Multiple operator integrals in perturbation theory

We start with the Birman--Solomyak approach to define double operator integrals and consider applications in estimating operator differences $f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the Birman--Solomyak approach to the Lifshits--Krein trace formula that is based on double operator integrals. We study the class of operator Lipschitz functions, operator differentiable functions, operator H\"older functions, obtain Schatten--von Neumann estimates for operator differences. Finally, we consider in Chapter 1 estimates of functions of normal operators and functions of $d$-tuples of commuting self-adjoint operators. In Chapter 2 we define multiple operator integrals with integrands in the integral projective tensor product of $L^\infty$ spaces. We consider applications of such multiple operator integrals to the problem of the existence of higher operator derivatives and to the problem of estimating higher operator differences. We also consider connections with trace formulae for functions of operators under perturbations of class $\boldsymbol{S}_m$, $m\ge2$. In the last chapter we define Haagerup-like tensor products of the first kind and of the second kind and we use them to study functions of noncommuting self-adjoint operators under perturbation. We show that for functions $f$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$ and for $p\in[1,2]$ we have a Lipschitz type estimate in the Schatten--von Neumann norm $\boldsymbol{S}_p$ for functions of pairs of noncommuting self-adjoint operators, but there is no such a Lipschitz type estimate in the norm of $\boldsymbol{S}_p$ with $p>2$ as well as in the operator norm. We also use triple operator integrals to estimate the trace norms of commutators of functions of almost commuting self-adjoint operators and extend the Helton--Howe trace formula for arbitrary functions in the Besov space $B_{\infty,1}^1({\Bbb R}^2)$.


Introduction
In this survey article we study the role of double operator integrals and multiple operator integrals in perturbation theory. Double operator integrals appeared in the paper [DK] by Yu.L. Daletskii and S.G. Krein. In that paper they considered the problem of differentiating the operator-valued function t → f (A + tK), where A and K are selfadjoint operators on Hilbert space. They discovered the following formula that expresses the derivative in terms of double operator integrals: for sufficiently nice functions f . Here E A stands for the spectral measure of A.
That time there was no rigorous theory of double operator integrals. Such a theory was developed later by Birman and Solomyak in [BS1], [BS2] and [BS4].
In general double operator integrals are expressions of the form Φ(x, y) dE 1 (x)T dE 2 (y), where Φ is a measurable function, T is a linear operator, and E 1 and E 2 are spectral measures on Hilbert space.
The Birman-Solomyak approach allows one to define such integrals in the case when T is a Hilbert Schmidt operator and Φ is an arbitrary bounded measurable function. This, in turn, permits us to define double operator integrals for arbitrary bounded linear operators T and for functions Φ satisfying certain assumptions (such functions are called Schur multipliers).
It turned out that double operator integrals play a very important role in perturbation theory. They appear naturally when estimating various norms of operator differences f (A) − f (B), where A is an unperturbed operator and B is a perturbed operator. In particular, double operator integrals are very helpful when studying the class of operator Lipschitz functions, i.e., functions f on R, for which (1.1) It turns out that if inequality (1.1) holds for all bounded self-adjoint operators A and B, then the same inequality holds for unbounded A and B once A − B is bounded. Roughly speaking, a functions f on R is operator Lipschitz if and only if the divided difference (x, y) → f (x)−f (y) x−y is a Schur multiplier. It is obvious that operator Lipschitz functions f must be Lipschitz, i.e., the inequality |f (x) − f (y)| ≤ const |x − y| must hold for x, y ∈ R. The question whether the converse is true was resolved in negative by Farforovskaya in [F1]. Later McIntosh [Mc] and Kato [Ka] proved that the function x → |x| is not operator Lischitz. Then in [JW] it was shown that operator Lipschitz functions must be differentiable everywhere on R (but not necessarily continuously differentiable, see [KSh2]. Later in [Pe2] necessary conditions for operator Lipschitzness were found in terms of Besov spaces and Carleson measures (see also [Pe5]).
In Chapter 1 we give an introduction to the theory of double operator integrals and define and characterize the class of Schur multipliers. Then we consider various applications of double operator integrals in perturbation theory. Namely, we study operator Lipschitz functions, operator Hölder functions, operator differentiable functions. We obtain sharp estimates for Schatten-von Neumann norms of operator differences f (A) − f (B) for functions f in the Hölder class Λ α (R). We present the Birman-Solomyak approach to the Lifshits-Krein trace formula that is based on double operator integrals. We also consider similar problems for functions of normal operators and for functions of m-tuples of commuting self-adjoint operators.
In Chapter 2 we proceed to multiple operator integrals, i.e., expressions of the form · · · m Φ(x 1 , x 2 , · · · , x m ) dE 1 (x 1 )T 1 dE 2 (x 2 )T 2 · · · T m−1 dE m (x m ). 3 We follow the approach to multiple operator integrals given in [Pe8] and define such multiple operator integrals in the case when the integrand Ψ belongs to the (integral) projective tensor products of the spaces L ∞ (E j ), 1 ≤ j ≤ m. We use this approach to study the problem of the existence of higher operator derivatives of the function t → f (A + tK) and express higher operator derivatives in terms of multiple operator integrals. We also use multiple operator integrals to obtain sharp estimates of higher operator differences m j=0 (−1) m−j m j f A + jK .
Finally, in the last section of Chapter 2 we apply multiple operator integrals to trace formulae for functions of self-adjoint operators of class S m with m ∈ Z, m ≥ 2. An alternative approach to multiple operator integrals is given in [JTT]. That approach is based on the Haagerup tensor product of L ∞ spaces. We define in Chapter 3 triple operator integrals whose integrands belong to the Haaherup tensor product of three L ∞ spaces. We study Schatten-von Neumann properties of such triple operator integrals and we see that their Schatten-von Neumann properties are not as nice as in the case of triple operator integrals with integrands in the integral projective tensor product.
We are going to use triple operator integrals to estimate functions of pairs of noncommuting self-adjoint operators under perturbation. It turns out that for our purposes none of the approaches based on the integral projective tensor product and on the Haagerup tensor product of L ∞ spaces works. We define new tensor products and call them Haagerup-like tensor products of the first kind and of the second kind. Then we define triple operator integrals with integrands in such Haagerup-like tensor products and use them to estimate the norms f (A 1 , B 1 ) − f (A 2 , B 2 ) , where (A 2 , B 2 ) is a perturbation of (A 1 , B 1 ) and f is a function in the Besov space B 1 ∞,1 (R 2 ). Note that functions f (A, B) for not necessarily commuting self-adjoint operators are defined as double operator integrals We show that for p ∈ [1, 2], we have a Lipschitz type estimate in the Schatten-von Neumann norm S p , but such Lipschitz type estimates do not hold in S p with p > 2 as well as in the operator norm. We conclude the chapter with estimating commutators of almost commuting self-adjoint operators (A and B are called almost commuting is AB − BA ∈ S 1 ). Such estimates allow us to extend the Helton-Howe trace formula for arbitrary functions in the Besov class B 1 ∞,1 (R 2 ). The results of the last chapter were obtained recently in [ANP1], [ANP2], [ANP3] and [AP9].
I am grateful to A.B. Aleksandrov for helpful remarks.

Preliminaries
In this section we collect necessary information on function spaces and operator ideals. Let w be an infinitely differentiable function on R such that w ≥ 0, supp w ⊂ 1 2 , 2 , and w(s) = 1 − w s 2 for s ∈ [1, 2]. (2.1) We define the functions W n , n ∈ Z, on R d by x j t j . Clearly, With each tempered distribution f ∈ S ′ R d , we associate the sequence {f n } n∈Z , (2. 2) The formal series n∈Z f n is a Littlewood-Paley type expansion of f . This series does not necessarily converge to f . Note that in this paper a significant role is played by the Besov spaces B 1 ∞,1 (R d and the series on the right converges uniformly. Initially we define the (homogeneous) Besov classḂ s p,q R d , s > 0, 1 ≤ p, q ≤ ∞, as the space of all f ∈ S ′ (R n ) such that According to this definition, the spaceḂ s p,q (R n ) contains all polynomials and all polynomials f satisfy the equality f B s p,q = 0. Moreover, the distribution f is determined by the sequence {f n } n∈Z uniquely up to a polynomial. It is easy to see that the series n≥0 f n converges in S ′ (R d ). However, the series n<0 f n can diverge in general. It can easily be proved that the series . Now the function f is determined uniquely by the sequence {f n } n∈Z up to a polynomial of degree less than r, and a polynomial g belongs to B s p,q R d if and only if deg g < r.
In the case when p = q we use the notation B s p (R d ) for B s p,p (R d We need a description of Λ α in terms of convolutions with de la Vallée Poussin type kernel V n .
To define a de la Vallée Poussin type kernel V n , we define the where w is the function defined by (2.1). We define V n , n ∈ Z, by In the definition of the classes Λ α (R d ), α > 0, we can replace the condition f n L ∞ ≤ const 2 −nα , n ∈ Z, with the condition (2.6) In the case of Besov classes B s ∞,q (R d ) the functions f n , defined by (2.2) have the following properties: f n ∈ L ∞ (R d ) and supp F f ⊂ {ξ ∈ R d : ξ ≤ 2 n+1 }. Such functions can be characterized by the following Paley-Wiener-Schwartz type theorem (see [R], Theorem 7.23 and exercise 15 of Chapter 7): Let f be a continuous function on R d and let M, σ > 0. The following statements are equivalent: Besov classes admit many other descriptions. We give here the definition in terms of finite differences. For h ∈ R d , we define the difference operator ∆ h , It is easy to see that B s p,q R d ⊂ L 1 loc R d for every s > 0 and B s p,q R d ⊂ C R d for every s > d/p. Let s > 0 and let m be the integer such that m − 1 ≤ s < m. The Besov 6 space B s p,q R d can be defined as the set of functions f ∈ L 1 loc R d such that However, with this definition the Besov space can contain polynomials of higher degree than in the case of the first definition given above. We refer the reader to [Pee] and [Tr] for more detailed information on Besov spaces.
2.2. Besov classes of periodic functions. Studying periodic functions on R d is equivalent to studying functions on the d-dimensional torus T d . To define Besov spaces on T d , we consider a function w satisfying (2.1) and define the trigonometric polynomials W n , n ≥ 0, by For a distribution f on T d we put and we say that f belongs the Besov class B s p,q (T d ), s > 0, 1 ≤ p, q ≤ ∞, if 2 n s f n L p n≥0 ∈ ℓ q . (2.7) Note that locally the Besov space B s p,q (R d ) coincides with the Besov space B s p,q of periodic functions on R d .

