| Russian Mathematical Surveys |
|
|
|
|
|
Besov spaces in operator theory V. V. Pellerabc a Saint Petersburg State University b St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences c Peoples' Friendship University of Russia
Russian version:
Uspekhi Matematicheskikh Nauk, 2024, Volume 79, Issue 1(475), DOI: https://doi.org/10.4213/rm10140 
Bibliographic databases:
   1. IntroductionIn 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 classesIn 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 circleLet $w$ be an infinitely differentiable function on $\mathbb{R}$ such that
$$
\begin{equation}
w\geqslant0,\quad \operatorname{supp} w\subset\biggl[\frac{1}{2}\,,2\biggr],\quad\text{and}\quad w(t)=1-w\biggl(\frac{t}{2}\biggr)\quad\text{for}\ \ t\in[1,2].
\end{equation}
\tag{2.1}
$$
We define the trigonometric polynomials $W_n$, $n\geqslant0$, by
$$
\begin{equation}
W_n(\zeta)\overset{\rm def}= \sum_{j\in\mathbb{Z}}w\biggl(\frac{|j|}{2^n}\biggr)\zeta^j,\quad n\geqslant1,\quad W_0(\zeta)\overset{\rm def}=\sum_{\{j:\,|j|\leqslant1\}}\zeta^j,\quad \zeta\in\mathbb{T}.
\end{equation}
\tag{2.2}
$$
If $f$ is a distribution (a generalized function) on $\mathbb{T}$, we put and we say that $f$ belongs to the Besov class $B_{p,q}^s(\mathbb{T})$, $s\in\mathbb{R}$, $0<p,q\leqslant\infty$, if
$$
\begin{equation}
\bigl\{2^{ns}\|f_n\|_{L^p}\bigr\}_{n\geqslant0}\in\ell^q.
\end{equation}
\tag{2.3}
$$
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
$$
\begin{equation*}
f\mapsto\displaystyle\int_{\mathbb{T}} fg\,\mathrm{d}\boldsymbol{m},
\end{equation*}
\notag
$$
where $\boldsymbol{m}$ is normalized Lebesgue measure on $\mathbb{T}$. 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
$$
\begin{equation}
\|f\|_{B_{p,q}^s(\mathbb{T})}= \bigl\|\{2^{ns}\|f_n\|_{L^p}\bigr\}_{n\geqslant0}\bigr\|_{\ell^p}.
\end{equation}
\tag{2.4}
$$
If $\min\{p,q\}<1$, the expression (2.4) is a quasinorm. Note that there are other natural norms (quasinorms) on Besov classes. 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:
$$
\begin{equation*}
\bigl(B_{p,q}^s(\mathbb{T})\bigr)^*=B_{p',q'}^{-s}(\mathbb{T}),\qquad p'=\frac{p}{1-p}\,,\quad q'=\frac{q}{1-q}\,,
\end{equation*}
\notag
$$
with respect to the bilinear form
$$
\begin{equation*}
(f,g)=\sum_{j\in\mathbb{Z}}\widehat f(j)\widehat g(j);
\end{equation*}
\notag
$$
here $f$ ranges over the dense subset of trigonometric polynomials in $B_{p,q}^s(\mathbb{T})$, while $g$ is an arbitrary function (distribution) in $B_{p',q'}^{-s}(\mathbb{T})$. Recall that if $g$ is a distribution on $\mathbb{T}$, the Fourier coefficient $\widehat g(j)$ is the value of the functional $g$ at the function $\overline z^j$. 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
$$
\begin{equation*}
(\Delta_\tau f)(\zeta)\overset{\rm def}= f(\tau\zeta)-f(-z),\qquad \zeta\in\mathbb{T},
\end{equation*}
\notag
$$
while the difference operator $\Delta^n_\tau$ is defined as the $n$th power of $\Delta_\tau$. 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:
$$
\begin{equation*}
\begin{aligned} \, B_{p,q}^s(\mathbb{T})&=\biggl\{f\in L^p(\mathbb{T})\colon\int_{\mathbb{T}} \frac{\|\Delta^n_\tau f\|_{L^p}^q}{|1-\tau|^{1+sq}}\, {\rm d}\boldsymbol{m}(\tau)<\infty\biggr\},\qquad q<\infty; \\ B_{p,\infty}^s(\mathbb{T})&=\biggl\{f\in L^p(\mathbb{T})\colon\sup_{\tau\ne1} \frac{\|\Delta^n_\tau f\|_{L^p}}{|1-\tau|^s}<\infty\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
In the case when $p=q=\infty$ and $0<s<1$, we obtain the Hölder class $\Lambda_s$:
$$
\begin{equation*}
f\in\Lambda_s=B_{\infty,\infty}^s(\mathbb{T})\quad\Longleftrightarrow\quad |f(\zeta_1)-f(\zeta_2)|\leqslant\mathrm{const}|\zeta_1-\zeta_2|^s,\ \ \zeta_1,\zeta_2\in\mathbb{T}.
\end{equation*}
\notag
$$
The classes $\Lambda_s\overset{\rm def}{=} B_\infty^s(\mathbb{T})=B_{\infty,\infty}^s(\mathbb{T})$ for $s>0$ form the so-called Hölder–Zygmund scale of spaces. In particular, $\Lambda_1$ is the Zygmund space:
$$
\begin{equation*}
f\in\Lambda_1\quad\Longleftrightarrow\quad |f(\zeta\tau)-2f(\zeta)+f(\zeta\overline\tau)|\leqslant \mathrm{const}|1-\tau|,\ \ \zeta,\tau\in\mathbb{T}.
\end{equation*}
\notag
$$
We denote by $\lambda_s$ the closure of the trigonometric polynomials in $\Lambda_s$, and we say that $\lambda_s$ is the separable Hölder–Zygmund space. 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
$$
\begin{equation}
\int_0^{1}(1-r)^{(m-s)q-1}\biggl\|\frac{\partial^m}{\partial r^m} \bigl((\mathcal{P} f)(r\zeta)\bigr)\biggr\|_{L^p(\mathbb{T})}^q\,{\rm d}r <\infty,\qquad q<\infty,
\end{equation}
\tag{2.5}
$$
or
$$
\begin{equation}
\sup_{r\in[0,1)}(1-r)^{m-s}\biggl\|\frac{\partial^m}{\partial r^m} \bigl((\mathcal{P} f)(r\zeta)\bigr)\biggr\|_{L^p(\mathbb{T})}<\infty,\qquad q=\infty,
\end{equation}
\tag{2.6}
$$
where $\mathcal{P} f$ denotes the Poisson integral of the distribution $f$. Consider the Riesz projections $\mathbb{P}_+$ and $\mathbb{P}_-$ defined on the class of all distributions on $\mathbb{T}$ by
$$
\begin{equation}
\mathbb{P}_+f=\sum_{j\geqslant0}\widehat f(j)z^j\quad\text{and}\quad \mathbb{P}_-f=\sum_{j<0}\widehat f(j)z^j.
\end{equation}
\tag{2.7}
$$
It follows easily from the definition of Besov classes in terms of convolutions with the $W_n$ that
$$
\begin{equation*}
\mathbb{P}_+B_{p,q}^s(\mathbb{T})\subset B_{p,q}^s(\mathbb{T})\quad\text{and}\quad \mathbb{P}_-B_{p,q}^s(\mathbb{T})\subset B_{p,q}^s(\mathbb{T}),\quad p,q>0,\quad s\in\mathbb{R}.
\end{equation*}
\notag
$$
We are going to use the notation $(B_{p,q}^s)_+$ for $\mathbb{P}_+B_{p,q}^s(\mathbb{T})$. Functions $f$ in $(B_{p,q}^s)_+$ (distributions for $s\leqslant0$) can be identified with the power series which makes the classes $(B_{p,q}^s)_+$ spaces of analytic functions in the disc. Such classes can be described in terms of conditions (2.5) and (2.6); in addition, partial derivatives of order $m$ can be replaced by the derivatives of order $m$ in the complex variable. In particular, for $p=q<\infty$ and $s<0$ the Besov classes $(B_p^s)_+\overset{\rm def}=(B_{p,p}^s)_+$ become weighted Bergman classes, that is, a function $f$ analytic in the unit disc $\mathbb{D}$ belongs to the Besov class $(B_p^s)_+$ if and only if
$$
\begin{equation*}
\int_\mathbb{D}(1-|\zeta|)^{-sp-1} |f(\zeta)|^p\,{\rm d}\boldsymbol{m}_2(\zeta)<\infty,\qquad p<\infty.
\end{equation*}
\notag
$$
The important Bloch space ${\mathfrak B}$ of functions $f$ analytic in $\mathbb{D}$ such that
$$
\begin{equation*}
\sup_{\zeta\in\mathbb{D}}|f'(\zeta)|(1-|\zeta|)<\infty,
\end{equation*}
\notag
$$
is also a Besov class of analytic functions, namely, $(B_\infty^0)_+$. Let us mention here the book [79], which contains an introduction to Besov spaces on the unit circle. 2.2. Besov classes on Euclidean spacesLet $d$ 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
$$
\begin{equation*}
\bigl(\mathscr{F} W_n\bigr)(x)=w\biggl(\frac{\|x\|}{2^n}\biggr),\qquad n\in\mathbb{Z}, \quad x=(x_1,\dots,x_d),\quad \|x\|\overset{\rm def}=\biggl(\,\sum_{j=1}^dx_j^2\biggr)^{1/2},
\end{equation*}
\notag
$$
where $\mathscr{F}$ is the Fourier transform, defined on $L^1(\mathbb{R}^d)$ by
$$
\begin{equation*}
\begin{gathered} \, (\mathscr{F} f)(t)=\int_{\mathbb{R}^d} f(x)e^{-{\rm i}(x,t)}\,{\rm d} x, \\ x=(x_1,\dots,x_d),\quad t=(t_1,\dots,t_d), \quad (x,t)\overset{\rm def}= \sum_{j=1}^dx_jt_j. \end{gathered}
\end{equation*}
\notag
$$
Clearly,
$$
\begin{equation*}
\sum_{n\in\mathbb{Z}}(\mathscr{F} W_n)(t)=1,\qquad t\in\mathbb{R}^d\setminus\{0\}.
\end{equation*}
\notag
$$
Note that the utilization of the same notation $W_n$ as in previous subsections will not lead to confusion. With each tempered distribution $f$ in ${\mathscr S}'(\mathbb{R}^d)$ we associate the sequence $\{f_n\}_{n\in\mathbb{Z}}$, The formal series $\sum_{n\in\mathbb{Z}}f_n$, being a Littlewood–Payley type expansion of $f$, does not necessarily converge to $f$. We start with the definition of the (homogeneous) Besov class $\dot B^s_{p,q}(\mathbb{R}^d)$, $s\in\mathbb{R}$, $0<p,q\leqslant\infty$, as the space of distributions $f$ of class ${\mathscr S}'(\mathbb{R}^n)$ (that is, tempered distributions) such that
$$
\begin{equation}
\{2^{ns}\|f_n\|_{L^p}\}_{n\in\mathbb{Z}}\in\ell^q(\mathbb{Z}),\quad \|f\|_{B^s_{p,q}}\overset{\rm def}= \bigl\|\{2^{ns}\|f_n\|_{L^p}\}_{n\in\mathbb{Z}}\bigr\|_{\ell^q(\mathbb{Z})}.
