Functions of noncommuting operators under perturbation of class $\boldsymbol{S}_p$

TL;DR

This paper demonstrates that for certain classes of functions and operators, small perturbations in the Schatten class can lead to larger changes in the functions of these operators, highlighting limitations in perturbation stability.

Contribution

The paper provides counterexamples showing that functions in the Besov class can amplify perturbations beyond the Schatten class for pairs and triples of operators.

Findings

Counterexamples for $p>2$ with pairs of self-adjoint operators

Extension of results to functions of contractions

Analogous results for triples of self-adjoint operators for all $p",

Abstract

In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty,1}^1({\Bbb R}^2)$ such that the differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class $\boldsymbol{S}_p$ but $f(A_2,B_2)-f(A_1,B_1)\not\in\boldsymbol{S}_p$. A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self-adjoint operators for any $p\ge1$

Equations

$$\Vert f(A)-f(B){\Vert }_{{\mathbf{S}}_1}\le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1}\Vert A-B{\Vert }_{{\mathbf{S}}_1}.$$

$$\Vert f(A)-f(B){\Vert }_{{\mathbf{S}}_p}\le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1}\Vert A-B{\Vert }_{{\mathbf{S}}_p}.$$

$$\Vert f(A)-f(B)\Vert \le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1}\Vert A-B\Vert$$

$$\Vert f(A)-f(B){\Vert }_{{\mathbf{S}}_p}\le c_p\Vert f{\Vert }_{\mathrm{Lip}}\Vert A-B{\Vert }_{{\mathbf{S}}_p}$$

$$\Vert f(A_2,B_2)-f(A_1,B_1){\Vert }_{{\mathbf{S}}_p}\le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1({\mathbb{R}}^2)}\max \{\Vert A_2-A_1{\Vert }_{{\mathbf{S}}_p},\Vert B_2-B_1{\Vert }_{{\mathbf{S}}_p}\}$$

$$\Vert f(T_2,R_2)-f(T_1,R_1){\Vert }_{{\mathbf{S}}_p}\le \mathrm{const}\Vert f{\Vert }_{(B_{\infty ,1}^1{)}_{+}({\mathbb{T}}^2)}\max \{\Vert T_2-T_1{\Vert }_{{\mathbf{S}}_p},\Vert R_2-R_1{\Vert }_{{\mathbf{S}}_p}\}$$

$$w\ge 0,\mathrm{supp}w\subset \left[\frac{1}{2},2\right],\text{mbox}andw(t)=1-w\left(\frac{t}{2}\right)\text{mbox}fort\in [1,2].$$

$$\big{(}{\mathscr{F}}W_{n}\big{)}(x)=w\left(\frac{\|x\|_{2}}{2^{n}}\right),\quad n\in{\mathbb{Z}},\quad x=(x_{1},\cdots,x_{d}),\quad\|x\|_{2}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\left(\sum_{j=1}^{d}x_{j}^{2}\right)^{1/2}.$$

$$f_n\stackrel{\mathrm{def}}{=}f\ast W_n.$$

$$\Vert f{\Vert }_{B_{\infty ,1}^1}\stackrel{\mathrm{def}}{=}\sum _{n\in \mathbb{Z}}{2}^n\Vert f_n{\Vert }_{L^{\infty }}<\infty .$$

$$\sum _{n<0}\frac{\partial f_n}{\partial x_j},1\le j\le d,$$

$$\frac{\partial f}{\partial x_j}=\sum _{n\in \mathbb{Z}}\frac{\partial f_n}{\partial x_j},1\le j\le d.$$

$$\Vert f{\Vert }_{B_{\infty ,1}^2}\stackrel{\mathrm{def}}{=}\sum _{n\in \mathbb{Z}}{2}^{2n}\Vert f_n{\Vert }_{L^{\infty }}<\infty$$

$$W_n(\zeta )\stackrel{\mathrm{def}}{=}\sum _{j\in {\mathbb{Z}}^d}w\left(\frac{|j|}{2^n}\right){\zeta }^j,n\ge 1,W_0(\zeta )\stackrel{\mathrm{def}}{=}\sum _{\{j:|j|\le 1\}}{\zeta }^j,$$

$$\zeta=(\zeta_{1},\cdots,\zeta_{d})\in{\mathbb{T}}^{d},\quad j=(j_{1},\cdots,j_{d}),\quad\mbox{and}\quad|j|=\big{(}|j_{1}|^{2}+\cdots+|j_{d}|^{2}\big{)}^{1/2}.$$

$$f_n=f\ast W_n,n\ge 0,$$

$$\sum _{n\ge 0}{2}^n\Vert f_n{\Vert }_{L^{\infty }}<\infty .$$

$$\big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{d})=\big{\{}f\in B_{\infty,1}^{1}({\mathbb{T}}^{d}):\widehat{f}(j_{1},j_{2},\cdots,j_{d})=0~{}{\rm whenever}~{}\min\{j_{1},j_{2},\cdots,j_{d}\}<0\big{\}}.$$

$$f(A,B)={\iint }_{\mathbb{R}\times \mathbb{R}}f(x,y)d{E}_A(x)d{E}_B(y),$$

$${\iint }_{\mathbb{R}\times \mathbb{R}}\Phi (x,y)d{E}_1(x)Qd{E}_1(y),$$

$$\Phi (x,y)={\int }_{\Omega }\phi (\omega ,x)\psi (\omega ,y)d\sigma (\omega ),$$

$${\int }_{\Omega }\Vert \phi (\omega ,\cdot ){\Vert }_{L_{E_1}^{\infty }}\Vert \psi (\omega ,\cdot ){\Vert }_{L_{E_2}^{\infty }}d\sigma (\omega )<\infty .$$

$${\iint }_{\mathbb{R}\times \mathbb{R}}\Phi (x,y)d{E}_1(x)Qd{E}_1(y)={\int }_{\Omega }\left(\int \phi (\omega ,x)d{E}_1(x)\right)Q\left(\int \psi (\omega ,y)d{E}_2(y)\right).$$

$$\Phi (x,y)=\sum _{n\ge 1}{\phi }_n(x){\psi }_n(y),$$

$$\sum _{n\ge 1}\Vert {\phi }_n{\Vert }_{L_{E_1}^{\infty }}\Vert {\psi }_n{\Vert }_{L_{E_2}^{\infty }}<\infty .$$

$$f(x,y)={\iint }_{\mathbb{R}\times \mathbb{R}}(\mathcal{F}f)(s,t)e^{\mathrm{i}sx}{e}^{\mathrm{i}ty}dsdt,$$

$$f(A,B)={\iint }_{\mathbb{R}\times \mathbb{R}}(\mathcal{F}f)(s,t)e^{\mathrm{i}sA}{e}^{\mathrm{i}tB}dsdt.$$