Operator ideals.
For a bounded linear operator T on Hilbert space, we consider its singular values s j (T ), j ≥ 0, Let S p , 0 < p < ∞, be the Schatten-von Neumann class of operators T on Hilbert space such that This is a normed ideal for p ≥ 1. The class S 1 is called trace class. For a linear operators T on a Hilbert space H its trace is defined by where {e j } j≥0 is an orthonormal basis in H . The right-hand side does not depend on the choice of a basis.

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The class S 2 is called the Hilbert-Schmidt class. It is a Hilbert space with inner product For p ∈ (1, ∞), the dual space (S p ) * can be isometrically identified with S p ′ with respect to the pairing The dual space to S 1 can be identified with the space of bounded linear operators, while the dual space to the space of compact operators can be identified with S 1 with respect to the same pairing.
We refer the reader to [GK] for detailed information on singular values and operator ideals.

Chapter 1 Applications of double operator integrals in perturbation theory
In the first chapter we give an introduction to the theory of double operator integrals that was developed by Birman and Solomyak. We discuss the problem of a representation for operator differences f (A) − f (B) in terms of double operator integrals. This allows us to obtain necessary conditions and sufficient conditions for a function on the real line to be operator Lipschitz. In particular, we show that if f belongs to the Besov class B 1 ∞,1 (R), then f is operator Lipschitz. It turns out that the same condition f ∈ B 1 ∞,1 (R) is also sufficient for operator differentiability. Next, we present the Birman-Solomyak approach to the Lifshits-Krein trace formula. Their approach is based on double operator integrals. We also discuss Hölder type estimates and Schatten-von Neumann estimates for operator differences.
Finally, we consider perturbations of functions of normal operators and perturbations of functions of m-tuples of commuting self-adjoint operators.

An introduction to double operator integrals
Double operator integrals appeared in the paper [DK] by Daletskii and S.G. Krein. It was Birman and Solomyak who developed later the beautiful theory of double operator integrals in [BS1], [BS2], and [BS4].
Note that the term Schur multiplier in the context of double operator integrals was introduced in [Pe2]. This is a generalization of the notion of a matrix Schur multiplier. Indeed, consider the very special case when the Hilbert space is the sequence space ℓ 2 and both spectral measures E 1 and E 2 are defined on the σ-algebra of all subsets of Z + as follows: E 1 (∆) = E 2 (∆) is the orthogonal projection onto the closed linear span of the vectors e n , n ∈ ∆, where {e n } n≥0 is the standard orthonormal basis of ℓ 2 . In this case a function Φ on Z + × Z + is a Schur multiplier if and only if the matrix {Φ(m, n)} m,n≥0 (for which we keep the notation Φ) is a matrix Schur multiplier, i.e., where B is the space of matrices that induce bounded linear operators on ℓ 2 and Φ ⋆ T is the Hadamard-Schur product of the matrices Φ and T . Recall that the Hadamard-Schur product A ⋆ B of matrices A = {a jk } j,k≥0 and B = {b jk } j,k≥0 is defined by It is easy to see that if a function Φ on X × Y belongs to the projective tensor product Clearly, the function is weakly measurable and It is easy to see that Another sufficient condition for a function to be a Schur multiplier can be stated in terms of the Haagerup tensor products of L ∞ spaces. The Haagerup tensor product over all representations of Φ of the form (1.1.5). Here .

It can easily be verified that
It is also easy to see that the series on the right converges in the weak operator topology and . As the following theorem says, the condition Φ ∈ L ∞ (E 1 ) ⊗ h L ∞ (E 2 ) is not only sufficient, but also necessary.
Theorem 1.1.1. Let Φ be a measurable function on X ×Y and let µ and ν be positive measures on X and Y that are mutually absolutely continuous with respect to E 1 and E 2 . The following are equivalent: (iv) there exist measurable functions ϕ on X × Ω and ψ on Y × Ω such that (1.1.3) holds and Ω |ϕ(·, w)| 2 dλ(w) if the integral operator f → k(x, y)f (y) dν(y) from L 2 (ν) to L 2 (µ) belongs to S 1 , then the same is true for the integral operator f → Ψ(x, y)k(x, y)f (y) dν(y).
The implications (iv)⇒(i)⇔(v) were established in [BS4]. In the case of matrix Schur multipliers the fact that (i) implies (ii) was proved in [Be]. We refer the reader to [Pe2] for the proof of the equivalence of (i), (ii), and (iv) and to [Pis] for the proof of the fact that (i) is equivalent to (iii).
Suppose that F 1 and F 2 are closed subsets of R. We denote by M F 1 ,F 2 the space of functions that belong to M(E 1 , E 2 ) for arbitrary spectral measures E 1 and E 2 such that supp E 1 ⊂ F 2 and supp E 2 ⊂ F 2 .
It is well known (see [KSh1] and [KSh3]) that if Φ is a continuous function on F 1 × F 2 and E 1 and E 2 are Borel spectral measures such that supp E 1 = F 2 and supp E 2 = F 2 , then Φ ∈ M F 1 ,F 2 if and only if Φ ∈ M(E 1 , E 2 ). The same conclusion under the weaker assumption that Φ is continuous in each variable was established in [AP5].
It is easy to see that conditions (i) -(iv) are also equivalent to the fact that Φ is a Schur multiplier of S 1 . It follows that if I is an operator ideal that is an interpolation ideal between the space of bounded linear operators and trace class S 1 and Φ satisfies one of the conditions (i) -(iv), then Φ is a Schur multiplier of I, i.e., In particular, this is true when I is the Schatten-von Neumann class S p , 1 < p < ∞.
If I is a separable (symmetrically normed) operator ideal (see [GK]) , we say that a function Φ is a Schur multiplier of I if the transformer T → Φ dE 1 T dE 2 defined on S 1 admits an extension to a bounded linear operator on I. In the case when I is an operator ideal dual to separable, we can define Schur multiplier of I by duality. We denote the space of Schur multipliers of I with respect to E 1 and E 2 by M I (E 1 , E 2 ).
Consider now the case when E 1 = E 2 = E and T ∈ S 1 . It follows easily from Theorem 1.1.1 that functions in the space M(E, E) of Schur multipliers have traces on the diagonal and the traces of functions in M (E, E) belong to L ∞ (E).
The following useful fact was established in [BS4].
Theorem 1.1.2. Let E be a spectral measure and Φ ∈ M(E, E). Suppose that T ∈ S 1 . Then where µ is the signed measure defined by Proof. It follows easily from Theorem 1.1.1 that it suffices to establish formula (1.1.7) in the case Φ(x, y) = ϕ(x)ψ(y), ϕ, ψ ∈ L ∞ (E). We have

A representation of operator differences in terms of double operator integrals
In the paper [DK] by Daletskii and S.G. Krein under certain assumptions on a function f on R the following formula was discovered: Consider the divided difference Df defined by It was established in [BS4] that in the case A − B ∈ S 2 , formula (1.2.1) holds for arbitrary Lipschitz functions f . To understand the right-hand side of (1.2.1) for Lipschitz functions, we have to define the divided difference Df on the diagonal ∆ def = {(x, x) : x ∈ R}. It turns out that no matter how we can define Df on the diagonal, formula (1.2.1) holds. If f is differentiable, it is natural to assume that (Df )(s, s) = f ′ (s). We can also define Df to be zero on the diagonal. Put 0, s = t.