\end{equation}
\tag{2.9}
$$
In accordance with this definition, $\dot B^s_{p,q}(\mathbb{R}^d)$ contains all polynomials; next, $\|f\|_{B^s_{p,q}}=0$ for an arbitrary polynomial $f$. 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 $\sum_{n\geqslant0}f_n$ converges in1 ${\mathscr S}'(\mathbb{R}^d)$. However, the series $\sum_{n<0}f_n$ can diverge in general. Nevertheless, it can be shown that the series
$$
\begin{equation}
\sum_{n<0}\frac{\partial^r f_n}{\partial x_1^{r_1}\cdots \partial x_d^{r_d}}\quad\text{for}\ \ r_j\geqslant0,\ \ 1\leqslant j\leqslant d,\ \ \sum_{j=1}^dr_j=r,
\end{equation}
\tag{2.10}
$$
converge uniformly in $\mathbb{R}^d$ if $r\in\mathbb{Z}_+$ and $r>s-d/p$. Note that for $q\leqslant1$ the series in (2.10) converge uniformly under the weaker condition $r\geqslant s-d/p$. 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
$$
\begin{equation*}
\frac{\partial^r f}{\partial x_1^{r_1}\dots\partial x_d^{r_d}}= \sum_{n\in\mathbb{Z}}\frac{\partial^r f_n}{\partial x_1^{r_1}\dots \partial x_d^{r_d}}\quad\text{for}\ \ r_j\geqslant0,\ \ 1\leqslant j\leqslant d,\ \ \sum_{j=1}^dr_j=r,
\end{equation*}
\notag
$$
in the space ${\mathscr S}'(\mathbb{R}^d)$, where $r$ is the minimum non-negative integer for which $r>s-d/p$ ($r\geqslant s-d/p$ if $q\leqslant1$). Now the function $f$ is uniquely determined by the sequence $\{f_n\}_{n\in\mathbb{Z}}$ up to a polynomial of degree less than $r$. Moreover, a polynomial $g$ belongs to $B^s_{p,q}(\mathbb{R}^d)$ if and only if $\deg g<r$. 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
$$
\begin{equation*}
\Lambda_\alpha(\mathbb{R}^d)\overset{\rm def}= B_\infty^\alpha(\mathbb{R}^d).
\end{equation*}
\notag
$$
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
$$
\begin{equation}
\int_{\mathbb{R}^d}|h|^{-d-sq} \|\Delta^m_h f\|_{L^p}^q\,{\rm d}h<\infty,\qquad q<\infty;
\end{equation}
\tag{2.11}
$$
or
$$
\begin{equation}
\sup_{h\ne 0}\frac{\|\Delta^m_h f\|_{L^p}}{|h|^s}<\infty,\qquad q=\infty.
\end{equation}
\tag{2.12}
$$
However, with this definition Besov classes can contain polynomials of degree higher than in the case of the definition in terms of convolutions with the functions $W_n$. 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,
$$
\begin{equation*}
P_d(x,t)=c_dt(|x|^2+t^2)^{-(d+1)/2},\qquad c_d=\pi^{-(d+1)/2}\Gamma\biggl(\frac{d+1}{2}\biggr).
\end{equation*}
\notag
$$
With each function $f\in L^1\bigl(\mathbb{R}^d,(\|x\|+1)^{-(d+1)}\,{\rm d} x\bigr)$ we can associate the Poisson integral $\mathcal{P} f$,
$$
\begin{equation*}
(\mathcal{P} f )(x,t)=\int_{\mathbb{R}^d}P_d(x-y,t)f(y)\,{\rm d} y.
\end{equation*}
\notag
$$
Then for each positive integer $m$ the following equality holds:
$$
\begin{equation*}
\frac{\partial^m (\mathcal{P} f)}{\partial t^m}(x,t)= \int_{\mathbb{R}^d}\frac{\partial^m P_d(x-y,t)}{\partial t^m}f(y)\,{\rm d}y.
\end{equation*}
\notag
$$
Note that the second integral makes sense for all $f$ in $L^1\bigl(\mathbb{R}^d, (\|x\|+ 1)^{-(d+m+1)}\,{\rm d} x\bigr)$. This allows one to define $\partial^m\mathcal{P}/\partial t^m$ for functions $f$ of the class $L^1\bigl(\mathbb{R}^d, (\|x\|+ 1)^{-(d+m+1)}\,{\rm d} x\bigr)$. 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
$$
\begin{equation}
\biggl(\int_0^{\infty}t^{(m-s)q-1}\biggl\|\biggl(\frac{\partial^m} {\partial t^m}\mathcal{P} f\biggr)(\,\cdot\,,t)\biggr\|_{L^p(\mathbb{R}^d)}^q\,{\rm d} t\biggr)^{1/q}<\infty,\qquad q <\infty;
\end{equation}
\tag{2.13}
$$
or
$$
\begin{equation}
\sup_{t>0}t^{m-s}\biggl\|\biggl(\frac{\partial^m}{\partial t^m} \mathcal{P} f\biggr)(\,\cdot\,,t)\biggr\|_{L^p(\mathbb{R}^d)} <\infty, \qquad q=\infty.
\end{equation}
\tag{2.14}
$$
Under this definition Besov classes can also contain polynomials of higher degrees than in the case of the definition in terms of convolutions with the functions $W_n$. Note also that under certain conditions and provisions the definition in terms of the Poisson integral can work even in the case when $p<1$ or ${q<1}$. 2.3. Inhomogeneous Besov classes on Euclidean spacesAs 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
$$
\begin{equation*}
\bigl(\mathscr{F} W^{[0]}\bigr)(t)=1- \sum_{n\geqslant1}(\mathscr{F} W_n)(t),\qquad t\in\mathbb{R}^d.
\end{equation*}
\notag
$$
It is easy to see that
$$
\begin{equation*}
\bigl(\mathscr{F} W^{[0]}\bigr)(t)=1,\quad\text{for}\ \ \|t\|\leqslant1\quad\text{and}\quad \operatorname{supp}\mathscr{F}W^{[0]}\subset\bigl\{t\in\mathbb{R}^d\colon \|t\|\leqslant2\bigr\}.
\end{equation*}
\notag
$$
Let $s>0$ and $p,q\in[1,\infty]$. We say that a tempered distribution $f$ on $\mathbb{R}^d$ belongs to the inhomogeneous Besov space $\textit{Б}^s_{p,q}(\mathbb{R}^d)$ if
$$
\begin{equation}
f^{[0]}\overset{\rm def}=f*W^{[0]}\in L^p(\mathbb{R}^d)\quad\text{and}\quad \bigl\{2^{ns}\|f_n\|_{L^p(\mathbb{R}^d)}\bigr\}_{n\geqslant1}\in\ell^q,
\end{equation}
\tag{2.15}
$$
where the functions $f_n$ are defined by (2.8). 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
$$
\begin{equation*}
\textit{Б}^s_{p,q}(\mathbb{R}^d)=B^s_{p,q}(\mathbb{R}^d)\cap L^p(\mathbb{R}^d).
\end{equation*}
\notag
$$
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 classesIn 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
$$
\begin{equation}
s_j(T)\overset{\rm def}=\inf\bigl\{\|T-R\|\colon \operatorname{rank} R\leqslant j\bigr\}.
\end{equation}
\tag{3.1}
$$
In the case when $T$ is a compact operator on Hilbert space, the sequence $\{s_j(T)\}_{j\geqslant0}$ coincides with the sequence of eigenvalues (counted with multiplicities) of the operator $(T^*T)^{1/2}$ arranged in non-increasing order. The Schatten–von Neumann class $\boldsymbol{S}_p$, $0<p<\infty$, consists by definition of the operators $T$ for which
$$
\begin{equation}
\|T\|_{\boldsymbol{S}_p}\overset{\rm def}= \biggl(\,\sum_{j\geqslant0}(s_j(T))^p\biggr)^{1/p}<\infty.
\end{equation}
\tag{3.2}
$$
For $p\geqslant1$ this is a normed ideal of operators on Hilbert space with norm (3.2). For $p<1$ the functional $\|\,{\cdot}\,\|_{\boldsymbol{S}_p}$ is not a norm: it is a $p$-norm, that is, the following inequality holds:
$$
\begin{equation}
\|T+R\|_{\boldsymbol{S}_p}^p\leqslant\|T\|_{\boldsymbol{S}_p}^p+ \|R\|_{\boldsymbol{S}_p}^p,\qquad T,R\in\boldsymbol{S}_p.
\end{equation}
\tag{3.3}
$$
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
$$
\begin{equation*}
\operatorname{trace} T\overset{\rm def}=\sum_{j\geqslant0}(Te_j,e_j),
\end{equation*}
\notag
$$
where $\{e_j\}_{j\geqslant0}$ is an orthonormal basis in $\mathscr{H}$. The right-hand side does not depend on the choice of a basis. The class $\boldsymbol{S}_2$ is called the Hilbert–Schmidt class. It forms a Hilbert space with inner product
$$
\begin{equation*}
(T,R)_{\boldsymbol{S}_2}\overset{\rm def}=\operatorname{trace}(TR^*).
\end{equation*}
\notag
$$
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
$$
\begin{equation*}
\langle T,R\rangle\overset{\rm def}=\operatorname{trace}(TR).
\end{equation*}
\notag
$$
Moreover, for $R\in\boldsymbol{S}_{p'}$ the norm of the linear functional $T\mapsto\operatorname{trace}(TR)$ on $\boldsymbol{S}_p$ coincides with the norm of $R$ in $\boldsymbol{S}_{p'}$. 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 classes4.1. Classical Hankel operators and Hankel matricesBesov classes were used to study Hankel operators in [70]. Hankel operators on the Hardy class2 $H^2$ 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
$$
\begin{equation*}
H_\varphi\overset{\rm def}=\mathbb{P}_-\varphi f,\qquad f\in\mathscr{P}_+
\end{equation*}
\notag
$$
(recall that the Riesz projections are defined by (2.7)). Such operators are not necessarily bounded. The boundedness criterion for Hankel operators is given by Nehari’s theorem: the operator $H_\varphi$ is bounded if and only if there exists a function $\psi$ in $L^\infty$ such that $H_\varphi=H_\psi$. The last equality is equivalent to the following one: In this case
$$
\begin{equation*}
\|H_\varphi\|=\operatorname{dist}_{L^\infty}(\psi,H^\infty).
\end{equation*}
\notag
$$
It follows from the theorems of Nehari and Ch. Fefferman that for $\varphi\in L^2$ the Hankel operator $H_\varphi$ is a bounded operator from $H^2$ to $H^2_-$ (to be more precise, it extends to a bounded operator from $H^2$ to $H^2_-$) if and only if the function $\mathbb{P}_-\varphi$ belongs to the class $\mathrm{BMO}$ of functions on $\mathbb{T}$ of bounded mean oscillation. Furthermore, the norm of $H_\varphi$ is equivalent to the norm of $\mathbb{P}_-\varphi$ in the space $\mathrm{BMO}$. 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
$$
\begin{equation*}
\begin{pmatrix} \widehat\varphi(-1)&\widehat\varphi(-2)&\widehat\varphi(-3)&\dots \\ \widehat\varphi(-2)&\widehat\varphi(-3)&\dots&\dots \\ \widehat\varphi(-3)&\vdots&\ddots \\ \vdots \end{pmatrix}.
\end{equation*}
\notag
$$
Such matrices are called Hankel matrices. We associate with a sequence of complex numbers $\{\alpha_j\}_{j\geqslant0}$ in $\ell^2$ the Hankel matrix
$$
\begin{equation}
\Gamma_\alpha=\begin{pmatrix} \alpha_0&\alpha_1&\alpha_2&\dots \\ \alpha_1&\alpha_2&\dots&\dots \\ \alpha_2&\vdots&\ddots \\ \vdots \end{pmatrix}
\end{equation}
\tag{4.1}
$$
and we are going to identify such matrices with operators on the space $\ell^2$ with respect to the natural basis in $\ell^2$. Clearly, the study of the metric properties of Hankel operators $H_\varphi$ is equivalent to the study of the metric properties of Hankel matrices $\Gamma_\alpha$. In particular, $\Gamma_\alpha$ induces a bounded operator on $\ell^2$ if and only if
$$
\begin{equation}
\alpha(z)\overset{\rm def}=\sum_{j\geqslant0}\alpha_jz^j\in \mathrm{BMOA}\overset{\rm def}=\mathbb{P}_+\mathrm{BMO}=\mathrm{BMO}\cap H^2,
\end{equation}
\tag{4.2}
$$
and in the case when the last condition is satisfied the following equality holds:
$$
\begin{equation*}
\|\Gamma_\alpha\|=\|\alpha\|_\mathrm{BMOA}\overset{\rm def}= \inf\{\|\psi\|_{L^\infty}\colon\psi\in L^\infty\ \text{and}\ \widehat\psi(j)=\alpha_j\ \text{for}\ j\geqslant0\}.
\end{equation*}
\notag
$$
On the other hand, the operator $\Gamma_\alpha$ is compact if and only if
$$
\begin{equation*}
\alpha(z)\overset{\rm def}=\sum_{j\geqslant0}\alpha_jz^j\in\mathrm{VMOA} \overset{\rm def}=\mathbb{P}_+{\rm VMO}={\rm VMO}\cap H^2.
