Russian Mathematical Surveys, 2024, Volume 79, Issue 1, Pages 1–52 DOI: https://doi.org/10.4213/rm10140e (Mi rm10140)

DOI: https://doi.org/10.4213/rm10140e

Funding agency	Grant number
Russian Science Foundation	23-11-00153
The research in § § 6 and 7 was supported by the Russian Science Foundation (grant no. 23-11-00153, https://rscf.ru/en/project/23-11-00153/).

Russian version:
Uspekhi Matematicheskikh Nauk, 2024, Volume 79, Issue 1(475), Pages 3–58
DOI: https://doi.org/10.4213/rm10140

Bibliographic databases:

1. Introduction

In this survey we discuss the important role played by the Besov classes in operator theory. We are going to demonstrate numerous situations when Besov classes are used in various issues of operator theory.

The Besov spaces were introduced in Besov’s paper [23]. It turned out that they play a very noticeable role in embedding theorems, approximation theory, interpolation theorems and many other divisions of function theory; see [24], [66], [67], and [105].

However, the importance of Besov spaces is not exhausted by the above. Starting in the 1980s, Besov classes have been playing a more and more important role in operator theory. It is the role of Besov spaces in operator theory which is the principal subject of the survey.

We start with the Besov spaces $B_p^{1/p}(\mathbb{T})$, $0<p<\infty$, of functions on the unit circle $\mathbb{T}$, which appear while describing the Hankel operators of Schatten–von Neumann classes $\boldsymbol{S}_p$. We also consider certain applications and generalizations of these results, such as commutators of multiplication operators and the Hilbert transform, Wiener–Hopf operators on a finite interval, and the Ibragimov–Solev problem in the theory of stationary ransom processes.

Then we consider the problem of estimating the norms of polynomials of power bounded operators on Hilbert space. This problem is closely related to certain other Besov spaces of functions on the circle $\mathbb{T}$ and to Hankel matrices in tensor products of the spaces $\ell^1$ and $\ell^\infty$. This leads in turn to a solution of Mazur’s Problems 8 and 88 from the Scottish Book [60].

The final part of the survey is devoted to applications of Besov spaces in the theory of perturbations of operators on Hilbert space. We are going to consider various problems related to the behaviour of functions of operators or functions of collections of operators under perturbations.

2. Definitions of Besov classes

In this survey we are going to deal with Besov space on the unit circle (in other words, with Besov spaces of periodic function on the real line) and with Besov spaces on Euclidean spaces. We start with the case of the unit circle. Moreover, we consider the case of the unit circle in slightly more detail than the case of Euclidean space because Besov spaces on Euclidean spaces were discussed in many textbooks and monographs. We also observe that the definitions and properties of Besov spaces on the unit circle $\mathbb{T}$ can easily be generalized to the case of Besov spaces on the torus $\mathbb{T}^d$ for an arbitrary natural number $d$.

2.1. Besov classes on the unit circle

Let $w$ be an infinitely differentiable function on $\mathbb{R}$ such that

We define the trigonometric polynomials $W_n$, $n\geqslant0$, by

Recall that a distribution on $\mathbb{T}$ is a continuous linear functional on the space of infinitely differentiable functions on the circle. The class $L^1(\mathbb{T})$ can naturally be embedded in the space of distributions: a function $g$ in $L^1(\mathbb{T})$ determines the linear functional

In the case when $p,q\geqslant1$, $B_{p,q}^s(\mathbb{T})$ is a Banach space. It can be endowed with various equivalent norms. In particular, we can put

If $p=q$, then it is common to use the notation $B_p^s(\mathbb{T})\overset{\rm def}{=} B_{p,p}^s(\mathbb{T})$.

The dual space of the Besov space $B_{p,q}^s(\mathbb{T})$ in the case when $p,q\geqslant1$ and $q<\infty$, can be described in the following way:

Let us proceed now to the description of the Besov spaces $B_{p,q}^s(\mathbb{T})$ for $s>0$ in terms of finite differences. Suppose for simplicity that $p\geqslant1$ and $q\geqslant1$. We define the difference operator $\Delta_\tau$, $\tau\in\mathbb{T}$, on the set of functions on $\mathbb{T}$ by

Suppose that $s>0$, $1\leqslant p\leqslant\infty$, and $1\leqslant q\leqslant\infty$. Then the following description of the spaces $B_{p,q}^s(\mathbb{T})$ holds:

We proceed now to the description of Besov spaces in terms of harmonic extension to the unit disc. Let $s\in\mathbb{R}$, and let $m$ be a non-negative integer such that $m>s$. Then a distribution $f$ on the circle $\mathbb{T}$ belongs to the space $B_{p,q}^s(\mathbb{T})$ if and only if

Consider the Riesz projections $\mathbb{P}_+$ and $\mathbb{P}_-$ defined on the class of all distributions on $\mathbb{T}$ by

It follows easily from the definition of Besov classes in terms of convolutions with the $W_n$ that

Let us mention here the book [79], which contains an introduction to Besov spaces on the unit circle.

2.2. Besov classes on Euclidean spaces

Let $d$ be a positive integer, and let $w$ be an infinitely smooth function on $\mathbb{R}$ satisfying requirements (2.1). We define the functions $W_n$, $n\in\mathbb{Z}$, on $\mathbb{R}^d$ by

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

Now we can define the modified (homogeneous) Besov space $B^s_{p,q}(\mathbb{R}^d)$. We say that a distribution $f$ belongs to the Besov class $B^s_{p,q}(\mathbb{R}^d)$ if condition (2.9) is satisfied and

In the case $p=q$ we use the notation $B_p^s(\mathbb{R}^d)$ for $B_{p,p}^s(\mathbb{R}^d)$.

Consider now the scale $\Lambda_\alpha(\mathbb{R}^d)$, $\alpha>0$, of Hölder–Zygmund classes. They can be defined by

The Besov classes admit many other descriptions. Let us state the description in terms of finite differences. For $h$ in $\mathbb{R}^d$ we define the difference operator $\Delta_h$ by

Let $s>0$, $m\in\mathbb{Z}$, and $m-1\leqslant s<m$. Let $p,q\in[1,+\infty]$. The Besov class $B_{p,q}^s(\mathbb{R}^d)$ can be defined as the space of functions $f$ in $L^1_{\rm loc}(\mathbb{R}^d)$ such that

The space $B_{pq}^s(\mathbb{R}^d)$ can be defined in terms of the Poisson integral. Let $P_d(x,t)$ be the Poisson kernel that solves the Dirichlet problem for the half-space $\mathbb{R}^{d+1}_+\overset{\rm def}=\{(x,t)\colon x\in\mathbb{R}^d, t>0\}$, that is,

Let $m\in\mathbb{Z}$, $s>0$, $m-1\leqslant s<m$, and $1\leqslant p,q\leqslant+\infty$. Then we can define the space $B_{pq}^s(\mathbb{R}^d)$ as the space of functions $L^1\bigl(\mathbb{R}^d,(\|x\|+1)^{-(d+m+1)}\,{\rm d} x\bigr)$ such that

2.3. Inhomogeneous Besov classes on Euclidean spaces

As a rule, it is homogeneous Besov spaces that appear in operator theory. However, in the recent papers [16] and [14] inhomogeneous Besov spaces also appear.