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Theorem 1.2.1. Suppose that A and B are (not necessarily bounded) self-adjoint operators such that A − B ∈ S 2 . Let f be a bounded Borel function on R 2 such that (1.2.2) Corollary 1.2.2. If A and B are self-adjoint operators such that A − B ∈ S 2 and f is a Lipschitz functions on R, Proof. It suffices to put f = D 0 f . We refer the reader to [BS4] for the proof of Theorem 1.2.1. Here we prove an analog of Theorem 1.2.1 in the case when A − B is a bounded operator.
Theorem 1.2.3. Suppose that A and B are (not necessarily bounded) self-adjoint operators such that the operator A − B is bounded. Let f be a function on R 2 such that f ∈ M(E A , E B ) and f R 2 \∆ = Df R 2 \∆ . Then formula (1.2.2) holds and Proof. Consider first the case when A an B are bounded operators. We have It is easy to see from the definition of double operator integrals in the Hilbert-Schmidt case that Suppose now that A and B are unbounded self-adjoint operators. Clearly, f must be a Lipschitz function. It follows easily that the domain of f (A) contains the domain of A and the same is true for the operator B.
is a densely defined operator. Let us prove that it extends to a bounded operator and its extension (for which we keep the same notion f (A) − f (B)) satisfies (1.2.2).
Consider the orthogonal projections and define bounded self-adjoint operators A [N ] and B [N ] by Obviously, in the strong operator topology. On the other hand, it is easy to see that because equality (1.2.2) holds for bounded self-adjoint operators. It remains to observe that Remark 1. Suppose now that I is a separable (or dual to separable) operator ideal of B(H ) equipped with a norm that makes it a Banach space. Let A and B be self-adjoint operators such that A − B ∈ I and let f be a Lipschitz function on R. As above one can show that if the divided difference Df can be extended to the diagonal ∆ and the resulting function f on R 2 belongs to the space M I (E A , E B ) of Schur multipliers of I, then formula (1.2.2) holds, We refer the reader to [BS4] for more detail.
Remark 2. Note also that the significance of formula (1.2.1) is in the fact that it allows us to linearize the nonlinear problem of estimating f (A) − f (B). Indeed, one can obtain desired estimates of f (A) − f (B) by studying properties of the linear transformer Remark 3. Similar results hold for functions of unitary operators, see [BS4]. Analogs of the above results can also be obtained for analytic functions of contractions and of dissipative operators, see [Pe3], [Pe9] and [AP6]. However, in the case of contractions and in the case of dissipative operators one has to consider double operator integrals with respect to semi-spectral measures.

Commutators and quasicommutators
In the previous section we have seen that operator differences f (A) − f (B) can be represented as double operator integrals with integrand equal to the divided difference Df . Birman and Solomyak observed (see [BS7]) that similar formula hold for commuta- . The proof of the following result is practically the same as the proof of Theorem 1.2.3. Theorem 1.3.1. Suppose that A and B are (not necessarily bounded) self-adjoint operators and Q is a bounded linear operator such that the operator AQ−QB is bounded.

Note that in the special case
Similar result holds in the case when AQ − QB belongs to the Hilbert Schmidt class or other operator ideals.
In the rest of the paper we discuss in details estimates of f (A) − f (B). Practically all the results are also valid for commutators and quasicommutators though we are not going to dwell on them.

Operator Lipschitz functions
In § 1.2 we have observed that if f is a differentiable function on R such that the divided difference Df belongs to the space of Schur multipliers M R,R , then f is operator Lipschitz. It turns out that the converse is also true. First of all, if f is an operator Lipschitz function on R, then f is differentiable everywhere on R which was established in [JW] (but not necessarily continuously differentiable: the function x → x 2 sin(1/x) is operator Lipschitz, see [KSh2]). On the other hand, it was shown in [Pe2] (see also [Pe4]) that if f is a differentiable operator Lipschitz function, then Df ∈ M R,R . Similar results hold for functions on the unit circle.
In this section we discuss some necessary conditions and sufficient conditions for a function to be operator Lipschitz.
We start with necessary conditions for functions on the unit circle. The following result was established in [Pe2].
Theorem 1.4.1. Let f be an operator Lipschitz function on T. Then f belongs to the Besov class B 1 1 (T). Proof. As we have discussed in § 1.2, the divided difference belongs to the space of Schur multipliers M T,T . Trivially, this implies that the function Consider the rank one operator P on L 2 (T) defined by By Theorem 1.1.1, the integral operator C f defined by belongs to trace class S 1 . An elementary calculation shows that where P + is the orthogonal projection from L 2 onto the Hardy class [Pe6], Ch. 1, § 1). It is easy to see that both Hankel operators H f and H f belong to S 1 . Recall that the Hankel operator H f : H 2 → H 2 − is defined by H f ϕ = P − f ϕ. By the trace class criterion for Hankel operators [Pe1] (see also [Pe6], Ch. 6, § 1), we find that f ∈ B 1 1 (T). The following stronger necessary condition was also obtained in [Pe2] is a Carleson measure on the unit disk D. Theorem 1.4.1 implies easily that a continuously differentiable function on T does not have to be operator Lipschitz. Indeed, it follows from (2.7) that the lacunary Fourier coefficients of the derivative of a function f in B 1 1 (T) must satisfy the condition while it is well known that an arbitrary sequence in ℓ 2 can be the sequence of lacunary coefficients of the derivative of a continuously differentiable function. The above results can be extended to functions on R. The analog of Theorem 1.4.1 is that if f is an operator Lipschitz function on R, then f belongs to the Besov space B 1 1 locally. An analog of Theorem 1.4.2 also holds as well as the characterization of the last necessary condition in terms of Carleson measures. We refer the reader to [Pe4] and [Pe3].
We proceed now to sufficient conditions for operator Lipschitzness. The following result was obtained in [Pe2].
We give here an idea of the proof of Theorem 1.4.3. It is easy to see that it suffices to prove the following inequality: suppose that ϕ is an analytic polynomial (i.e., a polynomial of z) of degree m, then the norm of Dϕ in the projective tensor product C(T)⊗C(T) admits the following estimate: (1.4.1) Note that the projective tensor product C(T)⊗C(T) can be defined in the same way as the projective tensor product of L ∞ spaces. It can easily be verified that Clearly, j,k≥0φ
A similar fact holds for functions on R. The following result was obtained in [Pe8].
Theorem 1.4.4. If f belongs to the Besov class B 1 ∞,1 (R), then f is an operator Lipschitz function on R.
It follows from the definition of B 1 ∞,1 (R) (see (2.3)) that to prove Theorem 1.4.4, it suffices to establish the following fundamental inequality: for an arbitrary bounded function f on R with Fourier transform supported in [−σ, σ].
for arbitrary self-adjoint operators A and B with bounded A − B and for an arbitrary function f in L ∞ (R) whose Fourier transform is supported in [−σ, σ]. In [AP4] it was shown that inequality (1.4.3) holds with constant 1 on the right. To prove inequality (1.4.2) we introduce the functions r u , u > 0, whose Fourier transforms F r u are defined by It is easy to show that r u ∈ L 1 (R) and r u L 1 ≤ const. It follows that the function 1 − r u is the Fourier transform of finite signed measure. We denote this measure by µ u . We have To prove inequality (1.4.2), we establish the following integral representation for the divided difference Df : Lemma 1.4.5. Let f be a bounded function on R whose Fourier transform has compact support in [0, ∞). Then the following representation holds: (1.4.4) To prove identity (1.4.4), we can first consider the special case when f is the Fourier transform of an L 1 function, in which case this is an elementary exercise, and then consider suitable approximation, see [Pe4] and [Pe8] for details.
Corollary 1.4.6. Let f be a bounded function on R whose Fourier transform is supported in [0, σ]. Then inequality (1.4.2) holds.
Proof. Clearly, f * µ u = 0 for u > σ. Representation (1.4.4) gives us the following estimates: To prove inequality (1.4.2) in the general case we represent f as the sum of f n = f * W n (see (2.2)). Consider now the function (f n ) + whose Fourier transform is equal to χ R + F f n . It remains to observe that (f n ) + L ∞ ≤ const f n L ∞ , see [Pe4] and [Pe8] for details.
Proof of Theorem 1.4.4. By (1.4.2), we have Note that inequality (1.4.3) and its version for Schatten-von Neumann norms will play a very important role in Hölder type inequalities, in Schatten-von Neumann estimates of operator differences, see Sections 1.7 and 1.8.
To conclude the section, I would like to mention that similar results also hold for functions of contractions and functions of dissipative operators, see [Pe3], [KSh4] and [AP6].