\end{equation*}
\notag
$$
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. Let $0<p<\infty$, and let $\varphi\in L^2$. Then the Hankel operators $H_\varphi$ belongs to the Schatten–von Neumann class $\boldsymbol{S}_p$ if and only if the function $\mathbb{P}_-\varphi$ belongs to the Besov class $B_p^{1/p}(\mathbb{T})$. Theorem 4.1 can be reformulated in the following way:
$$
\begin{equation*}
\Gamma_\alpha\in\boldsymbol{S}_p\quad\Longleftrightarrow\quad \sum_{j\geqslant0}\alpha_jz^j\in\bigl(B_p^{1/p}\bigr)_+,\qquad 0<p<\infty.
\end{equation*}
\notag
$$
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 $\boldsymbol{S}_p$. Theorem 4.2. Let $\alpha$ be given by a power series (4.2). Let $0<p<\infty$, and let $m$ be an integer such that $m>1/p$. Then 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 $p=1$. Proof of the sufficiency of the condition $\mathbb{P}_{-}\varphi\in B_p^{1/p}(\mathbb{T})$ for $p\leqslant1$. Let $\psi$ be a function analytic in the disc $\mathbb{D}$, and let $\psi_n=\psi*W_n$, where the analytic polynomials $W_n$ are defined by (2.2). Using inequality (3.3) (which is obviously also valid for $p=1$), we see that it follows from (2.3) and (2.4) that it suffices to establish the following assertion.
Let $\psi$ be an analytic polynomial of degree ${m-1}$. Then
$$
\begin{equation}
\|\Gamma_\psi\|_{\boldsymbol{S}_p}\leqslant 2^{1/p-1}m^{1/p}\|\psi\|_{L^p(\mathbb{T})}.
\end{equation}
\tag{4.3}
$$
Now, for an analytic polynomial $\psi$, $\deg\psi=m-1$, we define operators $A_j$, $0\leqslant j\leqslant2m-1$, of rank $1$ as follows. Consider the points
$$
\begin{equation*}
\zeta_j=e^{2\pi\mathrm{i} j/(2m)},\qquad 0\leqslant j\leqslant2m-1,
\end{equation*}
\notag
$$
on the circle $\mathbb{T}$. Let $f_j$ and $g_j$, $0\leqslant j\leqslant2m-1$, be sequences in $\ell^2$ defined by
$$
\begin{equation*}
f_j(k)=\begin{cases} \overline{\psi(\zeta_j)}\,\zeta_j^k, & 0\leqslant k<m, \\ 0, & k\geqslant m \end{cases}\quad\text{and}\quad g_j(n)=\begin{cases} \overline\zeta_j^n, & 0\leqslant k<m, \\ 0, & k\geqslant m. \end{cases}
\end{equation*}
\notag
$$
We define now the rank $1$ operator $A_j$, $0\leqslant j\leqslant2m-1$, by Then the following equality holds: To prove this equality we have to verify that the matrix entries of the operators on the left and on the right are the same. Clearly, in the case when $k\geqslant m$ or $n\geqslant m$, we have
$$
\begin{equation*}
(A_je_k,e_n)=0\quad\text{and}\quad (\Gamma_\psi e_k,e_n)=\widehat\psi(n+k)=0.
\end{equation*}
\notag
$$
Here $\{e_j\}_{j\geqslant0}$ is the standard orthonormal basis in $\ell^2$. Next, since $\psi$ is an analytic polynomial and $\deg\psi<m$, the following identity holds
$$
\begin{equation*}
\frac{1}{2m}\sum_{j=0}^{2m-1}\psi(\zeta_j)\overline\zeta_j^d= \widehat\psi(d)\quad\text{for}\ \ 0\leqslant d\leqslant 2m-1.
\end{equation*}
\notag
$$
Thus, for $n<m$ and $k<m$ we have $(A_je_k,e_n)=\psi(\zeta_j)\overline\zeta_j^{n+k}$, and so
$$
\begin{equation*}
\frac{1}{2m}\sum_{j=0}^{2m-1}(A_je_k,e_n)=\frac{1}{2m} \sum_{j=0}^{2m-1}\psi(\zeta_j)\overline\zeta_j^{n+k}=\widehat\psi(n+k).
\end{equation*}
\notag
$$
Now, representation (4.4), in combination with inequality (3.3), allows us to conclude that
$$
\begin{equation*}
\begin{aligned} \, \|\Gamma_\psi\|^p_{\boldsymbol{S}_p}&\leqslant\frac{1}{(2m)^p} \sum_{j=0}^{2m-1}\|A_j\|^p_{\boldsymbol{S}_p}= \frac{1}{(2m)^p}\sum_{j=0}^{2m-1}\|f_j\|^p\|g_j\|^p \\ &=\frac{1}{(2m)^p}\sum_{j=0}^{2m-1}\bigl(|\psi(\zeta_j)|m\bigr)^p, \end{aligned}
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\|\Gamma_\psi\|_{\boldsymbol{S}_p}\leqslant 2^{1/p-1}m^{1/p}\,\frac{1}{2m}\sum_{j=0}^{2m-1}|\psi(\zeta_j)|^p.
\end{equation}
\tag{4.5}
$$
To obtain (4.3) it suffices to use the well-known inequality
$$
\begin{equation*}
\frac{1}{2m}\sum_{j=0}^{2m-1}|\psi(\zeta_j)|^p\leqslant\|\psi\|_{L^p}^p.
\end{equation*}
\notag
$$
Incidentally, we can simply apply inequality (4.5) to the function $\psi_\tau$, $\tau\in\mathbb{T}$, $\psi_\tau(\zeta)\overset{\rm def}{=}\psi(\tau\zeta)$, observe that $\|\Gamma_{\psi_\tau}\|_{\boldsymbol{S}_p}=\|\Gamma_\psi\|_{\boldsymbol{S}_p}$, $\tau\in\mathbb{T}$, and integrate it with respect to the variable $\tau$, against Lebesgue measure $\boldsymbol{m}$ on $\mathbb{T}$. The proof of sufficiency is complete. 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
$$
\begin{equation}
(\mathcal{P} T)_{jk}\overset{\rm def}= \frac{1}{j+k+1}\sum_{m=0}^{j+k}(Te_m,e_{j+k-m}),\qquad j,k\geqslant0,
\end{equation}
\tag{4.6}
$$
is not a bounded operator on $\boldsymbol{S}_1$. Indeed, it is easy to observe that, as $T$ ranges over the set of projections of rank $1$, the Hankel matrices $\mathscr{P} T$ fill all matrices of the form $\Gamma_\alpha$, where $\alpha$ is an arbitrary analytic function in $\mathbb{D}$ with derivative in the Hardy class $H^1$, and so
$$
\begin{equation*}
\mathcal{P}\boldsymbol{S}_1=\bigl\{\Gamma_\varphi\colon\varphi'\in H^1\bigr\}.
\end{equation*}
\notag
$$
Nevertheless, $\{f\colon f'\in H^1\}\not\subset(B_1^1)_+$. Indeed, it is well known that the lacunary Fourier coefficients $\widehat g(2^k)$ of the functions $g$ in $H^1$ fill the whole space $\ell^2$, while the lacunary Fourier coefficients of the functions with derivatives in $(B_1^1)_+$ fill the space $\ell^1$. 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 matricesLet $\alpha$ and $\beta$ be real numbers, and let $\varphi$ be a function analytic in the disc $\mathbb{D}$. Consider generalized Hankel matrices of the form
$$
\begin{equation*}
\Gamma_\varphi^{\alpha,\beta}= \bigl\{(1+j)^\alpha(1+k)^\beta\bigr\}_{j,k\geqslant0}.
\end{equation*}
\notag
$$
The case $\alpha=\beta=0$ corresponds to the classical Hankel matrices. We are going to consider such matrices as operators on $\ell^2$. The following theorem describes bounded and compact operators of the form $\Gamma_\varphi^{\alpha,\beta}$ for $\alpha\geqslant0$ and $\beta\geqslant0$. Theorem 4.4. Let $\varphi$ be a function analytic in $\mathbb{D}$. The following statements hold: (i) if $\alpha>0$ and $\beta>0$, then $\Gamma_\varphi^{\alpha,\beta}$ is a matrix of a bounded operator on $\ell^2$ if and only if $\varphi$ belongs to the Hölder class $\Lambda_{\alpha+\beta}$; (ii) if $\alpha>0$ and $\beta>0$, then $\Gamma_\varphi^{\alpha,\beta}$ is a matrix of a compact operator on $\ell^2$ if and only if $\varphi$ belongs to the separable Hölder class $\lambda_{\alpha+\beta}$; (iii) if $\alpha>0$, then $\Gamma_\varphi^{\alpha,0}$ is a matrix of a bounded operator on $\ell^2$ if and only if the function
$$
\begin{equation*}
I_{-\alpha}\varphi\overset{\rm def}= \sum_{n\geqslant0}(1+n)^\alpha\widehat\varphi(n)z^n
\end{equation*}
\notag
$$
belongs to $\mathrm{BMO}$; (iv) if $\alpha>0$, then $\Gamma_\varphi^{\alpha,0}$ is a matrix of a compact operator on $\ell^2$ if and only if the function $I_{-\alpha}\varphi$ belongs to $\mathrm{VMO}$. The following theorem describes operators $\Gamma_\varphi^{\alpha,\beta}$ in the Schatten–von Neumann class $\boldsymbol{S}_p$. Theorem 4.5. Let $\varphi$ be a function analytic in $\mathbb{D}$, and let $0<p<\infty$. Suppose that $\alpha$ and $\beta$ are real numbers such that Then $\Gamma_\varphi^{\alpha,\beta}\in\boldsymbol{S}_p$ if and only if $\varphi$ belongs to the Besov class $(B_p^{1/p+\alpha+\beta})_+$. 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 approximationIn 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 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,
$$
\begin{equation*}
s_j(H_\varphi)=\inf\bigl\{\|H_\varphi-H_g\|\colon g\in L^\infty,\ \operatorname{rank} H_g\leqslant j\bigr\}.
\end{equation*}
\notag
$$
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 $\mathrm{BMO}$:
$$
\begin{equation*}
\rho_n(\varphi)\overset{\rm def}= \operatorname{dist}_{\operatorname{BMO}}(\varphi,{\mathcal R}_n).
\end{equation*}
\notag
$$
Theorem 5.1. Let $\varphi$ be a function $\mathbb{T}$ of class $\mathrm{BMO}$, and let $0<p<\infty$. Then $\varphi\in(B_p^{1/p})(\mathbb{T})$ if and only if 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 projectionFor 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
$$
\begin{equation*}
{\mathscr C}_\varphi=[M_\varphi,\mathbb{P}_+]f= \varphi\mathbb{P}_+f-\mathbb{P}_+\varphi f,\qquad f\in\mathscr{P}.
\end{equation*}
\notag
$$
Such commutators play an important role in harmonic analysis. It is well known that
$$
\begin{equation*}
(\mathbb{P}_+f)(\zeta)={\rm p.v.}\int_{\mathbb{T}} \frac{f(\tau)}{1-\overline\tau\zeta}\,{\rm d} \boldsymbol{m}(\tau)+ \frac{1}2f(\zeta),\qquad f\in\mathscr{P},\quad \zeta\in\mathbb{T}.
\end{equation*}
\notag
$$
Here the notation p.v. means the integral in the sense of principal value. It is easy to deduce from the above expression the following identity
$$
\begin{equation}
{\mathscr C}_\varphi f={\rm p.v.}\int_{\mathbb{T}} \frac{\varphi(\zeta)-\varphi(\tau)} {1-\overline\tau\zeta}f(\tau)\,{\rm d} \boldsymbol{m}(\tau),\qquad f\in\mathscr{P}.
\end{equation}
\tag{5.1}
$$
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 $f_-\overset{\rm def}{=}\mathbb{P}_-f$. Proof. We have
$$
\begin{equation*}
\begin{aligned} \, {\mathscr C}_\varphi f&=\varphi\mathbb{P}_+f-\mathbb{P}_+\varphi f= \varphi_+f_++\varphi_-f_+-\varphi_+f_+-\mathbb{P}_+\varphi_-f_+- \mathbb{P}_+\varphi_+f_- \\ &=\mathbb{P}_-\varphi_-f_+-\mathbb{P}_+\varphi_+f_-=H_{\varphi_-} f_+- \bigl(H_{\overline{\varphi_+}}\bigr)^*f_-, \end{aligned}
\end{equation*}
\notag
$$
which proves the theorem. 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. Theorem 5.3. Let $0<p<\infty$. Then the commutator ${\mathscr C}_\varphi$ defined by (5.1) belongs to the class $\boldsymbol{S}_p$ if and only if $\varphi$ belongs to the Besov class $(B_p^{1/p})(\mathbb{T})$. 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 transformFor 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
$$
\begin{equation*}
(\boldsymbol\varGamma_kf)(t)=\int_0^\infty k(s+t)f(s)\,{\rm d}s.