We define the function $W^{[0]}$ by

Note that this notation for inhomogeneous Besov classes was adopted in [16] and [14], and it is somewhat different from the notation commonly used. Nevertheless, it is convenient.

It is well known and easy to see that

We refer the reader to the books [67], [24], [66], and [105] for more detailed information on Besov spaces.

3. Introduction to Schatten–von Neumann classes

In this section we give a brief introduction to singular values of operators in Hilbert space and to Schatten–von Neumann classes. We refer the reader to [34] and [28] for the proofs of the facts presented below and for more detailed information.

For a bounded linear operator $T$ on Hilbert space, its singular values $s_j(T)$, $j\geqslant0$, are defined by

The Schatten–von Neumann class $\boldsymbol{S}_p$, $0<p<\infty$, consists by definition of the operators $T$ for which

For $p<1$ the functional $\|\,{\cdot}\,\|_{\boldsymbol{S}_p}$ is not a norm: it is a $p$-norm, that is, the following inequality holds:

The class $\boldsymbol{S}_1$ is called the trace class. If $T$ is a trace class operator on Hilbert space $\mathscr{H}$, then its trace $\operatorname{trace} T$ is defined by

The class $\boldsymbol{S}_2$ is called the Hilbert–Schmidt class. It forms a Hilbert space with inner product

For $p\in(1,\infty)$, the dual space $(\boldsymbol{S}_p)^*$ can isometrically be identified with the space $\boldsymbol{S}_{p'}$, $1/p+1/p'=1$, with respect to the bilinear form

The space dual to $\boldsymbol{S}_1$ can be identified with the space of all bounded linear operators with respect to the same bilinear form, while the space dual to the space of compact operators can be identified with $\boldsymbol{S}_1$.

4. Hankel operators and Besov classes

4.1. Classical Hankel operators and Hankel matrices

Besov classes were used to study Hankel operators in [70]. Hankel operators on the Hardy class^2[x]2We refer the reader to [33], [51], and [65] for the definitions and basic properties of the Hardy classes $H^p$ as well as the classes $\mathrm{BMO}$ and $\mathrm{VMO}$. $H^2$ of functions analytic in $\mathbb{D}$ can be defined in the following way. Let $\varphi$ be a function of class $L^2=L^2(\mathbb{T})$. On the set of analytic polynomials $\mathscr{P}_+$ the Hankel operator $H_\varphi$ from the Hardy space $H^2$ to the space $H^2_-\overset{\rm def}{=} L^2\ominus H^2$ is defined by

A compactness criterion for Hankel operators is given by Hartman’s theorem: if $\varphi\in L^\infty$, then $H_\varphi$ is compact if and only if $\varphi\in H^\infty+C$, that is, $\varphi$ can be represented as $\varphi=g+h$, where $g\in H^\infty$ and $g$ is a continuous function on $\mathbb{T}$. In combination with a theorem due to Sarason, this leads to the following criterion: the operator $H_\varphi$ is compact if and only if the function $\mathbb{P}_-\varphi$ belongs to the space $\mathrm{VMO}$ of functions on $\mathbb{T}$ of vanishing mean oscillation.

Detailed information on Hankel operators and their applications can be found in the book [79]; see also the book [65].

The Hankel operator $H_\varphi\colon H^2\to H^2_-$ in the orthonormal bases $\{z^j\}_{j\geqslant0}$ of the space $H^2$ and $\{\overline z^j\}_{j>0}$ of the space $H^2_-$ has the infinite matrix

We proceed now to conditions under which Hankel operators belong to the Schatten–von Neumann classes $\boldsymbol{S}_p$. The following theorem describes the Hankel operators $H_\varphi$ that belong to the Schatten–von Neumann classes $\boldsymbol{S}_p$.

Theorem 4.1 can be reformulated in the following way:

Applying the description of Besov classes (2.5) to Besov classes of analytic functions, we obtain the following description of the operators $\Gamma_\alpha$ of class $\boldsymbol{S}_p$.

In particular, taking $p=1$ we obtain the following description of trace class operators $\Gamma_\alpha$.

Note that the description of the trace class Hankel operators was obtained in [70]. This solved an old well-known problem; in particular, it was mentioned by Rosenblum [97], Krein [48], and Holland (see [18]).

The case $1<p<\infty$ was also covered in [70]. In the case $0<p<1$ a description of Hankel operators of class $\boldsymbol{S}_p$ was obtained in [72]. Note that other approaches in the case $0<p<1$ was offered by Pekarskii [68] and Semmes [103].

The framework of this survey does not allow us to give a complete proof of Theorem 4.1. We restrict ourselves to the proof of the sufficiency of the condition $\mathbb{P}_-\varphi\in B_p^{1/p}(\mathbb{T})$ for $0<p<1$, and we also discuss how it is possible to obtain the necessity of this condition in the case $p=1$.

For $p=1$, there are several approaches to the proof of the necessity of the condition $\alpha\in\big(B_1^1\big)_+$. One of them is selecting special bounded operators $B$ an $\ell^2$ and calculating the traces $\operatorname{trace}\Gamma_\alpha B$ (see [70] and [79]). Another one is constructing a certain special bounded projection ${\mathscr Q}$ from $\boldsymbol{S}_1$ onto the subspace of trace class Hankel operators (see [79], Chap. 6, § 5).

Note here that the averaging projection $\mathscr{P}$ onto the Hankel matrices defined by

We refer the reader to [72] and [79] for the proof of the necessity of the condition $\mathbb{P}_-\varphi\in B_p^{1/p}(\mathbb{T})$, $0<p<1$, in Theorem 4.1. The proof is rather difficult and is based on explicit computations of the $\boldsymbol{S}_p$-quasinorms of convolution operators on cyclic groups.