Operator differentiable functions
In the previous section we have shown that the condition f ∈ B 1 ∞,1 (R) is sufficient for f to be operator Lipschitz on R. It turns out that the same condition f ∈ B 1 ∞,1 (R) is also sufficient for operator differentiability.
exists in the operator norm for an arbitrary self-adjoint operator A and an arbitrary bounded self-adjoint operator K.
The following result can be found in [Pe4] and [Pe8].
Theorem 1.5.1. Let f be a function in B 1 ∞,1 (R). Then f is operator differentiable and whenever A is a self-adjoint operator and K is a bounded self-adjoint operator.
Formula (1.5.1) is called the Daletskii-Krein formula. It was established in [DK] under considerably stronger assumptions. Later Birman and Solomyak proved in [BS4] formula (1.5.1) under less restrictive assumptions.
Let me give an idea of the proof of Theorem 1.5.1. Under the hypotheses of the theorem, we have To establish formula (1.5.1), we can represent the divided difference Df as an element of the integral projective tensor product where σ is a σ-finite measure and Moreover, the functions ϕ x satisfy the following: for all x. This can be deduced easily from Lemma 1.4.5, see [Pe4] for details.
We have Clearly, the above conditions easily imply that which implies (1.5.1).
Remark 1. The problem of differentiability of the function t → f (A + tK) is the problem of the existence of the Gâteaux derivative of the map K → f (A + K) − f (A) defined on the real space of bounded self-adjoint operators. We have proved that this map is differentiable in the sense of Gâteaux for functions f in B 1 ∞,1 (R). The reasoning given above allows one to prove that under the same assumptions this map is differentiable in the sense of Fréchet and the differential of this map is the double operator integral Remark 2. The above argument shows that under the hypotheses of Theorem 1.5.1, the function t → f (A t ) is actually continuously differentiable in the operator norm.
Finally, I would like to mention that similar results hold for functions of unitary operators, functions of contractions and functions of dissipative operators, see [Pe2], [Pe9] and [AP6]. In particular, in the case of functions of unitary operators we can consider the problem of differentiability of the function t → f e itA U ), t ∈ R, where f is a function on the unit circle T, U is a unitary operator and A is a bounded self-adjoint operator. It was proved in [Pe2] that under the assumption f ∈ B 1 ∞,1 (T), the function 21 t → f e itA U ) is differentiable in the operator norm and its derivative is equal to the following double operator integral:

The Lifshits-Krein trace formula
The notion of the spectral shift function was introduced be I.M. Lifshits in [L]. He discovered in that paper a trace formula for f (A) − f (B) where A is the initial operator and B is a perturbed operator that involves the spectral shift function. Later M.G. Krein in [Kr] generalized the trace formula to a considerably more general situation when A is an arbitrary self-adjoint operator and B is a trace class perturbation of A.
Let A be a self-adjoint operator on Hilbert space and let B be a perturbed self-adjoint operator with A − B ∈ S 1 . It was shown in [Kr] that there exists a unique real function (1.6.1) whenever f is a differentiable function on R whose derivative is the Fourier transform of an L 1 function. The function ξ is called the spectral shift function associated with the pair (A, B). Moreover, it was shown in [Kr] that under the same assumptions The right-hand side of formula (1.6.1) is well defined for an arbitrary Lipschitz function f . Krein asked in [Kr] whether formula (1.6.1) holds for an arbitrary Lipschitz function f . It turns out, however, that the Lipschitzness of f does not imply that the operator f (A) − f (B) ∈ S 1 whenever A − B ∈ S 1 . This was first observed in [F2].
On the other hand, it can be shown that a function f preserves trace class perturbations, i.e., if and only if f is operator Lipschitz (the operators A and B do not have to be bounded). This implies that the necessary conditions for operator Lipschitzness mentioned in § 1.4 are also necessary for property (1.6.2). On the other hand, it was proved in [Pe4] that the condition f ∈ B 1 ∞,1 (R) sufficient for operator Lipschitzness (see § 1.4) is also sufficient for trace formula (1.6.1) to hold.
In this section we use the Birman-Solomyak approach [BS3] that is based on double operator integrals. Actually, their approach allows to prove the existence of a finite real signed Borel measure ν such that for sufficiently nice functions f . It follows from the results of [Kr] that ν is absolutely continuous with respect to Lebesgue measure and dν = ξ dm, where ξ is the spectral shift function. Moreover, we combine the Birman-Solomyak approach with the observation that for f ∈ B 1 ∞,1 (R), the function t → f (A + t(B − A)), t ∈ R, is continuously differentiable in the trace norm (this can be proved in the same way as Theorem 1.5.1) and the Daletskii-Krein formula holds for the derivative of this operator function. This allows us to prove the following extension of the Birman-Solomyak result: Theorem 1.6.1. Let A and B be self-adjoint operators on Hilbert space such that B − A ∈ S 1 . Then there exists a real signed Borel measure ν on R such that formula (1.6.3) holds for an arbitrary function This can be proved in exactly the same way as Theorem 1.5.1.
Since Df is a Schur multiplier (see § 1.4), it follows that where the signed measure ν s is defined on the Borels sets by Clearly, ν u ≤ T S 1 . It is easy to verify that where the signed measure ν is defined by Note that the function u → ν u is continuous in the space of measures equipped with the weak- * topology, and so integration makes sense. Clearly, ν ≤ T S 1 . We have already mentioned that dν = ξ dm, where ξ is the spectral shift function. This implies the following extension of the Krein theorem.
Theorem 1.6.2. Let A and B be self-adjoint operators such that B −A ∈ S 1 . Suppose that f ∈ B 1 ∞,1 (R). Then trace formula (1.6.1) holds. The original proof of Theorem 1.6.2 by a different method was obtained in [Pe4].

Operator Hölder functions. Arbitrary moduli of continuity
In this section we obtain norm estimates for f (A) − f (B), where A and B are selfadjoint operators and f is a Hölder function of order α, 0 < α < 1. Then we consider the more general problem of estimating f (A) − f (B) in terms of the modulus of continuity of f .
By analogy with the notion of operator Lipschitz functions. We say that a function f on R is operator Hölder of order α, 0 < α < 1, if The problem of whether a Hölder function of order α (recall that the class of such functions is denoted by Λ α (R), see § 2) is necessarily operator Hölder of order α remained open for 40 years and it was solved in [AP1] (see also [AP2] for a detailed presentation). The solution is given by the following theorem: Thus the term "an operator Hölder function of order α" turns out to be short-lived. We prove here Theorem 1.7.1 for bounded self-adjoint operators and refer the reader to [AP4] for details how to treat the case of unbounded operators.
Proof of Theorem 1.7.1. Let N be an integer.
and the series converges absolutely in the operator norm. Here f n = f * W n (see (2.2)) and V N is the de la Vallée Poussin type kernel defined by (2.5). Suppose that M < N . It is easy to see that Suppose now that N is the integer satisfying On the other hand, it follows from fundamental inequality (1.4.3) and from (2.3) that Suppose now that ω is an arbitrary modulus of continuity, i.e., ω is a continuous nondecreasing function on [0, ∞) such that ω(s + t) ≤ ω(s) + ω(t), s, t ≥ 0 and ω(0) = 0. We associate with ω the function ω * defined by It is easy to see that if ω * (x) < ∞ for some x > 0, then ω * (x) < ∞ for all x > 0 in which case ω * is also a modulus of continuity.
The following result was obtained in [AP1] and [AP2].
Theorem 1.7.2. Let ω be a modulus of continuity. Then for arbitrary self-adjoint operators A and B with bounded A − B.
The proof of Theorem 1.7.2 is similar to the proof of Theorem 1.7.1. Slightly weaker results were obtained independently in [FN2]. Theorem 1.7.2 implies the following result proved in [AP2]: Corollary 1.7.3. Suppose that A and B be self-adjoint operators with spectra in an interval [a, b]. Then for a continuous function f on [a, b] the following inequality holds: Theorem 1.7.3 improves earlier estimates obtained in [F3]. We refer the reader to [AP5] for more detailed information and more sophisticated estimates of f (A) − f (B).
Note that similar results hold for functions of unitary operators, contractions and dissipative operators, see [AP2] and [AP6].

Schatten-von Neumann estimates of operator differences
In this section we list several results on estimates of the norms of f (A) − f (B) in operator ideals and, in particular, in Schatten-von Neumann classes.
Fundamental inequality (1.4.2) together with formula (1.2.1) allows us to use Mityagin's interpolation theorem [Mi] to generalize and generalize inequality (1.4.3) to arbitrary separable (or dual to separable) ideals I: for arbitrary self-adjoint operators A and B with bounded A − B and for an arbitrary bounded function f on R whose Fourier transform is supported in [−σ, σ].
In particular, inequality (1.8.1) holds in the case I = S p , p ≥ 1. This implies the following result (see [Pe2] and [Pe4]).
Theorem 1.8.1. Let I be a separable (symmetrically normed) operator ideal or an operator ideal dual to separable and let f be a function in the Besov class B 1 ∞,1 (R).

Suppose that A and B are self-adjoint operators such that
where c p is a positive number that depends only on p.
We proceed now to estimating Schatten-von Neumann norms of f (A) − f (B) for functions f in the Hölder class Λ α (R), 0 < α < 1. For a nonnegative integer l and for p ≥ 1, we define the following norm on the space of bounded linear operators on Hilbert space: The following result obtained in [AP3] is crucial.
Theorem 1.8.3. Let 0 < α < 1. Then there exists a positive number c > 0 such that for every l ≥ 0, p ∈ [1, ∞), f ∈ Λ α (R), and for arbitrary self-adjoint operators A and B on Hilbert space with bounded A − B, the following inequality holds: for every j ≤ l.
Proof. Put f n def = f * W n , n ∈ Z, and fix an integer N . We have by (1.8.1) and (2.3), Clearly, for j ≤ l, (R) . To obtain the desired estimate, it suffices to choose the number N so that The following result can be deduced from Theorem 1.8.3. We refer the reader to [AP3] for details.
Theorem 1.8.4. Let 0 < α < 1 and 1 < p < ∞ and let f ∈ Λ α (R). Supposed that A and B are self-adjoint operators such that A − B ∈ S p . Then the operator f (A) − f (B) belongs to S p/α and where c α,p depends only on α and p.