\end{equation*}
\notag
$$
The operators $\boldsymbol\varGamma_k$ are integral analogues of the matrix operators $\Gamma_\alpha$ defined by (4.1). It turns out, however, that the operators $\boldsymbol\varGamma_k$ are not only integral analogues of Hankel operators $\Gamma_\alpha$, but also have Hankel matrices in the basis of Laguerre functions. We are going to discuss in this subsection how to define the operators $\boldsymbol\varGamma_k$ in the situation considerably more general than in the case $k\in L^1(\mathbb{R}_+)$, and we are also going to discuss the conditions of boundedness, compactness, and membership in the Schatten–von Neumann ideals for such operators. 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
$$
\begin{equation*}
\sup_{t\in\mathbb{R}}|f^{(m)}(t)|(1+|t|)^n<\infty\quad\text{for all}\ m\ \text{and}\ n\ \text{in}\ \mathbb{Z}_+.
\end{equation*}
\notag
$$
We can define now the space ${\mathcal S}'$ of tempered distributions as the space of continuous linear functionals on ${\mathcal S}$ endowed with the natural topology; see [102]. 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
$$
\begin{equation*}
(\mathscr{F} f)(s)=\int_\mathbb{R} e^{-2\pi{\mathrm{i}}ts}f(t)\,{\rm d}t \quad\text{and}\quad (\mathscr{F}^* f)(t)=\int_\mathbb{R} e^{2\pi{\mathrm{i}}ts}f(s)\,{\rm d}s
\end{equation*}
\notag
$$
map the Schwartz space ${\mathcal S}$ onto itself, which makes it possible to extend them by duality to linear maps of the space of tempered distributions ${\mathcal S}'$ onto itself. The following result holds. Theorem 5.4. Let $k\in{\mathcal D}'(0,\infty)$. The following statements are equivalent: (i) $\boldsymbol\varGamma_k$ is a bounded operator on $L^2[0,\infty)$; (ii) there exists a function $\varkappa$ of class $L^\infty(\mathbb{R})$ such that $(\mathscr{F}\varkappa)|(0,\infty)=k$; (iii) there exists a distribution $r$ in ${\mathcal S}'$ such that $r|(0,\infty)=k$, $\operatorname{supp} r\subset[0,\infty)$ and $\mathscr{F}^*r\in\mathrm{BMO}(\mathbb{R})$. 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
$$
\begin{equation*}
({\mathcal U}f)(t)=\frac{1}{\sqrt{\pi}}\, \frac{(f\circ\omega^{-1})(t)}{t+\mathrm{i}}\,,\qquad t\in\mathbb{R},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\omega(\zeta)\overset{\rm def}= \mathrm{i}\,\frac{1+\zeta}{1-\zeta}\,,\qquad \zeta\in\mathbb{D}.
\end{equation*}
\notag
$$
It is well known that ${\mathcal U}$ is a unitary operator from $L^2(\mathbb{T})$ onto $L^2(\mathbb{R})$; moreover, it maps the Hardy class $H^2$ on the disc onto the Hardy class $H^2(\mathbb{C}_+)$ on the half-plane. The following assertion holds. Theorem 5.5. The bounded operators $\boldsymbol\varGamma_k$ have Hankel matrices in the orthonormal basis $\{\mathscr{F}{\mathcal U}z^n\}_{n\geqslant0}$ in $L^2(\mathbb{R}_+)$. Moreover, the following identity holds: where the $L_n$ are the Laguerre polynomials defined by 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 $\boldsymbol{S}_p$. Theorem 5.6. Let $0<p<\infty$, and let $k$ be a distribution on $(0,\infty)$. Then $\boldsymbol\varGamma_k\in\boldsymbol{S}_p$ if and only if $k=(\mathscr{F}\psi)|(0,\infty)$ for a function $\psi$ in the Besov class $B_p^{1/p}(\mathbb{R})$. 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
$$
\begin{equation*}
(\boldsymbol{\mathcal C}_\varphi f)(y)=\text{v.p.}\int_\mathbb{R} \frac{\varphi(x)-\varphi(y)}{x-y}f(x)\,{\rm d}x,\qquad f\in L^2(\mathbb{R}).
\end{equation*}
\notag
$$
It is well known that the operator $\boldsymbol{\mathcal C}_f$ is bounded on $L^2(\mathbb{R})$ if and only if $\varphi\in\mathrm{BMO}(\mathbb{R})$. Theorem 5.7. Let $0<p<\infty$, and let $\varphi$ be a function of class $\mathrm{BMO}(\mathbb{R})$. Then the operator $\boldsymbol{\mathcal C}_\varphi$ belongs to the Schatten–von Neumann class $\boldsymbol{S}_p$ if and only if $\varphi$ belongs to the Besov class $B_p^{1/p}(\mathbb{R})$. 5.4. Wiener–Hopf operators on a finite intervalLet $\sigma>0$. 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
$$
\begin{equation*}
(W_{\sigma,k}f)(x)=\int_{-\sigma}^\sigma k(x-y)f(y)\,{\rm d}y,\qquad x\in(-\sigma,\sigma).
\end{equation*}
\notag
$$
Such operators are called Wiener–Hopf operators on a finite interval. In this subsection we discuss conditions under which such operators belong to $\boldsymbol{S}_p$ for $0<p<\infty$. 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
$$
\begin{equation*}
(\boldsymbol{\varGamma}_{\sigma,k}f)(x)= \int_{-\sigma}^\sigma k(x+y)f(y)\,{\rm d} y,\qquad x\in(-\sigma,\sigma).
\end{equation*}
\notag
$$
It is easy to see that $\boldsymbol{\varGamma}_{\sigma,k}=W_{\sigma,k}U$, where $U$ is the unitary operator 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:
$$
\begin{equation*}
\begin{alignedat}{2} \nu_j(x)&=w\biggl(\frac{x+2\sigma}{2^{j+1}\sigma}\biggr),&\qquad j&<0; \\ \nu_j(x)&=\nu_{-j}(-x),&\qquad j&>0; \\ \nu_0(x)&=1-\sum_{j\ne0}\nu_j(x),&& \end{alignedat}
\end{equation*}
\notag
$$
where $w$ is an infinitely smooth function satisfying (2.1). The following theorem was proved in [75] (note that in [96] the same result was established by different methods in the case when $p\geqslant1$). Theorem 5.8. Let $\sigma>0$ and $0<p<\infty$, and let $k$ be a distribution on $(-2\sigma,2\sigma)$. The following statements are equivalent: (i) $W_{\sigma,k}\in\boldsymbol{S}_p$; (ii)$\boldsymbol{\varGamma}_{\sigma,k}\in\boldsymbol{S}_p$; (iii) $ \quad\displaystyle \sum_{j\in\mathbb{Z}}2^{-|j|}\|\mathcal{F}(\nu_jk)\|_{L^p}^p<\infty; $ (iv) there exist functions $\psi_1$ and $\psi_2$ in the Besov class $B_p^{1/p}(\mathbb{R})$ such that andIn 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 processesThe 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 By the F. Riesz–Herglotz theorem (see [95], § 53) there exists a finite positive Borel measure $\mu$ on the unit circle $\mathbb{T}$ such that $c_n=\int_\mathbb{T} z^n\,d\mu$. The measure $\mu$ is called the spectral measure of the process. Then the closed linear span $\operatorname{clos}\operatorname{span}\{X_n\colon n\in\mathbb{Z}\}$ can isometrically be identified with the weighted space $L^2(\mu)$ via the map In addition, the past of the process is identified with the subspace
$$
\begin{equation*}
H^2_-(\mu)\overset{\rm def}= \operatorname{clos}\operatorname{span}\{z^j\colon j<0\},
\end{equation*}
\notag
$$
while the future, starting with the moment of time $n$ is identified with the subspace
$$
\begin{equation*}
z^nH^2(\mu)\overset{\rm def}= \operatorname{clos}\operatorname{span}\{z^j\colon j\geqslant n\}.
\end{equation*}
\notag
$$
We are going to consider regular stationary processes, that is, processes for which It is well known that a stationary process is regular if and only if its spectral measure $\mu$ is absolutely continuous with respect to Lebesgue measure and its density $w$ (it is called the spectral density of the process) satisfies the condition $\log w\in L^1(\mathbb{T})$.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]. Theorem 5.9. Let $1\leqslant p<\infty$, and let $w$ be the spectral density of a regular stationary process. The following are equivalent: (i) $\mathscr{P}_0\mathscr{P}^n\in\boldsymbol{S}_p$ for all $n\in\mathbb{Z}_+$; (ii) $\mathscr{P}_0\mathscr{P}^0\in\boldsymbol{S}_p$; (iii) $w$ admits a representation where $P$ is a polynomial with zeros on $\mathbb{T}$ and $\varphi$ is a real-valued function in the Besov class $B_P^{1/p}(\mathbb{T})$.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 BookThis 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 polynomialsWe 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, Here $c>1$ (the case $c=1$ corresponds to the case when $T$ is a contraction, that is, $\|T\|\leqslant1$). It is well known that for $c>1$ the operator $T$ does not have to be polynomially bounded, that is, for such operators $T$ the estimate
$$
\begin{equation*}
\|\varphi(T)\|\leqslant \mathrm{const}\max_{|\zeta|\leqslant1}|\varphi(\zeta)|
\end{equation*}
\notag
$$
for analytic polynomials $\varphi$ does not hold in general (this was established in [54]; see also [71] and the papers cited there). 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
$$
\begin{equation*}
|\mkern-1.5mu|\mkern-1.5mu| \varphi |\mkern-1.5mu|\mkern-1.5mu| _c\overset{\rm def}=\sup\bigl\{\|\varphi(T)\|\colon T\text{ satisfies the condition (6.1)}\bigr\}
\end{equation*}
\notag
$$
in an optimal way. 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 where $\{f_n(j)\}_{j\geqslant0}\in\ell^\infty$, $\{g_n(k)\}_{k\geqslant0}\in\ell^\infty$, and
$$
\begin{equation}
\sum_n\|f_n\|_{\ell^\infty}\|g_n\|_{\ell^\infty}<\infty.
\end{equation}
\tag{6.3}
$$
By the norm $\bigl\|\{\gamma_{jk}\}_{j,k\geqslant0}\bigr\|_{\ell^\infty \mathbin{\widehat\otimes}\ell^\infty}$ of the matrix $\{\gamma_{jk}\}_{j,k\geqslant0}$ in $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$ we understand the infimum of the left-hand side of inequality (6.3) over all representations of this matrix in the form (6.2). Equipped with this norm, the space $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$ becomes a Banach space. 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
$$
\begin{equation*}
(P_m\gamma)_{jk}=\begin{cases} \gamma_{jk},&j\leqslant m, \ k\leqslant m; \\ 0 & \text{otherwise}. \end{cases}
\end{equation*}
\notag
$$
We say that an infinite matrix $\gamma=\{\gamma_{jk}\}_{j,k\geqslant0}$ belongs to the space $V^2$ if
$$
\begin{equation*}
\sum_{m\in\mathbb{Z}_+}\|P_m\gamma\|_{\ell^\infty\mathbin{\widehat\otimes} \ell^\infty}<\infty.
\end{equation*}
\notag
$$
Here $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$ is a closed proper subspace of $V^2$. It is easy to observe that the identity matrix belongs to $V^2$ but does not belong to $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$. 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,
$$
\begin{equation*}
B=\{b_{jk}\}_{j,k\geqslant0}\in{\mathcal B}(\ell^2)\quad\Longrightarrow\quad A\star B\overset{\rm def}=\{a_{jk}b_{jk}\}_{j,k\geqslant0} \in{\mathcal B}(\ell^2).