In the case $1<p<\infty$ the proof of Theorem 4.1 given in [70] was obtained with the help of Marcinkiewicz’s interpolation theorem. Moreover, the reasoning given in the proof implies that the averaging projection $\mathscr{P}$ onto the Hankel matrices is bounded on $\boldsymbol{S}_p$ for $1<p<\infty$. We refer the reader to the book [79], in which all such questions are discussed in detail.

4.2. Generalized Hankel matrices

Let $\alpha$ and $\beta$ be real numbers, and let $\varphi$ be a function analytic in the disc $\mathbb{D}$. Consider generalized Hankel matrices of the form

The following theorem describes bounded and compact operators of the form $\Gamma_\varphi^{\alpha,\beta}$ for $\alpha\geqslant0$ and $\beta\geqslant0$.

The following theorem describes operators $\Gamma_\varphi^{\alpha,\beta}$ in the Schatten–von Neumann class $\boldsymbol{S}_p$.

We refer the reader to [79] for proofs of the above assertions and for more detailed discussions.

5. Some applications of the criterion of the membership of Hankel operators in classes $\boldsymbol{S}_p$

In this section we consider some applications of the description of the Hankel operators of class $\boldsymbol{S}_p$ in terms of Besov classes. The reader can learn applications in more detail in the book [79].

5.1. A characterization of Besov classes in terms of rational approximation

In this subsection we discuss how the Besov classes $(B_p^{1/p})(\mathbb{T})$ can be described in terms of rational approximation in the norm of the space $\mathrm{BMO}$.

First of all, we need the oldest theorem on Hankel matrices, that is, Kronecker’s theorem [50] on finite-rank Hankel matrices, which implies that a Hankel operator $H_\varphi\colon H^2\to H^2_-$ has rank $k$ if and only if $\mathbb{P}_-\varphi$ is a rational function of degree $k$.^3[x]3The degree $\deg p/q$ of the rational function $p/q$ in the case when $p$ and $q$ are polynomials that have no non-trivial common divisor is defined as $\max\{\deg p,\deg q\}$.

We are also going to use the profound Adamjan–Arov–Krein theorem [1], which asserts that in the case when $T$ is a Hankel operator, it suffices to consider exclusively Hankel operators $R$ of rank at most $j$ on the right-hand side of (3.1), that is,

A combination of Kronecker’s theorem with the Adamjan–Arov–Krein theorem, together with Nehari’ and Fefferman’s theorems mentioned above, allows us to obtain the following description of the Besov classes $(B_p^{1/p})(\mathbb{T})$ in terms of rational approximation in the norm of $\mathrm{BMO}$.

Let $\varphi\in L^\infty$ and $j\in\mathbb{Z}_+$. Consider the deviations $\rho_n(\varphi)$ of the function $\varphi$ from the set ${\mathcal R}_n$ of rational functions of degree at most $n$ with poles off $\mathbb{T}$ in the norm of $\mathrm{BMO}$:

Theorem 5.1 was obtained in [70] and [72]. We refer the reader to [79], Chap. 6, § 6, where these issues are discussed in detail and other related results are given.

5.2. Commutators of multiplication operators and the Riesz projection

For a function $\varphi$ of class $L^2$ the multiplication operator $M_\varphi$ is defined on the dense subset $\mathscr{P}$ of trigonometric polynomials by $M_\varphi f\overset{\rm def}{=}\varphi f$, $f\in\mathscr{P}$. Consider now the commutator ${\mathscr C}_\varphi$ of the operator $M_\varphi$ and the Riesz projection $\mathbb{P}_+$ defined by

Let us express now the commutator ${\mathscr C}_\varphi$ in terms of Hankel operators. It is convenient to use the notation $f_+\overset{\rm def}{=}\mathbb{P}_+f$ and $f_-\overset{\rm def}{=}\mathbb{P}_-f$.

Since the Hankel operator $H_{\varphi_-}$ acts from $H^2$ to $H^2_-$ and the operator $\bigl(H_{\overline{\varphi_+}}\bigr)^*$ acts from $H^2_-$ to $H^2$, it is easy to see that the commutator ${\mathscr C}_\varphi$ is bounded (compact) if and only if both the Hankel operators $H_{\varphi_-}$ and $H_{\overline{\varphi_+}}$ are bounded (compact). This implies some well-known results in harmonic analysis: the boundedness of the commutator ${\mathscr C}_\varphi$ is equivalent to the membership of $\varphi$ in the class $\mathrm{BMO}$, while the compactness of ${\mathscr C}_\varphi$ is equivalent to the membership of $\varphi$ in the class $\mathrm{VMO}$.

In the same way we can consider the problem of the membership of the commutator ${\mathscr C}_\varphi$ in the Schatten–von Neumann classes. Thus, the following criterion holds.

We refer the reader to the book [79] for more detailed information.

5.3. Hankel operators as integral operators on $L^2[0,\infty)$; commutators of multiplication operators and the Hilbert transform

For a function $k$ of class $L^1(\mathbb{R}_+)$ the integral Hankel operator $\boldsymbol\varGamma_k$ on the space $L^2(\mathbb{R}_+)$ is defined by

To state the results we need the notions of a distribution and a tempered distribution.

For an open subset $\Omega$ of the real line $\mathbb{R}$ we consider the space ${\mathcal D}(\Omega)$ of infinitely differentiable functions on $\Omega$ with compact support, and we equip this space with the natural topology of inductive limit. The space ${\mathcal D}'(\Omega)$ of distributions on $\Omega$ is defined as the space of continuous linear functionals on ${\mathcal D}(\Omega)$. We refer the reader to Schwartz’s book [102] for an introduction to the theory of distributions.

To define the space of tempered distributions, we consider the Schwartz space ${\mathcal S}$ of infinitely differentiable functions $f$ on $\mathbb{R}$ such that

It is well known that the Fourier transform $\mathscr{F}$ and the inverse Fourier transform $\mathscr{F}^*$ defined on the space $L^1(\mathbb{R})$ by

The following result holds.

We refer the reader to the book [79], Chap. 1, § 8, where such issues are discussed in detail and, in particular, a compactness criterion for the integral operators $\boldsymbol\varGamma_k$ is given.

Consider now the operator ${\mathcal U}$ defined on the space $L^2(\mathbb{T})$ by

The following assertion holds.

The proofs of the above statements can be found in [79], Chap. 1, § 8.

Let us now state a necessary and sufficient condition for an integral operator $\boldsymbol\varGamma_k$ to belong to the Schatten–von Neumann classes $\boldsymbol{S}_p$.

We refer the reader for details to [79], Chap. 6, § 7.

