying Lemma 5.7 (ii) with A = H(t), B = H(s), k = p and m = j, gives

H](s) , H](t) H](s) , H](t) j−1 j p+1 − j j+1 p+1 Tf [p] (B,...,B) Tf [p] (B,...,B) ] ] H(s)j−1,H(t),H(s), j+1H(t)p+1 ˜ − − ˜ = Tf [p+1] (Bj 1,H(t) H(s), jBp) ] ] H(s)j−1,H(t),H(s), j+1H(t)p+1 ˜ − − ˜ = Tf [p+1] (Bj 1, (t s)B, jBp) H](s) ,H(t),H(s), H](t) − j−1 j+1 p+1 = (t s) Tf [p+1] (B,...,B).

90 Combining the two previous displays, and applying Lemma 5.8 (ii) with p+1 < n yields

p+1 ∑p+1 ] ] 1 d 1 H(s)j−1,H(t),H(s),j+1H(t)p+1 f(H(t)) = lim T [p+1] (B,...,B) (p + 1)! dtp+1 p + 1 t→s f t=s j=1 p+1 1 ∑ = T H(s),...,H(s)(B,...,B) = T H(s),...,H(s)(B,...,B), p + 1 f [p+1] f [p+1] j=1 completing the proof. D

To prove an analog of (5.23) in the unitary case, we need the following analog of [86, Lemma 2.6], which was proved for operators on an inﬁnite dimensional Hilbert space and a smaller set of functions. The proof of the analog of [86, Lemma 2.6] in our case follows along the lines of the proof given in [86], with application of Lemmas 5.7 and 5.8 (ii), so it is omitted.

∈ n T ∈ 2 Lemma 5.10. Let f C ( ), let A and U B(ℓd) be a self-adjoint and a unitary operator, respectively. Denote U(t) = eitAU, t ∈ R. Then, for all 1 ≤ k ≤ n − 1 and all j1, . . . , jk ∈ N,

d ˜ Uk+1(t) j1 jk Tf [k] (A U(t),. ..,A U(t)) dt t=s ∑k+1 ˜ Uk+2(s) j1 jm−1 jm jk = i Tf [k+1] (A U(s),...,A U(s), AU(s),A U(s),...,A U(s)) m=1 ∑k ˜ Uk+1(s) j1 jm−1 jm+1 jm+1 jk + i Tf [k] (A U(s),...,A U(s),A U(s),A U(s),...,A U(s)), m=1 ˜ where Uk+1(t) is the tuple consisting of k + 1 copies of U(t).

The proof of the following theorem mainly repeats that of [86, Theorem 3.1], which was proved for a smaller set of f and operators A, B defined on a infinite- dimensional Hilbert space. However, for our purpose we need to calculate explicitly the coefficients cn,j1,...,jk from [86, Theorem 3.1].

∈ n T ∈ 2 Theorem 5.11. Let f C ( ), let A B(ℓd) be a self-adjoint operator and let ∈ 2 itA ∈ R U B(ℓd) be a unitary operator, U(t) = e U, t . Then

k ∑k ∑ 1 d 1 ˜ k Ul+1(s) j1 jl k f(U(t)) = i Tf [l] (A U(s),...,A U(s)), k! dt t=s j1! .. . jl! l=1 j1,...,jl≥1 j1+...+jl=k (5.24)

91 for all 1 ≤ k ≤ n − 1.

Proof. The formula (5.24) in the case k = 1 can be found in [30, (5)]. The formula (5.24) for an arbitrary 1 ≤ k ≤ n − 1 can be established by induction. We assume that it holds for k = p < n − 1 and verify below that it

j holds for k = p + 1. Denote Vj,t := A U(t). Applying Lemma 5.10 gives

p+1 ∑p ∑ d p! d ˜ f(U(t)) = ip T Ul+1(t)(V , ...,V ) p+1 f [l] j1,t jl,t dt t=s j1! . . . jl! dt t=s l=1 j1,...,jl≥1 j1+...+jl=p ∑p ∑ ∑l ˜ p+1 p! Ul+1(t) = i Tf [l] (Vj1,t,...,Vjm−1,t,Vjm+1,t,Vjm+1,t,...,Vjl,t) j1! . . . jl! l=1 j1,...,jl≥1 m=1 j1+...+jl=p ∑p ∑ ∑l+1 ˜ p+1 p! Ul+2(t) + i Tf [l+1] (Vj1,t,...,Vjm−1,t,V1,t,Vjm,t,...,Vjl,t) j1! . . . jl! l=1 j1,...,jl≥1 m=1 j1+...+jl=p

p+1 =: i (S1 + S2),

˜ where Uk+1(t) is the tuple consisting of k + 1 copies of U(t). Making the substitution ir = jr, 1 ≤ r ≠ m ≤ l and im = jm + 1 in S1, we obtain

∑p ∑l ∑ ˜ p! Ul+1(t) S1 = Tf [l] (Vi1,t,...,Vil,t). i1! . . . im−1!(im − 1)!im+1! . . . il! l=1 m=1 ir≥1,r≠ m,im≥2 i1+...+il=p+1

Relabeling the summands of S2 via the mapping l 7→ l − 1 and making the substitution ir = jr, 1 ≤ r ≤ m − 1, im = 1 and ir = jr−1, m + 1 ≤ r ≤ l, we obtain

∑p+1 ∑ ∑l ˜ p! Ul+1(t) S2 = Tf [l] (Vj1,t,...,Vjm−1,t,V1,t,Vjm,t,...,Vjl−1,t) j1! . . . jl−1! l=2 j1,...,jl−1≥1 m=1 j1+...+jl−1=p ∑p+1 ∑l ∑ ˜ p! Ul+1(t) = Tf [l] (Vi1,t,...,Vil,t). i1! . . . im−1!im+1! . . . il! l=2 m=1 i1,...,il≥1,im=1 i1+...+il=p+1

92 Thus,

U˜2(t) S1 + S2 = Tf [1] (Vp+1,t) p l ( ) ∑ ∑ ∑ ∑ p! + + i1! .. . im−1!(im − 1)!im+1! . . . il! l=2 m=1 ir≥1,r≠ m,im≥2 i1,...,il≥1,im=1 i1+...+il=p+1 i1+...+il=p+1 ˜ × Ul+1(t) Tf [l] (Vi1,t,...,Vil,t) ∑p+1 ∑ p! U˜p+2(t) + Tf [p+1] (Vi1,t,...,Vip+1,t), i1! .. . im−1!(im − 1)!im+1! . . . ip+1! m=1 i1,...,ip+1≥1,im=1 i1+...+ip+1=p+1 so

U˜2(t) S1 + S2 = Tf [1] (Vp+1,t) ∑p ∑l ∑ ˜ p! Ul+1(t) + Tf [l] (Vi1,t,...,Vil,t) i1! .. . im−1!(im − 1)!im+1! . . . il! l=2 m=1 i1,...,il≥1, i1+...+il=p+1

U˜p+2(t) + (p + 1)p!Tf [p+1] (V1,t,...,V1,t).

Since ∑l p! i + ... + i (p + 1)! = p! 1 l = , i ! .. . i − !(i − 1)!i ! . . . i ! i ! .. . i ! i ! .. . i ! m=1 1 m 1 m m+1 l 1 l 1 l it follows that ∑p ∑ ˜ U˜2(t) (p + 1)! Ul+1(t) S1 + S2 = Tf [1] (Vp+1,t) + Tf [l] (Vi1,t,...,Vil,t) i1! .. . il! l=2 i1,...,il≥1, i1+...+il=p+1 ∑p+1 ∑ ˜ ˜ Up+2(t) (p + 1)! Ul+1(t) +(p+1)!Tf [p+1] (V1,t,...,V1,t) = Tf [l] (Vi1,t,...,Vil,t), i1! .. . il! l=1 i1,...,il≥1, i1+...+il=p+1 proving the assertion. D

Recall that for bounded self-adjoint operators A and B and a unitary operator U on H, for f ∈ Cn(R) and g ∈ Cn(T) we set

− ∑n 1 k 1 d Rn,f,A(B) := f(A + B) − f(A + tB) (5.25) k! dtk k=0 t=0 and − ∑n 1 k iB 1 d itB Qn,g,U (B) := g(e U) − g(e U). (5.26) k! dtk k=0 t=0

93 The theorem below is justiﬁed in [29, Theorem 16] and [30, Theorem 6] in the particular case n = 2. In the self-adjoint case the extension to an arbitrary n ≥ 2 is relatively standard. However, the extension in the unitary case requires precise formulas obtained in the preceding theorem and the resulting formula (5.28) below appears to be new.

∈ n R ∗ ∗ ∈ 2 Theorem 5.12. (i) Suppose f C ( ), A = A ,B = B B(ℓd). Then, for the remainder deﬁned in (5.25), we have

A+B,A,...,A Rn,f,A(B) = Tf [n] (B,...,B). (5.27)

(ii) Suppose f ∈ Cn(T), A is a self-adjoint and U is a unitary operators in

2 B(ℓd). Then, for the remainder deﬁned in (5.26), we have n ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qn,f,U (A) = Tf [l] U, U,..., U . k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=n (5.28)

Proof. (i) Using (5.20) and Theorem 5.9, we have that

∑n−1 A+B,A − A,A,...,A Rn,f,A(B) = Tf [1] (B) Tf [k] (B,...,B). k=1 A+B,A − A,A Applying Lemma 5.7 (ii) with m = 1 to the diﬀerence Tf [1] (B) Tf [1] (B) on the right hand-side of the equality above, we obtain ∑n−1 A+B,A,A − A,A,...,A Rn,f,A(B) = Tf [2] (B,B) Tf [k] (B,...,B). k=2 Repeating the same argument (n − 1)-times, we arrive at (5.27). (ii) We will prove that for all 1 ≤ m ≤ n the formula

m ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qm,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=m (5.29) holds, using induction. The formula (5.28) will follow from (5.29) for m = n. For m = 1 by (5.20), we have ∞ ( ∑ k ) iA iA (iA) Q (A) = f(eiAU) − f(U) = T e U,U (eiAU − U) = T e U,U U , 1,f,U f [1] f [1] k! k=1

94 that is, (5.29) holds for m = 1. Assume that (5.29) holds for m = p < n, that is, we have p ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qp+1,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+... +jl=p 1 dp − itA p f(e U). p! dt t=0

Applying Theorem 5.11 and decomposing ip = ij1 · ... · ijl , we obtain

p ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qp+1,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=p p ( ) ∑ ∑ j1 jl − U,...,U (iA) (iA) Tf [l] U,..., U . j1! jl! l=1 j1,...,jl≥1 j1+...+jl=p Regrouping the terms, we obtain

p ( ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qp+1,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1 j +...+j =p 1 (l )) j1 jl − U,...,U (iA) (iA) Tf [l] U,..., U . j1! jl! The latter can be rewritten as p ( ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qp+1,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=p ( j j ) iA (iA) 1 (iA) l − T e U,...,U U,..., U f [l] j ! j ! ( 1 l ) ( )) j1 jl j1 jl eiAU,...,U (iA) (iA) − U,...,U (iA) (iA) + Tf [l] U,..., U Tf [l] U,..., U . j1! jl! j1! jl! Applying multilinearity of the multiple operator integral and the property (5.21), we obtain p ( ∞ ) ∑ ∑ ∑ k j2 jl eiAU,U,...,U (iA) (iA) (iA) Qp+1,f,U (A) = Tf [l] U, U,..., U k! j2! jl! l=1 j1,...,jl≥1 k=j1+1 j1+...+jl=p p ( ∞ ) ∑ ∑ ∑ k j1 jl eiAU,U,...,U (iA) (iA) (iA) + Tf [l+1] U, U,..., U k! j1! jl! l=1 j1,...,jl≥1 k=1 j1+...+jl=p

=: S1 + S2. (5.30)

95 Making a substitution i1 = j1 + 1, ir = jr for 2 ≤ r ≤ l in the ﬁrst sum on the right-hand side of the latter equality, we have

p ( ∞ ) ∑ ∑ ∑ k i2 il eiAU,U,...,U (iA) (iA) (iA) S1 = Tf [l] U, U, ..., U . k! i2! il! l=1 i1≥2,i2,...,il≥1, k=i1 i1+...+il=p+1

Relabeling the summands of S2 by the mapping l 7→ l − 1, and then making the substitution i1 = 1, ir = jr−1 for 2 ≤ r ≤ l we obtain

p+1 ( ∞ ) ∑ ∑ ∑ k j1 jl−1 eiAU,U,...,U (iA) (iA) (iA) S2 = Tf [l] U, U, ..., U k! j1! jl−1! l=2 j1,...,jl−1≥1 k=1 j1+...+jl−1=p p+1 ( ∞ ) ∑ ∑ ∑ k i2 il eiAU,U,...,U (iA) (iA) (iA) = Tf [l] U, U, ..., U . k! i2! il! l=2 i1=1,i2,...,il≥1, k=i1 i1+...+il=p+1 Combining the two previous equalities with (5.30), we arrive at

∞ ( ∑ k ) iA (iA) Q (A) = T e U,U,...,U p+1,f,U f [1] k! k=p+1 p ( ) ( ∞ ) ∑ ∑ ∑ ∑ k i2 il eiAU,U,...,U (iA) (iA) (iA) + + Tf [l] U, U, ..., U k! i2! il! l=2 i1=1,i2,...,il≥1, i1≥2,i2,...,il≥1, k=i1 i1+...+il=p+1 i1+...+il=p+1 ∞ ( ∑ k ) iA (iA) + T e U,U,...,U U, iAU, . . . , iAU f [p+1] k! k=1 p+1 ( ∞ ) ∑ ∑ ∑ k i2 il eiAU,U,...,U (iA) (iA) (iA) = Tf [l] U, U, ..., U , k! i2! il! l=1 i1,i2,...,il≥1, k=i1 i1+...+il=p+1 which proves (5.29). D

5.2 Multiple operator integrals in the setting of non-commutative Lp-spaces

In this thesis N always stands for a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. An unbounded densely defined operator on H is said to be affiliated with N if it commutes with all elements in the commutant of N . A closed densely defined operator X affiliated with N is called

τ-measurable if τ(E|X|(n, ∞)) → 0 as n → ∞, where E|X|(n, ∞) is the spectral

96 ∗ 1 projection of the self-adjoint operator |X| = (X X) 2 corresponding to the interval (n, ∞). We denote the complete topological ∗-algebra of all τ-measurable operators equipped with the measure topology by S(N , τ) (see [44,70]). The support projection supp(X) of an operator X ∈ S(N , τ) is defined as follows supp(X) := E|X|(0, ∞). We say that an operator X ∈ S(N , τ) is τ- finitely supported if τ(supp(X)) < ∞. For every X ∈ S(N , τ), real and imaginary part of X are defined by setting X + X∗ X − X∗ ℜX := , ℑX := . 2 2i

For a self-adjoint operator X ∈ S(N , τ), we write X+ = XEX ([0, ∞)), X− =

XEX ((−∞, 0)).

The notions of the distribution function n|X|, and that of the singular value function µ(X): t 7→ µ(t, X) for X ∈ S(N , τ), are deﬁned as follows

n|X|(t) := τ(E|X|(t, ∞)), t ∈ R; µ(t, X) := inf{s ≥ 0 : n|X|(s) ≤ t}, t ≥ 0.

It follows directly that the singular value function µ(X) is a decreasing, right- continuous function on the positive half-line [0, ∞). In the special case, when N is the von Neumann algebra L∞(0, ∞) equipped with the normal semiﬁnite trace given by integration with respect to the Lebesgue measure, the space S(N , τ) coincides with S(0, ∞), the space of all Lebesgue measurable functions f on (0, ∞) such that n|f|(t) < ∞ for some t ∈ R.

It is well-known (see e.g. [70, Corollary 2.3.16]) that for the operators X1,...,Xm ∈ S(N , τ), m ∈ N, we have that

( m ) m ( ) ∑ ∑ t µ t, X ≤ µ , X t > 0, (5.31) j m j j=1 j=1

( m ) m ( ) ∏ ∏ t µ t, X ≤ µ , X t > 0. (5.32) j m j j=1 j=1 For future references, we also record here the following simple estimate ∫ ( t ) 1 1 p µ(t, X) ≤ µ(s, X)pds , t > 0, 1 ≤ p < ∞,X ∈ S(N , τ). (5.33) t 0 We shall also need the notion of uniform submajorization introduced in [59], which plays an important role in the theory of singular traces (see [70, S 3.4]).

97 Let X, Y ∈ S(N , τ). We say that the operator Y is uniformly submajorized by X if there exists λ ∈ N such that ∫ ∫ b b µ(s, Y )ds ≤ µ(s, X)ds, λa a for all a, b > 0 such that λa < b. In this case we use the notation Y ▹ X. The following result is proved in [59, Lemma 8.4].

Lemma 5.13. For 0 ≤ X,Y ∈ S(N , τ), we have

X + Y ▹ µ(X) + µ(Y ).

∞ ∞ By σu : L (0, ∞) → L (0, ∞), u ∈ (0, ∞), we denote the dilation operator deﬁned by ( ) s (σ f)(s) = f , s ∈ (0, ∞) f ∈ L∞(0, ∞). (5.34) u u Lemma 5.14. For 0 ≤ X,Y ∈ S(N , τ), we have

µ(X) + µ(Y ) ▹ 2σ 1 µ(X + Y ). 2

Proof. Let 0 ≤ X,Y ∈ S(N , τ). The assertion of the lemma is known in the special case when X and Y are bounded operators. Indeed, it follows from [70, Lemma 3.4.4] that there exists λ ∈ N (more precisely, λ = 2), such that for 0 ≤ C,D ∈ N we have ∫ ∫ b b ( ) (µ(s, C) + µ(s, D))ds ≤ 2 µ 2s, C + D ds (5.35) λa a for all λa < b. For ﬁxed a, b > 0 such that λa < b consider the operators

X0 = min{X, µ(a, X)} ∈ N ,Y0 = min{Y, µ(a, Y )} ∈ N .

Observe that µ(s, X0) = µ(s, X) and µ(s, Y0) = µ(s, Y ) for all s ≥ a and µ(X0 +

Y0) ≤ µ(X + Y ). Applying (5.35) with C = X0, D = Y0, we obtain ∫ ∫ b b (µ(s, X) + µ(s, Y ))ds = (µ(s, X0) + µ(s, Y0))ds λa λa ∫ ∫ b b ≤ 2 µ(2s, X0 + Y0)ds ≤ 2 µ(2s, X + Y )ds. a a D

98 5.2.1 Classical non-commutative Lp-spaces and weak Lp- spaces

In this section we recall deﬁnitions and some important properties of the classical non-commutative Lp-spaces and the weak Lp-spaces. The noncommutative space Lp(N , τ), 1 ≤ p < ∞ is deﬁned as follows

p p L (N , τ) := {X ∈ S(N , τ): µ(|X|) ∈ L (0, ∞)}, ∥X∥Lp(N ,τ) := ∥µ(|X|)∥p,

p where (L (0, ∞), ∥ · ∥p) is the usual Lebesgue space. The uniform norm on ∞ L (N , τ) := N is denoted by ∥ · ∥∞. Recall that for 1 ≤ r, q ≤ ∞, the space (Lq + Lr)(N , τ) is deﬁned as

q r (L + L )(N , τ) := {X ∈ S(N , τ): ∥X∥(Lq+Lr)(N ,τ) < ∞}, where

q r ∥X∥(Lq+Lr)(N ,τ) := inf{∥Y ∥q +∥Z∥r : X = Y +Z,Y ∈ L (N , τ),Z ∈ L (N , τ)}.

We shall sometimes suppress the algebra (N , τ) from the notations. For example, we shall write ∥ · ∥Lq+Lr rather than ∥ · ∥(Lq+Lr)(N ,τ). This should not cause any confusion. The following simple lemma is technical, but essential for a subsequent exposition.

Lemma 5.15. Let k ∈ N and k < r < q < ∞. Let T be a multilinear operator

×k with dom(T ) ⊂ S(N , τ) . Suppose that for all 1 < p, pj < ∞, 1 ≤ j ≤ k, with

1 ∑k 1 = , p p j=1 j there exists a constant C(p) > 0 such that the operator T satisﬁes

∥T ∥Lp1 ×...×Lpk →Lp ≤ C(p). (5.36)

Then

q r ×k q r T ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)).

99 q r q Proof. By the deﬁnition, for X1, ..., Xk ∈ (L +L )(N , τ), there are Yj ∈ L (N , τ), r Zj ∈ L (N , τ) such that Xj = Yj + Zj, 1 ≤ j ≤ k.

∥T (X1,...,Xk)∥ q r = ∥T (Y1 + Z1,. ..,Yk + Zk)∥ q r L k +L k L k +L k ∑ = T (X1,A ,. ..,Xk,A ) q r L k +L k A ⊂{1,...,k} ∑ ≤ ∥T (X1,A ,. ..,Xk,A )∥ q r , (5.37) L k +L k A ⊂{1,...,k} where   Yj, j ∈ A Xj,A = , 1 ≤ j ≤ k.  Zj, j∈ / A Fix A ⊂ {1, . . . , k}. Setting

1 |A | k − |A | = + , pA r q we have that r q ≤ pA ≤ , k k q r pA and, therefore, L (N , τ) ⊂ (L k + L k )(N , τ). Thus,

∥T (X1,A ,...,Xk,A )∥ q r ≤ const ∥T (X1,A ,. ..,Xk,A )∥LpA . (5.38) L k +L k

Using (5.36), we obtain that ∏ ∏ ∥T (X1,A ,...,Xk,A )∥LpA ≤ c(pA ) ∥Yj∥Lq ∥Zj∥Lr j∈A j∈ /A ∏k ≤ c(pA ) (∥Yj∥Lq + ∥Zj∥Lr ). (5.39) j=1

Combining (5.37), (5.38) and (5.39), we arrive at ∑ ∥T (X1,...,Xk)∥ q r ≤ const ∥T (X1,A ,. ..,Xk,A )∥LpA L k +L k A ⊂{1,...,k} ∑ ∏k ≤ const c(pA ) (∥Yj∥Lq + ∥Zj∥Lr ). A ⊂{1,...,k} j=1

q Taking the inﬁmum over all representations Xj = Yj + Zj,Yj ∈ L (N , τ),Zj ∈ Lr(N , τ), 1 ≤ j ≤ k, we complete the proof. D

100 The space Lp,∞(N , τ), 1 ≤ p < ∞ is the set of all X ∈ S(N , τ) such that

′ 1 ∥ ∥ p ∞ X Lp,∞ := sup t µ(t, X) < + . t≥0

≤ ∞ p,∞ N ∥ · ∥′ If 1 p < , then the space L ( , τ) equipped with the quasi-norm Lp,∞ given above becomes a quasi-Banach space (see e.g. [70, Example 2.6.10] and [38]).

p,∞ Remark 5.16. For 1 0 0 which satisfies ′ p ′ ∥ · ∥ p,∞ ≤ ∥ · ∥ p,∞ ≤ ∥ · ∥ p,∞ . (5.40) L L p − 1 L In the special case, when N is the von Neumann algebra L∞(0, ∞) equipped with the normal semifinite trace given by integration with respect to the Lebesgue measure, the space Lp,∞(N , τ) coincides with the classical commutative weak Lp- space Lp,∞(0, ∞) (see e.g. [38,66,69]). Observe that if 1 < r < p < q < ∞, then Lp(N , τ),Lp,∞(N , τ) ⊂ (Lq + Lr)(N , τ) (see e.g. [69, Proposition 2.b.9 and p. 143]). ∥ · ∥′ The following result is the Höldertype inequality for the quasi-norm Lp,∞ and the norm ∥ · ∥Lp,∞ .