\end{equation*}
\notag
$$
It turns out that the converse also holds; see [21]. 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
$$
\begin{equation*}
\bigl\|\{\gamma_{jk}\}_{j,k\geqslant0}\bigr\|_{\ell^1 \mathbin{\check\otimes}\ell^1}= \sup\biggl\{\biggl|\,\sum_{j,k\geqslant0}\gamma_{jk}f_jg_k\biggr|\colon \|\{f_j\}_{j\geqslant0}\|_{\ell^\infty}\leqslant1, \|\{g_k\}_{k\geqslant0}\|_{\ell^\infty}\leqslant1\biggr\}.
\end{equation*}
\notag
$$
It well known that the dual space $(\ell^1\mathbin{\check{\otimes}}\ell^1)^*$ can be identified with the space $V^2$ with respect to the bilinear form
$$
\begin{equation*}
\bigl(\{\gamma_{jk}\}_{j,k\geqslant0},\{\beta_{jk}\}_{j,k\geqslant0}\bigr)= \sum_{j,k\geqslant0}\gamma_{jk}\beta_{jk},\qquad \{\gamma_{jk}\}_{j,k\geqslant0}\in\ell^1 \mathbin{\check\otimes}\ell^1,\quad\! \{\beta_{jk}\}_{j,k\geqslant0}\in V^2.
\end{equation*}
\notag
$$
Note also that the space $\ell^1\mathbin{\check{\otimes}}\ell^1$ can naturally be identified with the space of bounded linear operators from the space $c_0$ of sequences with zero limit to the space $\ell^1$. 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
$$
\begin{equation}
\widehat f(n)=\sum_{j+k=n}\gamma_{jk},\qquad n\geqslant0.
\end{equation}
\tag{6.4}
$$
The norm $\|f\|_{{\mathscr L}}$ of a function $f$ in the space ${\mathscr L}$ is defined as the infimum of the norms $\|\{\gamma_{jk}\}_{j,k\geqslant0}\|_{\ell^1\mathbin{\check{\otimes}}\ell^1}$ over all matrices $\{\gamma_{jk}\}_{j,k\geqslant0}$ in $\ell^1\mathbin{\check{\otimes}}\ell^1$ that satisfy (6.4). It is easy to show that
$$
\begin{equation*}
\|\varphi\|_{{\mathscr L}}=\sup\bigl\{|\langle\varphi,\psi\rangle|\colon \|\Gamma_\psi\|_{V^2}\leqslant1\bigr\},
\end{equation*}
\notag
$$
that is, the supremum is taken over all analytic functions $\psi$ in $\mathbb{D}$ for which the Hankel matrix $\Gamma_\psi$ belongs to the unit ball of the space $V^2$. Here
$$
\begin{equation*}
\langle\varphi,\psi\rangle\overset{\rm def}= \sum_{n\geqslant0}\widehat\varphi(n)\widehat\psi(n).
\end{equation*}
\notag
$$
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
$$
\begin{equation}
\biggl|\,\sum_{j,k\geqslant0}\gamma_{jk}(x_j,y_k)\biggr|\leqslant k_{\rm G} \|\{\gamma_{jk}\}_{j,k\geqslant0}\|_{\ell^1\mathbin{\check\otimes}\ell^1},
\end{equation}
\tag{6.5}
$$
where $k_{\rm G}$ is the so-called Grothendieck constant, that is, the least number for which the inequality holds. The following result was obtained in [71]. Theorem 6.1. Let $T$ be an operator on Hilbert space that satisfies (6.1). Then for every analytic polynomial $\varphi$. Proof. Let $x$ and $y$ be vectors in the unit ball of the Hilbert space, and let $\{\gamma_{jk}\}_{j,k\geqslant0}$ be a complex matrix whose entries, except for finitely many of them, are equal to zero and such that
$$
\begin{equation*}
\sum_{j+k=n}\gamma_{jk}=\widehat\varphi(n),\qquad n\geqslant0.
\end{equation*}
\notag
$$
We have
$$
\begin{equation*}
(\varphi(T)x,y)=\sum_{n\geqslant0}\widehat\varphi(n)(T^nx,y)= \sum_{j,k\geqslant0}\widehat\varphi(j+k)(T^jT^kx,y)= \sum_{j,k\geqslant0}\gamma_{jk}(T^kx,(T^*)^jy).
\end{equation*}
\notag
$$
Since $\|T^kx\|\leqslant c$ and $\|(T^*)^jy\|\leqslant c$, from Grothendieck’s inequality (6.5) we obtain which proves the theorem. Let us proceed now to more tangible estimates of the norms of operator polynomials. Definition 1. We say that a function $\psi$ analytic in $\mathbb{D}$ is a multiplier of the Hardy class $H^1$ if 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. Corollary 6.2. Let $T$ be a power bounded operator on Hilbert space. Then for every analytic polynomial $\varphi$. Let us define now the space $\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1$ of functions analytic in the disc $\mathbb{D}$ by
$$
\begin{equation*}
\begin{aligned} \, &{\rm VMOA}\mathbin{\widehat{\circledast}} H^1 \\ &\qquad=\biggl\{\,\sum_mf_m*g_m\colon f_m\in{\rm VMOA}, g_m\in H^1, \sum_m\|f_m\|_{\rm VMOA}\,\|g_m\|_{H^1}<\infty\biggr\}. \end{aligned}
\end{equation*}
\notag
$$
The norm in $\mathrm{VMOA}\mathbin{\widehat{\circledast}} H^1$ is defined by
$$
\begin{equation*}
\|\varphi\|_{{\rm VMOA}\mathbin{\widehat{\circledast}} H^1}\overset{\rm def}= \inf\biggl\{\,\sum_m\|f_m\|_{{\rm VMOA}}\,\|g_m\|_{H^1}\colon \sum_mf_m\ast g_m=\varphi\biggr\}.
\end{equation*}
\notag
$$
It is easy to see that
$$
\begin{equation}
\bigl({\rm VMOA}\mathbin{\widehat{\circledast}}H^1\bigr)^*={\mathfrak M}H^1
\end{equation}
\tag{6.8}
$$
with respect to the natural pairing. 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. Let $T$ be a power bounded operator on Hilbert space. Then for every analytic polynomial $\varphi$. 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 $(B^0_{\infty,1})_+$. Theorem 6.4. Let $T$ be a power bounded operator on Hilbert space. Then for every analytic polynomial $\varphi$. Proof. By Theorem 6.3 it suffices to prove that and the embedding is continuous.
Let $\varphi\in(B^0_{\infty,1})_+$. It is easy to see that Thus, The result follows now from the well-known fact that $\|W_n\|_{L^1}\leqslant\mathrm{const}$. The theorem is proved.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
$$
\begin{equation*}
\|\varphi(T)\|\leqslant\mathrm{const}\|\varphi\|_\flat
\end{equation*}
\notag
$$
for an arbitrary operator $T$ satisfying (6.1). Also, the constant on the right-hand side of (6.1) can depend on the number $c$ in (6.1). We say that this estimate is optimal if there exist $c>1$ and an operator $T$ satisfying (6.1) such that
$$
\begin{equation*}
\|\varphi(T)\|\geqslant\mathrm{const}\|\varphi\|_\flat.
\end{equation*}
\notag
$$
It is easy to see that in this case the completion $\mathscr{P}^\flat$ of the space of polynomials in the norm $\|\,{\cdot}\,\|_\flat$ is a Banach algebra with respect to pointwise multiplication. Moreover, this Banach algebra must be an operator algebra, that is, it must be isomorphic to a subalgebra of all operators on Hilbert space. 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
$$
\begin{equation*}
\widehat\varphi(n)=\sum_{j+k=n}\gamma_{jk},\qquad n\geqslant0.
\end{equation*}
\notag
$$
A natural question arises of whether the choice would be optimal if we put
$$
\begin{equation}
\gamma_{jk}=\frac{\widehat\varphi(n)}{n+1}\quad\text{in the case when}\ j+k=n+1.
\end{equation}
\tag{6.9}
$$
This leads to the question of describing the Hankel matrices of class $\ell^1\mathbin{\check{\otimes}}\ell^1$, which was answered by the following theorem obtained in [71]. Theorem 6.5. Let $\varphi$ be a function analytic in $\mathbb{D}$. Then $\Gamma_\varphi\in\ell^1\mathbin{\check{\otimes}}\ell^1$ if and only if $\varphi\in(B_{\infty,1}^1)_+$. Therefore, if we make the choice (6.9), then we obtain the estimate
$$
\begin{equation*}
\|\varphi(T)\|\leqslant\mathrm{const}\|\varphi\|_{B^0_{\infty,1}},
\end{equation*}
\notag
$$
which is not optimal as we have already mentioned. 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]). Corollary 6.7. The projection $\mathscr{P}$ is not a bounded operator on $\ell^1\mathbin{\check{\otimes}}\ell^1$, $\ell^\infty\mathbin{\widehat\otimes}\ell^\infty$, and $V^2$. 6.3. Problems 8 and 88 from the Scottish BookIn 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. Problem 8. Let $c$ be the space of convergent sequences. Consider the bilinear form ${\mathscr B}$ on $c\times c$ defined by where It is easy to see that ${\mathscr B}$ maps $c\times c$ to $c$. Problem 8 in the Scottish Book is whether ${\mathscr B}$ maps $c\times c$ onto the whole space $c$. Problem 88. Suppose that a Hankel matrix $\{\gamma_{j+k}\}$ belongs to the tensor product $\ell^1\mathbin{\check{\otimes}}\ell^1$. Problem 88 of the Scottish Book is whether this implies that that is, whether the sum of the matrix entries must be finite. 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:
$$
\begin{equation*}
{\mathscr A}Q=\{z_n\}_{n\geqslant0},\quad\text{where}\ \ z_n\overset{\rm def}=\frac{1}{n+1}\sum_{j+k=n}q_{jk}.
\end{equation*}
\notag
$$
Theorem 6.6 of this survey, obtained in [71], can be reformulated in the following way. Theorem 6.8. The operator ${\mathscr A}$ maps $V^2$ onto the space of Fourier coefficients of functions in the Besov class $(B^0_{1,\infty})_+$. This theorem implies that
$$
\begin{equation*}
{\mathscr A}\bigl(c\mathbin{\widehat\otimes} c\bigr)\subset {\mathscr A}\bigl(\ell^\infty\mathbin{\widehat\otimes} \ell^\infty\bigr) \subset{\mathscr A}(V^2)\subset\bigl\{\{f_n\}_{n\geqslant0}\colon f\in\bigl(B^0_{1,\infty}\bigr)_+\bigr\},
\end{equation*}
\notag
$$
and so
$$
\begin{equation*}
{\mathscr B}(c\times c)\subset\bigl\{\{f_n\}_{n\geqslant0}\colon f\in\bigl(B^0_{1,\infty}\bigr)_+\bigr\}.
\end{equation*}
\notag
$$
However, it is well known that
$$
\begin{equation*}
c\not\subset\bigl\{\{f_n\}_{n\geqslant0}\colon f\in\bigl(B^0_{1,\infty}\bigr)_+\bigr\}
\end{equation*}
\notag
$$
(see [81], where a detailed discussion of such issues is given). Let us now proceed to Problem 88. First of all, it follows from Theorem 6.5 in this survey that
$$
\begin{equation*}
\{\gamma_{j+k}\}_{j,k\geqslant0}\in\ell^1\mathbin{\check\otimes}\ell^1 \quad\Longleftrightarrow\quad \sum_{n\geqslant0}\gamma_nz^n\in\bigl(B^1_{\infty,1}\bigr)_+.
\end{equation*}
\notag
$$
A negative solution to Problem 88 follows from the following best possible estimate of the moduli of Fourier coefficients of functions in the Besov class $(B^1_{\infty,1})_+$:
$$
\begin{equation*}
\sum_{n\geqslant0}2^n\biggl(\,\sum_{k=2^n}^{2^{n+1}-1} |\widehat f(k)|^2\biggr)^{1/2}<\infty.
