Functions of perturbed pairs of noncommuting contractions
aa r X i v : . [ m a t h . F A ] A ug FUNCTIONS OF PERTURBED PAIRS OF NONCOMMUTINGCONTRACTIONS
A.B. ALEKSANDROV AND V.V. PELLER
Abstract.
We consider functions f ( T, R ) of pairs of noncommuting contractions onHilbert space and study the problem for which functions f we have Lipschitz typeestimates in Schatten–von Neumann norms. We prove that if f belongs to the Besovclass (cid:0) B ∞ , (cid:1) + ( T ) of nalytic functions in the bidisk, then we have a Lipschitz typeestimate for functions f ( T, R ) of pairs of not necessarily commuting contractions (
T, R )in the Schatten–von Neumann norms S p for p ∈ [1 , (cid:0) B ∞ , (cid:1) + ( T ) there are no Lipschitz such type estimates for p > Contents1. Introduction
2. Besov classes of periodic functions
3. Double and triple operator integralswith respect to semi-spectral measures
4. Lipschitz type estimates in Schatten–von Neumann norms
5. A representation of operator differencesin terms of triple operator integrals
6. Differentiability properties
7. The case p >
8. Open problems References Introduction
The purpose of this paper is to study the behavior of functions f ( T, R ) of (not nec-essarily commuting) contractions T and R under perturbation. We are going to obtainLipschitz type estimates in the Sachatten–von Neumann norms S p , 1 ≤ p ≤
2, for func-tions f in the Besov class (cid:0) B ∞ , (cid:1) + ( T ) of analytic functions. Note that functions f ( T, R )of noncommuting contractions can be defined in terms of double operator integrals withrespect to semi-spectral measures, see § The research of the first author is supposed by RFBR grant 17-01-00607. The publication was preparedwith the support of the RUDN University Program 5-100.Corresponding author: V.V. Peller; email: [email protected]. his paper can be considered as a continuation of the results of [Pe1]–[Pe7], [AP1]–[AP4], [AP6], [APPS], [NP], [ANP], [PS] and [KPSS] for functions of perturbed self-adjoint operators, contractions, normal operators, dissipative operators, functions ofcollections of commuting operators and functions of collections of noncommuting oper-ators.Recall that a Lipschitz function f on R does not have to be operator Lipschitz , i.e.,the condition | f ( x ) − f ( y ) | ≤ const | x − y | , x, y ∈ R , does not imply that k f ( A ) − f ( B ) k ≤ const k A − B k for arbitrary self-adjoint operators (bounded or unbounded, does not matter) A and B .This was first established in [F].It turned out that functions in the (homogeneous) Besov space B ∞ , ( R ) are operatorLipschitz; this was established in [Pe1] and [Pe3] (see [Pee] for detailed information aboutBesov classes). We refer the reader to the recent survey [AP4] for detailed information onoperator Lipschitz functions. In particular, [AP4] presents various sufficient conditionsand necessary conditions for a function on R to be operator Lipschitz. It is well knownthat if f is an operator Lipschitz function on R , and A and B are self-adjoint operatorssuch that the difference A − B belongs to the Schatten–von Neumann class S p , 1 ≤ p < ∞ , then f ( A ) − f ( B ) ∈ S p and k f ( A ) − f ( B ) k S p ≤ const k A − B k S p . Moreover, theconstant on the right does not depend on p . In particular, this is true for functions f inthe Besov class B ∞ , ( R ), i.e., k f ( A ) − f ( B ) k S p ≤ const k f k B ∞ , k A − B k S p , ≤ p ≤ ∞ . (1.1)However, it was discovered in [AP1] (see also [FN]) that the situation becomes quitedifferent if we replace the class of Lipschitz functions with the class Λ α ( R ) of H¨olderfunctions of order α , 0 < α <
1. Namely, the inequality | f ( x ) − f ( y ) | ≤ const | x − y | α , x, y ∈ R , implies that k f ( A ) − f ( B ) k ≤ const k A − B k α for arbitrary self-adjoint operators A and B . Moreover, it was shown in [AP2] that if A − B ∈ S p , p >
1, and f ∈ Λ a ( R ), then f ( A ) − f ( B ) ∈ S p/α and k f ( A ) − f ( B ) k S p/a ≤ const k A − B k α S p for arbitrary self-adjoint operators A and B .Analogs of the above results for functions of normal operators, functions of contrac-tions, functions of dissipative operators and functions of commuting collections of self-adjoint operators were obtained in [Pe2], [AP3], [APPS], [NP].Note that it was shown in [PS] that for p ∈ (1 , ∞ ), inequality (1.1) holds for arbitrary Lipschitz (not necessarily operator Lipschitz) functions f with constant on the right thatdepends on p . An analog of this result for functions of commuting self-adjoint operatorswas obtained in [KPSS].In [ANP] similar problems were considered for functions of two noncommuting self-adjoint operators (such functions can be defined in terms of double operator integrals,see [ANP]). It was shown in [ANP] that for functions f on R in the (homogeneous) esov class B ∞ , ( R ) and for p ∈ [1 , k f ( A , B ) − f ( A , B ) k S p ≤ const max (cid:8) k A − A k S p , k B − B k S p (cid:9) for arbitrary pairs ( A , B ) and ( A , B ) of (not necessarily commuting) self-adjointoperators.However, it was shown in [ANP] that for p > S p norm as well as in the operator norm. Moreover, it follows from theconstruction given in [ANP] that for p ∈ (2 , ∞ ] and for positive numbers ε, σ, M , thereexists a function f in L ∞ ( R ) with Fourier transform supported in [ − σ, σ ] × [ − σ, σ ] suchthat max (cid:8) k A − A k S p , k B − B k S p (cid:9) < ε while k f ( A , B ) − f ( A , B ) k S p > M. Here we use the notation k · k S ∞ for operator norm.This implies that unlike in the case of commuting operators, there cannot be anyH¨older type estimates in the norm of S p , p >
2, for H¨older functions f of order α .Moreover, for p >
2, there cannot be any estimate for k f ( A , B ) − f ( A , B ) k S p forfunctions in the Besov class B s ∞ ,q ( R ) for any q > s > k f ( A , B ) − f ( A , B ) k S for Lipschitz functions f on R , see [ANP].Finally, let us mention that in the case of functions of triples of noncommuting opera-tors there are no such Lipschitz type estimates for functions in the Besov class B ∞ , ( R )in the norm of S p for any p ∈ [1 , ∞ ]. This was established in [Pe7].In § f ( T, R ) of noncommuting contractions. We define the Haagerup and Haagerup-like tensor products of three copies of the disk-algebra C A and we define triple operatorintegrals whose integrands belong to such tensor products.Lipschitz type estimates in Schatten–von Neumann norm will be obtained in § p ∈ [1 ,