27
Note that for p = 1, the conclusion of Theorem 1.8.4 does not hold, see [AP3]. The example given in [AP3] is based on the S p criterion for Hankel operators, see [Pe1] and [Pe6].
Nevertheless, the conclusion of Theorem 1.8.4 can be obtained under stronger assumptions on f . The following result was obtained in [AP3].
Theorem 1.8.5. Let 0 < α ≤ 1 and let f be a function in the Besov class B α ∞1 (R). Supposed that A and B are self-adjoint operators such that A − B ∈ S 1 . Then the operator f (A) − f (B) belongs to S 1/α and Note that in [AP3] the above results were generalized to the case of considerably more general operator ideals.
Remark. As before, I would like to mention that similar results hold for functions of unitary operators, contractions and dissipative operators, see [AP3] and [AP6].

Functions of normal operators
We proceed now to the study of functions of normal operators under perturbation. The results of this section were obtained in [APPS]. Earlier weaker results were obtained in [FN1].
The spectral theorem allows us to define functions of a normal operator as integrals with respect to its spectral measure: Here N is a normal operator and E N is its spectral measure.
We are going to study estimates of f (N 1 ) − f (N 2 ) in terms of N 1 − N 2 . As in the case of functions of self-adjoint operators we can consider the divided difference and prove the formula (1.9.1) whenever N 1 and N 2 are normal operators with N 1 − N 2 ∈ S 2 and f is a Lipschitz function. Again, it does not matter how we define Df on the diagonal of C × C.
If f is a function on C such that the divided difference Df can be extended to the diagonal and the extension belongs to the space of Schur multipliers M C,C , then formula (1.9.1) as soon as N 1 and N 2 are normal operators with bounded difference. Moreover, such functions are necessarily operator Lipschitz, i.e., The trouble is that such functions are necessarily linear which follows from the results of [JW].
In [APPS] we used the following representation for f (N 1 ) − f (N 2 ): where A j = Re N j , B j = Im N j , x j = Re z j , y j = Im z j , j = 1, 2, and the divided differences D x f and D y f are defined by and It was established in [APPS] that for a function f in the Besov class B 1 ∞,1 (R 2 ), both divided differences D x f and D y f belong to the space of Schur multipliers M C,C . This follows from the following analog of fundamental inequality (1.4.2). Theorem 1.9.1. Let f be a bounded function on R 2 whose Fourier transform is supported in [−σ, σ] × [−σ, σ]. Then both D x f and D y f are Schur multipliers and The proof of Theorem 1.9.1 is based on the following lemma whose proved can be found in [APPS]. Lemma 1.9.2. Let f be a bounded function on R whose Fourier transform is supported in [−σ, σ]. Then n∈Z sin 2 σy (σy − πn) 2 = 1, x, y ∈ R. (1.9.4) Proof of Theorem 1.9.1. Clearly, it suffices to consider the case σ = 1. By Lemma 1.9.2, we have By Lemma 1.9.2, we have It remains to observe that (1.9.4) give us desired estimates of the norm of D x f and D y f in the Haagerup tensor product L ∞ ⊗ h L ∞ . The result follows nor from Theorem 1.1.1.
Theorem 1.9.1 implies the following result: ∞,1 . Moreover, if N 1 and N 2 are normal operators with bounded N 1 −N 2 , then formula (1.9.2) holds and ∞,1 (R 2 ), then f is an operator Lipschitz function. To prove that D x f, D y f ∈ M C,C , it suffices to apply Theorem 1.9.1 to each function f * W n (see (2.3)). Formula (1.9.2) can be proved by analogy with the proof of formula (1.2.1) for functions of self-adjoint operators. The operator Lipschitzness of f follows immediately from formula (1.9.2). We refer the reader to [APPS] for details.
As in the case of functions of self-adjoint operators, fundamental inequality (1.9.3) allows us to establish for functions of perturbed normal operators analogs of all the results of Sections 1.4-1.8 except for Theorem 1.8.2. In particular, a Hölder function of order α, 0 < α < 1, on R 2 must be operator Hölder of order α. An analog of Theorem 1.8.2 was obtained in [KPSS].
We refer the reader to [AP8] for more results on estimates of operator differences and quasicommutators for functions of normal operators.

Functions of commuting self-adjoint operators
In the previous section we considered the behavior of functions of normal operators under perturbation. This is equivalent to considering functions of pairs of commuting self-adjoint operators. Indeed, if N is a normal operator, then Re N and Im N are commuting self-adjoint operators. On the other hand, if A and B are commuting selfadjoint operators, then A + iB is a normal operator.
In this section we are going to study functions of d-tuples of commuting self-adjoint operators. It is natural to try to use the approach for functions of normal operators that has been used in the previous section. However, it turns out that it does not work for d ≥ 3.
Indeed, a natural analog of formula 1.9.2 for functions of triple of commuting selfadjoint operators would be the following formula: The methods of [APPS] that were outlined in § 1.9 allow us to prove that if f is a bounded function on R 3 with compactly supported Fourier transform, then D 1 f and D 3 f do belong to the space of Schur multipliers M R 3 ,R 3 . However, it turns out that the function D 2 f does not have to be in M R 3 ,R 3 , and so formula (1.10.1) cannot be used to prove that bounded functions on R 3 with compactly supported Fourier transform must be operator Lipschitz. This was established in [NP] where the following result was proved: Theorem 1.10.1. Suppose that g is a bounded continuous function on R such that the Fourier transform of g has compact support and is not a measure. Let f be the function on R 3 defined by (1.

10.2)
Then f is a bounded function on R 3 whose Fourier transform has compact support, but To construct a function g satisfying the hypothesis of Theorem 1.10.1, one can take, for example, the function g defined by Obviously, g is bounded and its Fourier transform F g satisfies the equality: c t for a nonzero constant c and sufficiently small positive t. It is easy to see that this implies that F g is not a measure.
Nevertheless, it was proved in [NP] by a different method that functions in the Besov class B 1 ∞,1 (R d ) are operator Lipschitz in the sense that The following result proved in [NP] plays the same role as fundamental inequality (1.4.2) in the case of functions of one self-adjoint operator.
Lemma 1.10.2 implies the following result (see [NP]) that considerably improves earlier estimates obtained in [F4].
If (A 1 , · · · , A d ) and (B 1 , · · · , B d ) are d-tuples of commuting self-adjoint operators, then where E A and E B are the joint spectral measures of the (A 1 , · · · , A d ) and (B 1 , · · · , B d ).
Lemma 1.10.2 allows us to obtain analogs of all the results of Sections 1.4-1.8 except for Theorem 1.8.2. An analog of Theorem 1.8.2 for d-tuples of commuting self-adjoint operators was obtained in [KPSS].

Chapter 2 Multiple operator integrals with integrands in projective tensor products and their applications
Multiple operator integrals were considered by several mathematicians, see [Pa], [St]. However, those definitions required very strong restrictions on the classes of functions that can be integrated. In [Pe8] multiple operator integrals were defined for functions that belong to the (integral) projective tensor product of L ∞ spaces. Later in [JTT] multiple operator integrals were defined for Haagerup tensor products of L ∞ spaces.
In this chapter we consider applications of multiple operator integrals with integrands in the integral projective tensor product of L ∞ spaces. Such multiple operator integrals have nice Schatten-von Neumann properties. In Chapter 3 we shall see that multiple operator integrals with integrands in the Haagerup tensor product of L ∞ spaces do not possess such properties.
We consider in this chapter applications of multiple operator integrals to higher operator derivatives and estimates of higher operator differences. We also consider connections between multiple operator integrals and trace formulae for perturbations of class S m , where m is positive integer greater than 1.

A brief introduction to multiple operator integrals
Multiple operator integrals are expressions of the form Here E 1 , · · · , E m are spectral measures on Hilbert space, Ψ is a measurable function, and T 1 , · · · , T m−1 are bounded linear operators on Hilbert space. The function Ψ is called the integrand of the multiple operator integral.
and Ψ is represented as in (2.1.1), the multiple operator integral is defined by 3) The following result shows that the multiple operator integral is well defined.
Theorem 2.1.1. The expression on the right-hand side of (2.1.3) does not depend on the choice of a representation of the form (2.1.1).
The following proof is based on the approach of [ACDS].
Proof. To simplify the notation, we assume that n = 3. In the general case the proof is the same. Consider the right-hand side of (2.1.3). It is easy to see that it suffices to prove its independence on the choice of (2.1.1) for finite rank operators T 1 and T 2 . It follows that we may assume that rank T 1 = rank T 2 = 1. Let T 1 = (·, u 1 )v 1 and T 2 = (·, u 2 )v 2 , where u 1 , v 1 , u 2 and v 2 are vectors in our Hilbert space. Suppose that w 1 and w 2 are arbitrary vectors. We are going to use the following notation: It is easy to verify that where ν 1 def = (E 1 v 1 , w 2 ), ν 2 def = (E 2 v 2 , u 1 ) and ν 3 def = (E 3 w 1 , u 2 ).