\end{equation*}
\notag
$$
Indeed, it is easy to see that this condition on the coefficients does not imply that We refer the reader to [81], where a detailed discussion of such issues is presented. 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 theory7.1. Functions of self-adjoint operators under perturbations of the operatorsLet $f$ be a continuous complex-valued function on the real line. Consider the problem of when the inequality holds for arbitrary (bounded) self-adjoint operators $A$ and $B$ on Hilbert space. Functions $f$ satisfying this property are called operator Lipschitz. We denote by $\operatorname{OL}(\mathbb{R})$ the space of operator Lipschitz functions on $\mathbb{R}$, and for $f\in\operatorname{OL}(\mathbb{R})$ we put
$$
\begin{equation*}
\|f\|_{\operatorname{OL}(\mathbb{R})}\overset{\rm def}= \sup\frac{\|f(A)-f(B)\|}{\|A-B\|}\,,
\end{equation*}
\notag
$$
where the supremum is taken over all pairs of self-adjoint operators $A$ and $B$ with bounded difference such that $A\ne B$. 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:
$$
\begin{equation*}
|f(x)-f(y)|\leqslant\mathrm{const}|x-y|,\qquad x,y\in\mathbb{R}.
\end{equation*}
\notag
$$
In the paper [32] by Farforovskaya it was shown that the converse is false. 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. Theorem 7.1. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R})$. Then inequality (7.1) holds for arbitrary self-adjoint operators $A$ and $B$. To prove Theorem 7.1 double operator integrals were used in [73] and [76]. Double operator integrals are expressions of the form
$$
\begin{equation}
\int_{\mathscr X}\int_{\mathscr Y} \Phi(x,y)\,\mathrm{d}E_1(x)Q\,\mathrm{d}E_2(y),
\end{equation}
\tag{7.2}
$$
where $E_1$ and $E_2$ are spectral measures on Hilbert space, $Q$ is a bounded linear operator on Hilbert space, and $\Phi$ is a bounded measurable functions of two variables. 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:
$$
\begin{equation*}
\biggl\|\int_{\mathscr X}\int_{\mathscr Y}\Phi(x,y)\,{\rm d} E_1(x)Q\,{\rm d} E_2(y)\biggr\|_{\boldsymbol{S}_2}\leqslant \|\Phi\|_{L^\infty}\|Q\|_{\boldsymbol{S}_2}.
\end{equation*}
\notag
$$
It turns out, however, that if $Q$ is an arbitrary bounded operator, then to define the double operator integral (7.2) we have to impose additional assumptions on the function $\Phi$. In [73] the natural maximal class of functions $\Phi$ for which the double operator integral (7.2) can be defined for arbitrary bounded operators $Q$ was named the class of Schur multipliers with respect to the spectral measures $E_1$ and $E_2$. There are several descriptions of the class of Schur multipliers; see [73]. We mention here the following one. Theorem 7.2. A bounded measurable function $\Phi$ is a Schur multiplier with respect to $E_1$ and $E_2$ if and only if $\Phi$ belongs to the Haagerup tensor product $L^\infty_{E_1}\mathbin{\otimes_{\rm h}}L^\infty_{E_2}$, that is, $\Phi$ admits a representation where $\varphi_n\in L^\infty_{E_1}$ and $\psi_n\in L^\infty_{E_2}$, and Note here that if $\Phi$ is a Schur multiplier with respect to spectral measures $E_1$ and $E_2$, then
$$
\begin{equation}
Q\in\boldsymbol{S}_p,\ \ 1\leqslant p<\infty\quad\Longrightarrow\quad \int_{\mathscr X}\int_{\mathscr Y}\Phi(x,y)\,{\rm d} E_1(x)Q\,{\rm d} E_2(y)\in\boldsymbol{S}_p.
\end{equation}
\tag{7.3}
$$
The role of double operator integrals in obtaining operator Lipschitz estimates can be demonstrated by the following result by Birman and Solomyak [25]. Theorem 7.3. Let $f$ be a differentiable function on $\mathbb{R}$, and let ${\mathfrak D} f$ be the corresponding divided difference, that is, Suppose that $A$ and $B$ are self-adjoint operators such that the operator $B-A$ is bounded and that ${\mathfrak D} f$ is a Schur multiplier with respect to the spectral measures $E_B$ and $E_A$ of the operators $B$ and $A$. Then $f(B)-f(A)$ is a bounded operator and 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]. Theorem 7.4. Let $f$ a function in the Besov class $B_{\infty,1}^1(\mathbb{R})$. Then ${\mathcal D}f$ is a Schur multiplier with respect to arbitrary pairs of spectral Borel measures on $\mathbb{R}$, $f$ is an operator Lipschitz function, and for arbitrary self-adjoint operators $A$ and $B$ with bounded difference identity (7.4) is satisfied and the following inequality holds: 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 inequalityIt 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
$$
\begin{equation}
\|f(A)-f(B)\|\leqslant\mathrm{const}\,\sigma\|f\|_{L^\infty}.
\end{equation}
\tag{7.6}
$$
In [6] (see also [10]) an alternative, more elementary approach to the operator Lipschitzness of functions of class $B_{\infty,1}^1(\mathbb{R})$ was proposed. The first step of this approach is an independent proof of the operator Bernstein-type inequality. Then the operator Lipschitzness of functions in $B_{\infty,1}^1(\mathbb{R})$ was deduced from it. Moreover, it was shown in [6] that inequality (7.6) actually holds with constant $1$. 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
$$
\begin{equation*}
{\mathscr E}_\sigma^\infty=\{f\in L^\infty(\mathbb{R})\colon \operatorname{supp}\mathscr{F} f\subset[-\sigma,\sigma]\}.
\end{equation*}
\notag
$$
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
$$
\begin{equation}
|f(z)|\leqslant e^{\sigma|\!\operatorname{Im} z|} \|f\|_{L^\infty(\mathbb{R})},\qquad z\in\mathbb{C}
\end{equation}
\tag{7.7}
$$
(see, for example, [55], p. 97). Bernstein’s inequality (see [22]) asserts that
$$
\begin{equation*}
\sup_{x\in \mathbb{R}}|f'(x)|\leqslant\sigma\sup_{x\in \mathbb{R}}|f(x)|
\end{equation*}
\notag
$$
for any function $f$ of class $L^\infty(\mathbb{R})$ whose Fourier transform has support in $[-\sigma,\sigma]$. It implies that
$$
\begin{equation}
|f(x)-f(y)|\leqslant\sigma\|f\|_{L^\infty(\mathbb R)}|x-y|,\qquad f\in{\mathscr E}_\sigma^\infty,\quad x,y\in\mathbb{R}.
\end{equation}
\tag{7.8}
$$
Bernstein also proved in [22] the following improvement of inequality (7.8):
$$
\begin{equation}
|f(x)-f(y)|\leqslant\beta(\sigma|x-y|)\|f\|_{L^\infty(\mathbb{R})},\qquad f\in{\mathscr E}_\sigma^\infty,\quad x,y\in\mathbb{R},
\end{equation}
\tag{7.9}
$$
where
$$
\begin{equation*}
\beta(t)\overset{\rm def}=\begin{cases} 2\sin \dfrac{t}{2}&\text{if}\ 0\leqslant t\leqslant\pi, \\ 2&\text{if}\ t>\pi. \end{cases}
\end{equation*}
\notag
$$
Obviously, $\beta(t)\leqslant\min(t,2)$ for all $t\geqslant0$. In [6] the following version of the operator Bernstein-type inequality was deduced from (7.9). Theorem 7.5. Let $f\in\mathscr {\mathscr E}_\sigma^\infty$. Then for arbitrary (bounded) self-adjoint operators $A$ and $B$. In particular, $\|f\|_{\operatorname{OL}(\mathbb{R})} \leqslant\sigma\|f\|_{L^\infty(\mathbb{R})}$. Proof. First of all, let us observe that inequality (7.9) implies the following vector version of it. Let $X$ be a complex Banach space. We denote by ${\mathscr E}_\sigma(X)$ the space of entire $X$-valued functions $f$ of exponential type at most $\sigma$, that is, satisfying the following condition: for each positive $\varepsilon$ there exists $c>0$ such that $\|f(z)\|_X\leqslant c e^{(\sigma+\varepsilon)|z|}$ for all $z\in\mathbb{C}$. Then the following inequality holds
$$
\begin{equation}
\|f(x)-f(y)\|_X\leqslant\beta(\sigma|x-y|)\|f\|_{L^\infty(\mathbb{R},X)} \leqslant \sigma\|f\|_{L^\infty(\mathbb{R},X)}|x-y|,\quad x,y\in\mathbb{R}.
\end{equation}
\tag{7.11}
$$
It is easy to see that inequality (7.11) reduces to (7.9) with the help of the Hahn–Banach theorem.
Now let $A$ and $B$ be self-adjoint operators in Hilbert space $\mathscr{H}$. Let us show that
$$
\begin{equation*}
\|f(A)-f(B)\|\leqslant\beta(\sigma\|A-B\|)\|f\|_{L^\infty}.
\end{equation*}
\notag
$$
Put $F(z)=f(A+z(B-A))$. Clearly, $F$ is an entire function that takes values in the space $\mathcal{B}(\mathscr{H})$ of bounded linear operators on $\mathscr{H}$ and $\|F(t)\|\leqslant\|f\|_{L^\infty(\mathbb{R})}$ for $t\in\mathbb{R}$. It follows from von Neumann’s inequality (see [104], Chap. I, § 8) that $F\in{\mathscr E}_{\sigma\|B-A\|}(\mathcal{B}(\mathscr{H}))$. To complete the proof it remains to apply the vector version of Bernstein’s inequality (7.11) to the vector-valued function $F$ for $x=0$ and ${y=1}$. The theorem is proved. Proof of inequality (7.5) (based on the operator Bernstein inequality). Let $f\in B_{\infty,1}^1(\mathbb{R})$. Consider the functions $f_n$ defined by (2.8). Clearly, $\operatorname{supp}\mathscr{F} f_n\subset[-2^{n+1},2^{n+1}]$. Then by (7.10) we have
$$
\begin{equation*}
\|f(A)-f(B)\|\leqslant\sum_{n\in\mathbb{Z}}\|f_n(A)-f_n(B)\|\leqslant \sum_{n\in\mathbb{Z}}2^{n+1}\|f_n\|_{L^\infty}\leqslant \mathrm{const}\|f\|_{B_{\infty,1}^1},
\end{equation*}
\notag
$$
which proves (7.5). Commentary. It can be shown that if $f$ is an operator Lipschitz function on $\mathbb{R}$, and $A$ and $B$ are self-adjoint operators such that $A-B\in\boldsymbol{S}_p$, $1\leqslant p<\infty$, then $f(A)-f(B)\in\boldsymbol{S}_p$. However, as shown in [93], for $1<p<\infty$ the conclusion $f(A)-f(B)\in\boldsymbol{S}_p$ holds under a weaker assumption on $f$; it suffices to assume that $f$ a Lipschitz condition. This solved a famous problem. 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 for arbitrary self-adjoint (no matter whether bounded or unbounded) operators $A$ and $B$. However, it turns out that, in contrast to the case of operator Lipschitz functions, a function $f$ is operator Hölder of order $\alpha$ if and only if it is a Hölder function of order $\alpha$, that is, belongs to $\Lambda_\alpha(\mathbb{R})$. This was established in [4]. Note that in the same paper [4] other estimates for function of operators under perturbation were found. Another approach to the proof of inequality (7.12) for functions $f$ in $\Lambda_\alpha(\mathbb{R})$ was found in [64].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
$$
\begin{equation*}
f(A)-f(B)\in\boldsymbol{S}_{p/\alpha}\quad\text{and}\quad \|f(A)-f(B)\|_{\boldsymbol{S}_{p/\alpha}}\leqslant \mathrm{const}\|f\|_{\Lambda_\alpha}\|A-B\|_{\boldsymbol{S}_p}^\alpha.