1 Lemma 5.17. Let m ∈ N and let 1 ≤ p, p1, . . . , pm < ∞ be such that = ∑ p m 1 . j=1 pj

1 p ∈ N (i) There is a constant c1(m, p) = m such that for all Xj Lpj ,∞( , τ), 1 ≤ j ≤ m, we have

∏m ′ ∥ · · ∥ ≤ ∥ ∥ p ,∞ X1 ... Xm Lp,∞ c1(m, p) Xj L j . j=1

p 1 ∞ p (ii) If 1 < p, p1, . . . pm < , then there is a constant c2(m, p) = p−1 m , such

pj ,∞ that for all Xj ∈ L (N , τ), 1 ≤ j ≤ m, we have

∏m ∞ ∥X1 · ... · Xm∥Lp,∞ ≤ c2(m, p) ∥Xj∥Lpj , . j=1

101 Proof. (i). It follows from (5.32) that for all t > 0,

∏m ( ) ∏m ( )− 1 ( )− 1 ∏m t t pj ′ t p ′ µ(t, X ·...·X ) ≤ µ ,X ≤ ∥X ∥ p ,∞ = ∥X ∥ p ,∞ . 1 m m j m j L j m j L j j=1 j=1 j=1 Hence, ∏m 1 1 ′ p · · ≤ p ∥ ∥ ∞ t µ(t, X1 . .. Xm) m Xj Lpj , . j=1 Taking the supremum over t > 0, and using (5.40), we arrive at ∏m ∏m ′ 1 ′ 1 ∥ · · ∥ ≤ p ∥ ∥ ∞ ≤ p ∥ ∥ p ,∞ X1 ... Xm Lp,∞ m Xj Lpj , m Xj L j , j=1 j=1 which proves (i). (ii). Using (5.40) and (i), we obtain

p ′ ∥X · ... · X ∥ p,∞ ≤ ∥X · .. . · X ∥ p,∞ 1 m L p − 1 1 m L ∏m ∏m p 1 ′ p 1 ∞ ≤ m p ∥X ∥ p ,∞ ≤ m p ∥X ∥ pj , . p − 1 j L j p − 1 j L j=1 j=1 D

It is straightforward that the dilation operator σu, u ∈ (0, ∞), deﬁned in (5.34) is a bounded linear operator on the Banach space Lp,∞(0, ∞) for 1 < p < ∞ with the norm 1 ∥σu∥Lp,∞→Lp,∞ = u p . (5.41)

5.2.2 Haagerup’s Lp-spaces

In this section we recall the construction of non-commutative Lp-spaces associated with an arbitrary von Neumann algebra. We use Haagerup’s deﬁnition [52], and Terp’s exposition of the subject [106]. The basics on von Neumann algebras and Tomita’s modular theory may be found in [56]. Let M be an arbitrary von Neumann algebra with a faithful normal semiﬁnite

ϕ0 weight ϕ0. We consider the one-parameter modular automorphism group σ = { ϕ0 } M σt t∈R (associated with ϕ0) on and obtain a semiﬁnite crossed product von N M o R Neumann algebra := σϕ0 which admits the canonical semiﬁnite trace τ and a trace-scaling dual action θ = {θs}s∈R such that

−s τ ◦ θs = e τ for all s ∈ R.

102 The original von Neumann algebra M can be identiﬁed with a θ-invariant von Neumann subalgebra L∞(M) of N . For 1 ≤ p < ∞, the noncommutative Lp-space Lp(M) is deﬁned as follows

p − s L (M) := {X ∈ S(N , τ): θs(X) = e p X for all s ∈ R}.

It is known from [106, Part II, Theorem 7] that there is a linear bijection ψ 7→ Xψ 1 between the predual space M∗ and L (M). Due to this correspondence we may deﬁne the trace tr : L1(M) → C as follows

1 tr(Xψ) := ψ(1),Xψ ∈ L (M). (5.42)

Given any X ∈ Lp(M), 1 ≤ p < ∞, we have the polar decomposition X = U|X|, where |X| is a positive operator in Lp(M) and U is a partial isometry contained in M. It is established in [106, Proposition 12] that |X|p ∈ L1(M). Thus, we can deﬁne a Banach norm (see [106, Corollary 27]) on Lp(M) by setting

p 1 p ∥X∥Lp := tr (|X| ) p , X ∈ L (M). (5.43)

∈ Lp M ∥ ∥ ∥ ∥′ The following lemma shows that if X ( ), then X Lp = X Lp,∞ and, therefore, Lp(M) is a closed linear subspace in Lp,∞(N , τ).

Lemma 5.18. [44, Lemma 1.7] Let 1 ≤ p < ∞. If X ∈ Lp(M), then

− 1 µ(t, X) = ∥X∥Lp · t p , t > 0.

The following result is proved in e.g. [106, Theorem 23]. ∑ k 1 Lemma 5.19. Let k ∈ N and 1 ≤ p, pj ≤ ∞, 1 ≤ j ≤ k, be such that = j=1 pj 1 ∈ pj M ≤ ≤ p . For every Xj L ( ), 1 j k, we have

∥X1 · ... · Xk∥Lp ≤ ∥X1∥Lp1 · ... · ∥Xk∥Lpk . (5.44)

Let now M be a semiﬁnite von Neumann algebra and τ0 be a faithful normal semiﬁnite trace on M. Let M⊗¯ L∞(R) be the von Neumann algebra tensor product of M and L∞(R) acting on the Hilbert space tensor product H⊗¯ L2(R) ′ equipped with the tensor product trace τ := τ0 ⊗ ν, where ν is the trace on L∞(R), given by ∫ ν(f) = f(s)e−sds, 0 ≤ f ∈ L∞(R). R

103 Recall that τ ′ is the unique faithful normal semiﬁnite trace on M⊗¯ L∞(R) satis- ′ ∞ fying τ (X ⊗ f) = τ0(X)ν(f),X ∈ M, f ∈ L (R). It is known that in the case when M is a semiﬁnite von Neumann algebra there exists a trace preserving ∗-isomorphism between the crossed product von Neumann algebra (N , τ) and (M⊗¯ L∞(R), τ ′) (see [32, Part II, Proposition 4.2]). Further in this subsection we identify (N , τ) with (M⊗¯ L∞(R), τ ′). It is also known that for all X ∈ M, f ∈ L∞(R), we have

θs(X ⊗ f) = X ⊗ ls(f), s ∈ R, (5.45)

where ls is the left translation by s (see e.g. [32, Part II, Proposition 4.2]). The following result is well-known (see e.g. [52, Theorem 2.1], [106, p. 62]). We present a short proof here for the reader’s convenience.

t Theorem 5.20. Let 1 ≤ p ≤ ∞ and ζp(t) = e p , t ∈ R. Let M be a semiﬁnite von Neumann algebra equipped with a faithful normal semiﬁnite trace τ0. The following statements hold.

∞ ′ (i) The operator X ⊗ ζp aﬃliated with M⊗¯ L (R) is τ -measurable for all X ∈ p L (M, τ0).

p p (ii) X ⊗ ζp ∈ L (M), for all X ∈ L (M, τ0).

(iii) The mapping

p X 7→ X ⊗ ζp,X ∈ L (M, τ0)

p p is an isometry from L (M, τ0) into L (M).

p Proof. (i). Take X ∈ L (M, τ0) and consider the operator X ⊗ ζp aﬃliated with M⊗¯ ∞ R ⊗ R → L ( ). Identifying the element X ζp with the Bochner function Xζp : p M · ∈ R L ( , τ0) given by Xζp (s) := X ζp(s), s (see e.g. [15, (5.1)]), we have

′ τ (E| ⊗ |(λ, ∞)) = (τ ⊗ ν)(E| ⊗ |(λ, ∞)) X ζp ∫ 0 X ζp ∫ −s − s −s s ∞ | | · p ∞ = τ0(E| |· p (λ, ))e ds = τ0(E X (λ e , ))e ds. R X e R

104 − s Making a substitution u = λ · e p in the latter integral and then integrating by parts, we obtain

∫ ∞ ′ ∞ ∞ p−1 −p τ (E|X⊗ζp|(λ, )) = p τ0(E|X|(u, ))u λ du 0 ∫ ∞ − −p p ∞ −p∥ ∥p = λ u d(τ0(E|X|(u, ))) = λ X Lp . (5.46) 0 ′ ∞ → → ∞ ⊗ Therefore, we have that τ (E|X⊗ζp|(λ, )) 0 as λ , i.e. the operator X ζp is τ ′-measurable. (ii). For all s ∈ R, using (5.45), we have

− s θs(X ⊗ ζp) = X ⊗ ls(ζp) = e p (X ⊗ ζp).

′ p Since by (i) the operator X ⊗ζp is τ -measurable, it follows that X ⊗ζp ∈ L (M). p p (iii). Fix X ∈ L (M, τ0). By (ii), we have that X⊗ζp ∈ L (M) and, therefore, from Lemma 5.18, we infer

− 1 µ(t, X ⊗ ζp) = t p ∥X ⊗ ζp∥Lp , t > 0.

On the other hand, from (5.46), it follows that

− 1 µ(t, X ⊗ ζp) = t p ∥X∥Lp , t > 0.

Thus, ∥X∥Lp = ∥X ⊗ ζp∥Lp . D

5.2.3 Multiple operator integrals

Let N be a semiﬁnite von Neumann algebra equipped with a faithful normal semiﬁnite trace τ.

Deﬁnition 5.21. Let k ∈ N and let 1 ≤ pj ≤ ∞, 1 ≤ j ≤ k be such that 0 ≤ ∑ k 1 k+1 ≤ 1. Let a bounded Borel function φ : R → C be ﬁxed. Let A0, ..., Ak j=1 pj

pj be self-adjoint operators aﬃliated with N . Suppose that for all Xj ∈ L (N , τ), 1 ≤ j ≤ k, for every m ∈ N the series

( ) ([ )) k ([ )) ∑ l l l l + 1 ∏ l l + 1 S (X , ..., X ) := φ 0 , ..., k E 0 , 0 X E j , j φ,m 1 k m m A0 m m j Aj m m l0,...,lk∈Z j=1

105 ∑ converges in the norm of Lp(N , τ), where 1 = k 1 and p j=1 pj

(X1, ..., Xk) 7→ Sφ,m(X1, ..., Xk), m ∈ N is a sequence from B(Lp1 (N , τ) × ... × Lpk (N , τ),Lp(N , τ)). If the sequence

{ } A0,...,Ak Sφ,m m≥1 converges pointwise to some multilinear operator Tφ , then, according to the Banach-Steinhaus theorem (see e.g. [107]), {Sφ,m}m≥1 is uniformly

A0,...,Ak ∈ p1 N × × pk N p N bounded and Tφ B(L ( , τ) ... L ( , τ),L ( , τ)). In this case,

A0,...,Ak the operator Tφ is called a multiple operator integral.

The definition above is different from the original Pavlov’s approach [76] and its modern modifications from [9, 81]. However, for a large class of functions φ all these definitions coincide (see e.g. [84, Lemma 3.5] or [92, Section 1], where a detailed comparison of these definitions is presented). In this thesis, in the infinite dimensional setting we shall work exclusively with multiple operator integrals defined by the procedure given in Definition 5.21. When we consider the multiple operator integrals on the product of sums of Lp-spaces, then we will use the following modified definition.

Deﬁnition 5.22. Let k ∈ N and let 1 ≤ qj ≤ rj ≤ ∞, 1 ≤ j ≤ k be such ∑ that 0 ≤ k 1 ≤ 1. Let a bounded Borel function φ : Rk+1 → C be ﬁxed. Let j=1 qj

A0, ..., Ak be self-adjoint operators aﬃliated with N . Suppose that for all Xj ∈

(Lqj + Lrj )(N , τ), 1 ≤ j ≤ k, for every m ∈ N the series

( ) ([ )) k ([ )) ∑ l l l l + 1 ∏ l l + 1 S (X , ..., X ) := φ 0 , ..., k E 0 , 0 X E j , j φ,m 1 k m m A0 m m j Aj m m l0,...,lk∈Z j=1 (5.47) ∑ ∑ converges in the norm of (Lq + Lr)(N , τ), where 1 = k 1 , 1 = k 1 , and q j=1 qj r j=1 rj

(X1, ..., Xk) 7→ Sφ,m(X1, ..., Xk), m ∈ N is a sequence from B((Lq1 +Lr1 )(N , τ)×...×(Lqk +Lrk )(N , τ), (Lq+Lr)(N , τ)). If

{ } A0,...,Ak the sequence Sφ,m m≥1 converges pointwise to some multilinear operator Tφ , then, according to the Banach-Steinhaus theorem (see e.g. [107]), {Sφ,m}m≥1 is

A0,...,Ak ∈ q1 r1 N × × qk rk N q uniformly bounded and Tφ B((L +L )( , τ) ... (L +L )( , τ), (L +

r N A0,...,Ak L )( , τ)). In this case, the operator Tφ is also called a multiple operator integral.

106 Lemma 5.23. Let k ∈ N be ﬁxed and let A0, ..., Ak be self-adjoint operators aﬃliated with N . Let k < r < q < ∞. Let φ : Rk+1 → C be such that × q r A0,...,Ak ∈ q r N k k k N Tφ B((L + L )( , τ) , (L + L )( , τ)). If A0, A1 are bounded, then q r for all X1,...,Xk ∈ (L + L )(N , τ) the following assertions hold.

(i) Setting

ψ0(x0, . . . , xk) := x0φ(x0, . . . , xk),

we have that

q r A0,A1,...,A q r ×k T k ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)) ψ0

and

T A0,A1,...,Ak (A X ,X ,...,X ) = T A0,A1,...,Ak (X ,X ,...,X ). φ 0 1 2 k ψ0 1 2 k

(ii) Setting

ψ1(x0, . . . , xk) := x1φ(x0, . . . , xk),

we have that

q r A0,A1,...,A q r ×k T k ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)) ψ1

and

T A0,A1,...,Ak (X A ,X ,...,X ) = T A0,A1,...,Ak (X ,X ,...,X ). φ 1 1 2 k ψ1 1 2 k

Proof. (i). For m ∈ N, directly from Deﬁnition 5.22, we have

Sφ,m(A0X1,X2,...,Xk) = Sψ0,m(X1,X2,...,Xk) + Sφ,m(A0,mX1,X2,...,Xk), (5.48) where ( ) ([ )) ∑ l l l + 1 A := A − E - , --- . 0,m 0 m A0 m m l∈Z

Since {Sφ,m}m∈N is a uniformly bounded sequence of operators and since ∥A0,m∥Lq+Lr → 0 as m → ∞, it follows that

∥Sφ,m(A0,mX1,X2,...,Xk)∥ q r L k +L k ∏k ≤ ∥ ∥ q ∥ ∥ ∥ ∥ → Sφ,m × r A0,m Lq+Lr Xj Lq+Lr 0, (Lq+Lr) k→L k +L k j=2

107 as m → ∞.

q r Since A0 is bounded, it follows that A0X1 ∈ (L + L )(N , τ). By Deﬁni- tion 5.22, we have that

→ A0,A1,...,Ak Sφ,m(A0X1,X2,...,Xk) Tφ (A0X1,X2,...,Xk)

q r q r in (L k + L k )(N , τ) as m → ∞. Taking the (L k + L k )(N , τ)-limit of (5.48), we complete the proof of (i). The proof of (ii) is similar to that of (i) and, therefore, is omitted. D

In the following lemma we collect simple properties of multiple operator integral, whose proofs follow immediately from Deﬁnition 5.21. For the reader’s convenience we demonstrate the short proof of item (i) in the case k = 1 below.

Lemma 5.24. Let k ∈ N be fixed and let A0, ..., Ak be self-adjoint operators affiliated with N . Let k < r < q < ∞. Let φ : Rk+1 → C be such that the × q r A0,...,Ak ∈ q r N k k k N ∈ operator Tφ B((L +L )( , τ) , (L +L )( , τ)). For all X1,...,Xk (Lq + Lr)(N , τ),X ∈ N and for any self-adjoint operator A affiliated with N , the following assertions hold.

(i) Setting

ψ0(x0, . . . , xk+1) := φ(x0, x2, . . . , xk+1),

we have that

q r A0,A,A1,...,A q r ×k T k ∈ B(N × (L + L )(N , τ) , (L k + L k )(N , τ)) ψ0

and

T A0,A,A1,...,Ak (X,X ,...,X ) = T A0,...,Ak (XX ,X ,...,X ). (5.49) ψ0 1 k φ 1 2 k

(ii) Setting

ψ1(x0, x1, . . . , xk+1) := φ(x1, . . . , xk+1),

we have that

q r A,A0,...,A q r ×k T k ∈ B(N × (L + L )(N , τ) , (L k + L k )(N , τ)) ψ1

and

T A,A0,...,Ak (X,X ,...,X ) = X · T A0,...,Ak (X ,...,X ). ψ1 1 k φ 1 k

108 Proof. (i). Let k = 1. Fix self-adjoint operators A0,A1 aﬃliated with N and

∞ R2 → C A0,A1 ∈ q 1 < r < q < . Let φ : be such that the operator Tφ B((L + Lr)(N , τ), (Lq + Lr)(N , τ)). In this case

ψ0(x0, x1, x2) = φ(x0, x2).

q r Take X1 ∈ (L + L )(N , τ),X ∈ N and a self-adjoint operator A aﬃliated with

N A0,A1 . Expanding the partial sums of the operator integral Tφ (XX1), we have

Sφ,m(XX1) ( ) ([ )) ([ )) ∑ l l l l + 1 l l + 1 = φ 0 , 1 E 0 , 0 XX E 1 , 1 m m A0 m m 1 A1 m m l ,l ∈Z ( 0 1 ) ([ )) ( ([ ))) ([ )) ∑ l l l l + 1 ∑ l l + 1 l l + 1 = φ 0 , 1 E 0 , 0 X E , X E 1 , 1 m m A0 m m A m m 1 A1 m m l ,l ∈Z l∈Z 0 1 ( ) ([ )) ([ )) ([ )) ∑ l l l l l + 1 l l + 1 l l + 1 = ψ 0 , , 1 E 0 , 0 XE , X E 1 , 1 0 m m m A0 m m A m m 1 A1 m m l0,l,l1∈Z ≥ = Sψ0,m(X, X1), m 1.

Taking the limit in (Lq + Lr)(N , τ), by Deﬁnition 5.21, we obtain (5.49).

Since {Sφ,m}m≥1 is a uniformly bounded sequence of linear operators from B((Lq + Lr)(N , τ), (Lq + Lr)(N , τ)), it follows that

∥ ∥ q r ∥ ∥ q r Sψ0,m(X,X1) L +L = Sφ,m(XX1) L +L

≤ const ∥XX1∥Lq+Lr ≤ const ∥X∥∞∥X1∥Lq+Lr , m ≥ 1.

{ } Thus, Sψ0,m m≥1 is a uniformly bounded sequence of bilinear operators from B(N × (Lq + Lr)(N , τ), (Lq + Lr)(N , τ)), and so its limit T A0,A,A1 also belongs ψ0 to B(N × (Lq + Lr)(N , τ), (Lq + Lr)(N , τ)). The proof for an arbitrary k > 1 follows the special case k = 1 verbatim. The proof of (ii) can be done similarly. D

Next we introduce a class of functions φ, for which the multiple operator

A0,...,Ak p integral Tφ is well-deﬁned on the direct product of L -spaces.

109 Integral momenta

Let k ∈ N and let { } ∑k k Rk := (s1, . . . , sk) ∈ R : sj ≤ 1, sj ≥ 0, j = 1, . . . , k . j=1 ∑ ∈ − k ∈ ∞ R For (s1, . . . , sk) Rk we denote s0 := 1 j=1 sj. Given h L ( ), we set ∫

φk,h (x0, . . . , xk) = h (s0x0 + ... + skxk) dvk, (5.50) Rk

k+1 k where (x0, . . . , xk) ∈ R and dvk is the Lebesgue measure on R . Following the terminology from [84] we shall call the function φk,h an integral momentum. This notion plays a crucial role in the approach here. Indeed, the functions φ :

Rk+1 → C A0,...,Ak for which we shall be considering multiple operator integral Tφ in this thesis are in fact of the form φk,h for suitable choice of h. Theorem 5.25 below presents a wide class of such functions.

Theorem 5.25 ( [84, Theorem 5.3]). Let h be a continuous bounded function and let k ≥ 1. Let φk,h be an integral momentum deﬁned in (5.50) and let

(A0,...,Ak) be an arbitrary (k + 1)-tuple of self-adjoint operators aﬃliated with 1 1 1 N . If 1 0, such that

T A0,...,Ak ≤ C(p) ∥h∥ . (5.51) φk,h ∞ Lp1 ×...×Lpk →Lp

Remark 5.26. Strictly speaking in the result above the operators T A0,...,Ak de- φk,h pend on the parameters p1, . . . , pk, p. However, it follows from the Deﬁnition 5.21 and the fact that Lp(N , τ), 1 ≤ p < ∞, is continuously embedded into S(N , τ) equipped with the measure topology, that these operators coincide on the intersection of their domains. Therefore, we may speak of a single multilinear operator ( ) × integral T A0,...,Ak with dom T A0,...,Ak ⊂ S(N , τ) k. φk,h φk,h

Let x0, x1, . . . , xk ∈ R and let f be a function on R. Recall that the divided difference f [j] (0 ≤ j ≤ k) is defined recursively as follows. The divided difference of the zeroth order f [0] is the function f itself. The

110 divided diﬀerence of the order k = 1, 2,... is deﬁned by   [k−1] [k−1]  f (x0,x2,...,xk)−f (x1,x2,...,xk) , if x0 ≠ x1, [k] x0−x1 f (x0, x1, . . . , xk) :=   d [k−1] f (x1, x2, . . . , xk), if x0 = x1, dx1

If f (k) ∈ L∞(R), then the function f [k] admits the following integral representation (see e.g. [92, (15)]), ∫ [k] (k) f (x0, . . . , xk) = f (s0x0 + ... + skxk) dvk, for every k ≥ 1. (5.52) Rk

Observe that the function f [k] is represented in (5.52) as a k-th integral mo-

(k) mentum φ = φk,h, with h = f . The following result is a modiﬁed version of [92, Lemma 10]. We provide a short proof for the reader’s convenience.