34
Thus It follows that W (T 1 , T 2 ) does not depend on the choice of a representation of the form (2.1.1).
The following result is an easy consequence of the above definitions.
. Suppose that T 1 , · · · , T m−1 are bounded linear operator. Then To simplify the notation, by S ∞ we mean the space of bounded linear operators on Hilbert space. The proof of the following result is also straightforward.
. Suppose that p j ≥ 1, 1 ≤ j ≤ m, and 1/p 1 + 1/p 2 + · · · + 1/p m ≤ 1. If T 1 , T 2 , · · · , T m are linear operators on Hilbert space such that T j ∈ S p j , 1 ≤ j ≤ m, then In particular, all the above facts hold for functions Ψ in the projective tensor product L ∞ (E 1 )⊗ · · ·⊗L ∞ (E m ) which consists of functions of the form

Higher operator derivatives
In § 1.4 we studied the problem of differentiability of the function t → f (A + tK), t ∈ R, for self-adjoint operators A and bounded self-adjoint operators K. In this section we are going to consider the problem of the existence of higher derivatives of this map.
In the paper [DK] Daletskii and Krein proved that in the case when the self-adjoint operator A is bounded for nice functions f the map t → f (A + tK) has m-th derivative and it can be expressed in terms a multiple operator integral whose integrand is a higher order divided difference of f .
Later in [Pe8] the existence of higher operator differences was proved under much less restrictive assumptions.
Definition. For a k times differentiable function f the divided differences D k f of order k are defined inductively as follows: (the definition does not depend on the order of the variables). Note that Dϕ = D 1 ϕ.
The following result was obtained in [Pe8]. To state it, we denote by B(R) the space of bounded Borel functions on R endowed with the norm Theorem 2.2.1. Let m be a positive integer and let f be a function in the Besov class B m ∞,1 (R). and Note that the integral projective tensor product of copies of B(R) can be defined in the same way as the integral projective tensor product of L ∞ spaces.
We sketch the proof of Theorem 2.2.1 in the special case m = 2. In the general case the proof is the same. Let f be a bounded functions on R whose Fouriesr transform is a compact subset of [0, ∞). The following identity is an analog of formula (1.4.5): u+v)s 3 e ius 1 e ivs 2 du dv.
(2.2.1) This is a simplified version of formula (5.6) in [Pe8]. As in § 1.5, it is easy to deduce from (2.2.1) the following estimate whenever f is a bounded function on R whose Fourier transform is supported in [0, σ].
The following theorem about the existence of the mth derivative of the function t → f (A t ), where A t def = A + tK, was obtained in [Pe8].
Theorem 2.2.2. Let m be a positive integer. Suppose that A is a self-adjoint operator and K is a bounded self-adjoint operator.
We refer the reader to [Pe8] for the proof.
Remark. Suppose that f ∈ B m ∞1 (R), m ≥ 2, but f does not necessarily belong to B 1 ∞1 (R). In this case we still can define the mth derivative of the function t → f (A t ) in the following way. We put where f n = f * W n , see (2.2). Then the series on the right-hand side of (2.2.3) converges absolutely in the norm. With this (natural) definition it can easily happen that the function t → f (A t ) can have mth derivative, but not necessarily the first derivative. We refer the reader to [Pe8] for details.
In a similar was one can consider the problem of taking higher operator derivatives for functions of unitary operators. Note that in [Pe8] the formula for the mth derivative of the function t → f (e itA )U has an error. A correct formula is an easy consequence of the results of Section 5 of [AP1].
Note also that similar results and similar formulae can be obtained for functions of contractions and for functions of dissipative operators, see [AP1] and [AP6].

Higher operator differences
In Chapter 1 we have seen that formula (1.2.1) plays a significant role in estimating various norms of the operator differences f (A) − f (B). In this section we are going to study higher order operator differences where A and K are self-adjoint operators on Hilbert space. We consider here only bounded self-adjoint operators A and K and refer the reader to [AP4] for a detailed study of the case when A is an unbounded self-adjoint operators.
As in the case of operator differences, an essential role is played by integral formulae for higher operator differences. In [AP2] it was shown that higher operator differences can be represented in terms of multiple operator integrals. This allowed one to obtain analogs of the results discussed in Chapter 1 for higher operator differences.
Recall that for functions f in the Besov class B m ∞,1 (R), the divided difference D m f of order m belongs to the integral projective product B(R)⊗ i · · ·⊗ i B(R) m+1 . The following formula was obtained in [AP2].
∞,1 (R) and let A and K be bounded self-adjoint operators on Hilbert space. Then Let us prove Theorem 2.3.1 in the special case m = 2.
Proof. Let f ∈ B 2 ∞,1 (R). We should prove the following formula: Put By (1.2.1), We have Similarly, The proof of the following result obtained in [AP2] is similar to the proof of Theorem 1.7.1.
Theorem 2.3.2. Let 0 < α < m and let f ∈ Λ α (R). Then there exists a constant c > 0 such that for every self-adjoint operators A and K on Hilbert space the following inequality holds: ∆ m K f (A) ≤ c f Λα(R) · K α . In particular, in the case α = 1, Theorem 2.3.2 means the following: let f be a function in the Zygmund class Λ 1 (R), i.e., f is a continuos function on R such that We refer the reader to [AP2] for an analog of Theorem 1.7.2 for higher order moduli of continuity.
To conclude this section, we also mention that the results of § 1.8 were generalized in [AP3] to the case higher order operator differences. We state here the following result whose proved can be found in [AP3].
Theorem 2.3.3. Let α > 0, m − 1 ≤ α < m, and m < p < ∞. There exists a positive number c such that for every f ∈ Λ α (R), for an arbitrary self-adjoint operator A, and an arbitrary self-adjoint operator K of class S p , the following inequality holds: Note also that similar results hold for functions of unitary operators, functions of contractions and functions of dissipative operators, see [AP2], [AP3] and [AP6].
2.4. Trace formulae for perturbations of class S m , m ≥ 2 In § 1.6 we have considered the Lifshits-Krein trace formula for f (A) − f (B) in the case when B is a trace class perturbation of A. In [Ko] Koplienko considered the case of Hilbert-Schmidt perturbations and he found a trace formula for the second order Taylor approximation Here A is a self-adjoint operator, K is a self-adjoint operator of class S 2 and η is a function in L 1 that is determined by A and K. It is called the spectral shift function of order 2. In [Ko] formula (2.4.1) was proved for rational functions with poles off R. Formula (2.4.1) was generalized in [Pe7] to the case when f is an arbitrary function in the Besov class B 2 ∞,1 (R). In [PSS] the authors considered the more general problem of perturbation of class S m , where m is an arbitrary positive integer and they obtained the following trace formula for the Taylor approximation T

(m)
A,K f of order m: They proved in [PSS] that there is a unique function η m in L 1 that depends only on A, K and m such that trace T (m) for functions f on R such that the derivatives f (j) are Fourier transforms of L 1 functions for 1 ≤ j ≤ m. The function η m is called the spectral shift function of order m. The results of [PSS] were improved in [AP7]. First, formula (2.4.2) was extended for arbitrary functions f in the Besov class B m ∞,1 (R). Secondly, much more general trace formulae for perturbations of class S m we obtained in [AP7].
It was shown in [AP7] that the Taylor approximation admits the following representation in terms of the multiple operator integral: Here A is a self-adjoint operator, K is a self-adjoint operator of class S 1 and f ∈ B m ∞,1 (R). In this formula by T

(m)
A,K f we mean where as usual f n = f * W n , see (2.2).
To establish formula (2.4.2), the authors of [PSS] proved the following inequality: Theorem 2.4.1. Let f ∈ B m ∞1 (R) and K ∈ S m . Then Note that the proof of Theorem 2.4.1 given in [AP7] contains an inaccuracy. We give here a corrected proof.
Proof. As before, it suffices to consider the case when f is a bounded function on R whose Fourier transform has compact support in (0, ∞). By Theorem 2.2.2, we have For simplicity we assume that m = 2. In the general case the proof is the same. Recall formula (2.2.1): (2.4.5) Let ω be a function in C ∞ (R) such that ω(0) = 1 and F ω is a nonnegative infinitely differentiable function with compact support. For ε > 0, we put f ε (x) def = ω(εx)f (x). Then supp F f ε is a compact and Hence, by Theorem 2.1 of [PSS], It is easy to see that f where C depends only on Φ. Moreover, supp F f ε is a compact subset of (0, ∞) for sufficiently small ε. It follows that By the spectral theorem, R in the norm of S 1 . The same is true about the second and the third integral on the right-hand side of (2.4.5). This proves that in the norm of S 1 , and so (2.4.4) holds with m = 2. Note that Theorem 2.4.1 was used in [AP7] to obtain considerably more general trace formulae. In particular trace formulae were found for and trace ∆ m K f (A).