$$\displaystyle\sum_{j,k\in{\mathbb{Z}}}\big{|}\hat{f}(j,k)\big{|}<\infty.$$

$$\displaystyle\sum_{j,k\in{\mathbb{Z}}}\big{|}\hat{f}(j,k)\big{|}\leq\sum_{n\geq 0}\sum_{j,k\in{\mathbb{Z}}}\big{|}\hat{f}_{n}(j,k)\big{|}$$

$$\le \mathrm{const}\sum _{n\ge 0}{2}^n\Vert f_n{\Vert }_{L^{\infty }}\le \mathrm{const}\Vert f{\Vert }_{B_{\infty ,1}^1}.$$

$$f(s,t)=\sum _{n\ge 1}\sum _{j=-2^n}^{2^n}{e}^{\mathrm{i}sj}\left(\sum _{j=-2^n}^{2^n}{\hat{f}}_{n-1}(j,k)e^{\mathrm{i}tk}\right),s,t\in \mathbb{R}.$$

Full text

The research of the first author is partially supported by RFBR grant 17-01-00607. The publication was prepared with the support of the RUDN University Program 5-100

Corresponding author: V.V. Peller; email: peller@math.msu.edu

1. Introduction

The paper is devoted to the study of Schatten–von Neumann properties of the increments $f(A_{2},B_{2})-f(A_{1},B_{1})$ under perturbation of pairs of not necessarily commuting bounded self-adjoint operators.

It was Farforovskaya who discovered in [F] that for Lipschitz functions $f$ on the real line ${\mathbb{R}}$ the condition that $A-B\in{\boldsymbol{S}}_{1}$ for self-adjoint operators $A$ and $B$ does not imply that $f(A)-f(B)\in{\boldsymbol{S}}_{1}$. Here ${\boldsymbol{S}}_{1}$ stands for trace class, see [GK].

Later it was shown in [Pe1] and [Pe2] for functions $f$ in the homogeneous Besov space $B_{\infty,1}^{1}({\mathbb{R}})$ (see § 2) the condition that $A-B\in{\boldsymbol{S}}_{1}$ does imply that $f(A)-f(B)\in{\boldsymbol{S}}_{1}$ and

[TABLE]

Moreover, it was shown there that under the same assumption this inequality also holds for all Schatten–von Neumann classes ${\boldsymbol{S}}_{p}$, $1\leq p\leq\infty$:

[TABLE]

Note that for $p=\infty$, by $\|\cdot\|_{{\boldsymbol{S}}_{p}}$ we mean operator norm.

It is well known (see [AP1]) that $f$ satisfies (1.1) for arbitrary self-adjoint operators with trace class difference if and only if $f$ is an opetrator Lipschitz function, i.e.,

[TABLE]

for arbitrary self-adjoint operators $A$ and $B$. We refer the reader to [AP1] for detailed information on operator Lipschitz functions.

It was shown in [PS] that for $p\in(1,\infty)$, there exists a positive number $c_{p}$ such that

[TABLE]

whenever $A$ and $B$ are self-adjoint operators with $A-B\in{\boldsymbol{S}}_{p}$.

Analogs of (1.2) for functions of $n$-tuples of self-adjoint operators were obtained in [APPS] for $n=2$ and in [NP] for $n>2$. We would also like to mention the paper [KPSS], in which an analog of (1.3) for $n$-tuples of self-adjoint operators was obtained.

Functions of pairs of noncommuting self-adjoint operators can be defined with the help of double operator integrals, see [ANP]. We discuss this issue in detail in § 3. In particular, the functions $f(A,B)$ can be defined for arbitrary bounded self-adjoint operators $A$ and $B$ and for every $f$ in the Besov class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$, see § 3.

It was shown in [ANP] that for $p\in[1,2]$, the following estimate of Lipschitz type holds in the norm of the Schatten–von Neumann ideal ${\boldsymbol{S}}_{p}$:

[TABLE]

for arbitrary pairs $(A_{1},B_{1})$ and $(A_{2},B_{2})$ of not necessarily commuting self-adjoint operators such that $A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$ and $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$. Moreover, an analog of this result for functions of unitary operators was also obtained there.

However, in the same paper [ANP] it was established that for $p>2$, there are no such Lipschitz type estimates as well as there are no such estimates in the operator norm.

Nevertheless, the authors of [ANP] failed to answer the important question of whether for $p\in(2,\infty)$, there exist a function $f$ of class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$ and pairs $(A_{1},B_{1})$ and $(A_{2},B_{2})$ of self-adjoint operators such that $A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$ and $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$ but $f(A_{2},B_{2})-f(A_{1},B_{1})\not\in{\boldsymbol{S}}_{p}$.

The main result of this paper (Theorem 4.7) gives an affirmative answer to this question.

In Section 5 of this article we consider a similar problem for functions of not necessarily commuting contractions. Recall that in [AP3] the following analog of inequality (1.4) was obtained for $p\in[1,2]$:

[TABLE]

for an arbitrary function $f$ in the Besov class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) (see § 2) of analytic functions and for arbitrary pairs of contractions $(T_{1},R_{1})$ and $(T_{2},R_{2})$ on Hilbert space satisfying $T_{2}-T_{1}\in{\boldsymbol{S}}_{p}$ and $R_{2}-R_{1}\in{\boldsymbol{S}}_{p}$. Recall that an operator $T$ is called a contraction if $\|T\|\leq 1$.

Note that it was shown in [Pe6] that inequality (1.5) holds for $p\in[1,\infty]$ in the case of pairs of commuting contractions $(T_{1},R_{1})$ and $(T_{2},R_{2})$ on Hilbert space.

It was shown in [AP3] that such an inequality of Lipschitz type does not hold for $p>2$ as well as in the operator norm.

The main result of Section 5 of this paper is that for $p\in(2,\infty)$, there exist a function $f$ of class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) and pairs $(T_{1},R_{1})$ and $(T_{2},R_{2})$ of contractions on Hilbert space such that $T_{2}-T_{1}\in{\boldsymbol{S}}_{p}$ and $R_{2}-R_{1}\in{\boldsymbol{S}}_{p}$ but $f(T_{2},R_{2})-f(T_{1},R_{1})\not\in{\boldsymbol{S}}_{p}$. Moreover, one can even construct such pairs of unitary operators $(T_{1},R_{1})$ and $(T_{2},R_{2})$.

Finally, in § 6 we obtain analogs of the results of § 4 for functions of triples of arbitrary bounded self-adjoint operators.