Theorem 5.6 allows us to obtain an analogue of Theorem 5.3 in the case of commutators of multiplication and the Hilbert transform. For a function $\varphi$ of class $\mathrm{BMO}(\mathbb{R})$ we consider the operator $\boldsymbol{\mathcal C}_f$ defined by

5.4. Wiener–Hopf operators on a finite interval

Let $\sigma>0$. We define the Wiener–Hopf operators on the space $L^2[-\sigma,\sigma]$ in the following way. Let $k$ be a locally integrable function on $(-2\sigma,2\sigma)$ (or even a distribution on $(-2\sigma,2\sigma)$). Consider the operator $W_{\sigma,k}$ defined on the dense subset of $L^2[-\sigma,\sigma]$ consisting of infinitely smooth functions $f$ with compact support in $(-\sigma,\sigma)$ by

Together with the operators $W_{\sigma,k}$ we are going to consider so-called truncated Hankel operators $\boldsymbol{\varGamma}_{\sigma,k}$ on $L^2[-\sigma,\sigma]$ defined by

To state the description of the operators $W_{\sigma,k}$ and $\boldsymbol{\varGamma}_{\sigma,k}$ of class $\boldsymbol{S}_p$, we define the following functions:

The following theorem was proved in [75] (note that in [96] the same result was established by different methods in the case when $p\geqslant1$).

In conclusion, we note that there are many generalizations of the results in §§ 4 and 5, in which Besov classes appear in the description of certain operators that belong to $\boldsymbol{S}_p$. Some of these results are listed in the references of [79]. However, apart from them, there are many other publications that appeared after the publication of [79].

5.5. The Ibragimov–Solev problem on stationary random processes

The Ibragimov–Solev problem stated in the note [39] concerns the regularity properties of stationary processes. The problem is to find out when the product of the projections onto the Past and onto the Future belongs to the Schatten–von Neumann class $\boldsymbol{S}_p$, $1\leqslant p<\infty$.

Without entering into details of the theory of stationary processes, we note that we understand by a stationary process (in the broad sense) a sequence of random variables $\{X_n\}_{n\in\mathbb{Z}}$ such that the mathematical expectation ${\mathsf E}X_jX_k$ depends only on the difference ${j-k}$. Let

We are going to consider regular stationary processes, that is, processes for which

Suppose that we have a regular stationary process with spectral density $w$. Consider now the orthogonal projections $\mathscr{P}_0$ and $\mathscr{P}^n$ onto the past and the future starting with moment $n$. The Ibragimov–Solev problem was to describe the spectral densities of regular stationary processes for which the operators $\mathscr{P}_0\mathscr{P}^n$ belong to the Schatten–von Neumann class $\boldsymbol{S}_p$. The following theorem gives a solution to this problem; see [70] and [87].

Note that for $p=2$ this assertion was proved earlier by Ibragimov and Solev [38], but the methods of [38] do not work for ${p\ne2}$.

6. Power bounded operators and the tensor products $\ell^\infty\mathbin{\widehat{\otimes}}\ell^\infty$ and $\ell^1\mathbin{\check{\otimes}}\ell^1$; Masur’s problems from the Scottish Book

This section is devoted to finding estimates of polynomials of power bounded operators on Hilbert space. It turns out that the resulting norms are closely related to the tensor products $\ell^\infty\mathbin{\widehat{\otimes}}\ell^\infty$ and $\ell^1\mathbin{\check{\otimes}}\ell^1$. Moreover, the problem of the description of Hankel matrices of class $\ell^1\mathbin{\check{\otimes}}\ell^1$ appears and we obtain such a description in terms of a Besov class. We also study the properties of the averaging projection onto the space of Hankel matrices in such norms. In conclusion we will see that such results lead to the solution of Mazur’s problems 8 and 88 from the Scottish Book.

6.1. Estimates of operator polynomials

We start this subsection with the problem of estimating functions of power bounded operators on Hilbert space. We are going to discuss some of such estimates. In particular, we will see that the norm of a polynomial of a power bounded operator can be estimated in terms of the norm of the polynomial in the Besov class $(B_{\infty,1}^0)_+$.

Thus, let $T$ be a power bounded operator on Hilbert space, that is,

The paper [71] was devoted to the problem of finding most optimal estimates for the norms $\|\varphi(T)\|$ under condition (6.1). In other words, the problem is to estimate the norm

To state the results we introduce some tensor products. We define the projective tensor product $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$ as the space of matrices $\{\gamma_{jk}\}_{j,k\geqslant0}$ that admit a representation

Consider now the tensor algebra $V^2$ (Varopoulos algebra) as the weak completion of the space $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$ in the following sense. We define the projections $P_m$, $m\in\mathbb{Z}_+$, onto the space of all infinite matrices by

It is easy to see that if $A=\{a_{jk}\}_{j,k\geqslant0}\in V^2$, then $A$ is a Schur multiplier of the space ${\mathcal B}(\ell^2)$ of matrices of bounded linear operators on $\ell^2$, that is,

Let us now define the injective tensor product $\ell^1\mathbin{\check{\otimes}}\ell^1$ of two copies of $\ell^1$ as the completion of the algebraic tensor product $\ell^1\otimes\ell^1$ in the norm

Let us now define the space ${\mathscr L}$ of functions $f$ analytic in the disc $\mathbb{D}$ for which there exists a matrix $\{\gamma_{jk}\}_{j,k\geqslant0}$ in $\ell^1\mathbin{\check{\otimes}}\ell^1$ such that

To present the estimate obtained in [71] of the norms of polynomials of power bounded operators in terms of the norm of the polynomial in ${\mathscr L}$, we recall Grothendieck’s inequality [35]. Let $\{\gamma_{jk}\}_{j,k\geqslant0}\in\ell^1\mathbin{\check{\otimes}}\ell^1$, and let $\{x_j\}_{j\geqslant0}$ and $\{y_k\}_{k\geqslant0}$ be vectors in the unit ball of a Hilbert space. Then

The following result was obtained in [71].

Let us proceed now to more tangible estimates of the norms of operator polynomials.

We denote by $\mathfrak{M}H^1$ the class of multipliers of $H^1$. By the norm of a function $\psi$ in $\mathfrak{M}H^1$ we mean the norm of the operator $f\mapsto\psi*f$ on $H^1$. It is easy to see that $\psi\in\mathfrak{M}H^1$ if and only if $\psi$ is a multiplier of the space $\mathrm{BMOA}$ (see (4.2)).

Theorem 6.1 implies easily the following estimate.

Let us define now the space $\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1$ of functions analytic in the disc $\mathbb{D}$ by

Now we are in a position to state the result of [71] on estimates of polynomials of power bounded operators in terms of the norms of polynomials in $\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1$.