Chapter 3 Triple operator integrals, Haagerup(-like) tensor products and functions of noncommuting operators
In this chapter we deal with triple operator integrals and we apply triple operator integrals to estimates of functions of perturbed noncommuting pairs of self-adjoint operators. It turns out that for this purpose it is not enough to consider triple operator integral whose integrands belong to the (integral) projective tensor product of L ∞ spaces. In [JTT] multiple operator integrals were defined for functions that belong to the Haagerup tensor product of L ∞ spaces. We define triple operator integrals for functions in the Haagerup tensor product in § 3.2. However, for our purpose we have to modify the notion of the Haagerup tensor product. We define in § 3.4 Haagerup-like tensor products of the first kind and of the second kind. We are going to use the following representation of f (A 1 , B 1 ) − f (A 2 , B 2 ) in terms of triple operator integrals: where the divided differences D [1] f and D [2] f are defined by Here f is a function in the Besov class B 1 ∞,1 (R 2 ) and (A 1 , B 1 ) and (A 2 , B 2 ) are pairs of (not necessarily commuting) self-adjoint operators.
It turns out that the divided differences do not have to belong to the integral projective tensor product of the L ∞ spaces. That is why we have to consider triple operator integrals defined for other classes of functions. In § 3.2 we define the Haagerup tensor product of L ∞ spaces and triple operator integrals for such functions. It turned out, however, that the divided differences do not have to belong to the Haagerup tensor product. To overcome the problems, we introduce in § 3.4 Haagerup-like tensor products of the first kind and of the second kind. We will see in § 3.5 that for functions f in B 1 ∞,1 (R 2 ), the divided difference D [1] f belongs to the Haagerup-like tensor product of the first kind, while the divided difference D [2] f belongs to the Haagerup-like tensor product of the second kind.
We obtain in § 3.6 Lipschitz type estimates for functions of noncommuting self-adjoint operators in the norm of S p with p ∈ [1, 2]. It turns out that such Lipschitz type estimates in the norm of S p for p > 2 and in the operator norm do not hold.
Finally, we use in § 3.8 triple operator integrals with integrands in Haagerup-like tensor products to estimates trace norms of commutators of functions of almost commuting operators.
In the first section of this chapter we define functions of noncommuting self-adjoint operators.

Functions of noncommuting self-adjoint operators
Let A and B be self-adjoint operators on Hilbert space and let E A and E B be their spectral measures. Suppose that f is a function of two variables that is defined at least on σ(A) × σ(B), where σ(A) and σ(B) are the spectra of A and B. If f is a Schur multiplier with respect to the pair (E A , E B ), we define the function f (A, B) of A and B by Note that the map f → f (A, B) is linear, but not multiplicative. If we consider functions of bounded operators, without loss of generality we may deal with periodic functions with a sufficiently large period. Clearly, we can rescale the problem and assume that our functions are 2π-periodic in each variable.
If f is a trigonometric polynomial of degree N , we can represent f in the form Thus f belongs to the projective tensor product L ∞⊗ L ∞ and It follows easily from (2.7) that every periodic function f on R 2 of Besov class B 1 ∞1 of periodic functions belongs to L ∞⊗ L ∞ , and so the operator f (A, B) is well defined by (3.1.1).
Note that the above definitions of functions of noncommuting operators is related to the Maslov theory, see [Ma]. If A and B are self-adjoint operators, we can consider the transformer L A of left multiplication by A and the transformer R B of right multiplication by B: Clearly, the transformers L A and R B commute.
We can consider the transformers L A and R B defined on the Hilbert Schmidt class S 2 . In this case they are commuting self-adjoint operators on S 2 and the spectral theorem allows us to define functions f (L A , R B ) for all bounded Borel functions f on R 2 .
If our Hilbert space is finite-dimensional, the definition of f (A, B) given by (3.1.1) is equivalent to the following one: where I is the identity operator, and so the definition of functions of noncommuting operators can be reduced to the functional calculus for the commuting self-adjoint operators on the Hilbert Schmidt class.
If our Hilbert space H is infinite-dimensional, we cannot apply f (L A , R B ) to the identity operator, which does not belong to the Hilbert Schmidt class. In this case we can consider the transformers L A and R B as commuting bounded linear operators on the space B(H ) of bounded linear operators on H . However, since B(H ) is not a Hilbert space and we cannot use the spectral theorem to define functions of L A and R B . Nevertheless, if f is a sufficiently nice function, we can define f (L A , R B ), in which case the functions f (A, B) defined by (3.1.1) coincide with f (L A , R B )I.

Haagerup tensor products and triple operator integrals
We proceed now to the approach to multiple operator integrals based on the Haagerup tensor product of L ∞ spaces. We refer the reader to the book [Pis] for detailed information about Haagerup tensor products of operator spaces. The Haagerup tensor product where α j , β jk , and γ k are measurable functions such that where B is the space of matrices that induce bounded linear operators on ℓ 2 and this space is equipped with the operator norm. In other words, over all representations of Ψ of the form (3.2.1).
It is well known that Without loss of generality we may assume c n def = ϕ n L ∞ ψ n L ∞ χ n L ∞ = 0 for every n.
We define α j , β j,k and γ k by and {β jk (x 2 )} j,k≥0 B ≤ 1. In [JTT] multiple operator integrals were defined for functions in the Haagerup tensor product of L ∞ spaces. Suppose that Ψ has a representation of the form (3.2.1) and can be defined in the following way. Consider the spectral measure E 2 . It is defined on a σ-algebra Σ of subsets of X 2 . We can represent our Hilbert space H as the direct integral Here µ is a finite measure on X 2 , x → G (x), is a measurable Hilbert family. The Hilbert space H consists of measurable functions f such that f (x) ∈ G (x), x ∈ X 2 , and Finally, for ∆ ∈ Σ, E(∆) is multiplication by the characteristic function of ∆. We refer the reader to [BS5], Ch. 7 for an introduction to direct integrals of Hilbert spaces. Suppose that Ψ belongs to the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ and (3.2.1) holds. The triple operator integral (3.2.3) is defined by Let us show that the series on the right converges in the weak operator topology. Let f and g be vectors in H . Put We consider the vectors v j and u k as elements of the direct integral (3.2.4), i.e., vector functions on X 2 .

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We have   j,k≥0 Keeping (3.2.6) in mind, we see that the last expression is equal to By properties of integrals with respect to spectral measures, Similarly, This implies that   j,k≥0 It follows that the series in (3.2.5) converges absolutely in the weak operator topology. The above inequalities show that Note that the triple operator integral is well defined by (3.2.5), i.e., the sum of the series in (3.2.5) does not depend on the choice of a representation (3.2.1), see [JTT] and [ANP3].
It is easy to verify that if Ψ is a function that belongs to the projective tensor product L ∞ (E 1 )⊗L ∞ (E 2 )⊗L ∞ (E 3 ), then the above definition coincides with the definition of the triple operator integral given in Chapter 2.
It turns out, however, that unlike in the case when the integrand belongs to the projective tensor product L ∞⊗ L ∞⊗ L ∞ (see Theorem 2.1.3), triple operator integrals with integrands in the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ do not possess the property with p < 2; this will be established in § 3.8. We will see in § 3.3 that for integrands Ψ in We do not know whether this can be true if 1/p + 1/q > 1/2.

Schatten-von Neumann properties
In this section we study Schatten-von Nemann properties of triple operator integrals with integrands in the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ . First, we consider the case when one of the operators is bounded and the other one belongs to the Hilbert-Schmidt class. Then we use an interpolation theorem for bilinear operators to a considerably more general situation.
The following result was established in [ANP1] and its detailed proof was published in [ANP3].
Theorem 3.3.1. Let E 1 , E 2 , and E 3 be spectral measures on Hilbert space and let Φ be a function in the Haagerup tensor product Suppose that T is a bounded linear operator and R is an operator that belongs to the Hilbert-Schmidt class S 2 . Then and It is easy to see that Theorem 3.3.1 implies the following fact: Corollary 3.3.2. Let E 1 , E 2 , E 3 , and Ψ satisfy the hypotheses of Theorem 3.3.1. If T is a Hilbert Schmidt operator and R is a bounded linear operator, then the operator W defined by (3.3.1) belongs to S 2 and Clearly, to deduce Corollary 3.3.2 from Theorem 3.3.1, it suffices to consider the adjoint operator W * .
Proof of Theorem 3.3.1. For simplicity we consider the case when E 3 is a discrete spectral measure and we refer the reader to [ANP3] for the general case. Under this assumption, there exists an orthonormal basis {e m } m≥0 , the spectral measure E 3 is defined on the σ-algebra of all subsets of Z + , and E 3 ({m}) is the orthogonal projection onto the one-dimensional space spanned by e m . In this case the function Ψ has the form We have Here E m is the spectral measure defined on the one point set {m} and the function Ψ m is defined on It is easy to see that It follows now from (3.2.7) that It follows that W ∈ S 2 and inequality (3.3.2) holds.
We are going to use Theorem 4.4.1 from [BL] on complex interpolation of bilinear operators. Recall that the Schatten-von Neumann classes S p , p ≥ 1, and the space of bounded linear operators B(H ) form a complex interpolation scale: This fact is well known. For example, it follows from Theorem 13.1 of Chapter III of [GK].
The following result was established in [ANP1] and its proof was published in [ANP3].
. Then the following holds: (i) if p ≥ 2, T ∈ B(H ), and R ∈ S p , then the triple operator integral in (3.3.1) belongs to S p and (3.3.5) (ii) if p ≥ 2, T ∈ S p , and R ∈ B(H ), then the triple operator integral in (3.3.1) belongs to S p and (iii) if 1/p + 1/q ≤ 1/2, T ∈ S p , and R ∈ S q , then the triple operator integral in (3.3.1) belongs to S r with 1/r = 1/p + 1/q and We will see in § 3.8 that neither (i) nor (ii) holds for p > 2.
Proof of Theorem 3.3.3. Let us first prove (i). Clearly, to deduce (ii) from (i), it suffices to consider W * .
Consider the bilinear operator W defined by By (3.2.7), W maps B(H ) × B(H ) into B(H ) and On the other hand, by Theorem 3.3.1, W maps B(H ) × S 2 into S 2 and It follows from the complex interpolation theorem for linear operators (see [BL], Theorem 4.1.2 that) W maps B(H ) × S p , p ≥ 2, into S p and Suppose now that 1/p + 1/q ≤ 1/2 and 1/r = 1/p + 1/q. It follows from statements (i) and (ii) (which we have already proved) that W maps B(H )×S r into S r and S r ×B(H ) into S r , and It follows from Theorem 4.4.1 of [BL] on interpolation of bilinear operators, W maps It remains to observe that for θ = r/p, which is a consequence of (3.3.4).