2. Besov spaces

In this article we deal with the homogeneous Besov class $B_{\infty,1}^{s}({\mathbb{R}}^{2})$, $s=1,2$, of functions on the real line and with the Besov class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) of functions on the two-dimensional torus that are analytic on the bidisk. We refer the reader to the book [Pee] and the papers [ANP] and [AP3] for information on these classes. Let $w$ be an infinitely differentiable function on ${\mathbb{R}}$ such that

[TABLE]

Consider the functions $W_{n}$, $n\in{\mathbb{Z}}$, on ${\mathbb{R}}^{d}$ such that

[TABLE]

With each tempered distribution f\in{\mathscr{S}}^{\prime}\big{(}{\mathbb{R}}^{d}\big{)}, we associate the sequence $\{f_{n}\}_{n\in{\mathbb{Z}}}$,

[TABLE]

The formal series $\sum_{n\in{\mathbb{Z}}}f_{n}$ is a Littlewood–Paley type expansion of $f$. This series does not necessarily converge to $f$.

Initially we define the (homogeneous) Besov class \dot{B}^{1}_{\infty,1}\big{(}{\mathbb{R}}^{d}\big{)} as the space of $f\in{\mathscr{S}}^{\prime}({\mathbb{R}}^{n})$ such that

[TABLE]

According to this definition, the space $\dot{B}^{1}_{\infty,1}({\mathbb{R}}^{n})$ contains all polynomials and all polynomials $f$ satisfy the equality $\|f\|_{B^{1}_{\infty,1}}=0$. Moreover, the distribution $f$ is determined by the sequence $\{f_{n}\}_{n\in{\mathbb{Z}}}$ uniquely up to a polynomial. It is easy to see that the series $\sum_{n\geq 0}f_{n}$ converges in ${\mathscr{S}}^{\prime}({\mathbb{R}}^{d})$. However, the series $\sum_{n<0}f_{n}$ can diverge in general. Obviously, the series

[TABLE]

converges uniformly on ${\mathbb{R}}^{d}$. Now we say that $f$ belongs to the homogeneous Besov class $B^{1}_{\infty,1}({\mathbb{R}}^{d})$ if (2.3) holds and

[TABLE]

A function $f$ is determined uniquely by the sequence $\{f_{n}\}_{n\in{\mathbb{Z}}}$ up to a a constant, and a polynomial $g$ belongs to B^{1}_{\infty,1}\big{(}{\mathbb{R}}^{d}\big{)} if and only if it is a constant.

In the definition of $B_{\infty,1}^{2}({\mathbb{R}}^{d})$ we should replace inequality (2.3) with the following one:

[TABLE]

and condition (2.4) with a similar condition that involves not only first order partial derivatives, but also second order.

Studying periodic functions on ${\mathbb{R}}^{d}$ is equivalent to studying functions on the $d$-dimensional torus ${\mathbb{T}}^{d}$. To define Besov spaces on ${\mathbb{T}}^{d}$, we consider a function $w$ satisfying (2.1) and define the trigonometric polynomials $W_{n}$, $n\geq 0$, by

[TABLE]

where

[TABLE]

For a distribution $f$ on ${\mathbb{T}}^{d}$ we put

[TABLE]

and we say that $f$ belongs the Besov class $B_{\infty,1}^{1}({\mathbb{T}}^{d})$ if

[TABLE]

Note that locally the Besov space $B_{p,q}^{s}({\mathbb{R}}^{d})$ coincides with the Besov space $B_{p,q}^{s}$ of periodic functions on ${\mathbb{R}}^{d}$.

We also define the Besov class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{d}) as the subspace of $B_{\infty,1}^{1}({\mathbb{T}}^{d})$ of holomorphic functions on the polydisk ${\mathbb{D}}^{d}$, i.e.,

[TABLE]

We refer the reader to [Pee] and [AP1] for more detailed information on Besov spaces.

3. Functions of noncommuting operators operators

In this section we describe briefly how to define functions of nocommuting operators.

Let $A$ and $B$ be not necessarily commuting bounded self-adjoint operators on Hilbert space. Functions $f(A,B)$ are defined by the following double operator integral

[TABLE]

where $E_{A}$ and $E_{B}$ are the spectral measures of $A$ and $B$, whenever the double operator integral on the right makes sense. Note that if the spectra $\sigma(A)$ and $\sigma(B)$ are contained in closed intervals ${\mathscr{I}}$ and $\mathscr{J}$, then $f(A,B)$ depends only on the restriction of $f$ to ${\mathscr{I}}\times\mathscr{J}$.

Recall that double operator integrals, i.e., expressions of the form

[TABLE]

appeared first in [DK]. Here $\Phi$ is a measurable function, $E_{1}$ and $E_{2}$ are spectral measures and $Q$ is a bounded linear operator. Later Birman and Solomyak developed in [BS1]–[BS3] a beautiful theory of double operator integrals. For the double operator integral (3.1) to be defined, the function $f$ has to satisfy certain assumptions.

The maximal class of functions $\Phi$, for which the double operator integral (3.1) can be defined is called the class of Schur multipliers with respect to the spectral measures $E_{1}$ and $E_{2}$. There are several characterizations of the class of Schur multipliers, see [Pe1].

We are going to use the following: $\Phi$ is a Schur multiplier with respect $E_{1}$ and $E_{2}$ if and only if $\Phi$ belongs to the integral projective tensor product $L^{\infty}_{E_{1}}\widehat{\otimes}_{\rm i}L^{\infty}_{E_{2}}$ of the $L^{\infty}$ spaces $L^{\infty}_{E_{1}}$ and $L^{\infty}_{E_{2}}$, i.e., $\Phi$ admits a representation

[TABLE]

where $(\Omega,\sigma)$ is a $\sigma$-finite measure space, and $\varphi$ and $\psi$ are measurable functions such that

[TABLE]

In this case

[TABLE]

In particular, $\Phi$ is a Schur multiplier if $\Phi$ belongs to the projective tensor product $L^{\infty}_{E_{1}}\widehat{\otimes}L^{\infty}_{E_{2}}$, i.e., $\Phi$ admits a representation

[TABLE]

where $\varphi_{n}$ and $\psi_{n}$ are measurable functions such that

[TABLE]

Let us also mention that $\Phi$ is a Schur multiplier if and only if it belongs to the Haagerup tensor product $L^{\infty}_{E_{1}}\otimes_{\rm h}L^{\infty}_{E_{2}}$, see [AP1], [Pe4] and [Pe1].