Theorem 6.3 follows easily from (6.7) and (6.8).

Finally, we are able to estimate the norms of polynomials of power bounded operators in terms of the norms of the polynomials in the Besov space $(B^0_{\infty,1})_+$.

A natural question arises of whether the above estimates for the norms of polynomials of power bounded operators are optimal. Suppose that $\|\,{\cdot}\,\|_\flat$ is a norm on the set of analytic polynomials and we have the estimate

Note that it was shown in [71] that $\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1$ and $(B^0_{\infty,1})_+$ are Banach algebras with respect to pointwise multiplication. Bennett showed in his review MR0658618 of [71], published in Mathematical Reviews, that the same is true for the space ${\mathscr L}$.

It turns out that $(B^0_{\infty,1})_+$ is not an operator algebra. Otherwise the norms $\|\,{\cdot}\,\|_{\mathscr L}$, $\|\,{\cdot}\,\|_{\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1}$, and $\|\,{\cdot}\,\|_{B^0_{\infty,1}}$ would be equivalent. It is easy to show (see [71]) that this would mean that $(B^0_{\infty,1})^*=B^0_{1,\infty}=\mathfrak{M}H^1$. However, it is well known (see the references in [71]) that $\mathfrak{M}H^1$ is a proper subset of the space $B^0_{1,\infty}$.

We would like to mention here the papers [20], [36], and [106], in which the functional calculus for power bounded operators on the Besov class $(B^0_{\infty,1})_+$ is generalized to semigroups of operators.

6.2. Hankel matrices and tensor products of the spaces $\ell^1$ and $\ell^\infty$

Let us now return to estimate (6.6). On the right-hand side of inequality (6.6) one has to minimize $\|\{\gamma_{jk}\}_{j,k\geqslant0}\|_{\ell^1\mathbin{\check{\otimes}}\ell^1}$ over all matrices $\{\gamma_{jk}\}_{j,k\geqslant0}$ in $\ell^1\mathbin{\check{\otimes}}\ell^1$ satisfying the condition

Therefore, if we make the choice (6.9), then we obtain the estimate

Let us return now to the averaging projection $\mathscr{P}$ onto the set of Hankel matrices, defined by (4.6). The following result obtained in [71] describes the image of the space $V^2$ under this projection.

This implies easily the following fact (see [71]).

6.3. Problems 8 and 88 from the Scottish Book

In this subsection we are going to see how the results of the previous subsection allow one to solve Problems 8 and 88 from the famous Scottish Book (see [60]). Both problems were posed by S. Mazur.

In [52] and [31] solutions to both problems were given.

It turned out, however, that it follows right away from the earlier paper [71] that both Problems 8 and 88 have negative solutions (see [81], in which a detailed discussion is presented).

Let us proceed to the discussion of the solution to Problem 8. Consider a linear operator ${\mathscr A}$ defined on the set of matrices $Q=\{q_{jk}\}_{j,k\geqslant0}$ and taking matrices to sequences of complex numbers:

Theorem 6.6 of this survey, obtained in [71], can be reformulated in the following way.

This theorem implies that

Let us now proceed to Problem 88. First of all, it follows from Theorem 6.5 in this survey that

Note also that in [69] certain quantitative estimates were obtained that are related to Problems 8 and 88, as well as some related problems were posed. In [81] it was shown that quantitative estimates also follow from the results of [71], while the problems posed in [69] can easily be solved with the help of the results of [71].

7. Besov classes in perturbation theory

7.1. Functions of self-adjoint operators under perturbations of the operators

Let $f$ be a continuous complex-valued function on the real line. Consider the problem of when the inequality

It is well known (see [10]) that if inequality (7.1) holds for all bounded self-adjoint operators, then it also holds for unbounded self-adjoint operators.

Obviously, operator Lipschitz functions must be Lipschitz functions, that is, the following inequality must be satisfied:

Moreover, as became clear later, operator Lipschitz functions are necessarily differentiable everywhere ([107], see also [10]) but not necessarily continuously differentiable ([41], see also [10]). In particular, the function $x\mapsto|x|$ is not operator Lipschitz.

Next, in [73] and [76] necessary conditions for operator Lipschitzness were obtained. In particular, if $f$ is an operator Lipschitz function, then $f$ belongs to the Besov class $B_1^1(\mathbb{R})$ locally. This necessary condition also shows that the conditions of continuous differentiability and the boundedness of the derivative are not sufficient for operator Lipschitzness. We refer the reader to [10] for other necessary conditions.

In the author’s papers [73] and [76] it was shown that the Besov class $B_{\infty,1}^1(\mathbb{R})$ consists of operator Lipschitz functions. In other words, the following assertion holds.

To prove Theorem 7.1 double operator integrals were used in [73] and [76]. Double operator integrals are expressions of the form

Double operator integrals appeared first in [30]. Later, in the series of papers [25]–[27] Birman and Solomyak created a beautiful theory of double operator integrals. In particular, their approach allows one to define double operator integrals of the form (7.2) for arbitrary operators $Q$ of Hilbert–Schmidt class $\boldsymbol{S}_2$; in this case the double operator integral must also belong to $\boldsymbol{S}_2$ and the following inequality holds:

There are several descriptions of the class of Schur multipliers; see [73]. We mention here the following one.

Note here that if $\Phi$ is a Schur multiplier with respect to spectral measures $E_1$ and $E_2$, then

The role of double operator integrals in obtaining operator Lipschitz estimates can be demonstrated by the following result by Birman and Solomyak [25].

It follows that if the divided difference ${\mathfrak D} f$ is a Schur multiplier with respect to all pairs of Borel spectral measures on $\mathbb{R}$, then $f$ is an operator Lipschitz function. The converse is also true; see [10] and the references given there.

The following result was obtained in [73].

Note here that the condition $f\in B_{\infty,1}^1(\mathbb{R})$ is not necessary for operator Lipschitzness. We refer the reader to [10], where the reader can find a detailed discussion of conditions for operator Lipschitzness.

7.2. An operator Bernstein-type inequality

It is easy to deduce from Theorem 7.4 the following so-called operator Bernstein-type inequality: let $f$ be a function of class $L^\infty(\mathbb{R})$ whose Fourier transform has support in the interval $[-\sigma,\sigma]$. Then for arbitrary self-adjoint operators $A$ and $B$ with bounded difference the following inequality holds

In this subsection we give a proof of the operator Bernstein-type inequality with constant $1$ and deduce from it inequality (7.5). We also give here a stronger operator Bernstein-type inequality obtained in [6], which is an operator version of an inequality obtained by Bernstein in [22].