2] and for a function f on T in the analytic Besov space (cid:0) B ∞ , (cid:1) + ( T ), the following Lipschitz type inequality holds: (cid:13)(cid:13) f ( T , R ) − f ( T , R ) (cid:13)(cid:13) S p ≤ const max (cid:8) k T − T k S p , k R − R k S p (cid:9) for arbitrary pairs ( T , T ) and ( R , R ) of contractions. Recall that similar inequalitywas established in [ANP] for functions of self-adjoint operators. However, to obtain thisinequality for functions of contractions, we need new algebraic formulae. Moreover, toobtain this inequality for functions of contractions, we offer an approach that does notuse triple operator integrals. To be more precise, we reduce the inequality to the caseof analytic polynomials f and we integrate over finite sets, in which case triple opera-tor integrals become finite sums. We establish explicit representations of the operatordifferences f ( T , R ) − f ( T , R ) for analytic polynomials f in terms of finite sums ofelementary tensors which allows us to estimate the S p norms.However, we still use triple operator integrals to obtain in § f in (cid:0) B ∞ , (cid:1) + ( T ). n § t f (cid:0) T ( t ) , R ( t ) (cid:1) for f in (cid:0) B ∞ , (cid:1) + ( T ) and contractive valued functions t T ( t ) and t R ( t ). Weobtain explicit formulae for the derivative in terms of triple operator integrals. Again,to prove the existence of the derivative, we do not need triple operator integrals.As in the case of functions of pairs self-adjoint operators (see [ANP]), there are noLipschitz type estimates in the norm of S p , p >
2, for functions of pairs of not necessarilycommuting contractions f ( T, R ), f ∈ (cid:0) B ∞ , (cid:1) + ( T ). This will be established in §
7. Notethat the construction differs from the construction in the case of self-adjoint operatorsgiven in [ANP].In § § m for normalized Lebesgue measure on the unit circle T and thenotation m for normalized Lebesgue measure on T .For simplicity we assume that we deal with separable Hilbert spaces.2. Besov classes of periodic functions
In this section we give a brief introduction to Besov spaces on the torus.To define Besov spaces on the torus T d , we consider an infinitely differentiable function w on R such that w ≥ , supp w ⊂ (cid:20) , (cid:21) , and w ( s ) = 1 − w (cid:16) s (cid:17) for s ∈ [1 , . Let W n , n ≥
0, be the trigonometric polynomials defined by W n ( ζ ) def = X j ∈ Z d w (cid:18) | j | n (cid:19) ζ j , n ≥ , W ( ζ ) def = X { j : | j |≤ } ζ j , where ζ = ( ζ , · · · , ζ d ) ∈ T d , j = ( j , · · · , j d ) , and | j | = (cid:0) | j | + · · · + | j d | (cid:1) / . For a distribution f on T d we put f n = f ∗ W n , n ≥ . (2.1)It is easy to see that f = X n ≥ f n ; (2.2)the series converges in the sense of distributions. We say that f belongs the Besov class B sp,q ( T d ), s >
0, 1 ≤ p, q ≤ ∞ , if (cid:8) ns k f n k L p (cid:9) n ≥ ∈ ℓ q . (2.3) he analytic subspace (cid:0) B sp,q (cid:1) + ( T d ) of B sp,q ( T d ) consists of functions f in B sp,q ( T d ) forwhich the Fourier coefficients b f ( j , · · · , j d ) satisfy the equalities: b f ( j , · · · , j d ) = 0 whenever min ≤ k ≤ d j k < . (2.4)We refer the reader to [Pee] for more detailed information about Besov spaces.3. Double and triple operator integralswith respect to semi-spectral measures3.1. Double operator integrals.
In this section we give a brief introduction todouble and triple operator integrals with respect to semi-spectral measures. Doubleoperator integrals with respect to spectral measures are expressions of the form
Z Z Φ( x, y ) dE ( x ) Q dE ( y ) , (3.1)where E and E are spectral measures, Q is a linear operator and Φ is a bounded mea-surable function. They appeared first in [DK]. Later Birman and Solomyak developedin [BS1]–[BS3] a beautiful theory of double operator integrals.Double operator integrals with respect to semi-spectral measures were defined in [Pe2],see also [AP4] (recall that the definition of a semi-spectral measure differs from thedefinition of a spectral measure by replacing the condition that it takes values in the setof orthogonal projections with the condition that it takes values in the set of nonnegativecontractions, see [AP4] for more detail).For the double operator integral to make sense for an arbitrary bounded linear operator T , we have to impose an additional assumption on Φ. The natural class of such functionsΦ is called the class of Schur multipliers , see [Pe1]. There are various characterizationsof the class of Schur multipliers. In particular, Φ is a Schur multiplier if and only if itbelongs to the Haagerup tensor product L ∞ ( E ) ⊗ h L ∞ ( E ) of L ∞ ( E ) and L ∞ ( E ), i.e.,it admits a representation of the formΦ( x, y ) = X j ϕ j ( x ) ψ j ( y ) , (3.2)where the ϕ j and ψ j satisfy the condition X j | ϕ j | ∈ L ∞ ( E ) and X j | ψ j | ∈ L ∞ ( E ) . (3.3)In this case Z Z Φ( x, y ) dE ( x ) Q dE ( y ) = X j (cid:16) Z ϕ j dE (cid:17) Q (cid:16) Z ψ j dE (cid:17) ; (3.4)the series converges in the weak operator topology. The right-hand side of this equalitydoes not depend on the choice of a representation of Φ in (3.2). ne can also consider double operator integrals of the form (3.1) in the case when E and E are semi-spectral measures . In this case, as in the case of spectral measures,formula (3.4) still holds under the same assumption (3.3).It is easy to see that if Φ belongs to the projective tensor product L ∞ ( E ) b ⊗ L ∞ ( E )of L ∞ ( E ) and L ∞ ( E ), i.e., Φ admits a representation of the form (3.2) with ϕ j and ψ j satisfying X j k ϕ j k L ∞ ( E ) k ψ j k L ∞ ( E ) < ∞ , then Φ is a Schur multiplier and (3.4) holds. Recall that if T is a contrac-tion (i.e., k T k ≤
1) on a Hilbert space H , then by the Sz.-Nagy dilation theorem (see[SNF]), T has a unitary dilation, i.e., there exist a Hilbert space K that contains H and a unitary operator U on K such that T n = P H U n (cid:12)(cid:12) H , n ≥ , where P H is the orthogonal projection onto H .Among all unitary dilations of T one can always select a minimal unitary dilation (ina natural sense) and all minimal unitary dilations are isomorphic, see [SNF].The existence of a unitary dilation allows us to construct the natural functional cal-culus f f ( T ) for functions f in the disk-algebra C A defined by f ( T ) = P H f ( U ) (cid:12)(cid:12) H = P H (cid:18)Z T f ( ζ ) dE U ( ζ ) (cid:19) (cid:12)(cid:12)(cid:12) H , f ∈ C A . where E U is the spectral measure of U .Consider the operator set function E T defined on the Borel subsets of the unit circle T by E T (∆) = P H E U (∆) (cid:12)(cid:12) H , ∆ ⊂ T . Then E T is a semi-spectral measure . It can be shown that it does not depend on thechoice of a unitary dilation. The semi-spectral measure E T is called the semi-spectralmeasure of T . Let f be a function on the torus T that belongs to the Haagerup tensor product C A ⊗ h C A , i.e., f admits a representationof the form f ( ζ, τ ) = X j ϕ j ( ζ ) ψ j ( τ ) , ζ, τ ∈ T , where ϕ j , ψ j are functions in C A such thatsup ζ ∈ T X j | ϕ j ( ζ ) | < ∞ and sup τ ∈ T X j | ψ j ( τ ) | < ∞ . For a pair (