Let us return to functions of pairs $A$ and $B$ of noncommuting self-adjoint operators. Suppose that $f\in L^{\infty}({\mathbb{R}}^{2})$ and its Fourier transform ${\mathscr{F}}f$ belongs to $L^{1}({\mathbb{R}}^{2})$. Then

[TABLE]

and so $f$ is a Schur multiplier with respect to arbitrary Borel measures on ${\mathbb{R}}$ and the operator $f(A,B)$ can be defined by

[TABLE]

In particular, we can define $f(A,B)$ for functions $f$ in the homogeneous Besov class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$. Indeed, it suffices to show that such a function locally coincides with a function whose Fourier transform is in $L^{1}({\mathbb{R}})$. This in turn can be reduced to the following fact for periodic functions: if $f\in B_{\infty,1}^{1}({\mathbb{T}}^{2})$, then

[TABLE]

Indeed,

[TABLE]

There is another way to define $f(A,B)$ for functions $f$ in $B_{\infty,1}^{1}({\mathbb{R}}^{2})$, see [ANP]. Without loss of generality we may assume that $\|A\|<\pi$, $\|B\|<\pi$ and $f$ is a $2\pi$-periodic function of Besov class $B_{\infty,1}^{1}$. Then we can represent $f$ as an element of the projective tensor product $C({\mathbb{T}})\hat{\otimes}C({\mathbb{T}})$ in the following way:

[TABLE]

Clearly, this representation allows us to estimate the tensor norm of $f$:

[TABLE]

by (2.3). Under these assumptions, the function $f(A,B)$ can be defined by

[TABLE]

Note here that in the case when $A$ and $B$ have finite spectra, we can define $f(A,B)$ for arbitrary functions $f$ on $\sigma(A)\times\sigma(B)$:

[TABLE]

where $P_{\lambda}$ is the orthogonal projection onto $\operatorname{Ker}(A-\lambda I)$ and $Q_{\mu}$ is the orthogonal projection onto $\operatorname{Ker}(B-\mu I)$.

Similarly, one can define functions of noncommuting unitary operators. Let $f$ be a function on ${\mathbb{T}}^{2}$ such that (3.2) holds. Then

[TABLE]

and so $f\in C({\mathbb{T}})\widehat{\otimes}C({\mathbb{T}})$ and

[TABLE]

for unitary operators $U$ and $V$.

It is also clear from above that functions $f$ in $B_{\infty,1}^{1}({\mathbb{T}}^{2})$ satisfy (3.2), and so one can define functions $f(U,V)$ for unitary $U$ and $V$ and for $f\in B_{\infty,1}^{1}({\mathbb{T}}^{2})$.

Let us proceed now to the case of functions of triples of not necessarily commuting self-adjoint operators.

For not necessarily commuting bounded self-adjoint operators $A$, $B$ and $C$ the functions $f(A,B,C)$ can be defined as triple operator integrals

[TABLE]

It was shown in [Pe3] that triple operator integrals were defined in the case when the integrand $f$ belongs to the integral projective tensor product $L^{\infty}_{E_{1}}\widehat{\otimes}_{\rm i}L^{\infty}_{E_{2}}\widehat{\otimes}_{\rm i}L^{\infty}_{E_{3}}$. Later triple operator integrals were defined in [JTT] for functions in the Haagerup tensor product $L^{\infty}_{E_{1}}\!\otimes_{\rm h}\!L^{\infty}_{E_{2}}\!\otimes_{\rm h}\!L^{\infty}_{E_{3}}$, see also [AP2] for properties of such triple operator integrals.

As in the case of functions of two noncommuting operators we can define functions $f(A,B,C)$ for functions $f$ whose Fourier transform is integrable:

[TABLE]

In this case

[TABLE]

However, using results of [Ki] (see also [N] and [dLKK]), one can show that unlike the case of two operators, a functions in $B_{\infty,1}^{1}({\mathbb{R}}^{3})$ does not have to coincide locally with a function with summable Fourier transform.

On the other hand by analogy with (3) one can show that functions in the Besov space $B_{\infty,1}^{2}({\mathbb{R}}^{3})$ coincide locally with functions with absolutely convergent Fourier transform.

As in the case of functions of two operators we can represent an arbitrary $2\pi$-periodic function of class $B_{\infty,1}^{2}$ as an element of the projective tensor product $C({\mathbb{T}})\widehat{\otimes}C({\mathbb{T}})\widehat{\otimes}C({\mathbb{T}})$ in the following way:

[TABLE]

This implies that

[TABLE]

Under the assumptions that $\|A\|<\pi$, $\|B\|<\pi$ and $\|C\|<\pi$, we can define now $f(A,B,C)$ by

[TABLE]

In the case when $A$, $B$ and $C$ have finite spectra, we can define $f(A,B,C)$ for arbitrary functions on $\sigma(A)\times\sigma(B)\times\sigma(C)$ by

[TABLE]

where $P_{\lambda}$ is the orthogonal projection onto $\operatorname{Ker}(A-\lambda I)$, $Q_{\mu}$ is the orthogonal projection onto $\operatorname{Ker}(B-\mu I)$ and $R_{\nu}$ is the orthogonal projection onto $\operatorname{Ker}(C-\nu I)$.

4. The case of self-adjoint operators

The main purpose of this section is for each $p>2$, to construct a function $f$ of class $B^{1}_{\infty,1}({\mathbb{R}}^{2})$ and pairs of bounded noncommuting self-adjoint operators $(A_{1},B_{1})$, $(A_{2},B_{2})$ such that $A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$, $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$ but $f(A_{2},B_{2})-f(A_{1},B_{1})\not\in{\boldsymbol{S}}_{p}$. Then we impose additional assumptions on the support of the Fourier transform of $f$.

Denote by $\mathscr{E}_{\sigma}^{\infty}({\mathbb{R}}^{d})$, $\sigma>0$, the space of bounded (continuous) functions $f$ on ${\mathbb{R}}^{d}$ whose Fourier transform ${\mathscr{F}}f$ is supported in $[-\sigma,\sigma]^{d}$. It is well known that $\mathscr{E}_{\sigma}^{\infty}({\mathbb{R}}^{2})$ equipped with the $L^{\infty}$-norm is a Banach space. It is easy to see that $f\in\mathscr{E}_{\bf 1}^{\infty}({\mathbb{R}}^{d})$ if and only if $f(\sigma x)\in\mathscr{E}_{\sigma}^{\infty}({\mathbb{R}}^{d})$.