\end{equation*}
\notag
$$
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 perturbationThe 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 for arbitrary unitary operators $U$ and $V$ on Hilbert space.In [73] the following result was obtained. Theorem 7.6. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{T})$. Then $f$ is an operator Lipschitz function on $\mathbb{T}$ and the following inequality holds for arbitrary unitary operators $U$ and $V$ on Hilbert space. 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:
$$
\begin{equation*}
f(U)-f(V)=\int_{\mathbb{T}}\int_{\mathbb{T}} \frac{f(\zeta)-f(\tau)}{\zeta-\tau}\,{\rm d} E_U(\zeta) (U-V)\,{\rm d}E_V(\tau)
\end{equation*}
\notag
$$
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: for any trigonometric polynomial $q$. Commentary. Note briefly that the problem of obtaining Lipschitz-type estimates for functions of contractions was considered first in the author’s paper [74] (an operator is called a contraction if its norm is at most $1$). In that paper, with the help of double operator integrals with respect to semispectral measure it was shown that if $f$ is a function in the Besov class $(B_{\infty,1}^1)_+$ of functions analytic in the disc $\mathbb{D}$ and $T$ and $R$ are contractions on Hilbert space, then 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 operatorsThe 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
$$
\begin{equation}
\|f(N_1)-f(N_2)\|\leqslant\mathrm{const}\|N_1-N_2\|
\end{equation}
\tag{7.14}
$$
holds for arbitrary normal operators $N_1$ and $N_2$ with bounded difference. 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
$$
\begin{equation*}
\begin{gathered} \, A_j\overset{\rm def}=\operatorname{Re} N_j=\frac{1}2(A_j+A_j^*),\qquad B_j\overset{\rm def}=\operatorname{Im} N_j=\frac{1}{2\mathrm{i}}(A_j-A_j^*), \\ \quad E_j\text{ is the spectral measure of } N_j,\ j=1,2. \end{gathered}
\end{equation*}
\notag
$$
In other words, $N_j=A_j+{\rm i}B_j$, $j=1,2$, where $A_j$ and $B_j$ are commuting self-adjoint operators. 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
$$
\begin{equation*}
\bigl({\mathfrak D}_xf\bigr)(z_1,z_2)\overset{\rm def}= \frac{f(x_1,y_2)-f(x_2,y_2)}{x_1-x_2}\,,\qquad z_1,z_2\in\mathbb{C},
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\bigl({\mathfrak D}_yf\bigr)(z_1,z_2)\overset{\rm def}= \frac{f(x_1,y_1)-f(x_1,y_2)}{y_1-y_2}\,,\qquad z_1,z_2\in\mathbb{C},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
x_j\overset{\rm def}=\operatorname{Re} z_j\quad\text{and}\quad y_j\overset{\rm def}=\operatorname{Im} z_j,\qquad j=1,2.
\end{equation*}
\notag
$$
Note that on the sets $\{(z_1,z_2)\colon x_1=x_2\}$ and $\{(z_1,z_2)\colon y_1=y_2\}$ the divided differences ${\mathfrak D}_xf$ and ${\mathfrak D}_yf$ are understood as the corresponding partial derivatives of $f$. 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:
$$
\begin{equation*}
{\mathscr E}_\sigma^\infty(\mathbb{R}^2)\overset{\rm def}= \bigl\{f\in L^\infty(\mathbb{R}^2)\colon\operatorname{supp}\mathscr{F} f\subset\{\zeta\in\mathbb{C}\colon|\zeta|\leqslant\sigma\}\bigr\},
\end{equation*}
\notag
$$
where $\sigma$ is a positive number. Lemma 7.7. Let $f$ be a function of class ${\mathscr E}_\sigma^\infty(\mathbb{R}^2)$. Then ${\mathfrak D}_xf$ and ${\mathfrak D}_yf$ are Schur multipliers for all Borel spectral measures and their norms in the space of Schur multipliers are less than or equal to $\mathrm{const}\,\sigma\|f\|_{L^\infty}$. Theorem 7.8. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$. Suppose that $N_1$ and $N_2$ are normal operators such that the operator $N_1-N_2$ is bounded. Then 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. Theorem 7.9. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R}^d)$. Then the following assertions hold: (i) there exists a positive number $K$ such that
$$
\begin{equation*}
\|f(A_1,\dots,A_d)-f(B_1,\dots,B_d)\|\leqslant K\|f\|_{B_{\infty,1}^1} \max_{1\leqslant j\leqslant d}\|A_j-B_j\|
\end{equation*}
\notag
$$
for arbitrary sets $\{A_1,\dots,A_d\}$ and $\{B_1,\dots,B_d\}$ of commuting self-adjoint operators such that the operators $A_j-B_j$, $1\leqslant j\leqslant d$, are bounded; (ii) if $\{A_1,\dots,A_d\}$ and $\{B_1,\dots,B_d\}$ are sets of commuting self-adjoint operators such that the operators $A_j-B_j$, $1\leqslant j\leqslant d$, belong to $\boldsymbol{S}_1$, then $f(A_1,\dots,A_d)-f(B_1,\dots,B_d)$ also belongs to $\boldsymbol{S}_1$. Commentary. Let us mention here the papers [86] and [13], in which analogues of these results for functions of pairs of commuting contractions and functions of pairs of commuting dissipative operators were obtained. 7.5. Operator differentiabilityWe proceed to the question of the differentiability of the map
$$
\begin{equation}
\begin{aligned} \, K\mapsto f(A+K)-f(A). \end{aligned}
\end{equation}
\tag{7.15}
$$
Here $f$ is a continuous function on $\mathbb{R}$, $A$ is a self-adjoint operator, and $K$ is a bounded self-adjoint operator. The function $f$ is called operator differentiable if this map, defined on the real Banach space of bounded self-adjoint operators, is differentiable for an arbitrary self-adjoint operator $A$. We refer the reader to [10] for a detailed discussion of operator differentiability and for an extensive bibliography; see also the papers [43] and [42]. 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. Theorem 7.10. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R})$. Then the map (7.15) is Gateaux differentiable for an arbitrary self-adjoint operator $A$. Moreover, its differential at $\boldsymbol{0}$ is the bounded linear transformer 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 obtain a formula for the $n$th-order derivative of this operator-valued function, we need the notions of higher-order divided differences and multiple operator integrals. Definition 2. Let $f$ be a function on $\mathbb{R}$ that has $k$ derivatives. We define the divided difference ${\mathfrak D}^k f$ of order $k$ by induction as follows: if $k\geqslant1$, then 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
$$
\begin{equation}
\psi(\lambda,\mu,\nu)=\int_Q f(\lambda,x)g(\mu,x)h(\nu,x)\,{\rm d}\sigma(x),
\end{equation}
\tag{7.17}
$$
where $(Q,\sigma)$ is a space with a $\sigma$-finite measure, $f$ is a measurable function on ${\mathscr X}\times Q$, $g$ is a measurable function on ${\mathscr Y}\times Q$, and $h$ is a measurable function on ${\mathscr Z}\times Q$; moreover,
$$
\begin{equation}
\int_Q\|f(\cdot,x)\|_{L^\infty(E)}\,\|g(\cdot,x)\|_{L^\infty(F)}\, \|h(\cdot,x)\|_{L^\infty(G)}\,{\rm d} \sigma(x)<\infty.
\end{equation}
\tag{7.18}
$$
The norm $\|\psi\|_{L^\infty\mathbin{\widehat\otimes}_{\rm i}L^\infty\mathbin{\widehat\otimes}_{\rm i}L^\infty}$ of $\psi$ in $L^\infty(E)\mathbin{\widehat\otimes}_{\rm i}L^\infty(F)\mathbin{\widehat\otimes}_{\rm i}L^\infty(G)$ is defined as the infimum of the left-hand side of (7.18) over all representations of the form (7.17). 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
$$
\begin{equation}
\begin{aligned} \, \nonumber &\int_{\mathscr X}\int_{\mathscr Y}\int_{\mathscr Z}\psi(\lambda,\mu,\nu) \,{\rm d} E(\lambda)T_1\,{\rm d} F(\mu)T_2\,{\rm d} G(\nu) \\ \nonumber &\qquad\overset{\rm def}=\int_Q\biggl(\,\int_{\mathscr X} f(\lambda,x)\,{\rm d} E(\lambda)\biggr)T_1 \biggl(\,\int_{\mathscr Y} g(\mu,x)\,{\rm d} F(\mu)\biggr) \\ &\qquad\qquad\times T_2 \biggl(\,\int_{\mathscr Z} h(\nu,x)\,{\rm d}G(\nu)\biggr)\,{\rm d}\sigma(x). \end{aligned}
\end{equation}
\tag{7.19}
$$
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]. Theorem 7.11. Let $m$ be a natural number. Suppose that $A$ is a bounded self- adjoint operator and $f$ is a function in the Besov class $B_{\infty,1}^m(\mathbb{R})$. Then the function ${\mathfrak D}^m f$ belongs to the integral projective tensor product 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 formulaIn 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
$$
\begin{equation}
\operatorname{trace}\bigl(f(B)-f(A)\bigr)= \int_\mathbb{R}f'(t)\xi(t)\,{\rm d}t
\end{equation}
\tag{7.20}
$$
holds for sufficiently nice functions $f$ on $\mathbb{R}$. Later, Krein [46] gave a rigorous justification of trace formula (7.20) and showed that it holds in the case when the derivative of $f$ is the Fourier transform of a complex Borel measure on $\mathbb{R}$. Krein also posed the problem of describing the maximal class of functions $f$ for which trace formula (7.20) holds for arbitrary self-adjoint operators $A$ and $B$ with trace class difference. In the author’s paper [76] the following result was obtained. Theorem 7.12. Let $A$ and $B$ be self-adjoint operators with trace difference and let $\xi$ be the spectral shift function for this pair. Then trace formula (7.20) holds for an arbitrary function $f$ in the Besov class $B_{\infty,1}^1(\mathbb{R})$. 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. Theorem 7.13. Let $f$ be a continuous function on $\mathbb{R}$. The following are equivalent: (i) trace formula (7.20) holds for an arbitrary pair of self-adjoint operators $A$ and $B$ with trace class difference and $\xi=\xi_{\{A,B\}}$; (ii) $f$ is an operator Lipschitz function. Commentary. Krein defined in [47] an integrable real-valued spectral shift function $\xi$ on the circle $\mathbb{T}$ for pairs of unitary operators $(U,V)$ with trace class difference and obtained the trace formula 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
$$
\begin{equation*}
\operatorname{trace}\bigl(f(T)-f(R)\bigr)= \int_{\mathbb{T}} f'(\zeta)\xi(\zeta)\,{\rm d} \zeta
\end{equation*}
\notag
$$
for an arbitrary operator Lipschitz function $f$ analytic in the unit disc. In the same papers [58] and [59] trace formulae were also obtained for functions of dissipative operators. 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 which has to be in the trace class under suitable assumptions and to find a trace formula for such operators.It was shown in [45] that there exists a unique function $\eta$ of class $L^1(\mathbb{R})$ such that
$$
\begin{equation}
\operatorname{trace}\biggl(f(B)-f(A)-\frac{\rm d}{{\rm d}s} f (A+sK)\Big|_{s=0}\biggr)=\int_\mathbb{R} f''(x)\eta(x)\,{\rm d} x
\end{equation}
\tag{7.21}
$$
for arbitrary rational functions $f$ with poles off $\mathbb{R}$. The function $\eta$ is called the generalized spectral shift function corresponding to the pair $(A,B)$. In this author’s paper [78] it was shown that trace formula (7.21) holds under a considerably milder assumption on $f$. Theorem 7.14. Let $A$ and $B$ be self-adjoint operators such that $K\overset{\rm def}{=} B-A\in\boldsymbol{S}_2$. Then trace formula (7.21) holds for an arbitrary function $f$ in the Besov class $B_{\infty,1}^2(\mathbb{R}^2)$. 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
$$
\begin{equation*}
{\mathscr T}^{(m)}_{A,K}f\overset{\rm def}= f(A+K)-f(A) -\frac{{\rm d}}{{\rm d} t}f(A+tK)\bigg|_{t=0}-\cdots-\frac{1}{(m-1)!}\, \frac{{\rm d}^{m-1}}{{\rm d} t^{m-1}}f(A+tK)\bigg|_{t=0}.
\end{equation*}
\notag
$$
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
$$
\begin{equation}
\mathscr{F} f^{(j)}\in L^1(\mathbb{R}),\qquad 0\leqslant j\leqslant m,
\end{equation}
\tag{7.22}
$$
the following trace formula holds:
$$
\begin{equation}
\operatorname{trace}({\mathscr T}^{(m)}_{A,K}f)= \int_\mathbb{R} f^{(m)}(x)\eta_m(x)\,{\rm d} x.