Lemma 5.27. Let h and h′ be bounded continuous functions on R. The formula

[1] ′ ∈ R φk+1,h (x0, . . . , xk+1) = φk,h(x0, . . . , xk+1), x0, . . . , xk+1 , (5.53)

[1] · holds, where φk,h is the first divided difference of the function φk,h( , x2, . . . , xk+1) with fixed variables x2, . . . , xk+1.

(k) Proof. Let f be such that f = h. Using (5.52) and (5.50) twice, for x0, . . . , xk+1 ∈ R, we have

[k+1] φk+1,h′ (x0, . . . , xk+1) = φk+1,f (k+1) (x0, . . . , xk+1) = f (x0, . . . , xk+1)

[k] [1] [1] [1] = (f ) (x0, . . . , xk+1) = φk,f (k) (x0, . . . , xk+1) = φk,h(x0, . . . , xk+1).

The following algebraic property extends [92, Theorem 11] to the case of τ- measurable operators. In its proof for simplicity we assume that the algebra N is atomless. However, the result may be proved without this assumption.

Theorem 5.28. Let N be atomless. Let k ∈ N, k + 1 < r < q < ∞. Let A, B ∈

q r (L + L )(N , τ) and A1,...,Ak be self-adjoint operators. Let h be a bounded continuous function on R such that h′ is also bounded and continuous. The following

111 equality

A,A1,...,Ak B,A1,...,Ak A,B,A1,...,Ak T (X1,...,Xk)−T (X1,...,Xk) = T (A−B,X1,...,Xk) φk,h φk,h φk+1,h′ (5.54)

q r holds, for all X1,...,Xk ∈ (L + L )(N , τ). If, in particular, f (k), f (k+1) are bounded continuous functions, then

A,A1,...,Ak − B,A1,...,Ak A,B,A1,...,Ak − Tf [k] (X1,...,Xk) Tf [k] (X1,...,Xk) = Tf [k+1] (A B,X1,...,Xk).

Proof. Observe ﬁrstly that due to Theorem 5.25 and Lemma 5.15, we have

A,A ,...,A B,A ,...,A q r ×k q r T 1 k ,T 1 k ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)) (5.55) φk,h φk,h and

A,B,A ,...,A q r ×(k+1) q r T 1 k ∈ B((L + L )(N , τ) , (L k+1 + L k+1 )(N , τ)). (5.56) φk+1,h′

Let us denote

ψ0(x0, . . . , xk+1) := x0 φk+1,h′ (x0, . . . , xk+1),

ψ1(x0, . . . , xk+1) := x1 φk+1,h′ (x0, . . . , xk+1);

φ0(x0, . . . , xk+1) := φk,h(x0, x2, . . . , xk+1),

φ1(x0, . . . , xk+1) := φk,h(x1, x2, . . . , xk+1).

By Lemma 5.27, we have

ψ0 − ψ1 = φ0 − φ1.

Take a projection Q such that τ(Q) < ∞. Since A, B ∈ (Lq + Lr)(N , τ), it

q r (m) follows that AQ, QB, AQ − QB ∈ (L + L )(N , τ). Let A = AEA([−m, m]) (m) and B = BEB([−m, m]). Using Lemmas 5.23 and 5.24, we obtain

(m) (m) A ,B ,A1,...,Ak (m) (m) T ((A Q − QB ),X1,...,Xk) φk+1,h′ (m) (m) (m) (m) A ,B ,A1,...,Ak (m) A ,B ,A1,...,Ak (m) = T (A Q, X1,...,Xk) − T (QB ,X1,...,Xk) φk+1,h′ φk+1,h′ L5.23 (m) (m) (m) (m) = T A ,B ,A1,...,Ak (Q, X ,...,X ) − T A ,B ,A1,...,Ak (Q, X ,...,X ) ψ0 1 k ψ1 1 k (m) (m) A ,B ,A1,...,Ak A,B,A1,...,Ak = T (Q, X ,...,X ) = T − (Q, X) ,...,X ) ψ0−ψ1 1 k φ0 φ1 1 k

(m) (m) (m) (m) A ,B ,A1,...,Ak − A ,B ,A1,...,Ak = Tφ0 (Q, X1,...,Xk) Tφ1 (Q, X1,...,Xk) L5.24 (m) (m) = T A ,A1,...,Ak (QX ,...,X ) − QT B ,A1,...,Ak (X ,...,X ). (5.57) φk,h 1 k φk,h 1 k

112 We have

(m) (m) A ,B ,A1,...,Ak (m) (m) T ((A Q − QB ),X1,...,Xk) φk+1,h′

A,B,A1,...,Ak (m) (m) = T (EA([−m, m])(A Q − QB )EB([−m, m]),X1,...,Xk). φk+1,h′

Clearly,

(m) (m) EA([−m, m])(A Q − QB )EB([−m, m])

= EA([−m, m])(AQ − QB)EB([−m, m]) → AQ − QB in (Lq + Lr)(N , τ) as m → ∞. By (5.56), we have

(m) (m) A ,B ,A1,...,Ak (m) (m) A,B,A1,...,Ak T ((A Q−QB ),X1,...,Xk) → T ((AQ−QB),X1,...,Xk) φk+1,h′ φk+1,h′

q r in (L k+1 + L k+1 )(N , τ) as m → ∞. We also have

(m) T A ,A1,...,Ak (QX ,...,X ) = E ([−m, m])T A,A1,...,Ak (QX ,...,X ). φk,h 1 k A φk,h 1 k

By (5.55), we have

(m) T A ,A1,...,Ak (QX ,...,X ) → T A,A1,...,Ak (QX ,...,X ) φk,h 1 k φk,h 1 k

q r in (L k + L k )(N , τ) as m → ∞. Similarly,

(m) QT B ,A1,...,Ak (X ,...,X ) → QT B,A1,...,Ak (X ,...,X ) φk,h 1 k φk,h 1 k

q r in (L k + L k )(N , τ) as m → ∞. Using these convergences and (5.57), we obtain

A,B,A1,...,Ak T ((AQ − QB),X1,...,Xk) φk+1,h′ = T A,A1,...,Ak (QX ,...,X ) − QT B,A1,...,Ak (X ,...,X ). (5.58) φk,h 1 k φk,h 1 k

Since τ is a semiﬁnite trace, it follows that there exists a sequence {Qn}n∈N of projections satisfying Qn ↑ 1 and τ(Qn) < ∞. Using (5.58), we have

A,B,A1,...,Ak T ((AQn − QnB),X1,...,Xk) φk+1,h′ = T A,A1,...,Ak (Q X ,...,X ) − Q T B,A1,...,Ak (X ,...,X ), n ∈ N. (5.59) φk,h n 1 k n φk,h 1 k

113 q r Since AQn −QnB → A−B as n → ∞, in (L +L )(N , τ) (see e.g. [28, Proposition 2.5], using that N is atomless), it follows that

A,B,A1,...,Ak A,B,A1,...,Ak T ((AQn − QnB),X1,...,Xk) → T ((A − B),X1,...,Xk) φk+1,h′ φk+1,h′

q r in (L k+1 +L k+1 )(N , τ) and so also with respect to the measure topology. Similarly,

q r since QnX1 → X1 in (L + L )(N , τ) and

Q T B,A1,...,Ak (X ,...,X ) → T B,A1,...,Ak (X ,...,X ) n φk,h 1 k φk,h 1 k

q r in (L k + L k )(N , τ), it follows that

T A,A1,...,Ak (Q X ,...,X ) − Q T B,A1,...,Ak (X ,...,X ) φk,h n 1 k n φk,h 1 k

→ T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ) φk,h 1 k φk,h 1 k

q r in (L k + L k )(N , τ) and so also with respect to the measure topology. Taking the limit in (5.59) with respect to the measure topology, we complete the proof of the theorem. D

Remark 5.29. Let 1 < r < q < ∞. Let A, B ∈ (Lq + Lr)(N , τ) be self-adjoint operators. Let h be a bounded continuous function on R such that h′ is also bounded and continuous. The version of Theorem 5.28 above for k = 0 is stated as follows − A,B − h(A) h(B) = Th[1] (A B). (5.60)

The formula (5.60) extends [9, Theorem 5.3] to unbounded perturbations and a substantially wider class of functions. The proof of (5.60) is similar to that of Theorem 5.28 (see also [91]).

Let g be a compactly supported function with g(0) = 0. For l ∈ N, we define   g(t) ̸ tl , t = 0 gl(t) :=  (5.61) 0, t = 0. and g0 := g. The following lemma presents important properties of the multiple operator integral defined with respect to the divided difference of the function gl, l ≥ 0.

114 Lemma 5.30. Let l ≥ 0, k ≤ 1 and g be a compactly supported function on R

′ (k) (k−1) with g(0) = 0, g (0) = 0. Let also (gl) , (gl+1) be bounded and continuous functions, where gl and gl+1 are deﬁned by (5.61). ∑ 1 k 1 If 1 < p < ∞ is such that = for some 1 < p1, . . . , pk < ∞, then p j=1 pj

0,A1,...,Ak ∈ p1 N × × pk N p N T [k] B(L ( , τ) ... L ( , τ),L ( , τ)), gl and p1p A1,...,A p p k ∈ 2 N × × k N p1−p N T [k−1] B(L ( , τ) ... L ( , τ),L ( , τ)). gl+1 Moreover,

0,A1,...,Ak · A1,...,Ak T [k] (X1,...,Xk) = X1 T [k−1] (X2,...,Xk), (5.62) gl gl+1

pj where Xj ∈ L (N , τ), 1 ≤ j ≤ k.

Proof. Fix l ≥ 0 and k ≥ 1. First we show that

[j] [j−1] ∈ R ≤ ≤ gl (0, x1, . . . , xj) = gl+1 (x1, . . . , xj), xj , 1 j k, (5.63) using the method of induction with respect to j. It is clear that

[1] gl (0, x1) = gl+1(x1).

Thus, (5.63) holds for j = 1. Suppose that (5.63) holds for some j such that

1 ≤ j < k. For x1 ≠ x2, we obtain

[j] − [j] [j+1] gl (0, x1, x3, . . . , xj+1) gl (0, x2, x3, . . . , xj+1) gl (0, x1, . . . , xj+1) = x1 − x2 [j−1] − [j−1] gl+1 (x1, x3, . . . , xj+1) gl+1 (x2, x3, . . . , xj+1) [j] = = gl+1(x1, x2, . . . , xj+1). x1 − x2

The case x1 = x2 is similar. Thus, (5.63) holds for j + 1, which completes the proof of (5.63). By Theorem 5.25 we have that

0,A ,...,A 1 j ∈ p1 N × × pk N p N T [k] B(L ( , τ) ... L ( , τ),L ( , τ)) gl and p1p A1,...,A p p k ∈ 2 N × × k N p1−p N T [k−1] B(L ( , τ) ... L ( , τ),L ( , τ)). gl+1

115 pj Fix Xj ∈ L (N , τ), 1 ≤ j ≤ k. Since E0(a, b) = 1, if and only if 0 ∈ (a, b), a < b, where E0(a, b) is the spectral projection of the operator 0, corresponding to the interval (a, b), it follows that

S [k] (X1,...,Xk) gl ,m ( ) ([ )) k ([ )) ∑ l l l l + 1 ∏ l l + 1 = g[k] 0 , ..., k E 0 , 0 X E j , j l m m 0 m m j Aj m m l0,...,lk∈Z j=1 ( ) k ([ )) ∑ l l ∏ l l + 1 = g[k] 0, 1 , ..., k X E j , j , m ≥ 1. l m m j Aj m m l1,...,lk∈Z j=1

Using (5.63) in the latter equality, we obtain

S [k] (X1,...,Xk) gl ,m ∑ ( ) ([ )) ∏k ([ )) − l l l l + 1 l l + 1 = X · g[k 1] 1 , ..., k E 1 , 1 X E j , j 1 l+1 m m A1 m m j Aj m m l1,...,lk∈Z j=2

= X1 · S [k−1] (X2,. ..,Xk), m ≥ 1. gl+1 ,m

Taking the limit in Lp(N , τ) on the both sides of the latter equality and appealing to Deﬁnition 5.21, we complete the proof. D

5.2.4 H¨older-type estimate

Let N be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. The main result of this section is the Hölder-type estimate given in Theorem 5.36, which may be compared with [92, Corollary 13 and Theorem 14]. In our proof of Theorem 5.36, we replace the interpolation argument (see [92, proof of Theorem 14]) with a different technique based on the operator Pq,r defined in (5.64) below. Observe that the operator Pq,r is a Calderón-type operator, which is a useful technical tool in many questions from interpolation theory (see e.g. [13, Chapter 3, Section 5]). We proceed with the definition of Pq,r.

For 1 < q, r < ∞ and X ∈ S(N , τ), we consider the operator Pq,r : S(N , τ) → S(0, ∞), given by the formula

( ∫ ) ( ∫ ∞ ) t 1/r 1/q 1 r 1 q (Pq,r(X))(t) := µ (s, X)ds + µ (s, X)ds , t > 0. (5.64) t 0 t t

116 r q Observe that if X ∈ (L + L )(N , τ), r < q, then the value Pq,r(X)(t) is a ﬁnite number for all t > 0. Observe also that there is t > 0 such that Pq,r(X)(t) = 0 if and only if X = 0.

Lemma 5.31. If 1 < r < q < ∞, then for p ∈ (r, q) there exists a constant c(p, q, r) > 0 such that

p,∞ ∥Pq,r(X)∥Lp,∞ ≤ c(p, q, r)∥X∥Lp,∞ ,X ∈ L (N , τ).

Proof. Let X ∈ Lp,∞(N , τ). For every t > 0, we have ( ∫ ) ( ∫ ∞ ) 1 t 1/r 1 1/q (P (X))(t) = µr(s, X)ds + µq(s, X)ds q,r t t 0 ∫ t ∫ (( t ) ( ∞ ) ) 1 1 − r 1/r 1 − q 1/q ≤ sup s p µ(s, X) · s p ds + s p ds s>0 t 0 t t (( ∫ ) ( ∫ ∞ ) ) t 1/r 1/q ′ 1 − r 1 − q ∥ ∥ p p = X Lp,∞ s ds + s ds t 0 t t r q (( − ) 1 ( − ) 1 ) p p ′ t r t q = ∥X∥ p,∞ + − L − r − q 1 p 1 p (( ) 1 ( ) 1 ) − 1 ′ p r p q p = t ∥X∥ p,∞ + . L p − r q − p Appealing to (5.40), we complete the proof. D

The following lemma may be viewed as a general result from the theory of multilinear operators on symmetric operator spaces (see e.g. [70]). It gives a useful estimate of a singular value function of such a multilinear operator in terms of the operator Pr,q. For simplicity, the result below is proved under assumption that N is atomless.

Lemma 5.32. Let k ∈ N and let N be atomless. Suppose that for every 1 < ∑ 1 k 1 p, p1, . . . , pk < ∞ with = , we have p j=1 pj

T ∈ B(Lp1 (N , τ) × ... × Lpk (N , τ),Lp(N , τ)) and

∥T ∥Lp1 ×...×Lpk →Lp ≤ C(p), (5.65) for some constant C(p) dependent on p only. Then for every k < r < q < ∞, there exists a constant C(k, q, r) > 0 such that ∏k µ(T (X1,...,Xk)) ≤ C(k, q, r) Pr,q(Xj), j=1

117 r q for Xj ∈ (L + L )(N , τ), 1 ≤ j ≤ k.

r q Proof. Fix k ∈ N, t > 0 and X1,...,Xk ∈ (L + L )(N , τ). Deﬁne the operators

Uj and Wj such that Xj = Uj + Wj, where µ(Uj) = µ(Xj)χ(0,t) and µ(s, Wj) =

µ(s + t, Xj), s ≥ 0, 1 ≤ j ≤ k (these operators can be deﬁned via [27, Lemma 1.3], using that N is atomless). For every subset A ⊂ {1, . . . , k}, we set   Uj, j ∈ A Xj,A = , 1 ≤ j ≤ k.  Wj, j∈ / A

Since T is a multilinear operator, it follows that ∑ T (X1,...,Xk) = T (X1,A ,...,Xk,A ). A ⊂{1,...,k}

By (5.31), we have that ( ) ∑ t µ(t, T (X ,...,X )) ≤ µ , T (X A ,...,X A ) . (5.66) 1 k 2k 1, k, A ⊂{1,...,k}

Next we estimate every summand on the right hand side of (5.66) separately. Let

A = {n1, . . . , nl} ⊂ {1, . . . , k} for l = |A | ≤ k and n1, . . . , nl ∈ {1, . . . , k}. Setting 1 l k − l = + , rA r q using (5.33) and assumption (5.65), we see that

( ) ( k ) 1 t 2 rA µ , T (X A ,...,X A ) ≤ ∥T (X A , ...,X A )∥ rA 2k 1, k, t 1, k, L ( k ) 1 2 rA ≤ C(rA )∥X A ∥X · . .. · ∥X A ∥X , t 1, 1,A k, k,A where   Lr(N , τ), j ∈ A Xj,A = , 1 ≤ j ≤ k.  Lq(N , τ), j∈ / A Since   ∥U ∥ , j ∈ A ∥ ∥ j r ≤ ≤ Xj,A X A = , 1 j k, j,  ∥Wj∥q, j∈ / A

118 it follows that

( ) ( k ) 1 ∏ ∏ t 2 rA µ ,T (X A ,...,X A ) ≤ C(rA ) ∥U ∥ · ∥W ∥ . 2k 1, k, t j r j q j∈A j∈ /A

Observing that

( ) 1 ∏ ∏ 1 r A ∥U ∥ · ∥W ∥ t j r j q ∈A ∈A j j /∫ ∫ ( ) l ∏ ( ∞ ) 1 ( ) k−l ∏ ( ∞ ) 1 1 r r 1 q q = µr(s, U )ds · µq(s, W )ds t j t j ∈A 0 ∈A 0 j ∫ j∫ / ∏ ( t ) 1 ∏ ( ∞ ) 1 1 r 1 q = µr(s, X )ds · µq(s, X )ds t j t j j∈A 0 j∈ /A t ∫ ∫ ∏k (( t ) 1 ( ∞ ) 1 ) 1 r 1 q ≤ µr(s, X )ds + µq(s, X )ds t j t j j=1 0 t ∏k = (Pr,q(Xj))(t), j=1 we obtain

( ) ∏k t k µ ,T (X A ,...,X A ) ≤ 2 rA C(rA ) (P (X ))(t). 2k 1, k, r,q j j=1

Using the latter estimate together with (5.66), we arrive at

k ∑ k ∏ r µ(t, T (X1,...,Xk)) ≤ 2 A C(rA ) (Pr,q(Xj))(t). A ⊂{1,...,k} j=1

Setting ∑ k C(k, q, r) := 2 rA C(rA ), A ⊂{1,...,k} we complete the proof. D

H¨older class Λα

By Λα we denote the class of all H¨olderfunctions of exponent 0 ≤ α ≤ 1, that is the functions h : R → C such that

|h(t ) − h(t )| ∥h∥ := sup 1 2 < +∞. Λα | − |α t1≠ t2 t1 t2

Observe that the class Λ1 coincides with the class of all Lipschitz functions on R.

119 ∫ Lemma 5.33. Fix a Schwartz function W such that R W (y)dy = 1. There exists a constant c(W ) > 0 such that for every h ∈ Λα, α ∈ [0, 1], and u > 0, we have

′ 1−α ∥ ∗ −1 ∥ ≤ · ∥ ∥ · (i) (h uσu W ) ∞ c(W ) h Λα u ,

−α ∥ − ∗ −1 ∥ ≤ · ∥ ∥ · (ii) h h uσu W ∞ c(W ) h Λα u , where σu−1 is the dilation operator deﬁned in (5.34) and ∗ denotes the standard convolution.

Proof. (i). Observe that

′ ′ 2 ′ (h ∗ uσu−1 W ) = h ∗ (uσu−1 W ) = h ∗ u σu−1 W . ∫ ′ ∈ R Since R W (us)ds = 0 for all u > 0, it follows that for all t ∫ 2 ′ 2 ′ (h ∗ u σu−1 W )(t) = u W (us)h(t − s)ds ∫R ∫ = u2W ′(us)(h(t − s) − h(t))ds + u2h(t) W ′(us)ds ∫R R = u2W ′(us)(h(t − s) − h(t))ds, u > 0. R Hence, substituting v = us, we obtain ∫ 2 ′ 2 ′ |(h ∗ u σu−1 W )(t)| ≤ u |W (us)| |h(t − s) − h(t)|ds ∫ R ∫ ≤ ∥ ∥ 2| ′ | | |α 1−α∥ ∥ · | ′ || |α h Λα u W (us) s ds = u h Λα W (v) v dv R ∫ R ≤ 1−α∥ ∥ · | ′ | { | |} · 1−α∥ ∥ u h Λα W (v) max 1, v dv =: c1(W ) u h Λα . R ∫ ∫ (ii). Using the assumption R W (y)dy = 1, we have that h(t) = R uW (us)h(t)ds and, therefore, ∫ ∫ h(t)−(h∗uσu−1 W )(t) = h(t)− uW (us)h(t−s)ds = uW (us)(h(t)−h(t−s))ds. R R Hence, substituting v = us, we obtain ∫

|h(t) − (h ∗ uσu−1 W )(t)| ≤ u|W (us)| |h(t) − h(t − s)|ds ∫ R ∫ ≤ ∥ ∥ | | | |α ≤ −α∥ ∥ · | | | |α h Λα u W (us) s ds u h Λα W (v) v dv R ∫ R ≤ −α∥ ∥ · | | { | |} · −α∥ ∥ u h Λα W (v) max 1, v dv =: c2(W ) u h Λα . R

Setting c(W ) = max{c1(W ), c2(W )}, we complete the proof. D

120 Recall that the function gl, l ≥ 0, is deﬁned in (5.61).

Lemma 5.34. Let m ≥ 0, l ≥ 0, 0 ≤ α ≤ 1 and let g be a compactly supported

(j) (m+l) (m) function with g (0) = 0, 0 ≤ j ≤ l − 1. If g ∈ Λα, then (gl) ∈ Λα. Moreover, ∥ (m)∥ ≤ ∥ (m+l)∥ (gl) Λα g Λα .