T, R ) of (not necessarily commuting contractions), the operator f ( T, R ) isdefined as the double operator integral
Z Z T × T f ( ζ, τ ) d E T ( ζ ) d E R ( τ ) = Z Z T × T f ( ζ, τ ) d E T ( ζ ) I d E R ( τ ) . ote that if f ∈ (cid:0) B ∞ , (cid:1) + ( T ), then f ∈ C A ⊗ h C A , and so we can take functions f ( T, R ) of contractions for an arbitrary function f in (cid:0) B ∞ , (cid:1) + ( T ). Indeed, if f is ananalytic polynomial in two variables of degree at most N in each variable, then we canrepresent f in the form f ( ζ, τ ) = N X j =0 ζ j N X k =0 b f ( j, k ) τ k ! . Thus f belongs to the projective tensor product C A ˆ ⊗ C A and k f k C A ˆ ⊗ C A ≤ N X j =0 sup τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =0 b f ( j, k ) τ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 + N ) k f k L ∞ (3.5)It follows easily from (2.3) that every function f of Besov class (cid:0) B ∞ , (cid:1) + ( T ) belongs toC A ˆ ⊗ C A , and so the operator f ( T, R ) is well defined. Clearly, f ( T, R ) = X n ≥ n +1 X j =0 T j n +1 X k =0 b f n ( j, k ) R k , (3.6)where f n is the polynomial defined by (2.1). It follows immediately from (3.5) and (2.3)that the series converges absolutely in the operator norm. Note that formula (3.6) canbe used as a definition of the functions f ( T, R ) of noncommuting contractions in thecase when f ∈ (cid:0) B ∞ , (cid:1) + ( T ). There are severalapproaches to multiple operator integrals. Triple operator integrals are expressions ofthe form W Φ def = Z Z Z Φ( x, y, z ) dE ( x ) X dE ( y ) Y dE ( z ) , where Φ is a bounded measurable function, E , E and E are spectral measures, and X and Y are bounded linear operators on Hilbert space.In [Pe4] triple (and more general, multiple) operator integrals were defined for func-tions Φ in the integral projective product L ∞ ( E ) ⊗ i L ∞ ( E ) ⊗ i L ∞ ( E ). For such func-tions Φ, the following Schatten–von Neumann properties hold: (cid:13)(cid:13)(cid:13)(cid:13)Z Z Z Φ dE X dE Y dE (cid:13)(cid:13)(cid:13)(cid:13) S r ≤ k Φ k L ∞ ⊗ i L ∞ ⊗ i L ∞ k X k S p k Y k S q , r = 1 p + 1 q , whenever 1 /p + 1 /q ≤
1. Later in [JTT] triple (and multiple) operator integrals weredefined for functions Φ in the Haagerup tensor product L ∞ ( E ) ⊗ h L ∞ ( E ) ⊗ h L ∞ ( E ).However, it turns out that under the assumption Φ ∈ L ∞ ⊗ h L ∞ ⊗ h L ∞ , the conditions X ∈ S p and Y ∈ S q imply that RRR Φ dE X dE Y dE ∈ S r , 1 /r = 1 /p + 1 /q , onlyunder the conditions that p ≥ q ≥
2, see [AP5] (see also [ANP]). Moreover, thefollowing inequality holds: (cid:13)(cid:13)(cid:13)(cid:13)Z Z Z Φ dE X dE Y dE (cid:13)(cid:13)(cid:13)(cid:13) S r ≤ k Φ k L ∞ ⊗ h L ∞ ⊗ h L ∞ k X k S p k Y k S q , r = 1 p + 1 q , henever p ≥ q ≥
2, see [AP5].Note also that to obtain Lipschitz type estimates for functions of noncommuting self-adjoint operators in [ANP], we had to use triple operator integrals with integrands Φthat do not belong to the Haagerup tensor product L ∞ ⊗ h L ∞ ⊗ h L ∞ . That is why we hadto introduce in [ANP] Haagerup-like tensor products of the first kind and of the secondkind.In this paper we are going to use triple operator integrals with integrands beingcontinuous functions on T that belong to Haagerup and Haagerup-like tensor productsof three copies of the disk-algebra C A . We briefly define such tensor products and discussinequalities we are going to use in the next section. Definition 1.
We say that a continuous function Φ on T belongs to the Haageruptensor product C A ⊗ h C A ⊗ h C A if Φ admits a representationΦ( ζ, τ, κ ) = X j,k ≥ α j ( ζ ) β jk ( τ ) γ k ( κ ) , ζ, τ, κ ∈ T , (3.7)where α j , β jk and γ k are functions in C A such thatsup ζ ∈ T X j ≥ | α j ( ζ ) | / sup τ ∈ T (cid:13)(cid:13) { β jk ( τ ) } j,k ≥ (cid:13)(cid:13) B sup κ ∈ T X k ≥ | γ k ( κ ) | / < ∞ . (3.8)Here k · k B stands for the operator norm of a matrix (finite or infinite) on the space ℓ oron a finite-dimensional Euclidean space. By definition, the norm of Φ in C A ⊗ h C A ⊗ h C A is the infimum of the left-hand side of (3.8) over all representations of Φ in the form of(3.7).Suppose that Φ ∈ C A ⊗ h C A ⊗ h C A and both (3.7) and (3.8) hold. Let T , T and T be contractions with semi-spectral measures E T , E T and E T . Then for bounded linearoperators X and Y , we can define the triple operator integral W Φ = Z Z Z Φ d E T X d E T Y d E T (3.9)as W Φ def = X j,k (cid:16) Z α j ( ζ ) d E T ( ζ ) (cid:17) X (cid:16) Z β jk ( τ ) d E T ( τ ) (cid:17) Y (cid:16) Z γ k ( κ ) d E T ( κ ) (cid:17) = X j,k α j ( T ) Xβ jk ( T ) Y γ k ( T ) . It is easy to verify that the series converges in the weak operator topology if we considerpartial sums over rectangles. It can be shown in the same way as in the case of tripleoperator integrals with respect to spectral measures that the sum on the right does notdepend on the choice of a representation of Φ in the form of (3.7), see Theorem 3.1 of[ANP]. e are going to use Lemma 3.2 of [AP5]. Suppose that { Z j } j ≥ is a sequence ofbounded linear operators on Hilbert space such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ Z ∗ j Z j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / ≤ M and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X j ≥ Z j Z ∗ j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) / ≤ M. (3.10)Let Q be a bounded linear operator. Consider the row R { Z j } ( Q ) and the column C { Z j } ( Q )defined by R { Z j } ( Q ) def = (cid:0) Z Q Z Q Z Q · · · (cid:1) and C { Z j } ( Q ) def = QZ QZ QZ ... . Then by Lemma 3.2 of [AP5], for p ∈ [2 , ∞ ], the following inequalities hold: (cid:13)(cid:13) R { Z j } ( Q ) (cid:13)(cid:13) S p ≤ M k Q k S p and (cid:13)(cid:13) C { Z j } ( Q ) (cid:13)(cid:13) S p ≤ M k Q k S p (3.11)whenever Q ∈ S p .It is easy to verify that under the above assumptions W Φ = R { α j ( T ) } ( X ) B C { γ j ( T ) } ( Y ) , (3.12)where B is the operator matrix { β jk ( T ) } j,k ≥ . Lemma 3.1.
