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S A B B k ratorielle. A 2 B 1 2 k c ( htcnb endin defined be can that 2 k 1 f x k et S − ( S 2 1 A ) 1 B ) B dE A ) + , 2 1 α lipschitzienne − k B k et S 0 , B − B ∞ 1 f 1 ( 1 .However, ). , ( y B B 1 − < α < B ) ( 2 f . R B ) u ne qui ( 2 k ∈ 2 ,B A, nous ) S k RS S 1 S ans 1 ≤ 1, 1 ). ) . L’int´egrale triple op´eratorielle est bien d´efini parce que la diff´erence divis´ee appartient au produit ∞ ∞ h ∞ tensoriel L (R) ⊗hL (R) ⊗ L (R) (voir la d´efinition dans la version anglaise). Plus pr´ecis´ement, si f est une fonction born´ee sur R2 dont la transform´ee de Fourier a un support dans le disque unit´e, nous obtenons la repr´esentation tensorielle suivante: f(x ,y) − f(x ,y) sin(x − jπ) sin(x − kπ) f(jπ,y) − f(kπ,y) 1 2 = 1 · 2 · . x − x x − jπ x − kπ jπ − kπ 1 2 j,k∈Z 1 2 X En outre, on a sin2(x − jπ) sin2(x − kπ) 1 = 2 =1, x x ∈ R, (x − jπ)2 (x − kπ)2 1 2 j∈Z 1 k∈Z 2 X X et f(jπ,y) − f(kπ,y) sup ≤ const kfkL∞(R), y∈R jπ − kπ j,k∈Z   B

ℓ2 j k f jπ,y f kπ,y jπ kπ −1 def ∂f jπ,y o`u B est l’espace d’op´erateurs born´es sur . Si = , alors ( ( )− ( ))( − ) = ∂x ( ). D’autre part, il se trouve que pour la mˆeme classe de fonctions c’est impossible d’obtenir une in´egalit´e du type lipschitzienne dans la norme op´eratorielle. Plus pr´ecis´ement, nous pouvons d´emontrer qu’il n’y a pas du nombre positif M pour lequel

kf(A1,B1) − f(A2,B2)k≤ MkfkL∞(R2)(kA1 − A2k + kB1 − B2k), chaque fois que f soit une fonction born´ee sur R dont la transform´ee de Fourier a un support dans le disque unit´eet A1, A2, B1, B2 soient des op´erateurs auto-adjoints du rang fini. En outre, on peut trouver des op´erateurs auto-adjoints du rang fini tels que kA1 − A2k ≤ 2π, 2 kB1 − B2k≤ 2π, et une fonction f sur R dont la transform´ee de Fourier a un support dans le disque unit´eet telle que kfkL∞(R) ≤ 1 pour lesquels la norme kf(A1,B1) − f(A2,B2)k soit la plus grande possible.

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1. Introduction

It was shown in [F] that a Lipschitz function f on R does not have to be operator Lipschitz, i.e., it does not have to satisfy the inequality kf(A) − f(B)k≤ const kA − Bk for self-adjoint operators A and 1 R B. In [Pe1] and [Pe2] it was shown that the condition f ∈ B∞,1( ) is sufficient for f to be operator s Lipschitz (see [Pee] and [Pe4] for information about Besov spaces Bp,q). It is also well known that f is operator Lipschitz if and only if f(A) − f(B) belongs to S1 whenever A − B ∈ S1 and kf(A) − f(B)kS1 ≤ const kA − BkS1 . It was shown in [AP] that if f is a H¨older function of order α, 0 <α< 1, then it is operator H¨older of order α, i.e., kf(A) − f(B)k≤ const kA − Bkα for self-adjoint operators A and B. Later the above results were generalized in [APPS] to functions of normal operators and in [NP] to functions of commuting n-tuples of self-adjoint operators. In this paper we are going to consider functions of noncommuting pairs of self-adjoint operators. Let A and B be self-adjoint operators on and let EA and EB be their spectral measures. Suppose that f is a function of two variables that is defined at least on σ(A) × σ(B). If f is a Schur multiplier with respect to the pair (EA, EB ), we define the function f(A, B) of A and B by

def f(A, B) = f(x, y) dEA(x) dEB (y) (1) ZZ (we refer the reader to [AP] for definitions of Schur multipliers and double operator integrals). Note that this f 7→ f(A, B) is linear, but not multiplicative. 2 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 N N f(x, y)= eijx fˆ(j, k)eiky . j −N k −N ! =X =X Thus f belongs to the projective tensor product L∞⊗ˆ L∞ and N N

ˆ iky ∞ kfkL∞⊗ˆ L∞ ≤ sup f(j, k)e ≤ (1+2N)kfkL x j=−N k=−N X X f B1 R2 L∞ ˆ L∞ f A, B It follows that every periodic function of class ∞1( ) belongs to ⊗ and the operator ( ) is well defined by (1).