(m+l) Proof. If l = 0, the assertion is trivial. Let m ≥ 0, l ≥ 1 be such that g ∈ Λα. First we show that ∫ ∏n (n) · · n−j ≤ ≤ gn(t) = g (ts1 ... sn) sj ds1 . . . dsn, 1 n l. (5.67) ×n [0,1] j=1

By the fundamental theorem of calculus and due to the assumption g(0) = 0, we trivially have ∫ 1 ′ g1(t) = g (ts1)ds1, 0 so (5.67) is valid for n = 1. Assume now that (5.67) holds for n − 1 instead of n. Calculating separately the integral with respect to sn, due to the assumption g(n−1)(0) = 0, we obtain

∫ ∏n (n) · · n−j g (ts1 ... sn) sj ds1 . . . dsn ×n [0,1] j=1 ∫ ∫ ( 1 ) ∏n (n) · · n−j = g (ts1 ... sn)dsn sj ds1 . . . dsn−1 [0,1]×(n−1) 0 ∏ j=1 ∫ ( ∫ · · ) n n−j ts1 ... sn−1 s (n) j=1 j = g (u)du · · ds1 . . . dsn−1 [0,1]×(n−1) 0 ts1 . .. sn−1 ∫ − 1 n∏1 (n−1) · · n−1−j = g (ts1 . .. sn−1) sj ds1 . . . dsn−1 t ×(n−1) [0,1] j=1 1 = · g − (t) = g (t). t n 1 n

By the induction, we complete the proof of (5.67). Now it follows from (5.67) (with n = l) that

∫ ∏l (m) (l+m) · · l+m−j (gl) (t) = g (ts1 ... sl) sj ds1 . . . dsl. (5.68) ×l [0,1] j=1

121 Hence, for t1 ≠ t2, we obtain

(m) (m) |(gl) (t1) − (gl) (t2)| ∫ ∏l ≤ | (l+m) · · − (l+m) · · | l+m−j g (t1s1 ... sl) g (t2s1 ... sl) sj ds1 . . . dsl ×l [0,1] j=1 ∫ ∏l ≤ ∥ (m+l)∥ | − |α α+l+m−j g Λα t1 t2 sj ds1 . . . dsl ×l [0,1] j=1 ∫ ∏l 1 ∥ (m+l)∥ | − |α α+l+m−j = g Λα t1 t2 sj dsj j=1 0 ≤ ∥ (m+l)∥ | − |α g Λα t1 t2 .

Proof of the estimate

The main technical tool in the proof of Theorem 5.36 is the estimate obtained in Theorem 5.35. The latter theorem may be compared with [92, Theorem 12]. However, our proof of Theorem 5.35 is diﬀerent from and simpler than that of [92, Theorem 12]. For instance, we completely avoid the use of the Besov classes. Recall that σu is the dilation operator deﬁned in (5.34).

Theorem 5.35. Let k ∈ N, let N be atomless and let k + 1 < r < q < ∞. There exists a constant C(k, q, r) > 0 such that for a pair of self-adjoint operators

r q A, B ∈ (L + L )(N , τ), for all self-adjoint operators A1,...,Ak aﬃliated with N and for any compactly supported function h ∈ Λα, α ∈ [0, 1], we have

µ(T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X )) φk,h 1 k φk,h 1 k ∏k ≤ ∥ ∥ − α C(k, q, r) h Λα σ3(Pr,q(A B)) σ3Pr,q(Xj), j=1

r q for all X1,...,Xk ∈ (L + L )(N , τ).

r q Proof. Fix self-adjoint operators A, B ∈ (L + L )(N , τ), A1,...,Ak aﬃliated with N , α ∈ [0, 1] and a compactly supported function h ∈ Λα. Observe that due to Theorem 5.25 and Lemma 5.15, we have

A,A ,...,A B,A ,...,A r q ×k r q T 1 k ,T 1 k ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)). φk,h φk,h

122 r q Take X1,...,Xk ∈ (L + L )(N , τ) and denote for brevity

D := (T A,A1,...,Ak − T B,A1,...,Ak )(X ,...,X ). φk,h φk,h 1 k ∫ Let W be a Schwartz function such that R W (y)dy = 1. Let t > 0 be ﬁxed and let u = u(t) > 0. Observing that the functions h ∗ uσu−1 W , h − h ∗ uσu−1 W are bounded and continuous, due to Theorem 5.25 and Lemma 5.15, we have

A,A ,...,A A,A ,...,A r q ×k r q T 1 k ,T 1 k , ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)) φk,h∗uσ W φk,h−h∗uσ W u−1 u−1 and similarly

B,A ,...,A B,A ,...,A r q ×k r q T 1 k ,T 1 k ∈ B((L + L )(N , τ) , (L k + L k )(N , τ)). φk,h∗uσ W φk,h−h∗uσ W u−1 u−1

Since φ = φ ∗ + φ − ∗ , it follows that k,h k,h uσu−1 W k,h h uσu−1 W

D = T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ) φk,h∗uσ W 1 k φk,h∗uσ W 1 k u−1 u−1 + T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ). (5.69) φk,h−h∗uσ W 1 k φk,h−h∗uσ W 1 k u−1 u−1

′ Observing that the function (h ∗ uσu−1 W ) is also bounded and continuous, by ∑ 1 k+1 1 Theorem 5.25, for all 1 < p, p1, . . . , pk+1 < ∞ with = , we have p j=1 pj

1 · A,B,A1,...,Ak ≤ ′ Tφ ∗ ′ C(p). (5.70) ∥ ∗ −1 ∥ k+1,(h uσ −1 W ) (h uσu W ) ∞ u Lp1 ×...×Lpk+1 →Lp In particular,

A,B,A ,...,A r q ×(k+1) r q T 1 k ∈ B((L + L )(N , τ) , (L k+1 + L k+1 )(N , τ)). φk+1,(h∗uσ W )′ u−1

Moreover, using (5.54) from Theorem 5.28, we obtain

T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ) φk,h∗uσ W 1 k φk,h∗uσ W 1 k u−1 u−1

A,B,A1,...,Ak = T (A − B,X1,...,Xk). (5.71) φk+1,(h∗uσ W )′ u−1

Combining (5.69) with (5.71), we have

D = T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ) φk,h∗uσ W 1 k φk,h∗uσ W 1 k u−1 u−1

A,B,A1,...,Ak + T (A − B,X1,...,Xk). φk+1,(h∗uσ W )′ u−1

123 Now, applying (5.31), we arrive at ( ) t A,B,A1,...,Ak µ(t, D) ≤ µ , T (A − B,X1,...,Xk) φk+1,(h∗uσ W )′ ( 3 u−1 ) ( ) t t + µ , T A,A1,...,Ak (X ,...,X ) + µ , T B,A1,...,Ak (X ,...,X ) . (5.72) φk,h−h∗uσ W 1 k φk,h−h∗uσ W 1 k 3 u−1 3 u−1

Next, we estimate every summand on the right hand side of (5.72) separately. Applying Lemma 5.32 with

1 T = · T A,B,A1,...,Ak , ′ φk+1,(h∗uσ W )′ ∥(h ∗ uσu−1 W ) ∥∞ u−1 and appealing to (5.70), we have that

A,B,A1,...,Ak µ(T (A − B,X1,...,Xk)) φk+1,(h∗uσ W )′ u−1 ∏k ′ ≤ C(k + 1, q, r)∥(h ∗ uσu−1 W ) ∥∞Pr,q(A − B) Pr,q(Xj) j=1 and, therefore, ( ) t A,B,A1,...,Ak µ , T (A − B,X1,...,Xk) φk+1,(h∗uσ W )′ 3 u−1 ( ) ∏k ( ) ′ t t ≤ C(k + 1, q, r)∥(h ∗ uσ −1 W ) ∥∞P (A − B) P (X ) . u r,q 3 r,q j 3 j=1

Similarly,

( ) k ( ) t ∏ t A,A1,...,Ak µ , T (X ,...,X ) ≤ C(k, q, r)∥h−h∗uσ −1 W ∥∞ P (X ) φk,h−h∗uσ W 1 k u r,q j 3 u−1 3 j=1 ( ) k ( ) t ∏ t B,A1,...,Ak µ , T (X ,...,X ) ≤ C(k, q, r)∥h−h∗uσ −1 W ∥∞ P (X ) . φk,h−h∗uσ W 1 k u r,q j 3 u−1 3 j=1

Combining the last three inequalities with (5.72) and setting

C′(k, q, r) := max{C(k + 1, q, r) + 2C(k, q, r)},

124 we infer that

( ) ∏k ( ) ′ t t µ(t, D) ≤ C(k + 1, q, r)∥(h ∗ uσ −1 W ) ∥∞P (A − B) P (X ) u r,q 3 r,q j 3 j=1 k ( ) ∏ t + 2C(k, q, r)∥h − h ∗ uσ −1 W ∥∞ P (X ) u r,q j 3 ( j=1 ( ) ′ ′ t ≤ C (k, q, r) ∥(h ∗ uσ −1 W ) ∥∞P (A − B) u r,q 3 ) k ( ) ∏ t + ∥h − h ∗ uσ −1 W ∥∞ P (X ) . u r,q j 3 j=1

Using Lemma 5.33 (i) and (ii), we obtain

( ( ) ) k ( ) t ∏ t µ(t, D) ≤ C′′(k, q, r)∥h∥ u1−αP (A − B) + u−α P (X ) , Λα r,q 3 r,q j 3 j=1 where C′′(k, q, r) = c(W ) · C′(k, q, r) and c(W ) is the constant from Lemma 5.33. ( ) −1 − t Taking u = Pr,q(A B) 3 , we arrive at (( ( )) ( ) t α−1 t µ(t, D) ≤ C′′(k, q, r)∥h∥ P (A − B) P (A − B) Λα r,q 3 r,q 3 ( ( )) ) k ( ) t α ∏ t + P (A − B) P (X ) r,q 3 r,q j 3 j=1 ( ( )) k ( ) t α ∏ t ≤ 2C′′(k, q, r)∥h∥ P (A − B) P (X ) , Λα r,q 3 r,q j 3 j=1 which completes the proof. D

The ﬁnal estimate of this section is given below.

Theorem 5.36. Let k ∈ N, let N be atomless and let k+1 0 such that for any compactly supported function h ∈ Λα, p,∞ α ∈ [0, 1], for all self-adjoint operators A, B ∈ L (N , τ) and all X1,...,Xk ∈ Lp,∞(N , τ), we have

A,A1,...,Ak B,A1,...,Ak − p Tφ (X1,...,Xk) Tφ (X1,...,Xk) ,∞ k,h k,h L k+α ( ∏k ) ≤ · ∥ ∥ ∥ − ∥α ∥ ∥ C(k, p) h Λα A B Lp,∞ Xj Lp,∞ . j=1

125 Proof. Let k ∈ N, k + 1 < p < ∞ and α ∈ [0, 1] be ﬁxed. Take h ∈ Λα, ∗ ∗ p,∞ p,∞ A = A ,B = B ∈ L (N , τ) and X1,...,Xk ∈ L (N , τ). 1 ∈ Setting r := 2 (p + k + 1) and q := 2p, observe that A, B, X1,...,Xk (Lr + Lq)(N , τ). Denote for brevity

D := T A,A1,...,Ak (X ,...,X ) − T B,A1,...,Ak (X ,...,X ). φk,h 1 k φk,h 1 k

By Theorem 5.35, we have that

∏k ≤ ∥ ∥ · − α · µ(D) C(k, q, r) h Λα σ3(Pr,q(A B)) σ3Pr,q(Xj). (5.73) j=1

p Taking the norm ∥ · ∥ ,∞ of both sides of (5.73), and applying Lemma 5.17 L k+α (ii), we obtain

∏k ′ α ∥ ∥ p ∞ ≤ ∥ ∥ − · D , C(k, q, r) C (k, p) h Λα σ3(Pr,q(A B)) p ∞ σ3Pr,q(Xj) , L k+α L α , Lp,∞ j=1 ( ) ′ p where C (k, p) = max c2 k + 1, and c2 is the constant from Lemma 5.17 0≤α≤1 k+α (ii). Recalling (5.41), we infer

∏k k+α ′ α ∥ ∥ p ∞ ≤ p ∥ ∥ − · D , 3 C(k, q, r)C (k, p) h Λα (Pr,q(A B)) p ∞ Pr,q(Xj) L k+α L α , Lp,∞ j=1 k α ∏ k+1 ′ ≤ p ∥ ∥ − · 3 C(k, q, r)C (k, p) h Λα Pr,q(A B) Pr,q(Xj) . Lp,∞ Lp,∞ j=1

Using Lemma 5.31, we arrive at

∏k k+1 ′ k+α α ∥ ∥ p ≤ p ∥ ∥ ∥ − ∥ · ∥ ∥ ∞ D ,∞ 3 C(k, q, r)C (k, p)c(p, q, r) h Λ A B p,∞ Xj Lp, L k+α α L j=1 ∏k ′′ α ≤ ∥ ∥ ∥ − ∥ ∞ · ∥ ∥ p,∞ C (k, p) h Λα A B Lp, Xj L , j=1

′′ k+1 ′ where C (k, p) := 3 p C(k, q, r)C (k, p) max c(p, q, r)k+α. D 0≤α≤1

The following assertion is more general than Theorem 5.36, it also extends the result [2, Theorem 5.8] to a multilinear setting. Its proof follows that of Theorem 5.36 mutatis mutandi. therefore, it is omitted.

126 Theorem 5.37. Let k ∈ N, let N be atomless and 1 < pj < ∞, 1 ≤ j ≤ k be ∑ k 1 such that < 1. There are constants c(k, p1, . . . , pk), c˜(k, p1, . . . , pk) > 0 j=0 pj such that for any compactly supported function h ∈ Λα, α ∈ [0, 1], and

∗ ∗ p0 pj (i) for all A = A ,B = B ∈ L (N , τ),Xj ∈ L (N , τ), 1 ≤ j ≤ k, we have

A,A1,...,Ak − B,A1,...,Ak Tφ (X1,...,Xk) Tφ (X1,...,Xk) k,h k,h Lp ( ∏k ) ≤ · ∥ ∥ ∥ − ∥α ∥ ∥ c(k, p1, . . . , pk) h Λα A B Lp0 Xj Lpj ; j=1

∗ ∗ p0,∞ pj ,∞ (ii) for all A = A ,B = B ∈ L (N , τ),Xj ∈ L (N , τ), 1 ≤ j ≤ k, we have

A,A1,...,Ak − B,A1,...,Ak Tφ (X1,...,Xk) Tφ (X1,...,Xk) k,h k,h Lp,∞ ( ∏k ) α ≤ · ∥ ∥ ∥ − ∥ ∞ ∥ ∥ ∞ c˜(k, p1, . . . , pk) h Λα A B Lp0, Xj Lpj , , j=1 ∑ where 1 = α + k 1 . p p0 j=1 pj

127 Chapter 6

Applications to operator Taylor remainders

In this chapter we work with bounded operators, so we can assume without loss of generality that the scalar function f is supported in some compact interval containing the spectra of operators to which the function is being applied. Let n R n R R Cc ( ) denote the subset of compactly supported functions in C ( ). Let Wn( ) be the set of all functions f ∈ Cn(R) such that f (j), Ff (j) ∈ L1(R), for j = 0, . . . , n, where as before Ff (j) denotes the Fourier transform of the function f (j).

n (n) (n+1) 2 The class Wn(R) includes the functions f ∈ C (R) for which f , f ∈ L (R) n+1 R ⊂ R [89, Lemma 7]. In particular, Cc ( ) Wn( ). Let n ∈ N and let f and g be n times continuously diﬀerentiable functions on the real line R and on the unit circle T, respectively. In this chapter we present the application of multiple operator integration theory to the problem associated with operator Taylor remainders. More precisely, for bounded self-adjoint operators A and B and for a unitary operator U, using multiple operator integration theory, we proved that the nth Taylor remainder

− ∑n 1 k 1 d Rn,f,A(B) := f(A + B) − f(A + tB) (6.1) k! dtk k=0 t=0 in the self-adjoint case or

− ∑n 1 k iB 1 d itB Qn,g,U (B) := g(e U) − g(e U) (6.2) k! dtk k=0 t=0

128 in the unitary case belongs to a Schatten class Sp/n provided that the perturbation B is in the Schatten class Sp with p > n, f ∈ Cn(R) and g ∈ Cn(T) (see Theorem 6.1 in the self-adjoint case and Theorem 6.2 in the unitary case). The second main result (Theorems 6.7 in the self-adjoint case and Theorem 6.15 in the unitary case) shows that the above assumption on B to be in Sp with p > n is sharp. This part of the thesis is based on [87] (a joint work with D. Potapov, A. Skripka and F. Sukochev).

6.1 Aﬃrmative results for Taylor remainders

In this section, we prove that the conditions f ∈ Cn(R) and g ∈ Cn(T) are sufficient for the Taylor remainders (6.1) and (6.2) to be in the Schatten class Sp/n when the perturbation is in Sp with p > n. In the proofs below we involve multiple operator integrals for infinite dimensional Hilbert spaces as defined in Definition 5.21. Observe that for f ∈ Wn(R) A,...,A all definitions of multiple operator integral Tf [n] coincide (see [84, Lemma 3.5 and Lemma 5.2]), in particular, the results from [9] are applicable to this multiple operator integral.

Theorem 6.1. Let 1 < n 0 ∈ n R ∗ ∈ H such that for f Cc ( ), A = A B( ),

(n) n ∗ p ∥Rn,f,A(B)∥Sp/n ≤ c(n, p)∥f ∥∞∥B∥Sp ,B = B ∈ S ,

where Rn,f,A(B) is deﬁned in (6.1).

∈ n R ≤ ∈ N Proof. Let f Cc ( ) and let Pr, 0 r < 1, be the Poisson kernel. For k deﬁne fk := P1/k ∗ f, where ∗ denotes the convolution. Then we have that

∥ (m) − (m)∥ → → ∞ fk f ∞ 0, k ,

≤ ≤ (m) ∗ (m) { }∞ ⊂ R for all 0 m n, since fk = P1/k f . Moreover, fk k=1 Wm( ) for every m = 1, . . . , n (see e.g. [102]). By [84, (5.29) and (5.30)], we have that ∫ 1 n−1 A+tB,...,A+tB Rn,f ,A(B) = n (1 − t) T (B,...,B) dt. k f [n] 0 k

129 From Theorem 5.25, it follows that

∥ ∥ ≤ ∥ (n)∥ ∥ ∥n Rn,fk,A(B) Sp/n c(n, p) fk ∞ B Sp .

(n) ∗ (n) ∥ ∥ ≤ Again using fk = P1/k f and Young’s inequality, since P1/k L1 1, for all k ∈ N, we obtain

∥ (n)∥ ≤ ∥ ∥ ∥ (n)∥ ≤ ∥ (n)∥ fk ∞ P1/k L1 f ∞ f ∞ and so ∥ ∥ ≤ ∥ (n)∥ ∥ ∥n Rn,fk,A(B) Sp/n c(n, p) f ∞ B Sp . (6.3)

We now prove that ( ) ( ) → → ∞ Tr Rn,fk,A(B)X Tr Rn,f,A(B)X as k (6.4) for every X ∈ S1. Indeed,

∥fk(A + B) − f(A + B)∥∞ ≤ ∥fk − f∥∞ → 0 and, similarly,

∥fk(A) − f(A)∥∞ ≤ ∥fk − f∥∞ → 0.

Moreover, for all 1 ≤ q ≤ n − 1, using subsequently [9, Theorem 5.7], Theo- ∈ ∩n−1 R rem 5.25, and the fact that f m=1Wm( ), we obtain

dq dq − A,...,A − A,...,A fk(A+tB) f(A+tB) = T [q] (B, ...,B) T [q] (B,...,B) Sp/q dtq t=0 dtq t=0 Sp/q f f k A,...,A ≤ (q) − (q) ∥ ∥q → → ∞ = T [q] (B,...,B) p/q C(p, q) f f B Sp 0, k . (fk−f) S k ∞

Along with H¨older’sinequality, the latter implies (6.4) for every X ∈ S1 ⊂ (Sp/q)∗. Finally, by the properties (6.3), (6.4), H¨older’sinequality, and denseness of S1 in (Sp/n)∗ = Sp/(p−n), we conclude

∥ ∥ ≤ · ∥ (n)∥ · ∥ ∥n Rn,f,A(B) Sp/n = sup Rn,fk,A(B)X C(n, p) f ∞ B Sp . ∈S1 ∥ ∥ ≤ X , X Sp/(p−n) 1

130 In the proof of the following theorem we use a notion of multiple operator integrals with respect to unitary operators. The deﬁnition of this operator integral can be found in e.g. [86]. We do not present it here and only observe that the idea is the same as in the deﬁnition of double operator integral (2.14) from Section 2.2.

Theorem 6.2. Let 1 < n 0 such that for f ∈ Cn(T), A = A∗ ∈ Sp, and a unitary U ∈ B(H),

∑n ∥ ∥ ≤ ∥ (l)∥ ∥ ∥n Qn,f,U (A) Sp/n C(n, p) f ∞ A Sp , l=1 where Qn,f,U (A) is deﬁned in (6.2).

n Proof. Let f ∈ C (T) and let Fk, k ∈ N be the Fejer kernel. By the deﬁnition of the Fejer kernel, the function fk := Fk ∗ f is a trigonometric polynomial (see, e.g., [43, Section 5]). Then we have that

∥ (l) − (l)∥ → → ∞ fk f ∞ 0, k , for all 0 ≤ l ≤ n. By [86, Theorem 3.1 and (4.2)], denoting U(t) = eitAU, we have that

Qn,fk,U (A) ∫ in 1 ∑n ∑ n−1 U(t),...,U(t) j1 jl = (1 − t) C(n, l, j1, . . . , jl) T (A U(t),...,A U(t)) dt, − f [l] (n 1)! 0 k l=1 j1,...,jl≥1 j1+...+jl=n where the constants C(n, l, j1, . . . , jl) do not depend on k, t, A, U(t). By [100, Theorem 3.6 (i)], there exists a constant C˜(n, p) > 0 such that for l = 1, . . . , n and j1, . . . , jl ∈ N satisfying j1 + ... + jl = n,

U(t),...,U(t) (l) j1 jl ≤ ˜ ∥ ∥ ∥ j1 ∥ ∥ jl ∥ T (A U(t),...,A U(t)) C(n, p) f ∞ A Sp/j1 ... A Sp/j f [l] k l k Sp/n ˜ ∥ (l)∥ ∥ ∥n = C(n, p) fk ∞ A Sp ,

for every k ∈ N. From Young’s inequality, since ∥Fn∥L1 ≤ 1 for all n ∈ N, it follows that ∥ (l)∥ ≤ ∥ ∥ ∥ (l)∥ ≤ ∥ (l)∥ fk ∞ Fn L1 f ∞ f ∞.