Under the above hypotheses, k B k ≤ sup τ ∈ T (cid:13)(cid:13) { β jk ( τ ) } j,k ≥ (cid:13)(cid:13) B . Proof.
Let U be a unitary dilation of the contraction T on a Hilbert space K , K ⊃ H . Clearly, we can consider the space ℓ ( H ) as a subspace of ℓ ( K ). It is easyto see that { β jk ( T ) } j,k ≥ = P ℓ ( H ) { β jk ( U ) } j,k ≥ (cid:12)(cid:12) ℓ ( H ) , where P ℓ ( H ) is the orthogonal projection onto ℓ ( H ). The result follows from theinequality k{ β jk ( U ) } j,k ≥ k ≤ sup τ ∈ T (cid:13)(cid:13) { β jk ( τ ) } j,k ≥ (cid:13)(cid:13) B , which is a consequence of thespectral theorem. (cid:4) It follows from Lemma 3.2 of [Pe8] that under the above assumptions, inequalities(3.10) hold for Z j = α j ( T ), j ≥
0, with M = sup ζ ∈ T (cid:16)P j ≥ | α j ( ζ ) | (cid:17) / and for Z j = γ j ( T ), j ≥
0, with M = sup ζ ∈ T (cid:16)P j ≥ | γ j ( ζ ) | (cid:17) / . This together with Lemma (3.1) nd inequalities (3.11) implies that under the above assumptions, (cid:13)(cid:13) R { α j ( T ) } ( X ) B C { γ j ( T ) } ( Y ) (cid:13)(cid:13) S r ≤ sup ζ ∈ T X j ≥ | α j ( ζ ) | / sup τ ∈ T (cid:13)(cid:13) { β jk ( τ ) } j,k ≥ (cid:13)(cid:13) B sup κ ∈ T X k ≥ | γ k ( κ ) | / (3.13)whenever p ≥ q ≥ /r = 1 /p + 1 /q .The following theorem is an analog of the corresponding result for triple operatorintegrals with respect to spectral measures, see [AP5]. It follows immediately from(3.13). Theorem 3.2.
Let T , T and T be contractions, and let X ∈ S p and Y ∈ S q , ≤ p ≤ ∞ , ≤ q ≤ ∞ . Suppose that Φ ∈ C A ⊗ h C A ⊗ h C A . Then W Φ ∈ S r , /r = 1 /p + 1 /q , and (cid:13)(cid:13)(cid:13)(cid:13)Z Z Z Φ d E T X d E T Y d E T (cid:13)(cid:13)(cid:13)(cid:13) S r ≤ k Φ k C A ⊗ h C A ⊗ h C A k X k S p k Y k S q . Recall that by S ∞ we mean the class of bounded linear operators. We define here Haagerup-like tensor prod-ucts of disk-algebras by analogy with Haagerup-like tensor products of L ∞ spaces, see[ANP]. Definition 2.
A continuous function Φ on T is said to belong to the Haagerup-liketensor product C A ⊗ h C A ⊗ h C A of the first kind if it admits a representationΦ( ζ, τ, κ ) = X j,k ≥ α j ( ζ ) β k ( τ ) γ jk ( κ ) , ζ, τ, κ ∈ T , (3.14)where α j , β k and γ jk are functions in C A such thatsup ζ ∈ T X j ≥ | α j ( ζ ) | / sup τ ∈ T X k ≥ | β k ( τ ) | / sup κ ∈ T (cid:13)(cid:13) { γ jk ( κ ) } j,k ≥ (cid:13)(cid:13) B < ∞ . Clearly, Φ ∈ C A ⊗ h C A ⊗ h C A if and only if the function( z , z , z ) Φ( z , z , z )belongs to the Haagerup tensor product C A ⊗ h C A ⊗ h C A .Similarly, we can define the Haagerup-like tensor product C A ⊗ h C A ⊗ h C A of the secondkind. Definition 3.
A continuous function Φ on T is said to belong to the Haagerup-liketensor product C A ⊗ h C A ⊗ h C A of the second kind if it admits a representationΦ( ζ, τ, κ ) = X j,k ≥ α jk ( ζ ) β j ( τ ) γ k ( κ ) , ζ, τ, κ ∈ T , (3.15) here α jk , β j and γ k are functions in C A such thatsup ζ ∈ T (cid:13)(cid:13) { α jk ( ζ ) } j,k ≥ (cid:13)(cid:13) B sup τ ∈ T X j ≥ | β j ( τ ) | / sup κ ∈ T X k ≥ | γ k ( κ ) | / < ∞ . Let us first consider the situation when Φ is defined by (3.14) or by (3.15) withsummation over a finite set. In this case triple operator integrals of the form (3.9) canbe defined for arbitrary bounded linear operators X and Y and for arbitrary contractions T , T and T .Suppose thatΦ( ζ, τ, κ ) = X j ∈ F X k ∈ F α j ( ζ ) β k ( τ ) γ jk ( κ ) , ζ, τ, κ ∈ T , α j , β k , γ jk ∈ C A , (3.16)where F and F are finite sets. We put Z Z Z Φ d E T X d E T Y d E T def = X j ∈ F X k ∈ F α j ( T ) Xβ k ( T ) Y γ jk ( T ) . (3.17)Suppose now thatΦ( ζ, τ, κ ) = X j ∈ F X k ∈ F α jk ( ζ ) β j ( τ ) γ k ( κ ) , ζ, τ, κ ∈ T , α jk , β j , γ k ∈ C A , (3.18)where F and F are finite sets. Then we put Z Z Z Φ d E T X d E T Y d E T def = X j ∈ F X k ∈ F α jk ( T ) Xβ j ( T ) Y γ k ( T ) . (3.19)The following estimate is a very special case of Theorem 3.4 below. However, we havestated it separately because its proof is elementary and does not require the definitionof triple operator integrals with integrands in Haagerup-like tensor products. Theorem 3.3.