Let $\sigma>0$. We say that an entire function $f$ is a function of exponential type at most $\sigma$ if for each positive number $\varepsilon$ there exists $c>0$ such that $|f(z)|\leqslant c e^{(\sigma+\varepsilon)|z|}$ for all $z\in\mathbb{C}$.

Denote by ${\mathscr E}_\sigma$ the set of all entire functions of exponential type at most $\sigma$. Put ${\mathscr E}_\sigma^\infty\overset{\rm def}{=}{\mathscr E}_\sigma\cap L^\infty(\mathbb{R})$. It is well known that

Note also that the space ${\mathscr E}_\sigma^\infty$ coincides with the set of entire functions $f$ such that $f\in L^\infty(\mathbb{R})$ and

Bernstein’s inequality (see [22]) asserts that

Bernstein also proved in [22] the following improvement of inequality (7.8):

In [6] the following version of the operator Bernstein-type inequality was deduced from (7.9).

By analogy with operator Lipschitz functions, we could introduce the notion of operator Hölder functions. Let $0<\alpha<1$. We say that a function $f$ on the real line $\mathbb{R}$ is operator Hölder of order $\alpha$ if the following inequality holds

Note also that in [5] the authors obtained estimates of operator differences $f(A)- f(B)$ in the case when $A$ and $B$ are self-adjoint operators with difference in $\boldsymbol{S}_p$ and $f$ is a function in the Hölder class $\Lambda_\alpha(\mathbb{R})$. In particular, it was shown in [5] that if $0<\alpha<1$, $p>1$, $f\in\Lambda_\alpha(\mathbb{R})$, and $A$ and $B$ are self-adjoint operators such that $A-B\in\boldsymbol{S}_p$, then

Let us also mention here the paper [61], in which the authors obtained estimates of the quasinorms $\|f(A)-f(B)\|_{\boldsymbol{S}_p}$ for self-adjoint operators $A$ and $B$ in the case $0<p<1$.

7.3. Functions of unitary operators under perturbation

The notion of operator Lipschitz functions can easily be generalized to functions on closed subsets of the complex plane $\mathbb{C}$. In particular, a continuous function $f$ on the unit circle $\mathbb{T}$ is called operator Lipschitz if

In [73] the following result was obtained.

In [73] Theorem 7.6 was proved with the help of a representation of the difference of functions of the operators in terms of the double operator integral:

Here $E_U$ and $E_V$ are the spectral measures of $U$ and $V$. More precisely, it was proved in [73] that the divided difference in the integrand is a Schur multiplier, and its norm in the space of Schur multipliers can be estimated in terms of the norm of $f$ in the Besov class $B_{\infty,1}^1(\mathbb{T})$.

As in the case of self-adjoint operators, in [6] an alternative approach to obtain operator Lipschitz estimate (7.13) was given. That approach is also based on an operator Bernstein-type inequality for functions of unitary operators:

Note also that an analogue of this result for functions of dissipative operators was obtained in [8].

7.4. Functions of normal operators under their perturbation. Functions of collections of commuting operators

The notion of operator Lipschitz functions can be generalized easily to functions defined on the complex plane $\mathbb{C}$. Let $f$ be a continuous complex-valued function on $\mathbb{C}$. We say that $f$ is an operator Lipschitz function if the inequality

As in the case of self-adjoint operators, if inequality (7.14) holds for all bounded normal operators, it also holds for unbounded normal operators; see [10].

It was shown in [17] that the functions in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$ are operator Lipschitz functions on $\mathbb{C}$ (here we identify naturally $\mathbb{C}$ with $\mathbb{R}^2$).

Let us introduce the following notation. Put

If $f$ is a function on $\mathbb{R}^2$ that has partial derivatives with respect to each variable, then consider the divided differences in each variable

The proof of the operator Lipschitzness of functions in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$ is based on the following fact. To state it, we introduce the following notation:

Note that the study of the behaviour of functions of normal operators under perturbation is equivalent to the problem of the behaviour of functions of pairs of commuting self-adjoint operators. Naturally, we arrive at the problem of the behaviour of functions of $d$ commuting self-adjoint operators. Such a problem was studied in the paper [62], in which the following result was obtained.

7.5. Operator differentiability

We proceed to the question of the differentiability of the map

In the definition of operator differentiability given above nothing is said whether we mean Gateaux or Fréchet differentiability. It turns out that in this case these notions are equivalent; see [10].

Let us give here a sufficient condition for operator differentiability.

We can also mention here the paper [82], in which the differentiability properties of functions of contractions are studied.

Let us proceed to operator differentiability of higher order. We start with the case when $A$ is a bounded self-adjoint operator. Then for a bounded self-adjoint operator $K$ and a continuous function $f$ on $\mathbb{R}$ we consider the problem of the existence of the $n$th-order derivative of the function

To write an expression for higher-order derivatives of operator functions, we need multiple operator integrals. There are several different approaches to the definition of multiple operator integrals. Here we are going to use an approach given in this author’s paper [80]. Note that for other problems it becomes useful to use other definitions of multiple operator integrals (see, in particular, the definitions given in § 7.8 of this survey).

For simplicity, we consider the case of triple operator integrals.

Let $({\mathscr X},E)$, $({\mathscr Y},F)$, and $({\mathscr Z},G)$ be spaces with spectral measures $E$, $F$, and $G$ on a Hilbert space $\mathscr{H}$. Suppose that a function $\psi$ on ${\mathscr X}\times{\mathscr Y}\times\mathscr{Z}$ belongs to the integral projective tensor product $L^\infty(E)\mathbin{\widehat\otimes}_{\rm i}L^\infty(F)\mathbin{\widehat\otimes}_{\rm i}L^\infty(G)$, that is, $\psi$ admits a representation

Suppose now that $T_1$ and $T_2$ are bounded linear operators on $\mathscr{H}$. For a function $\psi$ in $L^\infty(E)\mathbin{\widehat\otimes}_{\rm i}L^\infty(F)\mathbin{\widehat\otimes}_{\rm i}L^\infty(G)$ of the form (7.17), we put

It is well known (see, for example, [84] or [19]) that the right-hand side of (7.19) does not depend on the choice of representation (7.17); in other words, triple operator integrals are well defined.

The definition of multiple operator integrals in the general case can be given in the same way.

The following result was obtained in this author’s paper [80].