131 Therefore, there is a constant C(n, p) > 0 such that

∑n ∥ ∥ ≤ ∥ (l)∥ ∥ ∥n Qn,fk,U (A) Sp/n C(n, p) f ∞ A Sp (6.5) l=1 for every k ∈ N. The estimate (6.5) is an analog of the estimate (6.3) in the self-adjoint case. The rest of the proof follows in a manner similar to the proof of Theorem 6.1, with application of [86, Theorem 3.1] and [100, Theorem 3.6 (i)] instead of [9, Theorem 5.7] and Theorem 5.25, respectively. D

6.2 Counterexamples for Taylor remainders.

In this section we demonstrate that the results of Theorems 6.1 and 6.2 are sharp, that is, f ∈ Cn(R) (or f ∈ Cn(T)) is not sufficient for Taylor remainder (6.1) (or (6.2)) to map Sn to S1. The functions in the counterexamples arise from the function h:[−e−1, e−1] → R defined by  ( )  − 1 |x| log log |x| − 1 2 , x ≠ 0 h(x) :=  , (6.6) 0, x = 0 which was considered in [1]. {H }∞ Let m m=1 be a sequence of finite-dimensional Hilbert spaces, and consider the Hilbert direct sum H ⊕∞ H = m=1 m. ∈ H ∈ N ∥ ∥ ∞ For Am B( m), m , satisfying supm∈N Am ∞ < , we define the direct sum of operators ⊕∞ A = m=1Am by { }∞ A(ξ) := Am(ξm) m=1 { }∞ ∈ H { }∞ for every ξ = ξm m=1 . If Am m=1 is a sequence of self-adjoint operators, H { }∞ then A is a self-adjoint operator on ; if Am m=1 is a sequence of unitaries, then A is a unitary operator on H.

132 Let Am, Bm ∈ B(Hm), be self-adjoint operators and let Um ∈ B(Hm) be a unitary operator, m ∈ N. If f ∈ Cn(R) then, ⊕∞ ∞ R ⊕∞ (⊕ B ) = R (B ) (6.7) n,f, m=1Am m=1 m n,f,Am m m=1 and if f ∈ Cn(T), then ⊕∞ ∞ Q ⊕∞ (⊕ A ) = Q (A ). (6.8) n,f, m=1Um m=1 m n,f,Um m m=1

6.2.1 Self-adjoint case.

We extend h beyond [−e−1, e−1] so that it becomes a compactly supported function in C1(R) ∩ Cn(R \{0}) and satisfies (6.6) on the segment [−1/2 − e−1, e−1 + 1/2]. This extension is also denoted by h. We will apply h to operators with spectra in the segment [−e−1, e−1], so the values of h outside [−e−1, e−1] will not make any difference. For x ∈ R, we define

n−1 fn(x) := x h(x). (6.9)

n (n) (k) The following three lemmas prove that fn ∈ C (R), fn ∈ Cb(R), and fn (0) = 0, for 0 ≤ k ≤ n.

− 1 −1 Lemma 6.3. Deﬁne L(x) := (log(1 − log x)) 2 , for x ∈ (0, e + 1/2]. Then, for every j ∈ N, lim xjL(j)(x) = 0. (6.10) x→0+

Proof. For x ∈ (0, e−1 + 1/2], we have

′ 1 − 3 −1 −1 L (x) = (log(1 − log x)) 2 (1 − log x) x , 2 so, (6.10) holds for j = 1. Subsequent derivatives have the form ∑ (j) −αk −βk −j L (x) = ck,j(log(1 − log x)) (1 − log x) x , (6.11) k

133 where ck,j some constants, αk ≥ 3/2 and βk ≥ 1, for all k. This can be seen by induction. Indeed, let (6.11) hold for j. Then, we have ∑ (j+1) −αk −βk −j ′ L (x) = ck,j((log(1 − log x)) (1 − log x) x ) ∑k −αk−1 −βk−1 −j−1 = (ck,jαk(log(1 − log x)) (1 − log x) x k

−αk −βk−1 −j−1 + ck,jβk(log(1 − log x)) (1 − log x) x

−αk −βk −j−1 − jck,j(log(1 − log x)) (1 − log x) x ).

j (j) Renumbering the sum, we obtain (6.11). Evaluating limx→0+ x L (x) implies (6.10). D

Lemma 6.4. Let n ∈ N and fn be given by (6.9). Then,

(k) ≤ ≤ lim fn (x) = 0, 1 k n. x→0

(k) Proof. Let 1 ≤ k ≤ n be ﬁxed. We will only prove limx→0+ fn (x) = 0, and the (k) equality limx→0− fn (x) = 0 can be proved in a similar manner. Since for x ∈ (0, e−1 +1/2] we have | log x−1| = 1−log x, and so for 1 ≤ j ≤ k, it follows that   ∑j j h(j)(x) =   (x)(j−m)L(m)(x), m=0 m where L is deﬁned in Lemma 6.3. There are only two non-zero summands in the sum above, namely for m = j and m = j − 1. Thus, we have

h(j)(x) = xL(j)(x) + jL(j−1)(x). (6.12)

Let k = n. Applying (6.12) gives   ∑n n (n)   n−1 (n−j) (j) fn (x) = (x ) h (x) j j=0   ∑n n   j−1 (j) = cn,jx h (x) j j=1   ∑n n   j (j) j−1 (j−1) = cn,j(x L (x) + jx L (x)), (6.13) j=1 j

134 −1 for all x ∈ (0, e + 1/2] and some constants cn,j. Taking limx→0+ in the equality above and applying Lemma 6.3, we prove the assertion in the case k = n. Let now k < n. By (6.12), we have   ∑k k (k)   n−1 (k−j) (j) fn (x) = (x ) h (x) j j=0   ∑k k n−k−1   n−1−k+j (j) = cn,kx h(x) + cn,k,jx h (x) j j=1   ∑k k n−k−1   n−k j (j) j−1 (j−1) = cn,kx h(x) + cn,k,jx (x L (x) + jx L (x)). j=1 j

Evaluating limx→0+ in the equality above and applying Lemma 6.3, we complete the proof of the lemma. D

n (n) (k) Lemma 6.5. The function fn ∈ C (R). Moreover, fn is bounded and fn (0) = 0 for all 1 ≤ k ≤ n.

(k) Proof. The fact that fn (0) exists and equals 0 for every 1 ≤ k ≤ n follows from

− − ∫ f (k 1)(x) − f (k 1)(0) 1 x n n (k) ≤ | (k) | → = fn (u)du sup fn (u) 0 x x 0 u∈(0,x) as x → 0 by induction on k, which is a consequence of Lemma 6.4. The property (k) (k) fn (0) = 0 and Lemma 6.4 justify continuity of fn at 0, 1 ≤ k ≤ n. The (n) latter also implies boundedness of the derivative fn in a neighborhood of 0. (n) Boundedness of fn at inﬁnity follows from (6.11) and (6.13). D

The following lemma established an important connection between divided diﬀerences of fn and h.

Lemma 6.6. For x0, xn ∈ R and fn deﬁned in (6.9), we have

[n] [1] fn (x0, 0,..., 0, xn) = h (x0, xn).

Proof. We will prove the lemma by induction. If n = 1, by the deﬁnition of fn, we have

[1] [1] f1 (x0, x1) = h (x0, xn).

135 Let the assertion of the lemma be true for k < n. Let also x0 ≠ 0, xn ≠ 0 and x0 ≠ xn. Then, using the fact that fn(x) = I(x)fn−1(x), where I(x) = x, x ∈ R, we obtain [n−1] − [n−1] [n] (Ifn−1) (x0, 0,..., 0) (Ifn−1) (0,..., 0, xn) fn (x0, 0,..., 0, xn) = . x0 − xn (6.14) Evaluating the divided diﬀerence of the product, we obtain

∑n−1 [n−1] [l] [n−1−l] (Ifn−1) (x0, 0,..., 0) = I (x0, x1, . . . , xl)fn−1 (xl, . . . , xn−1) l=0 [n−1] [n−2] = x0fn−1 (x0, . . . , xn−1) + fn−1 (x1, . . . , xn−1), where x1 = ... = xn−1 = 0. Since for all 1 ≤ l ≤ n − 1, by (5.17) and Lemma 6.5, we have [n−1−l] 1 (n−1−l) f (x , . . . , x − ) = f (0) = 0, n−1 l n 1 (n − 1 − l)! n−1 it follows that [1] [n−1] [n−1] x0fn−1(x0, 0) [1] (Ifn−1) (x0, 0,..., 0) = x0fn−1 (x0, 0,..., 0) = n−2 = x0h (x0, 0). x0 (6.15) Similarly,

∑n−1 [n−1] [l] [n−1−l] (Ifn−1) (0,..., 0, xn) = fn−1(x1, . . . , xl+1)I (xl+1, . . . , xn), l=0 where x1 = ... = xn−1 = 0. Since for all 0 ≤ l ≤ n − 2, by (5.17) and Lemma 6.5, we have 1 f [l] (x , . . . , x ) = f (l) (0) = 0, n−1 1 l+1 l! n−1 it follows that

[n−1] [n−1] [1] (Ifn−1) (0,..., 0, xn) = xnfn−1 (0,..., 0, xn) = xnh (0, xn). (6.16)

Combining (6.15), (6.16) with (6.14), we arrive at [1] − [1] − [n] x0h (x0, 0) xnh (0, xn) h(x0) h(xn) [1] fn (x0, 0,..., 0, xn) = = = h (x0, xn). x0 − xn x0 − xn

If xn ≠ x0 and x0 = 0, then ∑n−1 [n−1] [l] [n−1−l] (Ifn−1) (0, 0,..., 0) = I (0,..., 0)fn−1 (0,..., 0) = 0, l=0

136 and so by (6.14) and (6.16), we obtain − [1] [n] 0 xnh (0, xn) [1] fn (0, 0,..., 0, xn) = = h (0, xn). xn

The case xn ≠ x0 and xn = 0 is similar.

Now let x0 = xn = 0. Then by (5.17) and Lemma 6.5, we have 1 f [n](0,..., 0) = f (n)(0) = 0 = h[1](0, 0). n n! n

The main goal of this subsection is proving the following result. Throughout the subsection we ﬁx n ∈ N and the functions h, fn deﬁned above.

Theorem 6.7. Let n ∈ N. There exist a separable Hilbert space H and self- adjoint operators A ∈ B(H) and B ∈ Sn(H) such that

− ∑n 1 k 1 d 1 fn(A + B) − fn(A + tB) ∈/ S (H), k! dtk k=0 t=0

n for the function fn ∈ C (R) deﬁned in (6.9).

The basis of the proof of this theorem is the following known estimate for the increment of an operator function, where the spectra of matrices stay within a ﬁxed segment as the dimension grows.

Theorem 6.8. [30, Theorem 7] For every d ∈ N, d ≥ 2, there exist non-zero self- ∈ 2 ⊂ − −1 −1 adjoint operators Ad, Bd B(ℓ2d) such that σ(Ad + Bd) = σ(Ad) [ e , e ],

0 ∈ σ(Ad) has multiplicity 2, any λ ∈ σ(Ad) \{0} has multiplicity 1, and

1 ∥h(Ad + Bd) − h(Ad)∥∞ ≥ const (log d) 2 ∥Bd∥∞, (6.17) where the constant is independent of d.

ˆ ˆ∗ Proof. It follows from the proof of [1, Corollary 3.8] that there exist Ad = Ad, ˆ ˆ∗ ∈ 2 ˆ Bd = Bd B(ℓ2d−2) satisfying (6.17) such that the set of eigenvalues of Ad consists of mutually distinct 2d − 2 non-zero real numbers from the segment −1 −1 ˆ ˆ ˆ [−e , e ] and σ(Ad + Bd) = σ(Ad). By changing the dimension from 2d − 2 to ˆ ˆ 2d and adding two zero rows and columns to the matrices Ad and Bd, one obtains the operators Ad and Bd satisfying all the conditions of Theorem 6.8. D

137 ∈ N ≥ ∈ 2 Theorem 6.9. Let n, d , d 2. For Ad,Bd B(ℓ2d) satisfying Theorem 6.8, ∈ 2 ∥ ∥ ≤ ≤ ∈ N there are operators X1,...Xn B(ℓ2d) with Xj Sn = 1, 1 j n such that

A +B ,A ,...,A 1 d d d d ≥ 2 T [n] (X1,...,Xn) S1 const (log d) , fn where fn is deﬁned in (6.9) and the constant const is independent of d.

Proof. By (5.20) and Theorem 6.8,

A +B ,A 1 ∥ d d d ∥∞ ∥ − ∥∞ ≥ 2 ∥ ∥∞ Th[1] (Bd) = h(Ad + Bd) h(Ad) const (log d) Bd .

Hence, by Theorem 5.5 (iii),

A +B ,A A +B ,A 1 d d d d d d ≥ 2 T [1] S1 →S1 = T [1] S∞→S∞ const (log d) . h 2d 2d h 2d 2d

Now, combining Theorem 5.5 (i) (with p1 = ... = pn = n and p = 1) and Lemma 6.6, we arrive at

··· Ad+Bd,Ad, ,Ad ≥ Ad+Bd,Ad T [n] Sn × ×Sn →S1 T [1] S1 →S1 fn 2d ... 2d 2d h 2d 2d 1 ≥ const (log d) 2 .

The result of the theorem follows immediately. D

We assume below that d ≥ 2 is ﬁxed and Ad,Bd are given by Theorem 6.8. The main part of the proof of Theorem 6.7 is contained in Lemmas 6.10–6.14 below.

∈ 2 Lemma 6.10. Let Ad,Bd B(ℓ2d) satisfy Theorem 6.8 and denote

⊗ ⊗ ∈ 2 Hd := E11 (Ad + Bd) + E22 Ad B(ℓ4d). (6.18)

˜ ˜ ∈ 2 ∥ ˜ ∥ ≤ ≤ There are operators X1,..., Xn B(ℓ4d) with Xj Sn = 1, 1 j n such that

H ,...,H 1 d d ˜ ˜ ≥ 2 T [n] (X1,..., Xn) S1 const (log d) , fn where the constant is independent of d.

Proof. Take X1,...,Xn as in Theorem 6.9 and set

˜ ˜ X1 := E12 ⊗ X1, Xj := E22 ⊗ Xj, 2 ≤ j ≤ n.

138 ˜ Clearly, ∥Xj∥Sn = ∥Xj∥Sn = 1, 1 ≤ j ≤ n. It follows from Lemma 5.6 that

Hd,...,Hd ˜ ˜ ⊗ Ad+Bd,Ad,...,Ad T [n] (X1,..., Xn) = E12 T [n] (X1,...,Xn). fn fn

Therefore, by Theorem 6.9, we have

H ,...,H A +B ,A ,...,A 1 d d ˜ ˜ d d d d ≥ 2 T [n] (X1,..., Xn) S1 = T [n] (X1,...,Xn) S1 const (log d) . fn fn

The following lemma establishes an analog of the result of [29, Lemma 3.2] for the higher dimensional case.

Lemma 6.11. For Hd deﬁned in (6.18), denote

˜ ⊕n+1 ∈ 2 Hd := j=1 Hd B(ℓ4d(n+1)); (6.19)

˜ ˜ ∈ 2 ℜ ˜ ℑ ˜ for X1,..., Xn B(ℓ4d) as in Lemma 6.10, denote X0,j := Xj and X1,j := Xj, ≤ ≤ (0) (0) ∈ { }n 1 j n. Then, there is (r1 , . . . , rn ) 0, 1 such that for the self-adjoint operator ∑n ( ) 1 (n+1) ⊗ (n+1) ⊗ Zd = Ej(j+1) Xr(0),j + E(j+1)j Xr(0),j (6.20) 2n j j j=1

Sn in the unit ball of 4d(n+1), we have

H˜ ,...,H˜ 1 d d ≥ 2 T [n] (Zd,...,Zd) S1 const (log d) , (6.21) fn where the constant is independent of d.

n Proof. Given an arbitrary n-tuple (r1, . . . , rn) ∈ {0, 1} , denote   0 X 0 ··· 0 0 0  r1,1     X 0 X ··· 0 0 0   r1,1 r2,2     0 X 0 ··· 0 0 0   r2,2   ......  Y :=  ......  , r1,...,rn  ......     − −   0 0 0 Xrn−2,n 2 0 Xrn−1,n 1 0     0 0 0 0 X − 0 X   rn−1,n 1 rn,n 

0 0 0 0 0 Xrn,n 0

139 × where every entry above is a 4d 4d-matrix. Observe that Yr1,...,rn is a self-adjoint 2 operator in B(ℓ4d(n+1)). We claim that

( ˜ ˜ ) Hd,...,Hd Hd,...,Hd T [n] (Xr1,1,...,Xrn,n) = P1,n+1 T [n] (Yr1,...,rn ,...,Yr1,...,rn ) , (6.22) fn fn

2 where P1,n+1 is a projection in the space B(ℓ4d(n+1)) on the subspace generated by 2 all elements from B(ℓ4d(n+1)) whose non-zero entries are located in the intersection { }m of the ﬁrst 4d-row and (n + 1)st 4d-column. Let λi i=1 be the set of distinct ≤ ˜ { }m eigenvalues of the operator Hd, m 4d. Clearly, Hd has the same set λi i=1 ˜ of distinct eigenvalues and the spectral projection of the operator Hd associated with λi equals n+1 E ˜ (λ ) = ⊕ E (λ ), 1 ≤ i ≤ m. Hd i j=1 Hd i Therefore, by (5.15), we have

˜ ˜ Hd,...,Hd T [n] (Yr1,...,rn ,...,Yr1,...,rn ) fn ∑m ( ) ( ) [n] ⊕n+1 ⊕n+1 = fn (λl0 , . . . , λln ) j=1 EHd (λl0 ) Yr1,...,rn ...Yr1,...,rn j=1 EHd (λln ) . l0,...,ln=1 Every matrix coeﬃcient in the sum above can be represented as ( ) ( ) ⊕n+1 E (λ ) Y ...Y ⊕n+1 E (λ ) j=1 Hd l0 r1,...,rn r1,...,rn j=1 Hd ln 0 E (λ ) X 0 ··· 0 0  Hd l0 r1,1  = E ˜ (λ ) Y ...Y E ˜ (λ ) ...... Hd l1 r1,...,rn r1,...,rn Hd ln ......   0 E (λ ) X E (λ ) 0 ··· 0 0  Hd l0 r1,1 Hd l1  = Y E ˜ (λ ) ...Y E ˜ (λ ) ...... r1,...,rn Hd l2 r1,...,rn Hd ln ......   ∗ 0 E (λ ) X E (λ )X E (λ ) ··· 0 0  Hd l0 r1,1 Hd l1 r2,2 Hd l2  = Y ...Y E ˜ (λ ) ...... r1,...,rn r1,...,rn Hd ln ......   ∗ ∗ ∗ · · · ∗ 0 E (λ ) X ...X E (λ )  Hd l0 r1,1 rn,n Hd ln  = ... = ...... , ...... where by ∗ we denote 4d × 4d matrices whose precise forms are irrelevant to the proof. Therefore,   ∗ ∗ ∗ · · · Hd,...,Hd ˜ ˜ 0 T [n] (Xr1,1,...,Xrn,n) Hd,...,Hd   T (Y ,...,Y ) = fn , [n] r1,...,rn r1,...,rn ...... fn ......

140 yielding (6.22).

Hd,...,Hd ˜ ≤ ≤ Using multilinearity of T [n] and the deﬁnition of Xj, 1 j n, it is fn straightforward to verify that ∑ Hd,...,Hd ˜ ˜ r1+...+rn Hd,...,Hd T [n] (X1,..., Xn) = i T [n] (Xr1,1,...,Xrn,n). fn fn r1,...,rn=0,1

Therefore,

Hd,...,Hd ˜ ˜ ≤ n Hd,...,Hd T [n] (X1,..., Xn) 2 max T [n] (Xr1,1,...,Xrn,n) . (6.23) fn S1 r1,...,rn=0,1 fn S1

(0) (0) ∈ { }n Take (r1 , . . . , rn ) 0, 1 such that

Hd,...,Hd Hd,...,Hd T [n] (Xr(0),1,...,Xr(0),n) = max T [n] (Xr1,1,...,Xrn,n) fn 1 n S1 r1,...,rn=0,1 fn S1 (6.24) and set 1 Zd := Yr(0),...,r(0) , 2n 1 n which coincides with (6.20). It follows directly from the deﬁnition of Zd and Lemma 6.10 that 1 ∑n ∥Z ∥Sn ≤ · 2 ∥X˜ ∥Sn = 1. d 2n n j=1 Moreover, applying subsequently (6.22), (6.24), (6.23) and Lemma 6.10, we obtain ( ) ˜ ˜ ˜ ˜ 1 1 Hd,...,Hd Hd,...,Hd T [n] (Zd,...,Zd) = T [n] Yr(0),...,r(0) ,. .., Yr(0),...,r(0) fn S1 fn 2n 1 n 2n 1 n S1

(6.22) 1 ≥ Hd,...,Hd n T [n] (Xr(0),1,. ..,Xr(0),n) (2n) fn 1 n S1

(6.24) 1 Hd,...,Hd = n max T [n] (Xr1,1,. ..,Xrn,n) (2n) r1,...,rn=0,1 fn S1

(6.23) 1 1 ≥ Hd,...,Hd ˜ ˜ n n T [n] (X1,. .., Xn) (2n) 2 fn S1

Lemma 6.10 1 ≥ const (log d) 2 , completing the proof. D

In the following lemma, we represent Zd as a commutator of some special matrices. Below [D,C] = DC − CD is the commutator of D and C.

141 Lemma 6.12. Let dm = m(n + 1), with m ∈ N, let {0, a1, ..., a2d−2} be the sequence of distinct eigenvalues of the matrix Ad given by Theorem 6.8, and deﬁne

D := diag{0,..., 0, a , . . . , a , . . . , a − , . . . , a − }. (6.25) dm | {z } |1 {z }1 |2dm 2 {z 2dm }2 4(n+1) 2(n+1) 2(n+1)

∥ ∥ n ≤ Let Zdm satisfy the assertions (6.20) and (6.21) of Lemma 6.11 as well as Zdm S 1. There is a self-adjoint C ∈ B(ℓ2 ) such that dm 4dm(n+1)

Zdm = i[Ddm ,Cdm ]. (6.26)

Proof. Denote

λ1 := 0, λj := aj−1, 2 ≤ j ≤ 2dm − 1.

First we will show that

E (λ )Z E (λ ) = 0, 1 ≤ j ≤ 2d − 1. (6.27) Ddm j dm Ddm j m

From the deﬁnition of Ddm , we derive

E (λ ) = E(dm) ⊗ I = E(m(n+1)) ⊗ I = E(n+1) ⊗ (E(m) ⊗ I ) Ddm 1 11 4(n+1) 11 4(n+1) 11 11 4(n+1) (n+1) ⊗ =: E11 Q.