Let X and Y be bounded linear operators and let T , T and T arecontractions. Suppose that F and F are finite sets. The following statements hold: (i) Let Φ be given by (3.16) . Suppose that q ≥ and /r def = 1 /p + 1 /q ∈ [1 / , . If X ∈ S p and Y ∈ S q , then the sum on the right of (3.17) belongs to S r and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈ F X k ∈ F α j ( T ) Xβ k ( T ) Y γ jk ( T ) ∈ S r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S r ≤ sup ζ ∈ T X j ∈ F | α j ( ζ ) | / sup τ ∈ T X k ∈ F | β k ( τ ) | / sup κ ∈ T (cid:13)(cid:13)(cid:8) γ jk ( κ ) (cid:9) j ∈ F ,k ∈ F (cid:13)(cid:13) B k X k S p k Y k S q . ii) Let Φ be given by (3.18) . Suppose that q ≥ and /r def = 1 /p + 1 /q ∈ [1 / , . If X ∈ S p and Y ∈ S q , then the sum on the right of (3.19) belongs to S r and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈ F X k ∈ F α jk ( T ) Xβ j ( T ) Y γ k ( T ) ∈ S r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S r ≤ sup ζ ∈ T (cid:13)(cid:13)(cid:8) α jk ( ζ ) (cid:9) j ∈ F ,k ∈ F (cid:13)(cid:13) B sup τ ∈ T X j ∈ F | β j ( τ ) | / sup κ ∈ T X k ∈ F | γ k ( κ ) | / k X k S p k Y k S q . Proof.
Let us prove (i). The proof of (ii) is the same. We are going to use a dualityargument. Suppose that Q ∈ S r ′ and k Q k S r ′ ≤
1, 1 /r + 1 /r ′ = 1. We havesup Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) trace Q X j ∈ F X k ∈ F α jk ( T ) Xβ j ( T ) Y γ k ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup Q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) trace X j ∈ F X k ∈ F γ k ( T ) Qα jk ( T ) Xβ j ( T ) Y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k Y k S q sup Q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X j ∈ F X k ∈ F γ k ( T ) Qα jk ( T ) Xβ j ( T ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S q ′ . Th result follows now from (3.12) and (3.13). (cid:4)
We define triple operator integrals with integrands in C A ⊗ h C A ⊗ h C A by analogywith triple operator integrals with respect to spectral measures, see [ANP] and [AP5].Let Φ ∈ C A ⊗ h C A ⊗ h C A and let p ∈ [1 , T , T and T are contractions.For an operator X of class S p and for a bounded linear operator Y , we define the tripleoperator integral W Φdef = Z Z Z Φ( ζ, τ, κ ) d E T ( ζ ) X d E T ( τ ) Y d E T ( κ ) (3.20)as the following continuous linear functional on S p ′ , 1 /p + 1 /p ′ = 1 (on the class ofcompact operators in the case p = 1): Q trace (cid:18)(cid:18)Z Z Z Φ( ζ, τ, κ ) dE T ( τ ) Y dE T ( κ ) Q dE T ( ζ ) (cid:19) X (cid:19) . Note that the triple operator integral
RRR Φ( ζ, τ, κ ) dE T ( τ ) Y dE T ( κ ) Q dE T ( ζ ) is welldefined as the integrand belongs to the Haagerup tensor product C A ⊗ h C A ⊗ h C A .Again, we can define triple operator integrals with integrands in C A ⊗ h C A ⊗ h C A byanalogy with the case of spectral measures, see [ANP] and [AP5]. Let Φ ∈ C A ⊗ h C A ⊗ h C A12 nd let T , T and T be contractions. Suppose that X is a bounded linear operator and Y ∈ S p , 1 ≤ p ≤
2. The triple operator integral W Φdef = Z Z Z Φ( ζ, τ, κ ) d E T ( ζ ) X d E T ( τ ) Y d E T ( κ ) (3.21)is defined as the continuous linear functional Q trace (cid:18)(cid:18)Z Z Z Φ( ζ, τ, κ ) dE ( κ ) Q dE ( ζ ) X dE ( τ ) (cid:19) Y (cid:19) on S p ′ (on the class of compact operators if p = 1).As in the case of spectral measures (see [AP5]), the following theorem can be proved: Theorem 3.4.
Suppose that T , T and T are contractions, and let X ∈ S p and Y ∈ S q . The following statements hold: (1) Let Φ ∈ C A ⊗ h C A ⊗ h C A . Suppose q ≥ and /r def = 1 /p + 1 /q ∈ [1 / , . If X ∈ S p and Y ∈ S q , then the operator W Φ in (3.20) belongs to S r and (cid:13)(cid:13)(cid:13)(cid:13) W Φ (cid:13)(cid:13)(cid:13)(cid:13) S r ≤ k Φ k C A ⊗ h C A ⊗ h C A k X k S p k Y k S q ; (3.22)(2) Let Φ ∈ C A ⊗ h C A ⊗ h C A . Suppose that p ≥ and /r def = 1 /p + 1 /q ∈ [1 / , . If X ∈ S p and Y ∈ S q , then the operator W Φ in (3.21) belongs to S r and (cid:13)(cid:13)(cid:13) W Φ (cid:13)(cid:13)(cid:13) S r ≤ k Φ k C A ⊗ h C A ⊗ h C A k X k S p k Y k S q . Lipschitz type estimates in Schatten–von Neumann norms
In this section we obtain Lipschitz type estimates in the Schatten–von Neumann classes S p for p ∈ [1 ,
2] for functions of contractions. To obtain such estimates, we are goingto use an elementary approach and obtain elementary formulae that involve only finitesums.Later we will need explicit expressions for operator differences, which will be obtainedin the next section in terms of triple operator integrals. Such formulae will be used in § f is a function that belongs to the Besov space (cid:0) B ∞ , (cid:1) + ( T ) of ana-lytic functions (see § f ( T, R ) for (not necessarily commuting) contractions T and R on Hilbert space by for-mula (3.6).For a differentiable function f on T , we use the notation D f for the divided difference:( D f )( ζ, τ ) def = f ( ζ ) − f ( τ ) ζ − τ , ζ = τf ′ ( ζ ) , ζ = τ, ζ, τ ∈ T . or a differentiable function f on T , we define the divided differences D [1] f and D [2] f by (cid:0) D [1] f (cid:1) ( ζ , ζ , τ ) def = f ( ζ , τ ) − f ( ζ , τ ) ζ − ζ , ζ = ζ ,∂f∂ζ (cid:12)(cid:12)(cid:12) ζ = ζ , ζ = ζ , ζ , ζ , τ ∈ T , and (cid:0) D [2] f (cid:1) ( ζ, τ , τ ) def = f ( ζ, τ ) − f ( ζ, τ ) τ − τ , τ = τ ,∂f∂τ (cid:12)(cid:12)(cid:12) τ = τ , τ = τ , ζ, τ , τ ∈ T . We need several elementary identities.Let Π m be the set of m th roots of 1:Π m def = { ξ ∈ T : ξ m = 1 } and let Υ m ( ζ ) def = ζ m − m ( ζ −
1) = 1 m m − X k =0 ζ k , ζ ∈ T . The following elementary formulae are well known. We give proofs for completeness.
Lemma 4.1.
Let f and g be analytic polynomials in one variable of degree less than m . Then Z T f g d m = 1 m X ξ ∈ Π m f ( ξ ) g ( ξ ) . In particular, Z T | f | d m = 1 m X ξ ∈ Π m | f ( ξ ) | . Proof.