2. Lipschitz type estimates in the trace norm

In this paper we use triple operator integrals to estimate functions of perturbed noncommuting operators in trace norm. Let E1, E2, and E3 be spectral measures on measurable spaces (X1, M1), (X2, M2), and (X3, M3). We say that a function Ψ on X1 × X2 × X3 belongs to the Haagerup tensor ∞ ∞ ∞ ∞ product of the spaces L (Ej ), j =1, 2, 3, (notationally, Ψ ∈ L (E1)⊗hL (E2)⊗hL (E3) ) if Ψ admits a representation

Ψ(x1, x2, x3)= αj (x1)βjk(x2)γk(x3); (2) j,k≥ X0 ∞ 2 ∞ here {αj}j≥0, {γk}k≥0 ∈ L (ℓ ) and {βjk}j,k≥0 ∈ L (B), where B is the space of infinite matrices that induce bounded linear operators on ℓ2. We refer the reader to [Pi] for Haagerup tensor products. It is well known (see [JTT]) that for a function Ψ satisfying (2) and for bounded linear operators T and R, one can define the triple operator integral

W = Ψ(x1, x2, x3) dE1(x1)T dE2(x2)R dE3(x3) (3) ZX1 ZX2 ZX3 and

∞ ∞ ∞ kW k≤kΨkL ⊗hL ⊗hL kT k·kRk, (4) where def ∞ ∞ ∞ ∞ 2 ∞ ∞ 2 kΨkL ⊗hL ⊗hL = inf k{αj}j≥0kL (ℓ )k{βjk}j,k≥0kL (B)k{γk}k≥0kL (ℓ ) , the infimum being taken over all representations of Ψ in the form (2). Note that this extends the definition of triple operator integrals given in [Pe3] for projective tensor products of L∞ spaces. We would like to define triple operator integrals of the form (3) in the case when one of the operators T or R is of trace class and find conditions on Ψ under which the operator W must be in S1. ∞ ∞ h ∞ Definition. A function Ψ is said to belong to the tensor product L (E1) ⊗h L (E2) ⊗ L (E3) if it admits a representation

Ψ(x1, x2, x3)= αj (x1)βk(x2)γjk(x3), xj ∈ Xj , (5) j,k≥ X0 ∞ 2 ∞ with {αj}j≥0, {βk}k≥0 ∈ L (ℓ ) and {γjk}j,k≥0 ∈ L (B). For a bounded linear operator R and for a trace class operator T , we define the triple operator integral

W = Ψ(x1, x2, x3) dE1(x1)T dE2(x2)R dE3(x3) ZX1 ZX2 ZX3 3 as the following continuous linear functional on the class of compact operators:

Q 7→ trace Ψ(x1, x2, x3) dE2(x2)R dE3(x3)Q dE1(x1) T . (6) ZX1 ZX2 ZX3   The fact that the linear functional (6) is continuous on the class of compact operators is a consequence of inequality (4), which also implies the following estimate:

∞ ∞ h ∞ kW kS1 ≤kΨkL ⊗hL ⊗ L kT kS1 kRk,

∞ ∞ h ∞ ∞ 2 ∞ 2 ∞ where kΨkL ⊗hL ⊗ L is the infimum of k{αj }j≥0kL (ℓ )k{βk}k≥0kL (ℓ )k{γjk}j,k≥0kL (B) over all representations in (5). ∞ h ∞ ∞ Similarly, suppose that Ψ ∈ L (E1) ⊗ L (E2) ⊗h L (E3), i.e., Ψ admits a representation

Ψ(x1, x2, x3)= αjk(x1)βj (x2)γk(x3) j,k≥ X0 ∞ 2 ∞ where {βj }j≥0, {γk}k≥0 ∈ L (ℓ ), {αjk}j,k≥0 ∈ L (B), T is a bounded linear operator, and R ∈ S1. Then the continuous linear functional