In the case when $A$ is not bounded, a problem arises which is caused by the fact that the condition $f\in B_{\infty,1}^m(\mathbb{R})$ for $m>1$ does not imply that $f$ is Lipschitz on the whole line. Therefore, to obtain the conclusion of the theorem in the case of unbounded operators we have to add the condition $f\in B_{\infty,1}^1(\mathbb{R})$.

7.6. Trace class perturbations and the Lifshitz–Krein formula

In his paper [56] Lifshits, while considering problems of quantum statistics and crystal theory, arrived to the following trace formula. Let $A$ and $B$ be self-adjoint operators in Hilbert space with trace-class difference. Then there exists an integrable real-valued function $\xi=\xi_{\{A,B\}}$ on $\mathbb{R}$ for which the trace formula

In the author’s paper [76] the following result was obtained.

However, the Besov class $B_{\infty,1}^1(\mathbb{R})$ is not the maximal class for which trace formula (7.20) is valid. The Krein problem was recently solved in this author’s paper [83]. The solution is given by the following theorem.

Let us briefly dwell on trace formulae for functions of contractions and functions of dissipative operators. The problem of finding an integrable spectral shift function for a pair of contractions with trace class difference remained open for a long time; see [59], where the history of this problem was discussed in detail. The problem was completely solved in [58] and [59] by different methods; see also an earlier paper [57], in which an additional restriction was imposed. Thus, in [58] and [59] it was proved that for an arbitrary pair of contractions $(T,R)$ on Hilbert space with trace class difference $T-R$ there exists an integrable function $\xi$ on the circle $\mathbb{T}$ such that

Note that earlier partial results in this direction were obtained by Langer [53], Adamjan–Neidhardt [2], Rybkin [98]–[101], and Krein [49]. A more detailed discussion of the history of this problem was presented in [59].

7.7. Trace formulae in the case of perturbations by operators of class $\boldsymbol{S}_m$, $m\geqslant2$

In Koplienko’s paper [45] the author introduced a generalized spectral shift function in the case of a perturbation of a self-adjoint operator by a self-adjoint operator of class $\boldsymbol{S}_2$ and obtained an appropriate trace formula.

Let $A$ and $B$ be self-adjoint operators such that $K\overset{\rm def}{=} B-A$ is an operator in the Hilbert–Schmidt class $\boldsymbol{S}_2$, and let $f$ be a sufficiently nice function on $\mathbb{R}$. Clearly, there is no hope to expect that the operator $f(B)-f(A)$ has to be of trace class. Koplienko’s idea was to consider the operator

It was shown in [45] that there exists a unique function $\eta$ of class $L^1(\mathbb{R})$ such that

In this author’s paper [78] it was shown that trace formula (7.21) holds under a considerably milder assumption on $f$.

Note here that the results of Koplienko were extended in [63] to the case of functions of unitary operators; in this author’s paper [78] these results were improved and an analogue of Koplienko’s trace formula for unitary operators was obtained in the case when the function on the unit circle belongs to the Besov class $B_{\infty,1}^2(\mathbb{T})$.

Note also that the results of [45] were generalized to the case of functions of contractions in [94].

In [90] Koplienko’s trace formula was extended to the case of perturbations by operators of class $\boldsymbol{S}_m$, where $m$ is an integer greater than $2$.

Let $A$ be a self-adjoint operator and let $K$ be a self-adjoint operator of class $\boldsymbol{S}_m$. To state the main result of [90] we define the Taylor operator polynomial ${\mathscr T}^{(m)}_{A,K}f$ for a sufficiently smooth function $f$ by

In [90] the authors established that for such operators $A$ and $K$ there exists an integrable function $\eta_m$ on $\mathbb{R}$ (a spectral shift function of order $m$ for the pair $A$ and $K$) such that for an arbitrary function $f$ on $\mathbb{R}$ satisfying the assumption

Next, it was shown in [7] that trace formula (7.23) also holds under a weaker condition than (7.22). Namely, the following result holds.

In [7] more general trace formulae were given in the case of perturbations by operators of class $\boldsymbol{S}_m$. Among the results of [7], we mention the following one.

The following result obtained in [7] shows that Theorem 7.14 is a special case of Theorem 7.16.

7.8. Functions of pairs of non-commuting operators

For not necessarily commuting self-adjoint operators $A$ and $B$ in Hilbert space and for a function $f$ on $\mathbb{R}^2$, we define the function $f(A,B)$ of the operators $A$ and $B$ as the double operator integral

In the case when the operators $A$ and $B$ are bounded, functions $f$ in the Besov class $B_{\infty,1}^1(\mathbb{R})$ are Schur multipliers with respect to the spectral measures $E_A$ and $E_B$ (see [3]), and so for such functions $f$ we can define the functions $f(A,B)$ of the operators $A$ and $B$.

We proceed now to the problem of the behaviour of functions of pairs of noncommuting self-adjoint operators under perturbation. In other words, we would like to estimate the norms of the difference $f(A_2,B_2)-f(A_1,B_1)$ in terms of the norms of $A_2-A_1$ and ${B_2-B_1}$. For this purpose we need triple operator integrals. Moreover, we need a definition of triple operator integrals, which differs from the one given in § 7.5.

Recall that triple operator integrals are expressions of the form

In [40] triple operator integrals were defined for functions $\Psi$ that belong to the Haagerup tensor product $L^\infty_{E_1}\mathbin{\otimes_{\rm h}}L^\infty_{E_2}\mathbin{\otimes_{\rm h}}L^\infty_{E_3}$, which, by definition, consists of the functions $\Psi$ that admit a representation

The norm of $\Psi$ in the Haagerup tensor product $L^\infty_{E_1}\mathbin{\otimes_{\rm h}}L^\infty_{E_2}\mathbin{\otimes_{\rm h}}L^\infty_{E_3}$ is defined as the infimum of the products

For such functions $\Psi$ the triple operator integral (7.24) is defined as the product of operator matrices

In addition, the following inequalities hold:

It turns out, however, that to obtain Lipschitz-type estimates for functions of two noncommuting self-adjoint operators under perturbation, the definition of triple operator integrals in the case when the integrand $\Psi$ belongs to the Haagerup tensor of $L^\infty$-spaces does not help.

In [3] the following Lipschitz-type estimate for functions of pairs of non- commuting self-adjoint operators was obtained.