It suffices to consider the case where f ( z ) = z j and g ( z ) = z k with 0 ≤ j, k Lemma 4.3. Let f and g be polynomials in two variables of degree less than m ineach variable. Then Z T f g d m = 1 m X ξ,η ∈ Π m f ( ξ, η ) g ( ξ, η ) . n particular, Z T | f | d m = 1 m X ξ,η ∈ Π m | f ( ξ, η ) | . Proof. It suffices to consider the case when f ( ζ, τ ) = ζ j τ j and g ( ζ, τ ) = ζ k τ k with 0 ≤ j , j , k , k < m . Then − m < j − k , j − k < m and X ξ,η ∈ Π m ξ j η j ξ k η k = ( , ( j , j ) = ( k , k ) m , ( j , j ) = ( k , k ) . (cid:4) (4.1)Suppose now that ( T , R ) and ( T , R ) are pairs of not necessarily commuting con-tractions. Theorem 4.4. Let f be an analytic polynomial in two variable of degree at most m in each variable. Then f ( T , R ) − f ( T , R ) = X ξ,η ∈ Π m Υ m ( ξT )( T − T ) Υ m ( ηT ) ( D [1] f )( ξ, η, R ) (4.2) and f ( T , R ) − f ( T , R ) = X ξ,η ∈ Π m ( D [2] f )( T , ξ, η ) Υ m ( ξR )( R − R ) Υ m ( ηR ) . (4.3)We are going to establish (4.2). The proof of (4.3) is similar.We need the following lemma. Lemma 4.5. Let ϕ be an analytic polynomial in one variable of degree at most m .Then ϕ ( T ) − ϕ ( T ) = X ξ,η ∈ Π m Υ m ( ξT )( T − T ) Υ m ( ηT )( D ϕ )( ξ, η ) . Proof of the lemma. Let 0 ≤ j, j , k, k < m . Then X ξ,η ∈ Π m ( ξT ) j ( ηT ) k ξ j η k = m T j T k , ( j , k ) = ( j, k ) , , ( j , k ) = ( j, k ) . Thus, X ξ,η ∈ Π m Υ m ( ξT ) Υ m ( ηT ) ξ j η k = T j T k if 0 ≤ j, k < n . Hence, X ξ,η ∈ Π m Υ m ( ξT ) T Υ m ( ηT ) ξ j η k = T j +11 T k and X ξ,η ∈ Π m Υ m ( ξT ) T Υ m ( ηT ) ξ j η k = T j T k +10 . It follows that X ξ,η ∈ Π m Υ m ( ξT )( T − T ) Υ m ( ηT ) ξ j η k = T j ( T − T ) T k henever 0 ≤ j, k < m .Let ϕ = m P s =0 a s z s . It is easy to see that( D ϕ )( z, w ) = X j,k ≥ ,j + k Clearly, it suffices to prove (4.2) in the case when f ( z , z ) = ϕ ( z ) z j , where ϕ is a polynomial of one variable of degree at most n and 0 ≤ j ≤ m .Clearly, in this case f ( T , R ) − f ( T , R ) = (cid:0) ϕ ( T ) − ϕ ( T ) (cid:1) R j . On the other hand, ( D f [1] )( ξ, η, R ) = ( D ϕ )( ξ, η ) R j . Identity (4.2) follows now from Lemma 4.5. (cid:4) For K ∈ L ( T ), we denote by I K the integral operator on L ( T ) with kernel function K , i.e., ( I K ϕ )( ζ ) = Z T K ( ζ, τ ) ϕ ( τ ) d m ( τ ) , ϕ ∈ L ( T ) . The following lemma allows us to evaluate the operator norm k I K k B ( L ) of this operatorfor polynomials K of degree less than m in each variable in terms of the operator normsof the matrix { K ( ζ, η ) } ζ,η ∈ Π m . Lemma 4.6. Let K be an analytic polynomial in two variables of degree less than m in each variable. Then k{ K ( ξ, η ) } ξ,η ∈ Π m k B = m k I K k B ( L ) . Proof. It is easy to see that k I K k B ( L ) = sup k ϕ k L ≤ , k ψ k L ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Z T × T K ( ζ, τ ) ϕ ( ζ ) ψ ( τ ) d m ( ζ ) d m ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) = sup k ϕ k L ≤ , k ψ k L ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z Z T × T K ( ζ, τ ) ϕ m ( z ) ψ m ( τ ) d m ( ζ ) d m ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) , here ϕ m ( z ) = m − P k =0 b ϕ ( k ) z k and ψ m ( z ) = m − P k =0 b ψ ( k ) z k . Hence, k I K k B ( L ) = sup (cid:12)(cid:12)(cid:12)(cid:12)Z Z T × T K ( ζ, w ) ϕ ( z ) ψ ( w ) d m ( ζ ) d m ( τ ) (cid:12)(cid:12)(cid:12)(cid:12) , where the supremum is taken over all polynomials ϕ and ψ in one variable of degreeless than m and such that k ϕ k L ≤ k ψ k L ≤ 1. Next, by Lemma 4.3, for arbitrarypolynomials ϕ and ψ with deg ϕ < m and deg ψ < m , we have Z Z T × T K ( ζ, τ ) ϕ ( z ) ψ ( w ) d m ( ζ ) d m ( τ ) = 1 m X ξ,η ∈ Π m K ( ξ, η ) ϕ ( ξ ) ψ ( η ) . It remains to observe that by Lemma 4.1, k ϕ k L ≤ P ξ ∈ Π m | ϕ ( ξ ) | ≤ m andthe same is true for ψ . (cid:4) Theorem 4.7. Let g be a polynomial in one variable of degree at most m . Then k{ ( D g )( ξ, η ) } ξ,η ∈ Π m k B ≤ m k g k L ∞ . Proof. The result follows from Lemma 4.6 and the inequality k I D g k B ( L ) ≤ k g k L ∞ , which is a consequence of the fact that k I D g k B ( L ) is equal to the norm of the Hankeloperator H ¯ g on the Hardy class H , see [Pe5], Ch. 1, Th. 1.10. (cid:4) Corollary 4.8. Let f be a trigonometric polynomial of degree at most m in eachvariable and let p ∈ [1 , . Suppose that T , R , T , R are contractions such that T − T ∈ S p and R − R ∈ S p . Then k f ( T , R ) − f ( T , R ) k S p ≤ m k f k L ∞ max (cid:8) k T − T k S p , k R − R k S p (cid:9) . Proof. Let us estimate k f ( T , R ) − f ( T , R ) k S p . The norm k f ( T , R ) − f ( T , R ) k S p can be estimated in the same way. The result is a consequence of formula (4.2), Theorem3.3, Theorem 4.7 and Corollare 4.2. (cid:4) Corollary 4.8 allows us to establish a Lipschitz type inequality for functions in (cid:0) B ∞ , (cid:1) + ( T ). Theorem 4.9. Let ≤ p ≤ and let f ∈ (cid:0) B ∞ , (cid:1) + ( T ) . Suppose that T , R , T , R are contractions such that T − T ∈ S p and R − R ∈ S p . Then k f ( T , R ) − f ( T , R ) k S p ≤ const k f k B ∞ , max (cid:8) k T − T k S p , k R − R k S p (cid:9) . Proof. Indeed, the result follows immediately from Corollary 4.8 and inequality (2.3). (cid:4) . A representation of operator differencesin