Haagerup-like tensor products and triple operator integrals
As we have mentioned in the introduction to this chapter, we are going to use a representation of f (A 1 , B 1 ) − f (A 2 , B 2 ) in terms of triple operator integrals that involve the divided differences D [1] f and D [2] f . However, we will see in § 3.8 that the divided differences D [1] f and D [2] f do not have to belong to the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ for an arbitrary function f in the Besov class B 1 ∞,1 (R 2 ). In addition to this, representation (3.0.6) involves operators of class S p with p ≤ 2. However, we will see in § 3.8 that statements (i) and (ii) of Theorem 3.3.3 do not hold for p < 2.
This means that we need a new approach to triple operator integrals. In this section we introduce Haagerup-like tensor products and define triple operator integrals whose integrands belong to such Haagerup-like tensor products. Note that the Haagerup-like tensor products were defined in [ANP1] and [AP9], see also [ANP3].
, for a bounded linear operator R, and for an operator T of class S p , we define the triple operator integral as the following continuous linear functional on S p ′ , 1/p + 1/p ′ = 1 (on the class of compact operators in the case p = 1): Clearly, the triple operator integral in (3.4.3) is well defined because the function . It follows easily from statement (i) of Theorem 3.3.3 that It is easy to see that in the case when Ψ belongs to the projective tensor product , the definition of the triple operator integral given above is consistent with the definition of the triple operator integral given in Chapter 2. Indeed, it suffices to verify this for functions Ψ of the form in which case the verification is obvious.
We also need triple operator integrals in the case when T is a bounded linear operator and R ∈ S p , 1 ≤ p ≤ 2.
Definition 3. A function Ψ is said to belong to the Haagerup-like tensor product the infimum being taken over all representations of the form (3.4.4).
T is a bounded linear operator, and R ∈ S p , 1 ≤ p ≤ 2. The continuous linear functional on the class S p ′ (on the class of compact operators in the case p = 1) determines an operator W of class S p , which we call the triple operator integral As above, in the case when Ψ ∈ L ∞ (E 1 )⊗L ∞ (E 2 )⊗L ∞ (E 3 ), the definition of the triple operator integral given above is consistent with the definition of the triple operator integral given in Chapter 2.
The following result can easily be deduced from Theorem 3.3.3, see [ANP3].
Theorem 3.4.1. Let Ψ ∈ L ∞ ⊗ h L ∞ ⊗ h L ∞ . Suppose that T ∈ S p and R ∈ S q , where 1 ≤ p ≤ 2 and 1/p + 1/q ≤ 1. Then the operator W in (3.4.2) belongs to S r , 1/r = 1/p + 1/q, and (3.4.6) If T ∈ S p , 1 ≤ p ≤ 2, and R is a bounded linear operator, then W ∈ S p and (3.4.7) In the same way we can prove the following theorem: Suppose that p ≥ 1, 1 ≤ q ≤ 2, and 1/p + 1/q ≤ 1. If T ∈ S p , R ∈ S q , then the operator W in (3.4.5) belongs to S r , 1/r = 1/p + 1/q, and If T is a bounded linear operator and R ∈ S p , 1 ≤ p ≤ 2, then W ∈ S p and 3.5. Conditions for D [1] f and D [2] f to be in Haagerup-like tensor products As we have already mentioned before, for functions f in B 1 ∞,1 (R 2 ), the divided differences D [1] f and D [2] f , do not have to belong to the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ . This will be seen in § 3.8. In this section we will see that for f ∈ B 1 ∞,1 (R 2 ), the divided difference D [1] f belongs to the tensor product L ∞ (E 1 )⊗ h L ∞ (E 2 )⊗ h L ∞ (E 3 ), while the divided difference D [2] f belongs to the tensor product L ∞ (E 1 )⊗ h L ∞ (E 2 )⊗ h L ∞ (E 3 ) for arbitrary Borel spectral measures E 1 , E 2 , and E 3 on R.
On the other hand, by Bernstein's inequality. This completes the proof of (3.5.3).
We refer the reader to [ANP3] for details.
The following result can be deduced easily from Theorem 3.5.1, see [ANP3].
Corollary 3.5.2. Let f be a bounded function on R 2 such that its Fourier transform is supported in {ξ ∈ R 2 : ξ ≤ σ}, σ > 0. Then the divided differences D [1] f and D [2] f have the following properties: for arbitrary Borel spectral measures E 1 , E 2 and E 3 . Moreover, (3.5.5) Corollary 3.5.2 implies in turn the following theorem that was established in [AP3].
Theorem 3.5.3. Let f ∈ B 1 ∞,1 (R 2 ). Then for arbitrary Borel spectral measures E 1 , E 2 and E 3 . Moreover, 3.6. Lipschitz type estimates in the case 1 ≤ p ≤ 2 In this section we discuss the results announced in [ANP1] and [ANP2] whose detailed proofs were given in [ANP3].
We will see in this section that for functions f in the Besov class B 1 ∞,1 (R 2 ), we have a Lipschitz type estimate for functions of noncommuting self-adjoint operators in the norm of S p with p ∈ [1, 2].
The following integral formula plays an important role.

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Then the following identity holds: (3.6.1) Note that by Theorem 3.5.3, the divided differences D [1] f and D [2] f belong to the corresponding Haagerup like tensor products, and so the triple operator integrals on the right make sense.
Proof. It suffices to prove that (3.6.3) Let us establish (3.6.2). Formula (3.6.3) can be proved in exactly the same way. Suppose first that the function D [1] f belongs to the projective tensor product L ∞ (E A 1 )⊗L ∞ (E A 2 )⊗L ∞ (E B 1 ). In this case we can write Note that the above equality does not make sense if D [1] f does not belong to L ∞⊗ L ∞⊗ L ∞ because the operators A 1 and A 2 do not have to be compact, while the definition of triple operator integrals with integrands in the Haagerup-like tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ assumes that the operators A 1 and A 2 belong to S 2 . It follows immediately from the definition of triple operator integrals with integrands in L ∞⊗ L ∞⊗ L ∞ that Thus Consider the functions f n defined by f n = f * W n , n ∈ Z, see (2.2). It is easy to see from the definition of the Besov class B 1 ∞,1 (R 2 ) that to prove (3.6.2), it suffices to show that f n (A 1 , B 1 ) − f n (A 2 , B 1 ) = D [1] f n (x 1 , x 2 , y) dE A 1 (x 1 )(A 1 − A 2 ) dE A 2 (x 2 ) dE B 1 (y).
As we have mentioned in § 2, the function f n is a restriction of an entire function of two variables to R × R. Thus it suffices to establish formula (3.6.2) in the case when f is an entire function. To complete the proof, we show that for entire functions f the divided differences D [1] f must belong to the projective tensor product a jk x j y k be an entire function and let R be a positive number such that the spectra σ(A 1 ), σ(A 2 ), and σ(B) are contained in [−R/2, R/2]. Clearly, where in the above expressions L ∞ means L ∞ [− R, R]. This completes the proof.
Proof. Let us first prove (ii). Let {g j } 1≤j≤N and {h j } 1≤j≤N be orthonormal systems in Hilbert space. Consider the rank one projections P j and Q j defined by P j v = (v, g j )g j and Q j v = (v, h j )h j , 1 ≤ j ≤ N.
We define the self-adjoint operators A 1 , A 2 , and B by A 1 = N j=1 2jP j , A 2 = N j=1 (2j + 1)P j , and B = N k=1 k Q k .
Note that (f (A 1 , B)h k , g j ) = τ jk (h k , g j ), 1 ≤ j, k ≤ N. Clearly, for every unitary matrix {u jk } 1≤j,k≤N , there exist orthonormal systems {g j } 1≤j≤N and {h j } 1≤j≤N such that (h k , g j ) = u jk . Put Obviously, {u jk } 1≤j,k≤N is a unitary matrix. Hence, we may find vectors {g j } N j=1 and {h j } N j=1 such that (h k , g j ) = u jk . Put τ jk = √ N u jk . Then Exactly the same construction works to prove (i). It suffices to replace in the above construction the S p norm with the operator norm and observe that A 1 − A 2 = 1 and f (A 1 , B) − f (A 2 , B) = √ N . Theorem 3.7.1 implies that there is no Lipschitz type estimate in the operator norm and in the S p norm with p > 2. Note that in the construction given in the proof the norms