Thus, we obtain

2n · E (λ )Z E (λ ) Ddm 1 dm Ddm 1 ( ∑n ( )) (n+1) ⊗ (n+1) ⊗ (n+1) ⊗ (n+1) ⊗ = (E11 Q) Ej(j+1) X (0) + E(j+1)j X (0) (E11 Q) rj ,j rj ,j j=1 (n+1) ⊗ (n+1) ⊗ = (E12 QX (0) )(E11 Q) = 0, r1 ,1

{ (0)}n ≤ ≤ − where rj j=1 is from Lemma 6.11. Note that, for 2 j 2dm 1, we have

E (λ ) = E(2dm) ⊗ I = E(2m(n+1)) ⊗ I . Ddm j (j+1)(j+1) 2(n+1) (j+1)(j+1) 2(n+1)

Since there exist 1 ≤ kj ≤ n + 1, 1 ≤ lj ≤ 2m such that

E(2m(n+1)) = E(n+1) ⊗ E(2m), (j+1)(j+1) kj kj lj lj it follows that

E (λ ) = E(n+1) ⊗ (E(2m) ⊗ I ) =: E(n+1) ⊗ P. Ddm j kj kj lj lj 2(n+1) kj kj

142 Hence,

2n · E (λ )Z E (λ ) Ddm j dm Ddm j ( ∑n ( )) (n+1) ⊗ (n+1) ⊗ (n+1) ⊗ (n+1) ⊗ = (Ek k P ) Ej(j+1) X (0) + E(j+1)j X (0) (Ek k P ) j j rj ,j rj ,j j j j=1 ( ) (n+1) (n+1) (n+1) = E ⊗ PX (0) + E − ⊗ PX (0) (E ⊗ P ) = 0. kj (kj +1) r ,kj kj (kj 1) r ,kj−1 kj kj kj kj−1

Therefore, (6.27) holds. We will demonstrate that ∑m C := i (λ − λ )−1 E (λ )Z E (λ ) dm k j Ddm j dm Ddm k j,k=1 j≠ k satisﬁes (6.26). Since

[D ,E (λ )Z E (λ )] dm Ddm j dm Ddm k = D · E (λ )Z E (λ ) − E (λ )Z E (λ ) · D dm Ddm j dm Ddm k Ddm j dm Ddm k dm = (λ − λ ) · E (λ )Z E (λ ), j k Ddm j dm Ddm k we obtain ∑m i[D ,C ] = E (λ )Z E (λ ). dm dm Ddm j dm Ddm k j,k=1 j≠ k By (6.27), the latter equals

∑m Z − E (λ )Z E (λ ) = Z , dm Ddm j dm Ddm j dm j=1 so (6.26) holds. D

Lemma 6.13. Let m ∈ N, dm = m(n + 1). Then, there are a self-adjoint ∈ 2 ∈ 2 operator Fdm B(ℓ4m(n+1)2 ) and a unitary operator Udm B(ℓ4m(n+1)2 ) such that

∥ ˜ ∥ n ≤ [Hdm ,Fdm ] S 1 and

˜ ˜ ( ) 1 Hd ,...,Hd −1 −1 T m m iU [H˜ ,F ]U , . . . , iU [H˜ ,F ]U ≥ const (log m) 2 , [n] dm dm dm dm dm dm dm dm fn S1 ˜ where Hdm satisﬁes (6.19) and the constant is independent of m.

˜ { } Proof. From the deﬁnition of Hdm it follows that 0, a1, ..., a2dm−2 is the sequence ˜ ≤ ≤ − of distinct eigenvalues of Hdm , where aj has multiplicity 2(n+1), 1 j 2dm 2

143 ∈ and 0 has multiplicity 4(n + 1). Thus, there exists a unitary operator Udm B(ℓ2 ) such that 4dm(n+1) H˜ = U D U −1, dm dm dm dm

where Ddm is deﬁned in (6.25) of Lemma 6.12. It follows from Lemma 6.12 that

Z = i[D ,C ] = iU −1[H˜ ,F ]U , dm dm dm dm dm dm dm where F = U C U −1. Thus, the claim of this lemma follows from Lemma 6.11. dm dm dm dm D

∈ N ∈ N Lemma 6.14. Let n . For m , let dm = m(n + 1), let Fdm ,Udm be as in Lemma 6.13, and deﬁne

˜ ˜ ∈ 2 Am := Hdm B(ℓ4m(n+1)2 ),

˜ where Hdm is given by (6.19). Then, there exists jm > m such that the self-adjoint operator ( ) − − 1 iFdm /jm iFdm /jm 2 B˜ := U e H˜ e − H˜ U ∈ B(ℓ 2 ) m dm dm dm dm 4m(n+1) satisﬁes ∑∞ ˜ n ∥Bm∥Sn < ∞, (6.28) m=1 ˜ ˜ −1 −1 σ(Am + Bm) ⊂ [−1/2 − e , e + 1/2], and

∑ ˜ ˜ − n−1 1 dk ˜ ˜ fn(Am + Bm) k=0 k! dtk t=0fn(Am + tBm) lim S1 = ∞, (6.29) m→∞ ˜ n ∥Bm∥Sn where fn is given by (6.9).

Proof. Fix m ∈ N and denote ( ) − − 1 iFdm /j ˜ iFdm /j − ˜ Wj,m : = Ud e Hdm e Hdm Udm m ( ) −1 iF /j iF /j −iF /j = U e dm H˜ − H˜ e dm e dm U , dm dm dm dm

144 for j ∈ N. By (6.1) and Theorem 5.12 (i),

˜ ˜ ˜ jnR (W ) = jnT Am+Wj,m,Am,...,Am (W ,...,W ) n,fn,A˜m j,m [n] j,m j,m fn

A˜m+Wj,m,A˜m,...,A˜m = T [n] (jWj,m, . . . , jWj,m). (6.30) fn

Note that

( iF /j iF /j ) − e dm − I I − e dm − 1 ˜ ˜ iFdm /j jWj,m = U Hd + Hd e Ud , dm 1/j m m 1/j m which approaches −i U −1[H˜ ,F ]U dm dm dm dm

→ ∞ ∥ ˜ ∥ n ≤ as j . Since, by Lemma 6.13, [Hdm ,Fdm ] S 1, there exists j0 such that

∥jWj,m∥Sn ≤ 1 for every j ≥ j0 and, hence,

1 ∥W ∥Sn ≤ . (6.31) j,m j

We also have ( ) −1 iF /j −iF /j A˜ + W = H˜ + U e dm H˜ e dm − H˜ U → H˜ , as j → ∞. m j,m dm dm dm dm dm dm

(n) [n] n+1 Since f ∈ Cb(R), we have fn ∈ Cb(R ) and it follows from Lemma 5.8 (i) that

A˜m+Wj,m,A˜m,...,A˜m T [n] (jWj,m, . . . , jWj,m) fn approaches

H˜ ,...,H˜ T dm dm (−iU −1[H˜ ,F ]U ,..., −iU −1[H˜ ,F ]U ) [n] dm dm dm dm dm dm dm dm fn as j → ∞. This result along with Lemma 6.13 and the representation (6.30) implies existence of ˜j0 ≥ 2, ˜j0 ≥ j0 such that for every j ≥ ˜j0,

n 1 ∥j R (W )∥S1 ≥ const (log m) 2 . (6.32) n,fn,A˜m j,m

˜ Since σ(Am) ⊂ [−1/e, 1/e] and ∥Wj,m∥Sn ≤ 1/2, we obtain

˜ −1 −1 σ(Am + Wj,m) ⊂ [−1/2 − e , e + 1/2].

145 Let jm ≥ max{˜j0, m}. Then, by (6.31), we have

1 B˜ := W ≤ m jm,m Sn m and, hence, (6.28) holds. It follows from (6.31) and jm ≥ j0 that

˜ 1 ∥Bm∥Sn ≤ . (6.33) jm

Combining (6.1), (6.32), and (6.33) yields (6.29). D

We now complete the proof of Theorem 6.7. The Hilbert space and operators satisfying Theorem 6.7 arise as direct sums of ﬁnite-dimensional Hilbert spaces ˜ and operators from Lemma 6.14, from which we have borrowed the notations Am ˜ and Bm, m ≥ 1.

Proof of Theorem 6.7. Denote

− ∑n 1 k ˜ ˜ 1 d ˜ ˜ βm := fn(Am + Bm) − fn(Am + tBm) , m ∈ N. k! dtk S1 k=0 t=0 { } ∥ ˜ ∥−n ∞ By (6.29) of Lemma 6.14, βm Bm Sn m=1 is an unbounded sequence. Hence, there exists a positive sequence {αm}m≥1 such that ∑∞ ∑∞ ˜ −n αm < ∞ and αmβm∥Bm∥Sn = ∞. (6.34) m=1 m=1 Set ⌊ ⌋ ˜ −n Nm = αm∥Bm∥Sn + 1.

We have both

˜ n ˜ n ˜ −n Nm∥Bm∥Sn ≤ αm + ∥Bm∥Sn and Nm ≥ αm∥Bm∥Sn .

Hence it follows from (6.34), (6.28) and (6.29) that ∑∞ ∑∞ ˜ n Nm∥Bm∥Sn < ∞ and Nmβm = ∞. m=1 m=1 Deﬁne

H ⊕Nm 2 ′ ⊕Nm ˜ ′ ⊕Nm ˜ m := j=1ℓ4m(n+1)2 ,Am := j=1Am,Bm := j=1Bm,

146 ˜ ˜ ′ ′ where Am and Bm are as in Lemma 6.14. The operators Am and Bm are self- H ∥ ′ ∥n ∥ ˜ ∥n adjoint elements in B( m). Since Bm Sn = Nm Bm Sn , we obtain ∑∞ ∥ ′ ∥n ∞ Bm Sn < . (6.35) m=1 We also have − ∑n 1 k ′ ′ − 1 d ′ ′ ≥ fn(Am + Bm) fn(Am + tBm) = Nmβm, m 1. k! dtk S1 k=0 t=0 Hence, ∞ − ∑ ∑n 1 k ′ ′ − 1 d ′ ′ ∞ fn(Am + Bm) fn(Am + tBm) = . (6.36) k! dtk S1 m=1 k=0 t=0 Consider

H ⊕∞ H ⊕∞ ′ ⊕∞ ′ := m=1 m,A := m=1Am,B := m=1Bm.

The operators A and B are self-adjoint elements of B(H), with σ(A) ⊂ [−e−1, e−1] and σ(A + B) ⊂ [−1/2 − e−1, e−1 + 1/2]. The property (6.35) ensures that B ∈ Sn(H) and ∑∞ ∥ ∥n ∥ ′ ∥n ∞ B Sn = Bm Sn < . m=1 By (6.7) and (6.36),

− ∑n 1 k 1 d fn(A + B) − fn(A + tB) = ∞, k! dtk S1 k=0 t=0 completing the proof of the theorem. D

6.2.2 Unitary case.

We extend the function h deﬁned on [−e−1, e−1] by (6.6) to a 2π-periodic function in C1(R) ∩ Cn(R \{0}), also denoted by h. For θ ∈ R, z ∈ T, deﬁne

iθ n−1 u(e ) := h(θ), φn(z) := (z − 1) u(z). (6.37)

Using the same argument as in the self-adjoint case (see Lemma 6.5), one can

n obtain that φn ∈ C (T). The following theorem is the principle result of this subsection.

147 Theorem 6.15. Let n ∈ N. There exist a separable Hilbert space H, a unitary operator U ∈ B(H), and a self-adjoint operator Z ∈ Sn(H) such that

− ∑n 1 k iZ 1 d itZ 1 φn(e U) − φn(e U) ∈/ S (H), (6.38) k! dtk k=0 t=0

n for the function φn ∈ C (T) deﬁned in (6.37).

Throughout the subsection we fix n ∈ N and the function φn defined above. Minor adjustment of the proof of Lemma 6.6 gives the following analog for the functions defined in (6.37).

Lemma 6.16. For every z0, zn ∈ T and u, φn deﬁned in (6.37),

[n] [1] φn (z0, 1,..., 1, zn) = u (z0, zn).

The next result is an immediate consequence of Lemma 6.16.

Corollary 6.17. Let ς : Tn+1 → C be given by

· · · · [n] ς(z0, . . . , zn) := z1 z2 ... zn−1 φn (z0, . . . , zn). (6.39)

For every z0, zn ∈ T,

[1] ς(z0, 1,..., 1, zn) = u (z0, zn).

Theorem 6.18. ( [30, Theorem 8]) For every integer d ≥ 3, there exist unitary ∈ 2 operators Hd,Kd B(ℓ2d+1) such that

Hd ≠ Kd, σ(Hd) = σ(Kd), 1 ∈ σ(Hd), and

1 ∥u(Kd) − u(Hd)∥∞ ≥ const (log d) 2 ∥Kd − Hd∥∞, (6.40) where u is deﬁned in (6.37) and the constant is independent of d.

Applying properties of multilinear Schur multipliers yields the following consequence of Theorem 6.18.

148 Theorem 6.19. Let d ∈ N, d ≥ 3. For ς deﬁned in (6.39) and unitary operators ∈ 2 Hd,Kd B(ℓ2d+1) satisfying Theorem 6.18, we have

1 Kd,Hd,...,Hd T ∥Sn × × n → 1 ≥ const (log d) 2 , ς 2d+1 ... S2d+1 S2d+1 where the constant is independent of d.

Proof. By Corollary 6.17 and Theorem 5.5 (ii) and (iii), we have

Kd,Hd,...,Hd ≥ Kd,Hd ∥ T n n 1 T [1] S1 → 1 ς S ×...×S →S u 2d+1 S2d+1 2d+1 2d+1 2d+1 Kd,Hd = T ∥S∞ → ∞ . u[1] 2d+1 S2d+1

Since Hd ≠ Kd,

Kd,Hd − T [1] (Kd Hd) ∞ Kd,Hd,...,Hd ≥ u Tς Sn ×...×Sn →S1 . 2d+1 2d+1 2d+1 ∥Kd − Hd∥∞ Since by (5.20),

Kd,Hd − − Tu[1] (Kd Hd) = u(Kd) u(Hd), the above inequality is equivalent to

∥u(Kd) − u(Hd)∥∞ Kd,Hd,...,Hd ≥ Tς Sn ×...×Sn →S1 . 2d+1 2d+1 2d+1 ∥Kd − Hd∥∞

Applying (6.40) completes the proof. D

The following estimate is essential in the proof of Theorem 6.15.

Corollary 6.20. For every integer d ≥ 3, there exist a self-adjoint operator ∈ 2 ∥ ∥ ≤ ∈ 2 Wd B(ℓ(4d+2)(n+1)) with Wd Sn 1 and a unitary operator Ud B(ℓ(4d+2)(n+1)) such that U ,...,U 1 d d ≥ 2 T [n] (WdUd,...,WdUd) const (log d) , (6.41) φn S1 where the constant is independent of d.

Proof. Let Hd and Kd be as in Theorems 6.18 and 6.19 and consider   K 0  d  ⊕n+1 Vd := ,Ud := j=1 Vd. 0 Hd

2 We note that Ud is a unitary operator acting on ℓ(4d+2)(n+1). By a verbatim ˜ repetition of the proof of Lemma 6.11 (replacing Hd in the construction with

149 Ud and denoting the outcome Zd by Wd), we obtain existence of a self-adjoint ∈ 2 ∥ ∥ ≤ operator Wd B(ℓ(4d+2)(n+1)) such that Wd Sn 1 and

U ,...,U 1 d d ≥ 2 Tς (Wd,...,Wd) S1 const (log d) . (6.42)

Therefore, (6.41) would follow from (6.42) and the representation

Ud,...,Ud Ud,...,Ud Tς (Wd,...,Wd) S1 = T [n] (WdUd,...,WdUd) S1 . (6.43) φn To prove (6.43), we set N = (4d + 2)(n + 1) and consider the spectral decom- ∑ N ∈ T position Ud = i=1 ziEUd (zi), zi . Then we have

Ud,...,Ud T [n] (WdUd,...,WdUd) φn ∑N [n] = φn (zr0 , . . . , zrn )EUd (zr0 )(WdUd)EUd (zr1 ) ... (WdUd)EUd (zrn ) r0,...,rn=1 ∑N (∑N ) [n] = φn (zr0 , . . . , zrn )EUd (zr0 )Wd zlEUd (zl) EUd (zr1 ) ...WdEUd (zrn )Ud r0,...,rn=1 l=1 ∑N [n] = zr1 . . . zrn−1 φn (zr0 , . . . , zrn )EUd (zr0 )Wd ...WdEUd (zrn )Ud r0,...,rn=1 ∑N (6.39) = ς(zr0 , . . . , zrn )EUd (zr0 )Wd ...WdEUd (zrn )Ud r0,...,rn=1

Ud,...,Ud = Tς (Wd,...,Wd)Ud.

Since Ud is unitary, the latter equality implies (6.43), completing the proof. D

Lemma 6.21. For every integer d ≥ 3, there exist a non-zero self-adjoint op- ∈ 2 ∈ 2 erator Zd B(ℓ(4d+2)(n+1)) and a unitary operator Ud B(ℓ(4d+2)(n+1)), such that ∑∞ n ∥Zd∥Sn < ∞, (6.44) d=1 and ∑ n−1 k iZd − 1 d itZd φn(e Ud) k=0 k! dtk t=0φn(e Ud) lim S1 = ∞, (6.45) →∞ n d ∥Zd∥Sn where φn is deﬁned in (6.37).

Proof. Let d be ﬁxed and let Wd and Ud be as in Corollary 6.20. Upon rescaling −1 Wd to ∥Wd∥Sn Wd, we assume that

∥Wd∥Sn = 1.

150 For m ∈ N, denote 1 W := W . m,d m d By Theorem 5.12 (ii),

n m Qn,φn,Ud (Wm,d) (6.46) n ( ∞ ) ∑ ∑ ∑ k j2 jl iWm,d n e Ud,Ud,...,Ud (iWm,d) (iWm,d) (iWm,d) = m T [l] Ud, Ud, ..., Ud φn k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=n n ( ∞ ) ∑ ∑ ∑ k j2 jl iWm,d (iW ) (iW ) (iW ) e Ud,Ud,...,Ud j1 m,d d d = T [l] m Ud, Ud, ..., Ud . φn k! j2! jl! l=1 j1,...,jl≥1 k=j1 j1+...+jl=n Note that ∑∞ (iW )k eiWd/m − I m m,d = −→ iW as m → ∞. k! 1/m d k=1 Hence, by Lemma 5.8 (i) and the representation (5.15), the summand in (6.46) with l = n, j1 = ... = jn = 1

∞ ( ∑ k ) iWm,d e Ud,Ud,...,Ud (iWm,d) T [n] m Ud, iWdUd, . . . , iWdUd φn k! k=1 approaches

Ud,Ud,...,Ud T [n] (iWdUd, . . . , iWdUd) φn as m → ∞. This result and Corollary 6.20 imply that for m large enough and for l = n and j1 = ... = jn = 1, we have

∞ ( ∑ k ) eiWm,d U ,U ,...,U (iWm,d) 1 d d d ≥ 2 T [n] m Ud, iWdUd, . . . , iWdUd const (log d) . φn k! S1 k=1 (6.47) We now turn to the analysis of the remaining terms on the right hand side of (6.46). Let l < n and let j1, . . . , jl ≥ 1 be such that j1 + ... + jl = n. In n particular, jk ≥ 2 for at least one value of k. Since φn ∈ C (T) and l < n, it follows from [101, Theorem 2] that there exists a constant K > 0 (depending on n and φn, but neither on d nor on the operators Ud and Wm,d) such that

iWm,d e Ud,Ud,...,Ud ≤ T [l] Sn/j1 × ×Sn/jl →S1 K. (6.48) φn (4d+2)(n+1) ... (4d+2)(n+1) (4d+2)(n+1)

151 Since ∞ ∑ k j1 (iWm,d) (iWd) mj1 −→ as m → ∞, k! j1! k=j1 it follows from (6.48) that

( ∞ ) ∑ k j2 jl iWm,d (iW ) (iW ) (iW ) e Ud,Ud,...,Ud j1 m,d d d T [l] m Ud, Ud,. .., Ud φn k! j2! jl! S1 k= j1 ∞ ∑ k j2 jl (iWm,d) (iWd) (iWd) ≤ j1 ≤ ˜ ∥ ∥n ≤ ˜ K m .. . K Wd Sn K k! j2! Sn/j2 jl! Sn/jl k=j1 Sn/j1 (6.49) for m large enough and l < n. Combining (6.47) and (6.49), we deduce from the identity (6.46) existence of m0 ∈ N and cn > 0 such that for every m ≥ m0,

n 1 ≥ 2 − ˜ m Qn,φn,U (Wm,d) const (log d) Kcn. (6.50) d S1

We may assume that m ≥ d, which ensures

1 ∥W ∥Sn ≤ . (6.51) m,d d

For m such that (6.50) and (6.51) hold, deﬁne

1 Z := W = W . d m,d m d

∥ ∥ ∥ ∥ 1 The inequality (6.51) implies (6.44). Since Wd Sn = 1, we have Zd Sn = m . Hence, the estimate (6.50) yields (6.45). D

The proof of Theorem 6.15 is completely analogous to that of Theorem 6.7, so we only outline its major steps.

Proof of Theorem 6.15. For d ∈ N, deﬁne

− ∑n 1 1 dk iZd itZd βd := φn(e Ud) − φn(e Ud) . k! dtk S1 k=0 t=0

Due to (6.45), there exists a positive sequence {αd}d≥1 such that ∑∞ ∑∞ −n αd < ∞ and αdβd∥Zd∥Sn = ∞. (6.52) d=1 d=1

152 Set ⌊ ⌋ −n Nd = αd∥Zd∥Sn + 1.

It follows from (6.52), (6.44) and (6.45) that ∑∞ ∑∞ n Nd∥Zd∥Sn < ∞ and Ndβd = ∞. d=1 d=1 Deﬁne

H ⊕Nd 2 ˜ ⊕Nd ˜ ⊕Nd d := j=1ℓ(4d+2)(n+1), Zd := j=1Zd, Ud := j=1Ud.

˜ ˜ Then, Zd is a self-adjoint and Ud a unitary operator in B(Hd), and ∑∞ ˜ n ∥Zd∥Sn < ∞. d=1 We have

∞ − ∞ ∑ ∑n 1 k ∑ ˜ 1 d ˜ iZd ˜ itZd ˜ φn(e Ud) − φn(e Ud) = Ndβd = ∞. k! dtk S1 d=1 k=0 t=0 d=1 Deﬁne H ⊕∞ H ⊕∞ ⊕∞ := d=1 d,Z := d=1Zd, U := d=1Ud. ∑ ∈ Sn H ∥ ∥n ∞ ∥ ˜ ∥n Note that Z ( ), with Z Sn = d=1 Zd Sn . By (6.8),

− ∑n 1 k iZ 1 d itZ φn(e U) − φn(e U) k! dtk S1 k=0 t=0 ∞ − ∑ ∑n 1 k ˜ 1 d ˜ iZd ˜ itZd ˜ = φn(e Ud) − φn(e Ud) = ∞, k! dtk S1 d=1 k=0 t=0 proving (6.38). D

153 Chapter 7

Diﬀerentiation of the norm of Haagerup Lp-space

Recall firstly that a map F between two normed spaces (X , ∥·∥X ) and (Y, ∥·∥Y ) is called Fréchetdifferentiable at a point x ∈ X if there exists a bounded linear ′ X Y operator Fx from into such that for every ε > 0 there is δ > 0, for every h ∈ X with ∥h∥X < δ, we have

∥ − − ′ ∥ ≤ ∥ ∥ F (x + h) F (x) Fx(h) Y ε h X .