It can be seen from the proof of Theorem 8.1 in [ANP] that the following result holds:

**      Lemma 4.1****.**

For each positive integer $N$, there exist a function $f$ of class ${\mathscr{E}}_{2\pi}^{\infty}({\mathbb{R}}^{2})$ with $\|f\|_{L^{\infty}({\mathbb{R}}^{2})}=1$ and positive self-adjoint operators $A_{1}$, $A_{2}$ and $B$ on ${\mathbb{C}}^{N}$ such that $\|A_{1}\|=2N$, $\|A_{2}\|=2N+1$, $\|B\|=N$, $f(A_{2},B)={\boldsymbol{0}}$ and

[TABLE]

**      Corollary 4.2****.**

For each positive integer $N$, there exist a function $f$ of class ${\mathscr{E}}_{N}^{\infty}({\mathbb{R}}^{2})$ with $\|f\|_{L^{\infty}({\mathbb{R}}^{2})}=1$ and (positive) self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that $f(A_{2},B)={\boldsymbol{0}}$ and

[TABLE]

**Proof. **Let us fix a positive integer $n$. Suppose that $f_{0}$, $A_{1}^{(0)}$, $A_{2}^{(0)}$ and $B^{(0)}$ satisfy the requirements of Lemma 4.1 with $N=n$. Put $f(x,y)=f_{0}((2n+1)x,(2n+1)y)$, $A_{1}=(2n+1)^{-1}A_{1}^{(0)}$, $A_{2}=(2n+1)^{-1}A_{2}^{(0)}$ and $B=(2n+1)^{-1}B^{(0)}$. Then $\|f\|_{L^{\infty}({\mathbb{R}}^{2})}=1$, $f\in\mathscr{E}_{2\pi(2n+1)}^{\infty}({\mathbb{R}}^{2})\subset\mathscr{E}_{20n}^{\infty}({\mathbb{R}}^{2})$, $A_{1}$, $A_{2}$ and $B$ are self-adjoint contractions on a finite-dimensional Hilbert space, $f(A_{2},B)={\boldsymbol{0}}$ and

[TABLE]

Thus, we have proved Corollary 4.2 for $N=20n$. The general case can easily be reduced to this special case. Let $N$ be an arbitary positive integer. I suffices to consider the case where $N>20$. Then $N=20n+k$ with $n\geq 1$ and $k$, $0\leq k<20$. It remains to observe that $20n>\operatorname{const}N$. $\blacksquare$

**      Corollary 4.3****.**

For each positive integer $N$, there exist a function $f$ of class $B_{\infty,1}^{1}({\mathbb{R}}^{2})$ with $\|f\|_{B_{\infty,1}^{1}({\mathbb{R}}^{2})}=1$ and (positive) self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that $f(A_{2},B)={\boldsymbol{0}}$ and

[TABLE]

**Proof. **It suffices to observe that $\|f\|_{B_{\infty,1}^{1}({\mathbb{R}}^{2})}\leq\operatorname{const}N\|f\|_{L^{\infty}({\mathbb{R}}^{2})}$ for all $f\in\mathscr{E}_{N}^{\infty}({\mathbb{R}}^{2})$. $\blacksquare$

Put B_{\infty,1}^{1}([-1,1]^{2})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{f\big{|}[-1,1]^{2}:~{}f\in B_{\infty,1}^{1}({\mathbb{R}}^{2})\} and

[TABLE]

for $g\in B_{\infty,1}^{1}([-1,1]^{2})$.

Let $p\in[1,\infty]$. Denote by ${\rm BOL}_{p}([-1,1]\times[-1,1])$ the set of functions $f$ in $B_{\infty,1}^{1}([-1,1]^{2})$ such that there exists a positive number $c$, for which

[TABLE]

whenever $A_{1}$, $A_{2}$, $B_{1}$ and $B_{2}$ are arbitrary self-adjoint contractions on a finite-dimensional Hilbert space. We denote by $\|f\|_{{\rm OL}_{{\boldsymbol{S}}_{p}}([-1,1]\times[-1,1])}$ the smallest constant $c$, for which (4.1) holds.

It is easy to see that $f\in{\rm BOL}_{p}([-1,1]\times[-1,1])$ if and only if there exists a constant $c$ such that

[TABLE]

for all self-adjoint contractions $A$, $A_{1}$, $A_{2}$, $B$, $B_{1}$ and $B_{2}$ on a finite-dimensional Hilbert space. Put

[TABLE]

Clearly, $\{f\in{\rm BOL}_{p}([-1,1]\times[-1,1]):f(0,0)=0\}$ equipped with norm $\|\cdot\|_{{\rm BOL}_{p}}$ is a Banach space.

**      Lemma 4.4****.**

Let $p\in(2,+\infty]$. Then there exists a function $f$ in $B_{\infty,1}^{1}({\mathbb{R}}^{2})$ such that f\big{|}[-1,1]^{2}\not\in{\rm BOL}_{p}([-1,1]\times[-1,1]).

**Proof. **Suppose that f\big{|}[-1,1]^{2}\in{\rm BOL}_{p}([-1,1]\times[-1,1]) for all $f\in B_{\infty,1}^{1}({\mathbb{R}}^{2})$. Then by the closed graph theorem there exists a constant $c$ such that $\|f\|_{{\rm BOL}_{p}([-1,1]\times[-1,1])}\leq c\|f\|_{B^{1}_{\infty,1}([-1,1]^{2})}$, and we get a contradiction with Corollary 4.3. $\blacksquare$

**      Corollary 4.5****.**

Let $p\in(2,+\infty]$. Then there exists a function $f$ in $B_{\infty,1}^{1}({\mathbb{R}}^{2})$ such that for every $M>0$, there exist self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that

[TABLE]

**      Theorem 4.6****.**

Let $p\in(2,\infty]$. Then there exists a function $f$ in $B_{\infty,1}^{1}({\mathbb{R}}^{2})$ such that for any positive numbers $M$ and $\delta$, there exist self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that

[TABLE]

**Proof. **It follows from Corollary 4.5 that there are self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space that satisfy the right inequality in (4.2).

We can select a positive integer $N$ such that $\frac{1}{N}\|A_{1}-A_{2}\|_{{\boldsymbol{S}}_{p}}<\delta$. Put now $A^{(j)}=A_{1}+\frac{j}{N}(A_{2}-A_{1})$, $0\leq j<N$. It remains to observe that

[TABLE]

at least for one $j$, $0\leq j<N$. $\blacksquare$

**      Theorem 4.7****.**

For each $p$ in $(2,\infty)$, there exist a function $f$ in $B^{1}_{\infty,1}({\mathbb{R}}^{2})$ and self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on Hilbert space such that $A_{1}-A_{2}\in{\boldsymbol{S}}_{p}$ but $f(A_{1},B)-f(A_{2},B)\not\in{\boldsymbol{S}}_{p}$.