Q 7→ trace Ψ(x1, x2, x3) dE3(x3)Q dE1(x1)T dE2(x2) R ZX1 ZX2 ZX3   on the class of compact operators determines a trace class operator

def W = Ψ(x1, x2, x3) dE1(x1)T dE2(x2)R dE3(x3). ZX1 ZX2 ZX3 Moreover,

∞ h ∞ ∞ kW kS1 ≤kΨkL ⊗ L ⊗hL kT k·kRkS1 . Note that the above definitions of triple operator integrals extend the definition given in [Pe3] in terms of the projective tensor product of the L∞ spaces. We would like to obtain a sufficient condition on a function f on R2 under which

kf(A1,B1) − f(A2,B2)kS1 ≤ const(kA1 − A2kS1 + kB1 − B2kS1 ), whenever (A1,B1) and (A2,B2) are pairs of (not necessarily commuting) self-adjoint operators such that A1 − A2 ∈ S1 and B1 − B2 ∈ S1. Clearly, it suffices to verify the following inequalities:

kf(A1,B) − f(A2,B)k≤ const kA1 − A2kS1 and kf(A, B1) − f(A, B2)k≤ const kB1 − B2kS1 . Theorem 2.1. Let f be a bounded function on R2 whose Fourier transform is supported in the ball {ξ ∈ R2 : kξk≤ 1}. Then f(x ,y) − f(x ,y) sin(x − jπ) sin(x − kπ) f(jπ,y) − f(kπ,y) 1 2 = 1 · 2 · . x − x x − jπ x − kπ jπ − kπ 1 2 j,k∈Z 1 2 X Moreover, sin2(x − jπ) sin2(x − kπ) 1 = 2 =1, x x ∈ R, (x − jπ)2 (x − kπ)2 1 2 j∈Z 1 k∈Z 2 X X and f(jπ,y) − f(kπ,y) sup ≤ const kfkL∞(R). y∈R jπ − kπ j,k∈Z   B

−1 ∂f Note that if j = k, we assume that (f(jπ,y) − f(kπ,y ))(jπ − kπ) = ∂x (jπ,y). Theorem 2.1 implies the following result: 4 Theorem 2.2. Let f be a bounded function on R2 whose Fourier transform is supported in the ball {ξ ∈ R2 : kξk≤ σ}. Then the function Ψ defined by

f(x1,y) − f(x2,y) Ψ(x1, x2,y)= , x1, x2, y ∈ R, x1 − x2 ∞ ∞ h ∞ belongs to the tensor product L (R) ⊗h L (R) ⊗ L (R),

∞ ∞ h ∞ ∞ 2 kΨkL ⊗hL ⊗ L ≤ const σkfkL (R ) and

f(x1,y) − f(x2,y) f(A1,B) − f(A2,B)= dEA1 (x1)(A1 − A2) dEA2 (x2) dEB (y), (7) R R R x − x Z Z Z 1 2 whenever A1, A2, and B are self-adjoint operators such that A1 − A2 ∈ S1. In a similar way we can prove the following fact: Theorem 2.3. Under the same hypotheses on f, the function Ψ defined by

f(x, y1) − f(x, y2) Ψ(x, y1,y2)= , x,y1, y2 ∈ R, y1 − y2

∞ R h ∞ R ∞ R ∞ h ∞ ∞ ∞ 2 belongs to the tensor product L ( ) ⊗ L ( ) ⊗h L ( ), kΨkL ⊗ L ⊗hL ≤ const σkfkL (R ), and

f(x, y1) − f(x, y2) f(A, B1) − f(A, B2)= dEA(x) dEB1 (x1)(B1 − B2) dEB2 (y2), (8) R R R y − y Z Z Z 1 2 whenever A, B1, and B2 are self-adjoint operators such that B1 − B2 ∈ S1. Theorems 2.2 and 2.3 imply the main result of this section: 1 R2 Theorem 2.4. Let f ∈ B∞,1( ). Then

1 kf(A ,B ) − f(A ,B )kS1 ≤ const kfk (kA − A kS1 + kB − B kS1 ), (9) 1 1 2 2 B∞,1 1 2 1 2 whenever (A1,B1) and (A2,B2) are pairs of self-adjoint operators, A1 − A2 ∈ S1 and B1 − B2 ∈ S1. 1 R2 We have defined functions f(A, B) for f in B∞,1( ) only for bounded self-adjoint operators A and B. However, formulae (7) and (8) allow us to define the difference f(A1,B1) − f(A2,B2) in the case 1 R2 when f ∈ B∞,1( ) and the self-adjoint operators A1, A2, B1, B2 are possibly unbounded once we know that the pair (A2,B2) is a trace class perturbation of the pair (A1,B1). Moreover, inequality (9) also holds for such operators.