′ Note that if the map F is a bounded linear operator, then obviously Fx(h) = F (h), ′ ∈ X that is Fx = F for every x . The derivative F ′ is an operator from X into B(X , Y). Thus we can consider ′′ X → X X Y ′′ ∈ X X Y the second derivative F : B( ,B( , )), that is Fx B( ,B( , )) = B(X × X , Y). The map F is called twice Fr´echetdiﬀerentiable at a point x ∈ X if there ′′ X × X Y exists a bounded bilinear operator Fx from into such that for every

ε > 0 there is δ > 0, for every h ∈ X with ∥h∥X < δ, we have

′ 1 ′′ 2 ∥F (x + h) − F (x) − F (h) − - F (h, h)∥Y ≤ ε∥h∥ . x 2 x X

Definition 7.1. The map F is called m-times Fréchet differentiable at a point

(m) ×m x ∈ X if there exists a bounded multilinear operator Fx from X into Y such

154 that for every ε > 0 there is δ > 0, for every h ∈ X with ∥h∥X < δ, we have

∑m 1 (k) m ∥F (x + h) − F (x) − - F (h, . . . , h)∥Y ≤ ε∥h∥ . k! x X k=1 For more relevant definitions and terminology concerning differentiability of abstract functions we refer to [72]. Let 1 ≤ p < ∞ and let M ⊆ B(H) be an arbitrary von Neumann algebra. By Lp(M) we denote the Haagerup Lp-space on the algebra M equipped with the norm ∥ · ∥Lp (see definition in Section 5.2.2). This chapter resolves the question concerning differentiability of the norms of non-commutative Lp-space Lp(M) associated with M suggested by G. Pisier and Q. Xu in their survey [83]. The main result of this chapter is stated in Theorem 7.2 below.

7→ ∥ ∥p ∈ Lp M Theorem 7.2. The function A A p,A ( ) is

(i) infinitely many times Fréchetdifferentiable, whenever p is an even integer;

(ii) (p − 1)-times Fr´echetdiﬀerentiable, whenever p is an odd integer;

(iii) ⌊p⌋-times Fr´echetdiﬀerentiable, whenever p is not an integer.

The result of Theorem 7.2 is sharp (see Remark 7.17 below). We define Fréchet derivatives in Theorem 7.2 via singular traces on the weak non-commutative L1-space associated with a semifinite (non-finite) von Neumann algebra. We develop this theory first.

7.1 Traces on L1,∞(N , τ)

Let N be a semifinite von Neumann algebra equipped with a faithful normal semifinite trace τ. In this subsection we adopt some terminology from the theory of singular traces on symmetric operator spaces (see [70]) for traces on L1,∞(N , τ) (see also [42, Section 6]). A trace ϕ on L1,∞(N , τ) is a linear functional, which is unitarily invariant, i.e. ϕ : L1,∞(N , τ) → C satisfies ϕ(UXU ∗) = ϕ(X) for all X ∈ L1,∞(N , τ) and all unitaries U ∈ N .

155 A trace ϕ on L1,∞(N , τ) is said to be normalized if ϕ(X) = 1, for any 0 ≤ X ∈ L1,∞(N , τ) with µ(t, X) = t−1, t > 0. Let M be a von Neumann algebra. Recall that (N , τ) denotes the crossed product von Neumann algebra and that tr is the trace on L1(M) deﬁned by (5.42). The proof of the following lemma is immediate from Lemma 5.18 and existence of the Jordan decomposition of elements from L1(M) see [5, Theorem 6].

Lemma 7.3. For any normalized trace ϕ on L1,∞(N , τ) we have that

ϕ(X) = tr(X),X ∈ L1(M).

Fix a free ultraﬁlter ω on N. The limit with respect to the ultraﬁlter ω is denoted by limn→ω. In the following lemma we introduce a particular trace on L1,∞(N , τ), which

1,∞ is essential in the proof of our main result. By (L (N , τ))+ we denote the set of all positive operators from L1,∞(N , τ).

1,∞ Lemma 7.4. The functional ϕ :(L (N , τ))+ → R+, given by ∫ 1 n ϕ(X) := lim µ(t, X)dt, 0 ≤ X ∈ L1,∞(N , τ). (7.1) → n ω log(1 + n) 1 extends to a positive normalized trace on L1,∞(N , τ).

Proof. First we show that ϕ is a linear functional on all positive elements of L1,∞(N , τ). The equality

ϕ(αX) = αϕ(X), α > 0, 0 ≤ X ∈ L1,∞(N , τ) is obvious. Let 0 ≤ X,Y ∈ L1,∞(N , τ) and let n > 1 be ﬁxed. We show that ϕ(X + Y ) = ϕ(X) + ϕ(Y ). By Lemma 5.13, there exists λ ∈ N satisfying ∫ ∫ n n µ(t, X + Y )dt ≤ (µ(t, X) + µ(t, Y ))dt, λa a for all λa < n. Taking a = 1/λ we infer that ∫ ∫ n n µ(t, X + Y )dt ≤ (µ(t, X) + µ(t, Y ))dt 1 1/λ∫ ∫ n 1 = (µ(t, X) + µ(t, Y ))dt + (µ(t, X) + µ(t, Y ))dt 1 1/λ

156 Dividing both parts of the latter inequality by log(1 + n) and taking limn→ω, we obtain that ϕ(X + Y ) ≤ ϕ(X) + ϕ(Y ).

Conversely, by Lemma 5.14, there exists λ ∈ N such that ∫ ∫ n n (µ(t, X) + µ(t, Y ))dt ≤ 2 µ(2t, X + Y )dt, λa < n. λa a Without loss of generality, we may assume that λ > 2. Taking again a = 1/λ, we obtain ∫ ∫ ∫ n n 2n (µ(t, X) + µ(t, Y ))dt ≤ 2 µ(2t, X + Y )dt = µ(t, X + Y )dt 1 ∫ 1/λ∫ ∫2/λ 1 n 2n = µ(t, X + Y )dt + µ(t, X + Y )dt + µ(t, X + Y )dt. (7.2) 2/λ 1 n Observe that ∫ ∫ 2n 2n ≤ ∥ ∥′ −1 ∥ ∥′ µ(t, X + Y )dt X + Y L1,∞ t dt = X + Y L1,∞ log 2. n n

Dividing (7.2) by log(1 + n) and taking limn→ω, we obtain that

ϕ(X) + ϕ(Y ) ≤ ϕ(X + Y ).

1,∞ Thus, ϕ is additive and positively homogeneous on (L (N , τ))+. We extend ϕ on L1,∞(N , τ) by linearity and denote this extension by ϕ again. Since µ(UXU ∗) = µ(X) for all X ∈ L1,∞(N , τ) and all unitaries U ∈ N , it follows that ϕ is unitarily invariant. The fact that ϕ is positive and normalized follows directly from deﬁnition (7.1). D

Lemma 7.5. The extension of ϕ deﬁned in Lemma 7.4 is a bounded linear functional on L1,∞(N , τ).

1,∞ Proof. If X ∈ (L (N , τ))+, then ∫ 1 n |ϕ(X)| ≤ lim µ(t, X)dt → n ω log(1 + n) 1 ∫ n ′ 1 −1 ′ ≤ ∥X∥ 1,∞ lim t dt = ∥X∥ 1,∞ . L → L n ω log(1 + n) 1

157 For every X ∈ L1,∞(N , τ), we have

X = ℜX + iℑX = (ℜX)+ − (ℜX)− + i(ℑX)+ − i(ℑX)−,

1,∞ where (ℜX)+, (ℜX)−, (ℑX)+, (ℑX)− ∈ (L (N , τ))+. Moreover, we have that ∥ ℜ ∥′ ∥ ℜ ∥′ ∥ ℑ ∥′ ∥ ℑ ∥′ ≤ ∥ ∥′ ( X)+ L1,∞ , ( X)− L1,∞ , ( X)+ L1,∞ , ( X)− L1,∞ c0 X L1,∞ , where c0 ∥ · ∥′ is the modulus of concavity of the quasi-norm L1,∞ , (see e.g. [58, Chapter 1]). Therefore, if X ∈ L1,∞(N , τ), then

|ϕ(X)| = |ϕ((ℜX)+) − ϕ((ℜX)−) + iϕ((ℑX)+) − iϕ((ℑX)−)|

( ) 1 2 2 2 = (ϕ((ℜX)+) − ϕ((ℜX)−)) + (ϕ((ℑX)+) − ϕ((ℑX)−))

( ) 1 2 2 2 2 2 ≤ max{ϕ((ℜX)+) , ϕ((ℜX)−) } + max{ϕ((ℑX)+) , ϕ((ℑX)−) }

( ) 1 ≤ {∥ ℜ ∥′2 ∥ ℜ ∥′2 } {∥ ℑ ∥′2 ∥ ℑ ∥′2 } 2 max ( X)+ L1,∞ , ( X)− L1,∞ + max ( X)+ L1,∞ , ( X)− L1,∞ √ ≤ ∥ ∥′ 2c0 X L1,∞ .

Lemma 7.6. If X ∈ L1,∞(N , τ) is such that τ(supp(X)) < ∞, then ϕ(X) = 0, where ϕ is deﬁned in Lemma 7.4.

Proof. Let 0 ≤ X ∈ L1,∞(N , τ) be such that τ(supp(X)) = c for some c > 0.

By the deﬁnition of the distribution function n|X| we have that n|X|(0) = c. Since

µ(X) is the right inverse function of n|X|, it follows that µ(c, X) = 0. Therefore, ∫ ∫ n c µ(t, X)dt = µ(t, X)dt, for all n ≥ c. 1 1

Dividing the latter integral by log(1 + n) and taking limn→ω, we obtain that ϕ(X) = 0. D

7.2 First Fr´echet derivative

The next lemma is, essentially, a restatement of [64, Lemma 3.1].

Lemma 7.7. Let M be a von Neumann algebra. For every 1 < p < ∞, the norm

p ∥ · ∥Lp : L (M) → R is Fréchetdifferentiable. Its Fréchetderivative ∂(∥A∥Lp ):

158 Lp(M) → R at the point 0 ≠ A ∈ Lp(M) is given by the formula

1−p p−1 p ∂(∥A∥Lp )(X) = ∥A∥Lp tr(X · |A| sgn(A)),X ∈ L (M).

Here, tr is the trace on L1(M) deﬁned by (5.42).

Proof. It is enough to show the assertion for self-adjoint A, X ∈ Lp(M) (see

∗ p Remark 7.16 (i) below). Let A = A ∈ L (M) be such that ∥A∥Lp = 1. We claim that the hyperplane

SH := {X : tr(X · |A|p−1sgn(A)) = 1} (7.3) is a supporting hyperplane at A for the unit ball of Lp(M) (see e.g. p. 193 in [65] for the deﬁnition of this notion). Indeed,

tr(|A|p−1sgn(A) · A) = tr(|A|p) = 1 and

p−1 p−1 |tr(X · |A| sgn(A))| ≤ ∥X∥Lp · ∥|A| ∥ p ≤ 1, L p−1 for all X from the unit ball of Lp(M). By [63, Theorem 4.2.1 and Theorem A], we have that Lp(M) is uniformly smooth. Hence, the norm of Lp(M) is Fréchet differentiable (see e.g. [65, (6), p. 364]). Thus, at every point of the unit ball there is a unique supporting hyperplane (a tangent plane), i.e. the hyperplane plane SH defined in (7.3) (see [65, (12), p. 349]). Again invoking [65, (11), p. 349], we complete the proof. D

p Lemma 7.8. For every 1 < p < ∞, the ﬁrst Fr´echetderivative ∂(∥A∥Lp ): p p p L (M) → R of the function A 7→ ∥A∥Lp at the point A ∈ L (M) is given by

p p−1 p ∂(∥A∥Lp )(X) = p · tr(X · |A| sgn(A)),X ∈ L (M).

Proof. For A = 0 the assertion is trivial. Let 0 ≠ A ∈ Lp(M). Using Lemma 7.7 and the chain rule (see e.g. [72, p. 462]), we have

p p−1 1−p p−1 ∂(∥A∥Lp )(X) = p · ∥A∥Lp ∥A∥Lp tr(X · |A| sgn(A)) = p · tr(X · |A|p−1sgn(A)),X ∈ Lp(M).

159 7.3 Main estimate

In this section we obtain the key estimate guaranteeing existence of the Fr´echet derivatives in Theorem 7.2. The main result of this section is stated in the following theorem. Throughout the section we ﬁx 2 < p < ∞. By considering N ⊗ L∞(0, 1) instead of N , we may assume without loss of generality that N is atomless.

Theorem 7.9. Let p ∈ (m, m+1], m ≥ 2. There exists a constant c11(p) > 0 such that for any compactly supported function g on R with g(j)(0) = 0, 0 ≤ j ≤ m−1, (m−1) ∈ ≤ ≤ − ∗ ∗ ∈ and g Λp−m, for every 1 k m 1, and A0 = A0,...,Ak = Ak Lp,∞(N , τ), we have

( k ) ∑ p−k−1 A0,...,Ak (m−1) p ≤ · ∥ ∥ ∥ ∥ p,∞ T [k] ,∞ c11(p) g Λp−m Aj L . (7.4) g (Lp,∞)×k→L--p−1 j=0

Our approach consists in a usage of two-dimensional induction presented in the following lemma.

Lemma 7.10. Let m ∈ N, m ≥ 2. Suppose, we have the statements Stk,l, k ≥ 1, l ≥ 0, k + l ≤ m − 1. If the following conditions

(i) Stk,l holds when k + l = m − 1

(ii) Stk+1,l & Stk−1,l+1 ⇒ Stk,l, whenever k + l < m − 1 and k ≥ 2.

(iii) St2,l implies St1,l whenever l ≤ m − 2. hold, then every statement Stk,l holds for all k ≥ 1, l ≥ 0, k + l ≤ m − 1.

We modify the estimate (7.4), introducing an additional parameter l ≥ 0 as given in the statement below. Fix m ≥ 2 such that p ∈ (m, m + 1] and ﬁx a compactly supported function g with g(j)(0) = 0, 0 ≤ j ≤ m − 1, and

(m−1) g ∈ Λp−m. For every k + l ≤ m − 1, k ≥ 1, l ≥ 0, the statement Stk,l is given as follows.

160 Statement Stk,l. There is a constant c12(p) ≥ 1, deﬁned in (7.12) below, such ∗ ∗ ∈ p,∞ N that for every A0 = A0,...,Ak = Ak L ( , τ), we have

A0,...,Ak p T [k] ---,∞ p,∞ ×k p−l−1 gl (L ) →L ( k ) ∑ p−k−l−1 2p−k−2l (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ (1 + 2p) c12(p) g Λp−m Aj L , (7.5) j=0 where gl is given by (5.61).

Observe that the assertion of Theorem 7.9 coincides with Stk,0. Using Lemma 7.10, we prove below that Stk,l holds for all pairs (l, k) such that k ≥ 1, l ≥ 0, k + l ≤ m − 1.

The next lemma shows that Stk,l satisﬁes the condition (i) of Lemma 7.10.

Lemma 7.11. Statement Stk,l holds for every k ≥ 1, l ≥ 0, such that k + l = m − 1.

p,∞ Proof. Fix k ≥ 1 and l ≥ 0 such that k+l = m−1. Take X1,...,Xk ∈ L (N , τ).

0,A1,...,Ak Adding and subtracting T [k] (X1,...,Xk) and then using Lemma 5.30, we gl have

A0,...,Ak T (X1,...,Xk) g[k] l ( ) A0,...,Ak 0,A1,...,Ak 0,A1,...,Ak = T − T (X1,...,Xk) + T (X1,...,Xk) g[k] g[k] g[k] (l l ) l A0,...,Ak − 0,A1,...,Ak · A1,...,Ak = T [k] T [k] (X1,...,Xk) + X1 T [k−1] (X2,...,Xk). gl gl gl+1

· 0,A2,...,Ak Adding and subtracting X1 T [k−1] (X2,...,Xk) to the latter sum, and again gl+1 applying Lemma 5.30, we obtain ( ) A0,...,Ak A0,...,Ak 0,A1,...,Ak T (X1,...,Xk) = T − T (X1,...,Xk) g[k] g[k] g[k] l ( l ) l · A1,...,Ak − 0,A2,...,Ak · A2,...,Ak + X1 T [k−1] T [k−1] (X2,...,Xk) + X1X2 T [k−2] (X3,...,Xk). gl+1 gl+1 gl+2

Repeating this argument k times and observing that

0,Ak · T [1] (Xk) = Xk gl+k(Ak), gl+k−1

161 we arrive at ( ) A0,...,Ak A0,...,Ak − 0,A1,...,Ak T [k] (X1,...,Xk) = T [k] T [k] (X1,...,Xk) gl gl gl ∑k−1 ( ) Aj ,...,Ak 0,Aj+1,...,Ak + X1 · ... · Xj · T − − T − (Xj+1,...,Xk) g[k j] g[k j] j=1 l+j l+j

+ X1 · ... · Xk · gl+k(Ak). (7.6)

Next, we estimate every summand in (7.6) separately.

(m−1) Since g ∈ Λp−m and k + l = m − 1, it follows from Lemma 5.34 that

∥ (k)∥ ≤ ∥ (m−1)∥ (gl) Λp−m g Λp−m .

(k) Now, applying Theorem 5.36 with h = (gl) , α = p − m, A = A0 and B = 0, p p and observing that p−l−1 = k+(p−m) , we have ( )

A0,...,Ak 0,A1,...,Ak − p T [k] T [k] (X1,...,Xk) ,∞ p−l−1 gl gl L ∏k (k) p−m ≤ · ∥ ∥ · ∥ ∥ ∞ ∥ ∥ p,∞ C(k, p) gl Λp−m A0 Lp, Xj L j=1 ( k ) k ∑ p−m ∏ (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ ∥ ∥ p,∞ C(k, p) g Λp−m Aj L Xj L , (7.7) j=0 j=1 where C(k, p) is the constant from Theorem 5.36. Next, we estimate the second summand on the right hand side of (7.6). Fix 1 ≤ j ≤ k − 1. By Lemma 5.17 (ii), we have that ( ) Aj ,...,Ak 0,Aj+1,...,Ak · · · − p X1 ... Xj T [k−j] T [k−j] (Xj+1,...,Xk) ,∞ g g p−l−1 l+(j l+j ) L p ≤ ∥ ∥ ∞ · · ∥ ∥ ∞ c2 j + 1, − − X1 Lp, . .. Xj Lp, ( p l 1 ) Aj ,...,Ak 0,Aj+1,...,Ak · − p T [k−j] T [k−j] (Xj+1,...,Xk) ,∞ , (7.8) p−j−l−1 gl+j gl+j L ( ) p where c2 j + 1, p−l−1 is the constant from Lemma 5.17 (ii). (m−1) Since g ∈ Λp−m and k − j + l + j = m − 1, it follows from Lemma 5.34 that ∥ (k−j)∥ ≤ ∥ (m−1)∥ (gl+j) Λp−m g Λp−m .

162 (k−j) Applying Theorem 5.36 with h = (gl+j) , α = p − m, A = Aj and B = 0, and p p observing that p−j−l−1 = k−j+(p−m) , we have ( ) Aj ,...,Ak 0,Aj+1,...,Ak − p T [k−j] T [k−j] (Xj+1,...,Xk) ,∞ p−j−l−1 gl+j gl+j L (k−j) p−m ≤ − · ∥ ∥ · ∥ ∥ ∞ ∥ ∥ p,∞ · · ∥ ∥ p,∞ C(k j, p) gl+j Λp−m Aj Lp, Xj+1 L ... Xk L ( k ) ∑ p−m (m−1) ≤ − · ∥ ∥ · ∥ ∥ p,∞ ∥ ∥ p,∞ · · ∥ ∥ p,∞ C(k j, p) g Λp−m Aj L Xj+1 L ... Xk L , j=0 (7.9) where C(k − j, p) is from Theorem 5.36. Combining (7.9) with (7.8) and setting ( ) p − c(j, k, p, l) := c2 j + 1, p−l−1 C(k j, p), we arrive at ( ) Aj ,...,Ak 0,Aj+1,...,Ak · · · − p X1 ... Xj T [k−j] T [k−j] (Xj+1,...,Xk) ,∞ p−l−1 gl+j gl+j L ( k ) k ∑ p−m ∏ (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ ∥ ∥ p,∞ c(j, k, p, l) g Λp−m Aj L Xj L . (7.10) j=0 j=1 Finally, we estimate the third summand on the right hand side of (7.6). Ob- serving that

gm−1(t) p−m p−m | − | | | ≤ ∥ − ∥ | | sup gm 1(t) = sup p−m t gm 1 Λp−m t t=0̸ t=0̸ t

p−l−1 p−m k and appealing to Lemma 5.17 (ii) (with p = p + p ) and Lemma 5.34, we arrive at

( ) k p ∏ ∥ · · · ∥ p ≤ ∥ ∥ p ∥ ∥ ∞ X1 ... Xk gl+k(Ak) ,∞ c2 k+1, gm−1(Ak) ,∞ Xj Lp, L p−l−1 p − l − 1 L p−m j=1 ( ) ∏k p p−m ≤ ∥ ∥ ∥| | ∥ p ∥ ∥ ∞ c2 k + 1, gm−1 Λ − Ak ,∞ Xj Lp, p − l − 1 p m L p−m j=1 ( ) ∏k p (m−1) p−m ≤ c k + 1, ∥g ∥ ∥A ∥ p,∞ ∥X ∥ p,∞ 2 p − l − 1 Λp−m k L j L j=1 ( ) ( k ) k ∑ p−m ∏ p (m−1) ≤ c k + 1, ∥g ∥ ∥A ∥ p,∞ ∥X ∥ p,∞ (7.11) 2 p − l − 1 Λp−m j L j L j=0 j=1 Setting

{ ∑k−1 ( )} −2p+k+2l p c12(p) := (1+2p) max 1+C(k, p)+ c(j, k, p, l)+c2 k+1, , 1≤k≤m−1 p − l − 1 j=1 (7.12)

163 and combining (7.7), (7.10) and (7.11) with (7.6), we see that (7.5) holds. This completes the proof of the lemma. D

Proof of Theorem 7.9. We use Lemma 7.10 to prove Theorem 7.9.