Let $1\leqslant p\leqslant2$. Then there exists a positive number $C$ such that

It is easy to deduce from the definition of Besov classes given in § 2 that to prove inequality (7.27) it suffices to establish that if $f\in L^\infty(\mathbb{R}^2)$ and $\operatorname{supp}\mathscr{F}f\subset\{\xi\in \mathbb{R}^2\colon \|\xi\|\leqslant1\}$, then for $1\leqslant p\leqslant2$ the following inequality holds:

To establish inequality (7.28) the authors of [3] used the following representation of the operator difference $f(A_1,B_1)-f(A_2,B_2)$ in terms of triple operator integrals:

It turns out (see [3]) that under the condition $f\in{\mathscr E}_1^\infty(\mathbb{R}^2)$ the divided differences ${\mathfrak D}^{[1]}f$ and ${\mathfrak D}^{[2]}f$ admit the representations

Similar facts can easily be stated for functions of class ${\mathscr E}_\sigma^\infty(\mathbb{R}^2)$ for an arbitrary positive number $\sigma$.

It would seem, the representations of the divided differences ${\mathfrak D}^{[1]}$ and ${\mathfrak D}^{[2]}$ in the form (7.32) and (7.33) remind a representation of a function of three variables in the form of the Haagerup tensor product. However, upon closer examination, it turns out that this is not the case. The thing is that in the definition of the Haagerup tensor product of three $L^\infty$-spaces we see that in representation (7.25) the matrix-valued function $y\mapsto\{\beta_{jk}(y)\}_{j,k}$ is a function of the second variable while in representations (7.32) and (7.33) the matrix-valued functions are functions of the first and third variables. What is more, this distinction is very essential.

Therefore, the need to define triple operator integrals for the functions ${\mathfrak D}^{[1]}$ and ${\mathfrak D}^{[2]}$ led in [3] to notions of Haagerup-like tensor products (see also [12]).

Now it is the turn to define triple operator integrals whose integrands belong to $L^\infty(E_1)\mathbin{\otimes_{\rm h}}L^\infty(E_2)\mathbin{\otimes^{\rm h}}L^\infty(E_3)$.

Then the triple operator integral in (7.36) is well defined because the function

Let us now define triple operator integrals in the case of integrands in a Haagerup- like tensor product of the second kind.

Moreover, the following inequality holds:

These definitions of triple operator integrals were given in the paper [3]; we also refer the reader to [12], where triple operator integrals were defined in a more general situation.

Now we are ready to state the theorem on Lipschitz-type estimates for functions of non-commuting self-adjoint operators under perturbation.

Theorem 7.18 was established in [3]. In the same paper it was shown that for $p>2$, as well as in the operator norm, such a Lipschitz estimate does not hold.

Note also that it was shown in [85] that such Lipschitz-type estimates cannot be generalized to the case of functions of three non-commuting self-adjoint operators.

7.9. Functions of pairs of unbounded non-commuting operators

We proceed now to the case of pairs of unbounded not necessarily commuting self-adjoint operators.

Let $1\leqslant p\leqslant 2$, and let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R})$. Suppose now that $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of unbounded not necessarily commuting self-adjoint operators such that $A_2-A_1\in\boldsymbol{S}_p$ and $B_2-B_1\in\boldsymbol{S}_p$.

Note, first of all, that both triple operator integrals on the right-hand side of representation (7.29) are still well defined as in the case of pairs of bounded operators and determine operators of class $\boldsymbol{S}_p$, while their norms in $\boldsymbol{S}_p$ can still be estimated in terms of the right-hand side of (7.39).

The problem is how to define functions of operators $f(A_1,B_1)$ and $f(A_2,B_2)$ as unbounded densely defined operators. It is to this problem that the papers [15] and [16] are devoted.

In § 7.8 we defined the functions $f(A,B)$ of not necessarily commuting self- adjoint operators in the case when $f\in L^\infty_{E_A}\mathbin{\otimes_{\rm h}}L^\infty_{E_B}$. Now let $f$ be a function of two variables, and let $f_\sharp$ be the function defined by

It was established in [14] and [16] that if $f\in{\mathscr E}_\sigma^\infty(\mathbb{R}^2)$, $\sigma>0$, then $f_\sharp\in L^\infty_{E_1}\mathbin{\otimes_{\rm h}} L^\infty_{E_2}$ for arbitrary Borel spectral measures $E_1$ and $E_2$. Therefore, if $f\in{\mathscr E}_\sigma^\infty(\mathbb{R}^2)$, then the operator $f_\sharp(A,B)$ is bounded, while $f(A,B)$ is a not necessarily bounded densely defined operator with domain ${\rm D}(B)$. In addition, the following inequality holds

Thus, for an arbitrary pair $(A,B)$ of not necessarily bounded self-adjoint operators and for a function $f$ in the inhomogeneous Besov class $\textit{�}_{\infty,1}^1(\mathbb{R}^2)$, we can define the operator $f(A,B)$ as a densely defined operator. Moreover, the following result holds (see [15] and [16]).

Note here that an analogue of this result for functions of pairs of non-commuting dissipative operators was obtained in [14].

7.10. Functions of pairs of almost commuting operators

Let $A$ and $B$ be bounded self-adjoint operators. They are said to be almost commuting if their commutator $[A,B]\overset{\rm def}{=} AB-BA$ is a trace-class operator.

Polynomial calculus for a pair $(A,B)$ of almost commuting self-adjoint operators is defined in the following way. If $\varphi$ is a polynomial of the form $\varphi(s,t)=\sum_{j,k}a_{jk}s^jt^k$, then the operator $\varphi(A,B)$ is defined by

In [37] Helton and Howe obtained a trace formula for commutators of polynomials of almost commuting self-adjoint operators. Let $A$ and $B$ be almost commuting self-adjoint operators, and let $\varphi$ and $\psi$ be polynomials of two real variables. Then

Polynomial calculus was extended in [29] to calculus on the class of functions $\varphi=\mathscr{F}\omega$ that are the Fourier transforms of complex Borel measures $\omega$ on $\mathbb{R}^2$ such that

Next, in this author’s paper [77] such results were obtained for a yet wider class of functions. Moreover, in the same paper [77] it was shown that it is impossible to extend such functional calculus to the class of all continuously differentiable functions on $\mathbb{R}^2$ that satisfy certain natural requirements and $[\varphi(A,B),\psi(A,B)]\in\boldsymbol{S}_1$ for arbitrary continuously differentiable functions $\varphi$ and $\psi$.

Most advanced results in this direction were obtained in the paper [11], in which the desired functional calculus was constructed on the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$ and it was shown that this functional calculus is almost multiplicative.

Let us proceed now to the main results of [11]. Note that these results use triple operator integrals for Haagerup-like tensor products of the $L^\infty$-spaces that were introduced in [3] and the estimates of the trace norms of such operator integrals that were found in [3]; see § 7.8.

Now we can extend the Helton–Howe trace formula to the case of functions in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$.

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