**Proof. **By Theorem 4.6, for each positive integer $k$, there exist self-adjoint contractions $A_{1}^{(k)}$, $A_{2}^{(k)}$ and $B^{(k)}$ such that

[TABLE]

for every $k$. It remains to put $A_{1}=\bigoplus\limits_{k=1}^{\infty}A_{1}^{(k)}$, $A_{2}=\bigoplus\limits_{k=1}^{\infty}A_{2}^{(k)}$ and $B=\bigoplus\limits_{k=1}^{\infty}B^{(k)}$. $\blacksquare$

Let $\Lambda$ be a closed subset of ${\mathbb{R}}^{2}$. Denote by \big{(}B^{1}_{\infty,1}\big{)}_{\Lambda}\big{(}{\mathbb{R}}^{2}\big{)} the set of all functions $f$ in B^{1}_{\infty,1}\big{(}{\mathbb{R}}^{2}\big{)} such that $\operatorname{supp}\mathscr{F}f\subset\Lambda$. The seminorm $\|\cdot\|_{B^{1}_{\infty,1}({\mathbb{R}}^{2})}$ is a norm on \big{(}B^{1}_{\infty,1}\big{)}_{\Lambda}\big{(}{\mathbb{R}}^{2}\big{)} if and only if $0\notin\Lambda$.

Let $\Lambda$ be a closed subset of ${\mathbb{R}}^{2}$ and $r>0$ . Put $\tau_{\Lambda}(r)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\inf R$, where the infimum is taken over all $R>r$ such that the set $\Lambda\cap R{\mathbb{D}}$ contains a disk of radius $r$. In particular, $\tau_{\Lambda}(r)=\infty$ if $\Lambda$ contains no disk of radius $r$.

**      Theorem 4.8****.**

Let $\Lambda\subset{\mathbb{R}}^{2}$ and $2<p<\infty$. Suppose that $\liminf\limits_{r\to\infty}r^{\frac{1}{p}-\frac{3}{2}}\tau_{\Lambda}(r)=0$. Then there exists a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{R}}^{2})$ and self-adjoint operators $A_{1}$, $A_{2}$ and $B$ such that $A_{1}-A_{2}\in{\boldsymbol{S}}_{p}$ but $f(A_{1},B)-f(A_{2},B)\not\in{\boldsymbol{S}}_{p}$.

We need the following lemma.

**      Lemma 4.9****.**

Let $R>r\geq 1$ and let $D$ be a disk of radius $r$, $D\subset R{\mathbb{D}}$. Then there exist a function $f\in\mathscr{E}_{R}^{\infty}({\mathbb{R}}^{2})$ with $\operatorname{supp}\mathscr{F}f\subset D$ and positive self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that

[TABLE]

**Proof. **It suffices to consider the case when $r$ is a positive integer. Applying Corollary 4.2 to $N=r$, we get a function $f_{0}\in\mathscr{E}_{r}^{\infty}({\mathbb{R}}^{2})$ with $\|f\|_{L^{\infty}({\mathbb{R}}^{2})}=1$ and positive self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on a finite-dimensional Hilbert space such that $f_{0}(A_{2},B)={\boldsymbol{0}}$ and

[TABLE]

We can take $a\in{\mathbb{R}}^{2}$ such that $|a|<R$ in such a way that $a+r{\mathbb{D}}\subset D$. Put $f\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{{\rm i}(x,a)}f_{0}$. Then

[TABLE]

It remains to observe that $\|f\|_{B^{1}_{\infty,1}({\mathbb{R}}^{2})}\leq\operatorname{const}R\|f\|_{L^{\infty}({\mathbb{R}}^{2})}=\operatorname{const}R$. $\blacksquare$

**      Lemma 4.10****.**

Let $\Lambda\subset{\mathbb{R}}^{2}$ and let $2<p<\infty$. Suppose that $\liminf\limits_{r\to\infty}r^{\frac{1}{p}-\frac{3}{2}}\varkappa_{\Lambda}(r)=0$. Then there exists a function $f$ in $(B_{\infty,1}^{1})_{\Lambda}({\mathbb{R}}^{2})$ such that f\big{|}[-1,1]^{2}\not\in{\rm BOL}_{p}([-1,1]\times[-1,1]).

**Proof. **To prove the result, we can repeat the the argument in the proof of Lemma 4.4 and use Lemma 4.9 instead of Corollary 4.3. $\blacksquare$

This lemma implies Theorem 4.8 in the same a way as Lemma 4.4 implies Theorem 4.7.

Theorem 4.8 readily implies the following theorem.

**      Theorem 4.11****.**

Let $\Lambda\subset{\mathbb{R}}^{2}$ and $1\leq\alpha<\frac{3}{2}$. Suppose that $\tau_{\Lambda}(r)\leq\operatorname{const}r^{\alpha}$ for all sufficiently large $r$. Then for each $p>(\frac{3}{2}-\alpha)^{-1}$, there exists a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{R}}^{2})$ and self-adjoint operatrs $A_{1}$, $A_{2}$ and $B$ such that $A_{1}-A_{2}\in{\boldsymbol{S}}_{p}$ but $f(A_{1},B)-f(A_{2},B)\not\in{\boldsymbol{S}}_{p}$.

**      Corollary 4.12****.**

Let $\Lambda$ be a nondegenerate angle in ${\mathbb{R}}^{2}$. Then for each $p>2$, there exists a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{R}}^{2})$ and self adjoint operators $A_{1}$, $A_{2}$ and $B$ such that $A_{1}-A_{2}\in{\boldsymbol{S}}_{p}$ but $f(A_{1},B)-f(A_{2},B)\not\in{\boldsymbol{S}}_{p}$.

The following theorem is a version of Theorem 4.7 in the case $p=\infty$.

**      Theorem 4.13****.**

There exist a function $f$ in $B^{1}_{\infty,1}({\mathbb{R}}^{2})$ and self-adjoint contractions $A_{1}$, $A_{2}$ and $B$ on Hilbert space such that $A_{1}-A_{2}$ is compact but $f(A_{1},B)-f(A_{2},B)$ is not compact.

**Proof. **By Theorem 4.6, for each positive integer $k$, there exist self-adjoint contractions $A_{1}^{(k)}$, $A_{2}^{(k)}$ and $B^{(k)}$ on a finite-dimensional Hilbert space $\mathcal{H}_{k}$ such that such that

[TABLE]

for every $k$. It remains to put $A_{1}=\bigoplus\limits_{k=1}^{\infty}A_{1}^{(k)}$, $A_{2}=\bigoplus\limits_{k=1}^{\infty}A_{2}^{(k)}$, $B=\bigoplus\limits_{k=1}^{\infty}B^{(k)}$ and observe that $A_{1}-A_{2}$ is compact but $f(A_{1},B)-f(A_{2},B)$ is not compact. $\blacksquare$

In the same way we can get a version of Theorem 4.8 in the case $p=\infty$. The same is true about Theorem 4.11 and Corollary 4.12, which are special cases of Theorem 4.8.

5. The case of contractions

The purpose of this section is to obtain analogs of the results of § 4 for functions of pairs of noncommuting contractions. We obtain analogs of Theorems 4.7, 4.8, 4.11 and 4.13.