3. Lipschitz type estimates in the operator norm

The main result of this section shows that unlike in the case of commuting pairs of self-adjoint 1 R2 operators, the condition f ∈ B∞,1( ) does not imply Lipschitz type estimates in the operator norm. Theorem 3.1. There is no positive number M such that

kf(A1,B1) − f(A2,B2)k≤ MkfkL∞(R2)(kA1 − A2k + kB1 − B2k) for all bounded functions f on R2 with Fourier transform supported in [−1, 1]2 and for all finite rank self-adjoint operators A1, A2, B1, B2.

Construction. Let {fj}1≤j≤N and {gk}1≤k≤N be orthonormal systems. Consider the rank one projections Pj and Qj defined by

Pj u = (u,fj)fj and Qju = (u,gj)gj , 1 ≤ j ≤ N. 5 We define the self-adjoint operators A1, A2, and B = B1 = B2 by N N N

A1 = 4jπPj , A2 = (4j + 2)πPj , and B = 4kπQk. j=1 j=1 k X X X=1 Clearly, kA1 − A2k≤ 2π. Put 1 − cos x 1 − cos y ϕ(x, y)=4 · · , x,y ∈ R. x2 y2

Given a matrix {τjk}1≤j,k≤N , we define the function f by

f(x, y)= τjkϕ(x − 4πj,y − 4πk). ≤j,k≤N 1 X It is easy to see that

kfkL∞(R2) ≤ const max |τjk|, f(4jπ, 4kπ)= τjk, f((4j + 2)π, 4kπ)=0, 1 ≤ j, k ≤ N, 1≤j,k≤N and the Fourier transform of f is supported in [−1, 1] × [−1, 1]. One can easily verify that

f(A1,B)= τjkPj Qk, while f(A2,B)= 0. ≤j,k≤N 1 X It can easily be shown that the supremum of kf(A1,B)k over all Pj and Qk is the Schur multiplier norm of the matrix {τjk}1≤j,k≤N . This norm can be made arbitrarily large as N → ∞, while assumption |τjk|≤ 1, 1 ≤ j, k ≤ N, can still hold. Remark. The above construction shows that there exist a function f on R2 whose Fourier transform 2 is supported in [−1, 1] such that kfkL∞(R) ≤ 1 and self-adjoint operators of finite rank A1, A2, B1, B2 such that kA1 − A2k≤ 2π and kB1 − B2k≤ 2π, but kf(A1,B1) − f(A2,B2)k is greater than any given positive number. In particular, the fact that f is a H¨older function of order α ∈ (0, 1) does not imply α α the estimate kf(A1,B1) − f(A2,B2)k≤ const(kA1 − A2k + kB1 − B2k ). We conclude the paper with a theorem that can be deduced from the results of this section.

Theorem 3.2. There are spectral measure E1, E2 and E3 on Borel subsets of R, a function Ψ in ∞ ∞ ∞ the Haagerup tensor product L (E1) ⊗h L (E2) ⊗h L (E3) and an operator Q in S1 such that

Ψ(x1, x2, x2) dE1(x1) dE2(x2)Q dE3(x3) 6∈ S1. ZZZ References

[AP] A.B. Aleksandrov and V.V. Peller, Operator H¨older–Zygmund functions, Advances in Math. 224 (2010), 910- 966. [APPS] A.B. Aleksandrov, V.V. Peller, D. Potapov and F. Sukochev, Functions of normal operators under per- turbations, Advances in Math. 226 (2011), 5216-5251. [F] Yu.B. Farforovskaya, The connection of the Kantorovich-Rubinshtein metric for spectral resolutions of selfadjoint operators with functions of operators, Vestnik Leningrad. Univ. 19 (1968), 94–97. (Russian). [JTT] K. Juschenko, I.G. Todorov and L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc. 361 (2009), 4683-4720. [NP] F.L. Nazarov and V.V. Peller Functions of n-tuples of commuting self-adjoint operators, J. Funct. Anal. 266 (2014), 5398-5428. [Pee] J. Peetre, New thoughts on Besov spaces, Duke Univ. Press., Durham, NC, 1976. [Pe1] V.V. Peller, Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funktsional. Anal. i Prilozhen. 19:2 (1985), 37–51 (Russian). English transl.: Funct. Anal. Appl. 19 (1985), 111–123. [Pe2] V.V. Peller, Hankel operators in the perturbation theory of of unbounded self-adjoint operators. Analysis and partial differential equations, 529–544, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990. [Pe3] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006), 515–544. [Pe4] V.V. Peller, Hankel operators and their applications, Springer-Verlag, New York, 2003. [Pi] G. Pisier, Introduction to operator space theory, London Math. Society Lect. Notes series 294, Cambridge University Press, 2003.

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