terms of triple operator integrals In this section we obtain an explicit formula for the operator differences f ( T , R ) − f ( T , R ), f ∈ (cid:0) B ∞ , (cid:1) + ( T ), in terms of triple operator integrals. Theorem 5.1. Let f ∈ (cid:0) B ∞ , (cid:1) + ( T ) . Then D [1] f ∈ C A ⊗ h C A ⊗ h C A and D [2] f ∈ C A ⊗ h C A ⊗ h C A . Lemma 5.2. Let f be an analytic polynomial in two variables of degree at most m ineach variable. Then (cid:0) D [1] f (cid:1) ( ζ , ζ , τ ) = X ξ,η ∈ Π m Υ m ( ζ ξ ) Υ m ( ζ η ) (cid:0) D [1] f (cid:1) ( ξ, η, τ ) (5.1) and (cid:0) D [2] f (cid:1) ( ζ, τ , τ ) = X ξ,η ∈ Π m ( D [2] f )( ζ, ξ, η )Υ m ( τ ξ )Υ m ( τ η ) . (5.2) Proof. Both formulae (5.1) and (5.2) can be verified straightforwardly. However, wededuce them from Theorem 4.4.Formula (5.1) follows immediately from formula (4.2) if we consider the special casewhen T , T and R are the operators on the one-dimensional space of multiplication by ζ , ζ and τ . Similarly, formula (5.2) follows immediately from formula (4.3). (cid:4) Corollary 5.3. Under the hypotheses of Lemma , (cid:13)(cid:13) D [1] f (cid:13)(cid:13) C A ⊗ h C A ⊗ h C A ≤ m k f k L ∞ and (cid:13)(cid:13) D [2] f (cid:13)(cid:13) C A ⊗ h C A ⊗ h C A ≤ m k f k L ∞ . Proof. The result is a consequence of Lemma 5.2, Theorem 4.7, Corollary 4.2 andDefinitions 2 and 3 in § (cid:4) Proof of Theorem 5.1. The result follows immediately from Corollary 5.3 andinequality (2.3). (cid:4) Theorem 5.4. Let p ∈ [1 , . Suppose that T , R , T , R are contractions such that T − T ∈ S p and R − R ∈ S p . Then for f ∈ (cid:0) B ∞ , (cid:1) + ( T ) , the following formula holds: f ( T , R ) − f ( T , R )= Z Z Z (cid:0) D [1] f (cid:1) ( ζ , ζ , τ ) dE T ( ζ )( T − T ) dE T ( ζ ) dE R ( τ ) , + Z Z Z (cid:0) D [2] f (cid:1) ( ζ, τ , τ ) dE T ( ζ ) dE R ( τ )( R − R ) dE R ( τ ) . (5.3) Proof. Suppose first that f is an analytic polynomial in two variables of degree atmost m in each variable. In this case equality (5.3) is a consequence of Theorem 4.4,Lemma 5.2 and the definition of triple operator integrals given in Subsection 3.5. n the general case we represent f by the series (2.1) and apply (5.3) to each f n . Theresult follows from (2.3). (cid:4) Differentiability properties In this section we study differentiability properties of the map t f (cid:0) T ( t ) , R ( t ) (cid:1) (6.1)in the norm of S p , 1 ≤ p ≤ 2, for functions t T ( t ) and t R ( t ) that take contractivevalues and are differentiable in S p .We say that an operator-valued function Ψ defined on an interval J is differentiable in S p if Φ( s ) − Φ( t ) ∈ S p for any s, t ∈ J , and the limitlim h → h (cid:0) Ψ( t + h ) − Ψ( t ) (cid:1) def = Φ ′ ( t )exists in the norm of S p for each t in J . Theorem 6.1. Let p ∈ [1 , and let f ∈ (cid:0) B ∞ , (cid:1) + ( T ) . Suppose that t T ( t ) and t R ( t ) are operator-valued functions on an interval J that take contractive values andare differentiable in S p . Then the function (6.1) is differentiable on J in S p and ddt f (cid:0) T ( t ) , R ( t ) (cid:1)(cid:12)(cid:12)(cid:12) t = s = Z Z Z (cid:0) D [1] f (cid:1) ( ζ , ζ , τ ) dE T ( s ) ( ζ ) T ′ ( s ) dE T ( s ) ( ζ ) dE R ( s ) ( τ )+ Z Z Z (cid:0) D [2] f (cid:1) ( ζ, τ , τ ) dE T ( s ) ( ζ ) dE R ( s ) ( τ ) R ′ ( s ) dE R ( s ) ( τ ) ,s ∈ J . Proof. As before, it suffices to prove the result in the case when f is an analyticpolynomial of degree at most m in each variable. Suppose that f is such a polynomial.Put F ( t ) def = f (cid:0) T ( t ) , R ( t ) (cid:1) . We have F ( s + h ) − F ( s )= X ξ,η ∈ Π m Υ m (cid:0) ξT ( s + h ) (cid:1)(cid:0) T ( s + h ) − T ( s ) (cid:1) Υ m (cid:0) ηT ( s ) (cid:1)(cid:0) D [1] f (cid:1)(cid:0) ξ, η, R ( s + h ) (cid:1) + X ξ,η ∈ Π m (cid:0) D [2] f (cid:1)(cid:0) T ( s ) , ξ, η (cid:1) Υ m (cid:0) ξR ( s + h ) (cid:1)(cid:0) R ( s + h ) − R ( s ) (cid:1) Υ m (cid:0) ηR ( s ) (cid:1) . Clearly,lim h → h (cid:0) T ( s + h ) − T ( s ) (cid:1) = T ′ ( s ) and lim h → h (cid:0) R ( s + h ) − R ( s ) (cid:1) = R ′ ( s ) n the norm of S p . On the other hand, it is easy to see thatlim h → Υ m (cid:0) ξT ( s + h ) (cid:1) = Υ m (cid:0) ξT ( s ) (cid:1) , lim h → (cid:0) D [1] f (cid:1)(cid:0) ξ, η, R ( s + h ) (cid:1) = (cid:0) D [1] f (cid:1)(cid:0) ξ, η, R ( s ) (cid:1) and lim h → Υ m (cid:0) ξR ( s + h ) (cid:1) = Υ m (cid:0) ξR ( s ) (cid:1) in the operator norm. Hence, F ′ ( s ) = X ξ,η ∈ Π m Υ m (cid:0) ξT ( s ) (cid:1) T ′ ( s )Υ m (cid:0) ηT ( s ) (cid:1)(cid:0) D [1] f (cid:1)(cid:0) ξ, η, R ( s ) (cid:1) + X ξ,η ∈ Π m (cid:0) D [2] f (cid:1)(cid:0) T ( s ) , ξ, η (cid:1) Υ m (cid:0) ξR ( s ) (cid:1) R ′ ( s )Υ m (cid:0) ηR ( s ) (cid:1) . It follows now from Lemma 5.2 and from the definition of triple operator integrals givenin § Z Z Z (cid:0) D [1] f (cid:1) ( ζ , ζ , τ ) dE T ( s ) ( ζ ) T ′ ( s ) dE T ( s ) ( ζ ) dE R ( s ) ( τ )+ Z Z Z (cid:0) D [2] f (cid:1) ( ζ, τ , τ ) dE T ( s ) ( ζ ) dE R ( s ) ( τ ) R ′ ( s ) dE R ( s ) ( τ )which completes the proof. (cid:4) The case p > In this section we show that unlike in the case p ∈ [1 , S p in the case when p > f ( T, R ), f ∈ (cid:0) B ∞ , (cid:1) + ( T ),of not noncommuting contractions. In