Actually, we are going to work with unitary operators and obtain analogs of Theorems 4.7, and 4.13 for unitary operators and functions that belong to the Besov class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) of functions analytic in the bidisk. Since unitary operators are contractions, we obtain thereby analogs of Theorems 4.7 and 4.13 for functions of contractions.

Let $p\in[1,\infty]$. Denote by ${\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}})$ the set of the functions $f$ of class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) such that

[TABLE]

for arbitrary unitary operators $U_{1}$, $U_{2}$, $V_{1}$ and $V_{2}$ on a finite-dimensional Hilbert space. We denote by $\|f\|_{{\rm OL}_{{\boldsymbol{S}}_{p}}({\mathbb{T}}\times{\mathbb{T}})}$ the smallest constant $c$, for which (5.1) holds.

It is easy to see that $f\in{\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}})$ if and only if there exists a constant $c$ such that

[TABLE]

for all unitary operators $U$, $U_{1}$, $U_{2}$, $V$, $V_{1}$ and $V_{2}$ on a finite-dimensional Hilbert space. Put

[TABLE]

Clearly, the space $\{f\in{\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}}):f(1,1)=0\}$ equipped with the norm $\|\cdot\|_{{\rm BOL}_{p}}$ is a Banach space.

**      Lemma 5.1****.**

Let $p\in(2,+\infty]$. Then there exists a function $f$ of class \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) such that $f\not\in{\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}})$.

**Proof. **Suppose that \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2})\subset{\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}}). It is easy to deduce from the closed graph theorem there exists a constant $c$ such that $\|f\|_{{\rm BOL}_{p}({\mathbb{T}}\times{\mathbb{T}})}\leq c\|f\|_{B^{1}_{\infty,1}({\mathbb{T}}^{2})}$. This contradicts Theorem 7.3 of [AP3]. $\blacksquare$

**      Corollary 5.2****.**

Let $p\in(2,+\infty]$. Then there exists a function f\in\big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) such that for every $M>0$, there exist unitary operators $U_{1}$, $U_{2}$ and $V$ on a finite-dimensional Hilbert space such that

[TABLE]

**      Lemma 5.3****.**

Let $U$ be a unitary operator such that $U-I\in{\boldsymbol{S}}_{p}$, where $p\in[1,\infty]$. Then there exists a self-adjoint operator $A$ such that $e^{{\rm i}A}=U$ and

[TABLE]

**Proof. **We can take a self-adjoint operator $A$ such that $e^{{\rm i}A}=U$ and $\|A\|\leq\pi$. Clearly, $\|e^{{\rm i}A}-I\|=|e^{{\rm i}\|A\|}-1|$. That implies the required result for $p=\infty$. Now let $p<\infty$. Clearly, $A$ is a compact self-adjoint operator. Let $\{s_{k}\}_{k=1}^{\infty}$ be a sequence of all eigenvalues taking in account the multiplicity. Then

[TABLE]

**      Corollary 5.4****.**

Let $1\leq p\leq\infty$ and let $U_{1}$ and $U_{2}$ be unitary operators such that $U_{1}-U_{2}\in{\boldsymbol{S}}_{p}$. Then for each positive integer $N$ there exists a sequence \big{\{}U^{[k]}\big{\}}_{k=0}^{N} of unitary operators such that $U^{[0]}=U_{1}$, $U^{[N]}=U_{2}$ and \big{\|}U^{[k]}-U^{[k-1]}\big{\|}_{{\boldsymbol{S}}_{p}}\leq\dfrac{\pi}{2N}\|U_{1}-U_{2}\|_{{\boldsymbol{S}}_{p}} for $k=1,2,\cdots,N$.

**Proof. **By Lemma 5.3, there exists a self-adjoint operator $A$ such that $e^{{\rm i}A}=U_{1}^{-1}U_{2}$ and $\|A\|_{{\boldsymbol{S}}_{p}}\leq\dfrac{\pi}{2}\|U_{2}U_{1}^{-1}-I\|_{{\boldsymbol{S}}_{p}}=\dfrac{\pi}{2}\|U_{1}-U_{2}\|_{{\boldsymbol{S}}_{p}}$. It remains to put $U^{[k]}=U_{1}e^{{\rm i}\frac{k}{N}A}$. $\blacksquare$

**      Theorem 5.5****.**

Suppose that $2<p\leq\infty$. Then there exists a function $f$ in \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) such that for any positive numbers $M$ and $\delta$, there exist a finite-dimensional Hilbert space and unitary operators $U_{1}$, $U_{2}$ and $V$ on it such that

[TABLE]

**Proof. **By Corollary 5.2 there exists a function f\in\big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) such that for every $M>0$, there exist unitary operators $U_{1}$, $U_{2}$ and $V$ such that

[TABLE]

We can take an integer $N$ such that $\dfrac{\pi}{2N}<\delta$. Applying Corollary 5.4, we get a sequence $\{U^{[k]}\}_{k=0}^{N}$ of unitary operators such that $U^{[0]}=U_{1}$, $U^{[N]}=U_{2}$ and

[TABLE]

It remains to prove that \big{\|}f\big{(}U^{[k]},V\big{)}-f\big{(}U^{[k-1]},V\big{)}\big{\|}_{{\boldsymbol{S}}_{p}}>M\big{\|}U^{[k]}-U^{[k-1]}\big{\|}_{{\boldsymbol{S}}_{p}} for some integer $k$ such that $1\leq k\leq N$.

Assume the contrary. Then

[TABLE]

and we get a contradiction. $\blacksquare$

**      Theorem 5.6****.**

Suppose that $2<p<\infty$. Then there exists a function $f$ in \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) and unitary operators $U_{1}$, $U_{2}$ and $V$ such that $U_{1}-U_{2}\in{\boldsymbol{S}}_{p}$ but $f(U_{1},V)-f(U_{2},V)\not\in{\boldsymbol{S}}_{p}$.

**Proof. **By Theorem 5.5, for each positive integer $k$, there exist unitary operators $U_{1,k}$, $U_{2,k}$ and $V_{k}$ such that

[TABLE]

for every $k$. It remains to put $U_{1}=\bigoplus\limits_{k=1}^{\infty}U_{1,k}$, $U_{2}=\bigoplus\limits_{k=1}^{\infty}U_{2,k}$ and $V=\bigoplus\limits_{k=1}^{\infty}V_{k}$. $\blacksquare$

Let $\Lambda$ be a subset of ${\mathbb{Z}}^{2}$. Put $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\{f\in B^{1}_{\infty,1}({\mathbb{T}}^{2}):\operatorname{supp}\widehat{f}\subset\Lambda\}$. For a positive integer $m$ we, denote by $\varkappa_{\Lambda}(m)$ the least positive integer $N$ such that there exist $n_{1},n_{2}\in{\mathbb{Z}}$ satisfying the following conditions

[TABLE]

Put $\varkappa_{\Lambda}(m)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\infty$ if there is no such $N$.