If k + l = m − 1, then the statement Stk,l holds by Lemma 7.11. This shows (i) of Lemma 7.10. Step 1: We show that condition (ii) of Lemma 7.10 holds. Suppose k ≥ 2 and k + l < m − 1 and suppose that Stk+1,l and Stk−1,l+1 hold. (k) (k+1) We claim that (gl) , (gl) are bounded and continuous. Indeed, since g(m−1) is continuous and k +l, k +l+1 ≤ m−1, it follows that the functions g(l+k) and g(l+k+1) are continuous. Now, using representation (5.68), we obtain that (k) (k+1) gl , gl are also continuous. By the assumption that g is compactly supported, (k) (k+1) we obtain that gl , gl are also compactly supported and so bounded. ∈ p,∞ N 0,A1,...,Ak Let X1,...,Xk L ( , τ). Adding and subtracting T [k] (X1,...,Xk), gl and then applying Theorem 5.28 and Lemma 5.30, we have ( ) A0,...,Ak A0,...,Ak − 0,A1,...,Ak 0,A1,...,Ak T [k] (X1,...,Xk) = T [k] T [k] (X1,...,Xk) + T [k] (X1,...,Xk) gl gl gl gl

A0,0,A1,...,Ak A1,...,Ak = T [k+1] (A0,X1,...,Xk) + X1T [k−1] (X2,...,Xk). gl gl+1 (7.13)

p,∞ It follows from Stk+1,l (applied to the operators A0, 0,A1,...,Ak ∈ L (N , τ)) that

A0,0,A1,...,Ak p T [k+1] ,∞ p,∞ ×(k+1) p−l−1 gl (L ) →L ( k ) ∑ p−k−l−2 2p−k−2l−1 (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ (1 + 2p) c12(p) g Λp−m Aj L . (7.14) j=0

p,∞ It follows from Stk−1,l+1 (applied to the operators A1,...,Ak ∈ L (N , τ)) that

A1,...,Ak p T [k−1] ,∞ p,∞ ×(k−1) p−l−2 gl+1 (L ) →L ( k ) ∑ p−k−l−1 2p−k−2l−1 (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ (1 + 2p) c12(p) g Λp−m Aj L . (7.15) j=1

164 A0,...,A k p In order to estimate the norm ∥T (X ,...,X )∥ ,∞ , we use (7.13), [k] 1 k p−l−1 gl L together with the triangle inequality and Lemma 5.17 (ii), which yield

A0,...,Ak A0,0,A1,...,Ak p ≤ p T [k] (X1,...,Xk) ,∞ T [k+1] (A0, X1,...,Xk) ,∞ g p−l−1 g p−l−1 l ( L ) l L

p A1,...,Ak ∥ ∥ p,∞ p + c2 2, X1 L T [k−1] (X2, ...,Xk) ,∞ . (7.16) p−l−2 p − l − 1 gl+1 L

Observing that ( ) p c 2, ≤ 2p (7.17) 2 p − l − 1 and combining (7.14) and (7.15) with (7.16), we infer that

A0,...,Ak p T [k] (X1,...,Xk) ,∞ p−l−1 gl L ( ( k ) ∑ p−k−l−2 2p−k−2l−1 (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ ∥ ∥ p,∞ (1 + 2p) c12(p) g Λp−m A0 L Aj L j=0 ( k ) ) k ∑ p−k−l−1 ∏ + 2p ∥Aj∥Lp,∞ ∥Xj∥Lp,∞ j=1 j=1 ( k ) k ∑ p−k−l−1 ∏ 2p−k−2l (m−1) ≤ · ∥ ∥ · ∥ ∥ p,∞ ∥ ∥ p,∞ (1 + 2p) c12(p) g Λp−m Aj L Xj L , j=0 j=1 which proves Stk,l. Step 2: We show (iii) of Lemma 7.10. Suppose that 0 ≤ l ≤ m − 2 and suppose that St2,l holds. Arguing as at the beginning of Step 1, we have that (1) (2) p,∞ the functions (gl) , (gl) are bounded and continuous. Let X1 ∈ L (N , τ) be ﬁxed. Applying Theorem 5.28 and Lemma 5.30, we obtain ( ) A0,A1 A0,A1 − 0,A1 0,A1 T [1] (X1) = T [1] (X1) T [1] (X1) + T [1] (X1) gl gl gl gl

A0,0,A1 = T [2] (A0,X1) + X1gl+1(A1). gl

Hence, using the triangle inequality and Lemma 5.17 (ii) (see also (7.17)), we infer that

A0,A1 p T [1] (X1) ,∞ g p−l−1 l L

A0,0,A1 ≤ p ∥ ∥ p,∞ ∥ ∥ p T [2] (A0, X1) ,∞ + 2p X1 L gl+1(A1) ,∞ . (7.18) p−l−1 p−l−2 gl L L

Next, we estimate every summand on the right hand side of (7.18) separately.

165 p,∞ It follows from St2,l (applied to the operators A0, 0,A1 ∈ L (N , τ)) that

A0,0,A1 p T [2] (A0,X1) ,∞ g p−l−1 l L ( ) p−l−3 2p−2−2l (m−1) ≤ ∥ ∥ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ (1+2p) c12(p) g Λp−m A0 L + A1 L A0 L X1 L . (7.19)

In order to estimate the second summand on the right hand side of (7.18), we ﬁrst observe that

| | L.5.34 gl+1(t) g(t) gm−1(t) (m−1) ≤ ∥ − ∥ ≤ ∥ ∥ sup p−l−2 = sup p−1 = sup p−m gm 1 Λp−m g Λp−m , t=0̸ t t=0̸ |t| t=0̸ t and, therefore,

gl+1(t) p−l−2 p−l−2 p p ∥g (A )∥ ,∞ ≤ sup ∥|A | ∥ ,∞ ≤ ∥g − ∥ ∥A ∥ p,∞ . l+1 1 p−l−2 p−l−2 1 p−l−2 m 1 Λp−m 1 L L t=0̸ t L (7.20) Combining (7.19) and (7.20) with (7.18), we arrive at

A0,A1 p T [1] (X1) ,∞ g p−l−1 l L ( ) p−l−3 2p−2−2l (m−1) ≤ ∥ ∥ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ (1+2p) c12(p) g Λp−m A0 L + A1 L A0 L X1 L

(m−1) p−l−2 ∥ ∥ ∥ ∥ ∞ ∥ ∥ p,∞ + 2p g Λp−m A1 Lp, X1 L ( )( ) p−l−2 (m−1) 2p−2−2l ≤ ∥ ∥ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ g Λp−m c12(p)(1 + 2p) + 2p A0 L + A1 L X1 L ( ) p−l−2 2p−1−2l (m−1) ≤ ∥ ∥ ∥ ∥ p,∞ ∥ ∥ p,∞ ∥ ∥ p,∞ c12(p)(1 + 2p) g Λp−m A0 L + A1 L X1 L .

This proves St1,l.

It follows from Lemma 7.10 that the statement Stk,l holds for all k ≥ 1, l ≥ 0, k + l ≤ m − 1. In particular, for l = 0 and 1 ≤ k ≤ m − 1, we obtain (7.4). This

2p completes the proof of Theorem 7.9 (with c11(p) = (1 + 2p) c12(p)). D

p p 7.4 Taylor expansion for A 7→ ∥A∥Lp,A ∈ L .

In this section we prove Theorem 1.4. Throughout the section we fix a von Neumann algebra M (not necessary semifinite) equipped with a faithful normal N M o R semifinite weight ϕ0. Let be the semifinite von Neumann algebra σϕ0 equipped with the canonical semifinite trace τ. Recall that tr denotes the trace

166 1 on L (M) (see (5.42)). In this section all operators A, X, X1,...,Xk are assumed to be self-adjoint (see Remark 7.16), unless explicitly specified otherwise. The following lemma shows that the function A 7→ tr(|A|p),A ∈ Lp(M) is in fact infinitely many times Fréchet differentiable, provided that p is an even integer, and so it proves Theorem 7.2 (i).

Lemma 7.12. Let p be an even integer. If A, X ∈ Lp(M), then

∑p ( ) p p 1 A ∥A + X∥ p − ∥A∥ p − ∆ X,...,X = 0, (7.21) L L k! k,p | {z } k=1 k-times

A Lp M ×k where ∆k,p are symmetric multilinear bounded functionals on ( ) deﬁned by ( ) ∑ ( ∑ ) A s0 s1 · · sk ∆k,p X1,...,Xk := tr A Xπ(1)A ... Xπ(k)A ,

π∈Sk s0+...+sk=p−k, s0,...,sk≥0

p where Sk is the set of all permutations of {1, . . . , k} and Xj ∈ L (M), 1 ≤ j ≤ k, 1 ≤ k ≤ p. Moreover, ( ) A p! p−k k ∆ X,...,X ≤ ∥A∥ p ∥X∥ p , 1 ≤ j ≤ k, 1 ≤ k ≤ p. (7.22) k,p | {z } (p − k)! L L k-times Proof. The equality (7.21) follows from the binomial expansion of |A + X|p and deﬁnition (5.43). The estimate (7.22) is a straightforward consequence of the H¨olderinequality (5.44). D

Next we proceed with the proof of Theorem 7.2 (ii) and (iii). Since for 1 < p < 2 the result is known from Lemma 7.8 and for p = 2 the result is proved in Lemma 7.12, we assume henceforth that 2 0 and f is smooth outside the interval [−ε, ε]. Since p > 2, it follows that the function f p is a continuously diﬀerentiable compactly supported function. Observe also that if m ≥ 2 is such that p ∈ (m, m + 1], then (f p)(k) is a bounded continuous compactly supported function for 1 ≤ k ≤ m. Clearly,

p (m) (f ) ∈ Λp−m.

167 Deﬁnition 7.13. Let A ∈ Lp,∞(N , τ), p ∈ (m, m + 1]. Let ϕ be a normalized

1,∞ p,∞ positive trace on L (N , τ) deﬁned in (7.1). For X1,...,Xm ∈ L (N , τ) we deﬁne   p ′ ϕ(X1 · (f ) (A)), k = 1 δA (X ,...,X ) := ∑ ( ) k,p,ϕ 1 k  1 · A,··· ,A ≤ ≤  k! ϕ Xπ(1) T((f p)′)[k−1] (Xπ(2), ...,Xπ(k)) , 2 k m, π∈Sk (7.23) where Sk is a set of permutations of {1, . . . , k}.

A p,∞ N ×k The following lemma shows that δk,p,ϕ is well-deﬁned on L ( , τ) .

∈ p,∞ N A Lemma 7.14. If A L ( , τ), then δk,p,ϕ is a symmetric multilinear bounded functional on Lp,∞(N , τ)×k for every 1 ≤ k ≤ m.

A Proof. The fact that δk,p,ϕ is multilinear and symmetric follows immediately from deﬁnition (7.23) and linearity of ϕ (see Lemma 7.4). p ∞ p,∞ p−1 − , Let A, X1,...,Xm ∈ L (N , τ). It is clear that sgn (A)|A| ∈ L p 1 (N , τ) p−1 1,∞ (see [106, Chapter II, Proposition 12]) and X1 · sgn (A)|A| ∈ L (N , τ). We claim that

˜ p ′ p−1 A = X1 · (f ) (A) − X1 · p sgn(A)|A| (7.24) is a τ-ﬁnitely supported operator. Indeed, let f p(t) = |t|p + β(t), t ∈ R, where β is a smooth function such that β(t) = 0 for t ∈ (−ε, ε). We have that (f p)′(t) = p−1 ′ ′ ˜ ′ p sgn (t)|t| + β (t), t ∈ R and β (t) = 0, t ∈ (−ε, ε). Therefore, A = X1 · β (A). p,∞ Since, A ∈ L (N , τ), it follows that τ(E|A|(ε, ∞)) < ∞. Hence,

˜ ′ τ(suppA) = τ(supp(X1 · β (A)))

′ ≤ τ(suppβ (A)) = τ(E|β′(A)|(0, ∞)) = τ(EA(−∞, −ε)) + τ(EA(ε, ∞)) < ∞.

Now, using Lemmas 7.4 and 7.6, we infer that

p−1 p ′ p ϕ(X1 · sgn (A)|A| ) = ϕ(X1 · (f ) (A))

By Lemma 5.17 (i) and Lemma 7.5, we have

A p−1 p−1 · · | | ≤ ∥ ∥ ∞ ∥ | | ∥ p δ (X1) = p ϕ(X1 sgn (A) A ) const X1 Lp, sgn (A) A ,∞ 1,p,ϕ L p−1

168 A p,∞ N and, therefore, δ1,p,ϕ is bounded on L ( , τ). p ′ Applying Theorem 7.9 with g = (f ) , α = p − m, and A0 = ... = Ak−1 = A, we have

A,...,A p (m) (p−k)k p ≤ · ∥ ∥ ∥ ∥ ∞ T p ′ [k−1] ,∞ c12(p) (f ) Λp−m A Lp, ((f ) ) (Lp,∞)×(k−1)→L p−1 for all 2 ≤ k ≤ m. Fixing a permutation π ∈ Sk, using again Lemma 5.17 (i) and Lemma 7.5, we obtain

( ) · A,··· ,A ϕ Xπ(1) T p ′ [k−1] (Xπ(2),...,Xπ(k)) ((f ) ) A,··· ,A ≤ ∥ ∥ ∥ ∥ p,∞ p ϕ c1(2, 1) Xπ(1) L T p ′ [k−1] (Xπ(2),...,Xπ(k)) ,∞ ((f ) ) L p−1 p (m) (p−k)k ≤ ∥ ∥ ∥ ∥ p,∞ ∥ ∥ ∥ ∥ ∞ ∥ ∥ p,∞ ∥ ∥ p,∞ ϕ c1(2, 1)c12(p) Xπ(1) L (f ) Λp−m A Lp, Xπ(2) L ... Xπ(k) L

p (m) (p−k)k ∥ ∥ ∥ ∥ ∥ ∥ ∞ ∥ ∥ p,∞ ∥ ∥ p,∞ = ϕ c1(2, 1)c12(p) (f ) Λp−m A Lp, X1 L ... Xk L .

It follows that for 2 ≤ k ≤ m,

A p (m) (p−k)k ≤ ∥ ∥ ∥ ∥ ∥ ∥ ∞ ∥ ∥ p,∞ ∥ ∥ p,∞ δk,p,ϕ(X1,...,Xk) ϕ c1(2, 1)c12(p) (f ) Λp−m A Lp, X1 L ... Xk L , which completes the proof of the lemma. D

The following theorem proves Theorem 7.2 (ii) and (iii).

∈ ∈ Lp M A Theorem 7.15. Let p (m, m + 1]. If A ( ), then the restriction of δk,p,ϕ ×k ×k to Lp(M) is a symmetric multilinear bounded functional on Lp(M) for every 1 ≤ k ≤ m. Moreover, for A, X ∈ Lp(M), we have ∑m ( ) p p 1 A p ∥A + X∥ p − ∥A∥ p − δ X,...,X = O(∥X∥ p ). (7.25) L L k k,p,ϕ | {z } L k=1 k-times

A Proof. By Lemma 7.14, δk,p,ϕ is a symmetric multilinear bounded functional on Lp,∞(N , τ)×k for every 1 ≤ k ≤ m. Recalling that Lp(M) is a closed linear subspace in Lp,∞(N , τ) (see comments preceding Lemma 5.18), we obtain that A L M ×k A the restriction of δk,p,ϕ to p( ) (denoted again by δk,p,ϕ) is a symmetric ×k multilinear bounded functional on Lp(M) for every 1 ≤ k ≤ m. We proceed to prove (7.25). Let A, X ∈ Lp(M). Applying the fundamental p theorem of calculus to the function t 7→ ∥A + tX∥Lp , we have that ∫ 1 p p d p ∥A + X∥Lp − ∥A∥Lp = ∥A + tX∥Lp dt. (7.26) 0 dt

169 By Lemma 7.7, we have

d p p−1 ∥A + tX∥ p = tr(X · p|A + tX| sgn(A + tX)). (7.27) dt L

− p Fix t ∈ [0, 1]. It is clear that |A + tX|p 1sgn(A + tX) ∈ L p−1 (M) (see [106, Chapter II, Proposition 12]) and, therefore, |A+tX|p−1sgn(A+tX)·X ∈ L1(M). By Lemma 7.3, we have that

tr(X · p|A + tX|p−1sgn(A + tX)) = ϕ(X · p|A + tX|p−1sgn(A + tX)). (7.28)

Similarly to (7.24), we obtain

X· |A + tX|p−1sgn(A + tX) − X · (f p)′(A + tX) is a τ-ﬁnitely supported operator. Hence, using Lemmas 7.4 and 7.6, we infer

ϕ(X · p|A + tX|p−1sgn(A + tX)) = ϕ(X · (f p)′(A + tX)). (7.29)

Combining (7.26)-(7.29), we arrive at ∫ 1 p p p ′ ∥A + X∥Lp − ∥A∥Lp = ϕ(X · (f ) (A + tX))dt. (7.30) 0 Next we claim that

m∑−1 p ′ p ′ k A,··· ,A (f ) (A + tX) = (f ) (A) + t Tφ (X,...,X) k,(fp)(k+1) ( k=1 ) m−1 A+tX,A,··· ,A − A,··· ,A + t Tφ (X,...,X) Tφ (X,...,X) . (7.31) m−1,(fp)(m) m−1,(fp)(m)

Indeed, by (5.60), we have that

(f p)′(A + tX) − (f p)′(A) = tT A+tX,A(X). φ1,(fp)′′

It follows from Theorem 5.28 (see (i)) that ( ) (f p)′(A + tX) − (f p)′(A) = tT A,A (X) + t T A+tX,A(X) − T A,A (X) φ1,(fp)′′ φ1,(fp)′′ φ1,(fp)′′

A,A 2 A+tX,A,A = tTφ p ′′ (X) + t Tφ (X,X). 1,(f ) 2,(fp)(3)

Repeating this process m − 1 times, we obtain (7.31).

170 Now, we are in a position to prove the Taylor expansion (7.25). Plugging (7.31) into (7.30), we obtain that ∫ m∑−1 1 p p p ′ k A,··· ,A ∥A+X∥ p −∥A∥ p = ϕ(X ·(f ) (A))+ t ϕ(X ·T (X,...,X))dt L L φ p (k+1) 0 k,(f ) ∫ k=1 1 ( ( )) + tm−1 · ϕ X · T A+tX,A,··· ,A(X,...,X) − T A,··· ,A (X,...,X) dt φ − p (m) φ − p (m) 0 m 1,(f ) m 1,(f ) − m∑1 1 · p ′ · A,··· ,A = ϕ(X (f ) (A)) + ϕ(X Tφ (X,...,X)) k + 1 k,(fp)(k+1) ∫ k=1 1 ( ( )) + tm−1 · ϕ X · T A+tX,A,··· ,A(X,...,X) − T A,··· ,A (X,...,X) dt. φ − p (m) φ − p (m) 0 m 1,(f ) m 1,(f ) Using Deﬁnition 7.13, we arrive at

∑m ( ) p p 1 A ∥A + X∥ p − ∥A∥ p − δ X,...,X L L k k,p,ϕ | {z } k=1 ∫ k-times 1 ( ( )) = tm−1 · ϕ X · T A+tX,A,··· ,A(X,...,X) − T A,··· ,A (X,...,X) dt. φ − p (m) φ − p (m) 0 m 1,(f ) m 1,(f ) (7.32)

Next, we estimate the integral on the right hand side of (7.32). Using Lemma 7.5, Lemma 5.17 (i) and Theorem 5.36 with k = m − 1, α = p − m, h = (f p)(m), A = A + tX, B = A, we obtain ( ( ))

· A+tX,A,··· ,A − A,··· ,A ϕ X Tφ (X,...,X) Tφ (X,...,X) m−1,(fp)(m) m−1,(fp)(m) A+tX,A,··· ,A A,··· ,A ≤ ∥ ∥ ∥ ∥ p,∞ − p ϕ c1(2, 1) X L Tφ (X,...,X) Tφ (X,...,X) ,∞ m−1,(fp)(m) m−1,(fp)(m) L p−1 p (m) p−m m−1 ≤ ∥ ∥ − ∥ ∥ p,∞ ∥ ∥ ∥ ∥ ∞ ∥ ∥ ∞ ϕ c1(2, 1)C(m 1, p) X L (f ) Λp−m tX Lp, X Lp,

p (m) p−m p ∥ ∥ − ∥ ∥ ∥ ∥ ∞ = ϕ c1(2, 1)C(m 1, p) (f ) Λp−m t X Lp, .

Therefore, ∫ ( ( )) 1 tm−1 · ϕ X · T A+tX,A,··· ,A(X,...,X) − T A,··· ,A (X,...,X) dt φ − p (m) φ − p (m) 0 m 1,(f ) m 1,(f ) ∫ 1 p (m) p p−1 ≤ ∥ ∥ − ∥ ∥ ∥ ∥ ∞ ϕ c1(2, 1)C(m 1, p) (f ) Λp−m X Lp, t dt 0 1 p (m) p p = ∥ϕ∥c (2, 1)C(m − 1, p)∥(f ) ∥ ∥X∥ p,∞ ≤ const ∥X∥ p , p 1 Λp−m L L where the constant is independent of A, X. This completes the proof of (7.25). D

171 Remark 7.16. (i) It is suﬃcient to establish the assertion of Theorem 7.2 only for self-adjoint operators. Indeed, for every self-adjoint operators A and X Lp M A from ( ) the existence of δk,p,ϕ’s satisfying (7.25) is established in The- orem 7.15. For an arbitrary X ∈ Lp(M) consider the mapping   1  0 X α(X) = 1/p . 2 X∗ 0

Introducing the von Neumann algebra A := M⊗¯ M2, where M2 is the von Neumann algebra of all 2 × 2 complex matrices, and observing that

p p p L (M⊗M2) = L (M)⊗L (M2, Tr) (see e.g. [55, p. 70]), we have that α is an isometric embedding of Lp(M) into the (real) subspace of all self-adjoint operators from Lp(A).

p Finally, for arbitrary operators A, X1, ..., Xk ∈ L (M), we set

1 δA (X ,...,X ) := δα(A)(α(X ), . . . , α(X )), 1 ≤ k ≤ m. (7.33) k,p,ϕ 1 k 2 k,p,ϕ 1 k

A It is straightforward that δk,p,ϕ’s are bounded symmetric multilinear forms satisfying (7.25).

(ii) Note that since α is a linear operator over the ﬁeld of real numbers, it

A follows that the δk,p,ϕ deﬁned by (7.33) are multilinear functionals over the ﬁeld of real numbers.

Remark 7.17. We claim that Theorem 7.2 is sharp. Indeed, if M is a von Neumann algebra of type I, then Lp(M) contains an isometric copy of the space ℓp. If M is not an algebra of type I, then Lp(M) contains an isometric copy of

p L (R) for the hyperﬁnite II1 factor R (see [83, Theorem 3.5]). It is clear that Lp(R) contains an isometric copy of Lp(0, 1). The sharpness of Theorem 7.2 follows now from the diﬀerentiability properties of the norms of the classical spaces Lp(0, 1) and ℓp (see [26,105]).

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