particular, there are no such Lipschitz type esti-mates for functions f ∈ (cid:0) B ∞ , (cid:1) + ( T ) in the operator norm. Moreover, we show that for p > 2, such Lipschitz type estimates do not hold even for functions f in (cid:0) B ∞ , (cid:1) + ( T )and for pairs of noncommuting unitary operators .Recall that similar results were obtained in [ANP] for functions of noncommutingself-adjoint operators. However, in this paper we use a different construction to obtainresults for functions of unitary operators. Lemma 7.1. For each matrix { a ξ η } ξ,η ∈ Π m , there exists an analytic polynomial f intwo variables of degree at most m − in each variable such that f ( ξ, η ) = a ξ η for all ξ, η ∈ Π m and k f k L ∞ ( T ) ≤ sup ξ,η ∈ Π m | a ξ η | . Proof. Put f ( z, w ) def = X ξ,η ∈ Π m a ξ η Υ m ( zξ )Υ m ( wη ) . learly, f ( ξ, η ) = a ξ η for all ξ, η ∈ Π m and | f ( z, w ) | ≤ sup ξ,η ∈ Π m | a ξ η | X ξ,η ∈ Π m | Υ m ( zξ ) | | Υ m ( wη ) | = sup ξ,η ∈ Π m | a ξ η | X ξ ∈ Π m | Υ m ( zξ ) | X η ∈ Π m | Υ m ( wη ) | = sup ξ,η ∈ Π m | a ξ η | by Corollary 4.2. (cid:4) Lemma 7.2. For each m ∈ N , there exists an analytic polynomial f in two variablesof degree at most m − in each variable, and unitary operators U , U and V such that k f ( U , V ) − f ( U , V ) k S p > π − m − p k f k L ∞ ( T ) k U − U k S p for every p > . Proof. One can select orthonormal bases { g ξ } ξ ∈ Π m and { h η } η ∈ Π m in an m -dimensionalHilbert space H such that | ( g ξ , h η ) | = m − for all ξ, η ∈ Π m . Indeed, let H bethe subspace of L ( T ) of analytic polynomials of degree less than m . We can put g ξ def = √ m Υ m ( zξ ) and h η = z k , where η = e π i k/m , 0 ≤ k ≤ m − { P ξ } ξ ∈ Π m and { Q η } η ∈ Π m defined by P ξ v = ( v, g ξ ) g ξ , ξ ∈ Π m , and Q η v = ( v, h η ) h η , η ∈ Π m . We define the unitary operators U , U , and V by U = X ξ ∈ Π m ξP ξ , U = e π i m U and V = X η ∈ Π m ηQ η . By Lemma 7.1, there exists an analytic polynomial f in two variables of degree at most4 m − f ( ξ, η ) = √ m ( g ξ , h η ) for all ξ, η ∈ Π m , f ( ξ, η ) = 0 forall ξ ∈ Π m \ Π m , η ∈ Π m and k f k L ∞ ( T ) = 1. Clearly, f ( U , V ) = and f ( U , V ) = P ξ,η ∈ Π m f ( ξ, η ) P ξ Q η . We have( f ( U , V ) h η , g ξ ) = f ( ξ, η )( h η , g ξ ) = 1 √ m . Hence, rank f ( U , V ) = 1 and k f ( U , V ) − f ( U , V ) k S p = k f ( U , V ) k S p = k f ( U , V ) k S = √ m. It remains to observe that k U − U k S p = (cid:12)(cid:12) − e π i m (cid:12)(cid:12) m p < πm p − . (cid:4) Remark. If we replace the polynomial f constructed in the proof of Lemma 7.2 withthe polynomial g defined by g ( z , z ) = z m − z m − f ( z , z ) , it will obviously satisfy the same inequality: k g ( U , V ) − g ( U , V ) k S p > π − m − p k g k L ∞ ( T ) k U − U k S p . (7.1)It is easy to deduce from (2.3) that for such polynomials gc m k g k L ∞ ( T ) ≤ k g k B ∞∞ , ≤ c m k g k L ∞ ( T )21 or some constants c and c .This together with (7.1) implies the following result: Theorem 7.3. Let M > and < p ≤ ∞ . Then there exist unitary operators U , U , V and an analytic polynomial f in two variables such that k f ( U , V ) − f ( U , V ) k S p > M k f k B ∞ , ( T ) k U − U k S p . Open problems In this section we state open problems for functions of noncommuting contractions. Functions of triples of contractions. Recall that it was shown in [Pe7] that for f ∈ B ∞ , ( R ), there are no Lipschitz type estimates in the norm of S p for any p > f ( A, B, C ) of triples of noncommuting self-adjoint operators. We conjecturethat the same must be true in the case of functions of triples of not necessarily commutingcontractions. Note that the construction given in [Pe7] does not generalize to the caseof functions of contractions. Lipschitz functions of noncommuting contractions. Recall that an unknownreferee of [ANP] observed that for Lipschitz functions f on the real line there are noLipschitz type estimates for functions f ( A, B ) of noncommuting self-adjoint operatorsin the Hilbert–Schmidt norm. The construction is given in [ANP]. We conjecture thatthe same result must hold in the case of functions of noncommuting contractions. Lipschitz type estimates for p > It follows fromresults of [ANP] that in the case of functions of noncommuting self-adjoint operatorsfor any s > q > p > 2, there exist pairs of self-adjoint operators ( A , A )and ( B , B ) and a function f in the homogeneous Besov space B s ∞ ,q ( R ) such that k f ( A , B ) − f ( A , B ) k S p can be arbitrarily large while max {k A − A k S p , k B − B k S p } can be arbitrarily small. In particular, the condition f ∈ B s ∞ ,q ( R ) does not imply anyLipschitz or H¨older type estimates in the norm of S p , p > 2, for any positive s and q .It is easy to see that in the case of contractions the situation is different: for any q > p ≥ 1, there exists s > f ∈ B s ∞ ,q guarantees a Lipschitztype estimate for functions of not necessarily commuting contractions in S p .It would be interesting to find optimal conditions on f that would guarantee Lipschitzor H¨older type estimates in S p for a given p . References [ANP] A.B. Aleksandrov, F.L. Nazarov and V.V. 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