The same technique as above allows us to obtain the following result:

**      Theorem 5.7****.**

Let $\Lambda\subset{\mathbb{Z}}^{2}$ and $2<p<\infty$. Suppose that $\liminf\limits_{m\to\infty}m^{\frac{1}{p}-\frac{3}{2}}\varkappa_{\Lambda}(m)=0$. Then there exists a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})$ and unitary operators $U_{1}$, $U_{2}$ and $V$ such that $U_{1}-U_{2}\in{\boldsymbol{S}}_{p}$ but $f(U_{1},V)-f(U_{2},V)\not\in{\boldsymbol{S}}_{p}$.

To prove this theorem, we need the following result which is essentially Lemma 7.2 in [AP3]; see also its proof in [AP3].

**      Lemma 5.8****.**

For each $m\in\mathbb{N}$, there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable, and unitary operators $U_{1}$, $U_{2}$ and $V$ in ${\mathbb{C}}^{m}$ such that

[TABLE]

for every $p>0$.

Remark. Put $g(z_{1},z_{2})=z_{1}^{n_{1}}z_{2}^{n_{2}}f(z_{1},z_{2})$, where $n_{1},n_{2}\in{\mathbb{Z}}$. If $f$ satisfies (5.2), then $g$ also satisfies (5.2).

This remark implies the following result:

**      Lemma 5.9****.**

Let $\Lambda\subset{\mathbb{Z}}^{2}$ and let $2<p<\infty$. Suppose that

[TABLE]

for some integers $n_{1}$, $n_{2}$ such that $-N\leq n_{1},n_{2}\leq N-m$ and

[TABLE]

for some positive integers $m$ $N$. Then there exist a trigonometric polynomial $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})$ and unitary operators $U_{1}$, $U_{2}$ and $V$ on a finite-dimensional Hilbert space such that

[TABLE]

Sketch of the proof of Lemma 5.9. It suffices consider the case where $m\in 16\mathbb{N}$. Let $m=16m_{0}$. Note that there exists a “square”

[TABLE]

that is contained in the “square”

[TABLE]

and such that $\operatorname{dist}((0,0),Q^{\prime})\leq\operatorname{const}m$. It remains to apply the remark after Lemma 5.8 and to observe that $\|g\|_{B^{1}_{\infty,1}}\leq\operatorname{const}N\|g\|_{L^{\infty}({\mathbb{T}}^{2})}$ for any trigonometrical polynomial $g$ with $\operatorname{supp}\widehat{g}\subset Q^{\prime}$. $\blacksquare$

**      Corollary 5.10****.**

Under the hypotheses of Theorem 5.7, for every $M>0$, there exist a function $f\in(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})$ and unitary operators $U_{1}$, $U_{2}$ and $V$ on a finite-dimensional Hilbert space such that

[TABLE]

Theorem 5.7 can be deduced now from Corollary 5.10 in the same way as Theorem 5.6 was deduced from Theorem 7.3 in [AP3].

Theorem 5.7 readily implies the following result:

**      Theorem 5.11****.**

Let $\Lambda\subset{\mathbb{Z}}^{2}$ and $1\leq\alpha<\frac{3}{2}$. Suppose that $\varkappa_{\Lambda}(m)\leq\operatorname{const}m^{\alpha}$ for all $m\geq 1$. Then for each $p>(\frac{3}{2}-\alpha)^{-1}$, there exists a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})$ and unitary operators $U_{1}$, $U_{2}$ and $V$ such that $U_{1}-U_{2}\in{\boldsymbol{S}}_{p}$ but $f(U_{1},V)-f(U_{2},V)\not\in{\boldsymbol{S}}_{p}$.

**      Corollary 5.12****.**

Let $X$ be a nondegenerate angle in ${\mathbb{R}}^{2}$. Let $\Lambda\subset{\mathbb{Z}}^{2}$. Suppose that $\Lambda\supset X\cap{\mathbb{Z}}^{2}$. Then for each $p>2$, there exist a function $f$ in $(B^{1}_{\infty,1})_{\Lambda}({\mathbb{T}}^{2})$ and unitary operators $U_{1}$, $U_{2}$ and $V$ such that $U_{1}-U_{2}\in{\boldsymbol{S}}_{p}$ but $f(U_{1},V)-f(U_{2},V)\not\in{\boldsymbol{S}}_{p}$.

In the same way as in the case of self-adjoint operators (see § 4) all main results of this section have natural versions in the case $p=\infty$. For example, the corresponding version of Theorem 5.6 in the case $p=\infty$ can be formulated as follows.

**      Theorem 5.13****.**

There exist a function $f$ in \big{(}B_{\infty,1}^{1}\big{)}_{+}({\mathbb{T}}^{2}) and unitary operators $U_{1}$, $U_{2}$ and $V$ such that $U_{1}-U_{2}$ is compact but $f(U_{1},V)-f(U_{2},V)$ is not compact.

6. The case of triples of noncommuting self-adjoint operators

A slight modification of the proof of Theorem 4.1 of [Pe5] shows that if $1\leq p\leq\infty$, for $p\in[1,\infty]$, there exist a sequence $\{f_{N}\}$ of functions in ${\mathscr{E}}_{2\pi}^{\infty}({\mathbb{R}}^{3})$, sequences \big{\{}A^{(N)}\big{\}}, \big{\{}B^{(N)}\big{\}}, \big{\{}C_{1}^{(N)}\big{\}}, and \big{\{}C_{2}^{(N)}\big{\}} of self-adjoint operators of rank at most $N$ such that

[TABLE]

and

[TABLE]

Recall that we have defined in § 3 functions $f(A,B,C)$ of arbitrary bounded self-adjoint operators $A$, $B$ and $C$ for functions of class $B_{\infty,1}^{2}({\mathbb{R}}^{3})$.

The method used in § 4 allows us to prove the following result.

**      Theorem 6.1****.**

Let $1<p<\infty$. Then there exist a function $f$ in $B_{\infty,1}^{2}({\mathbb{R}}^{3})$ and triples self-adjoint operators $(A_{1},B_{1},C_{1})$ and $(A_{2},B_{2},C_{2})$ such that $A_{2}-A_{1}\in{\boldsymbol{S}}_{p}$, $B_{2}-B_{1}\in{\boldsymbol{S}}_{p}$, $C_{2}-C_{1}\in{\boldsymbol{S}}_{p}$ but

[TABLE]

Note that as in § 4, one can prove a similar result if we replace ${\boldsymbol{S}}_{p}$ with the class of compact operators.