\end{equation}
\tag{7.23}
$$
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. Theorem 7.15. Let $m\geqslant3$. Suppose that $A$ is a self-adjoint operator and $K$ is a self-adjoint operator of class $\boldsymbol{S}_m$. Then trace formula (7.23) holds for an arbitrary function $f$ in the Besov class $B_{\infty,1}^m(\mathbb{R})$. 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. Theorem 7.16. Let $m$ be a natural number, and let $A$ and $K$ be self-adjoint operators such that $K\in\boldsymbol{S}_m$. Suppose that $\mu$ is a complex Borel measure on $\mathbb{R}$. Then there exists a complex Borel measure $\mu_{\{A,K\}}$ such that the equality holds for an arbitrary function $f$ in the Besov class $B^m_{\infty1}(\mathbb{R})$. Moreover, if the measure $\mu$ is absolutely continuous (with respect to Lebesgue measure), then the measure $\mu_{\{A,K\}}$ is also absolutely continuous. The following result obtained in [7] shows that Theorem 7.14 is a special case of Theorem 7.16. Theorem 7.17. Let $m$ be a positive integer, and let $A$ and $K$ be self-adjoint operators such that $K\in\boldsymbol{S}_m$. Consider the absolutely continuous measure $\mu$ on $\mathbb{R}$ defined by Then the formula holds for an arbitrary function $f$ in the Besov class $B_{\infty1}^m(\mathbb{R})$. Commentary. Note that the results of [90] were extended to the case of functions of contractions in [91] and to the case of functions of unitary operators in [92]. 7.8. Functions of pairs of non-commuting operatorsFor 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
$$
\begin{equation*}
f(A,B)=\int_\mathbb{R}\int_\mathbb{R}f(s,t)\,{\rm d} E_A(s)\,{\rm d} E_B(t)= \int_\mathbb{R}\int_\mathbb{R} f(s,t)\,{\rm d} E_A(s)I\,{\rm d} E_B(t)
\end{equation*}
\notag
$$
in the case when the function $f$ is a Schur multiplier with respect to the spectral measures $E_A$ and $E_B$ of the operators $A$ and $B$. 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
$$
\begin{equation}
\int_{\mathscr X}\int_{\mathscr Y}\int_{\mathscr Z}\Psi(x,y,z)\,{\rm d} E_1(x)T\,{\rm d} E_2(y)R\,{\rm d} E_3(z),
\end{equation}
\tag{7.24}
$$
where $E_1$, $E_2$, and $E_3$ are spectral measures on Hilbert space, and $T$ and $R$ are bounded linear operators. 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
$$
\begin{equation}
\Psi(x,y,z)=\sum_{j,k\geqslant0}\alpha_j(x)\beta_{jk}(y)\gamma_k(z),
\end{equation}
\tag{7.25}
$$
where the functions $\alpha_j$, $\beta_{jk}$, and $\gamma_k$ satisfy the requirements
$$
\begin{equation*}
\sum_{j\geqslant0}|\alpha_j|^2\in L^\infty_{E_1}\quad\text{and}\quad \sum_{k\geqslant0}|\gamma_k|^2\in L^\infty_{E_3}
\end{equation*}
\notag
$$
and the function
$$
\begin{equation*}
y\mapsto\bigl\|\{\beta_{jk}(y)\}_{j,k}\bigr\|_{\mathcal B}
\end{equation*}
\notag
$$
belongs to $L^\infty_{E_2}$. Here the symbol ${\mathcal B}$ denotes the class of infinite matrices that determine bounded linear operators on $\ell^2$. In addition, the norm of a matrix in ${\mathcal B}$ is defined as the norm of the corresponding operator from $\ell^2$ to $\ell^2$. 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
$$
\begin{equation*}
\|\{\alpha_j\}_{j\geqslant0}\|_{L^\infty_{E_1}(\ell^2)}\, \|\{\beta_{jk}\}_{j,k\geqslant0}\|_{L^\infty_{E_2}({\mathcal B})}\, \|\{\gamma_k\}_{k\geqslant0}\|_{L^\infty_{E_3}(\ell^2)}
\end{equation*}
\notag
$$
over all representations of $\Psi$ of the form (7.25). For such functions $\Psi$ the triple operator integral (7.24) is defined as the product of operator matrices
$$
\begin{equation*}
\begin{pmatrix} A_0&A_1&A_2&\dots& \end{pmatrix}\begin{pmatrix} B_{00}&B_{01}&B_{02}&\dots& \\ B_{10}&B_{11}&B_{12}&\dots \\ B_{20}&B_{21}&B_{22}&\dots \\ \vdots&\vdots&\vdots&\ddots \end{pmatrix}\begin{pmatrix} G_0 \\ G_1 \\ G_2 \\ \vdots \end{pmatrix},
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
A_j\overset{\rm def}=\int\alpha_j\,{\rm d} E_1,\quad B_{jk}\overset{\rm def}=\int\beta_{jk}\,{\rm d} E_2\quad\text{и}\quad G_k\overset{\rm def}=\int\gamma_k\,{\rm d} E_3.
\end{equation*}
\notag
$$
In addition, the following inequalities hold:
$$
\begin{equation*}
\biggl\|\int\!\!\!\int\!\!\!\int\Psi\,{\rm d} E_1T\,{\rm d} E_2R\,{\rm d} E_3\biggr\|\leqslant \|\Psi\|_{L^\infty\mathbin{\otimes_{\rm h}}L^\infty\mathbin{\otimes_{\rm h}}L^\infty}\, \|T\|\,\|R\|
\end{equation*}
\notag
$$
(see [40]) and
$$
\begin{equation}
\begin{gathered} \, \biggl\|\int\!\!\!\int\!\!\!\int\Psi\,{\rm d} E_1T\,{\rm d} E_2R\,{\rm d} E_3\biggr\|_{\boldsymbol{S}_r}\leqslant\|\Psi\|_{L^\infty \mathbin{\otimes_{\rm h}}L^\infty\mathbin{\otimes_{\rm h}}L^\infty}\, \|T\|_{\boldsymbol{S}_p}\,\|R\|_{\boldsymbol{S}_q}, \\ \nonumber \frac{1}{r}=\frac{1}{p}+\frac{1}{q}\,, \end{gathered}
\end{equation}
\tag{7.26}
$$
in the case when $2\leqslant p\leqslant\infty$ and $2\leqslant q\leqslant\infty$ (see [12]). 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
$$
\begin{equation}
\|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p}\leqslant C\|f\|_{B_{\infty,1}^1} \max\bigl\{\|A_1-A_2\|_{\boldsymbol{S}_p}, \|B_1-B_2\|_{\boldsymbol{S}_p}\bigr\}
\end{equation}
\tag{7.27}
$$
for an arbitrary function $f$ in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$ and for arbitrary bounded (not necessarily commuting) self-adjoint operators $A_1$, $A_2$, $B_1$, and $B_2$ satisfying the assumptions $A_2-A_1\in\boldsymbol{S}_p$ and $B_2-B_1\in\boldsymbol{S}_p$. 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:
$$
\begin{equation}
\|f(A_1,B_1)-f(A_2,B_2)\|_{\boldsymbol{S}_p}\leqslant \mathrm{const}\|f\|_{L^\infty}\max\bigl\{\|A_1-A_2\|_{\boldsymbol{S}_p}, \|B_1-B_2\|_{\boldsymbol{S}_p}\bigr\}
\end{equation}
\tag{7.28}
$$
for arbitrary bounded self-adjoint operators $A_1$, $A_2$, $B_1$, and $B_2$ such that $A_2-A_1\in\boldsymbol{S}_p$ and $B_2-B_1\in\boldsymbol{S}_p$. 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:
$$
\begin{equation}
\begin{aligned} \, &f(A_1,B_1)-f(A_2,B_2) \nonumber \\ &\qquad=\int\!\!\!\int\!\!\!\int\bigl({\mathfrak D}^{[1]}f\bigr)(x_1,x_2,y) \,{\rm d} E_{A_1}(x_1)(A_1-A_2)\,{\rm d} E_{A_2}(x_2)\,{\rm d} E_{B_1}(y) \nonumber \\ &\qquad\qquad+\int\!\!\!\int\!\!\!\int\bigl({\mathfrak D}^{[2]}f\bigr)(x,y_1,y_2)\,{\rm d} E_{A_2}(x)\,{\rm d} E_{B_1}(y_1)(B_1-B_2)\,{\rm d} E_{B_2}(y_2), \end{aligned}
\end{equation}
\tag{7.29}
$$
where
$$
\begin{equation}
\bigl({\mathfrak D}^{[1]}f\bigr)(x_1,x_2,y)\overset{\rm def}= \frac{f(x_1,y)-f(x_2,y)}{x_1-x_2}
\end{equation}
\tag{7.30}
$$
and
$$
\begin{equation}
\bigl({\mathfrak D}^{[2]}f\bigr)(x,y_1,y_2)\overset{\rm def}= \frac{f(x,y_1)-f(x,y_2)}{y_1-y_2}\,.
\end{equation}
\tag{7.31}
$$
Of course, we have to clarify in which sense we understand 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
$$
\begin{equation}
\bigl({\mathfrak D}^{[1]}f\bigr)(x_1,x_2,y)=\sum_{j,k\in\mathbb{Z}} \frac{\sin(x_1-j\pi)}{x_1-j\pi}\cdot\frac{\sin(x_2-k\pi)}{x_2-k\pi} \cdot\frac{f(j\pi,y)-f(k\pi,y)}{j\pi-k\pi}
\end{equation}
\tag{7.32}
$$
and
$$
\begin{equation}
\bigl({\mathfrak D}^{[2]}f\bigr)(x,y_1,y_2)=\sum_{j,k\in\mathbb{Z}} \frac{f(j\pi,x)-f(k\pi,x)}{j\pi-k\pi}\cdot \frac{\sin(y_1-j\pi)}{y_1-j\pi}\cdot\frac{\sin(y_2-k\pi)}{y_2-k\pi};
\end{equation}
\tag{7.33}
$$
also, in the case $j=k$ we assume that
$$
\begin{equation*}
\frac{f(j\pi,y)-f(k\pi,y)}{j\pi-k\pi}= \frac{\partial f(x,y)}{\partial x}\bigg|_{(j\pi,y)}.
\end{equation*}
\notag
$$
It is well known that
$$
\begin{equation*}
\sum_{j\in\mathbb{Z}}\frac{\sin^2(x_1-j\pi)}{(x_1-j\pi)^2}= \sum_{k\in\mathbb{Z}}\frac{\sin^2(x_2-k\pi)}{(x_2-k\pi)^2}=1,\qquad x_1,x_2\in\mathbb{R},
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\sup_{y\in\mathbb{R}}\biggl\|\biggl\{\frac{f(j\pi,y)- f(k\pi,y)}{j\pi-k\pi}\biggr\}_{j,k\in\mathbb{Z}}\biggr\|_{\mathcal{B}} \leqslant\mathrm{const}\|f\|_{L^\infty(\mathbb{R})}.
\end{equation*}
\notag
$$
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]). Definition 3. We say that a bounded measurable function $\Psi$ on ${\mathscr X}\times{\mathscr Y}\times Z$ belongs to the Haagerup-like tensor product of the first kind $L^\infty(E_1)\mathbin{\otimes_{\rm h}}L^\infty(E_2)\mathbin{\otimes^{\rm h}}L^\infty(E_3)$ if $\Psi$ admits a representation of the form Now it is the turn to define triple operator integrals whose integrands belong to $L^\infty(E_1)\mathbin{\otimes_{\rm h}}L^\infty(E_2)\mathbin{\otimes^{\rm h}}L^\infty(E_3)$. Definition 4. Let $1\leqslant p\leqslant2$. For functions $\Psi$ in $L^\infty(E_1)\mathbin{\otimes_{\rm h}}L^\infty(E_2)\mathbin{\otimes^{\rm h}}L^\infty(E_3)$, for a bounded linear operator $R$ and an operator $T$ of class $\boldsymbol{S}_p$, we define the triple operator integral as the following continuous linear functional on $\boldsymbol{S}_{p'}$, $1/p+1/p'=1$ (on the class of compact operators in the case $p=1$): Then the triple operator integral in (7.36) is well defined because the function belongs to the Haagerup tensor product $L^\infty(E_2)\mathbin{\otimes_{\rm h}}L^\infty(E_3)\mathbin{\otimes_{\rm h}}L^\infty(E_1)$. It is easy to deduce from (7.26) that
$$
\begin{equation*}
\|W\|_{\boldsymbol{S}_p}\leqslant\|\Psi\|_{L^\infty\mathbin{\otimes_{\rm h}} L^\infty\mathbin{\otimes^{\rm h}}L^\infty}\|T\|_{\boldsymbol{S}_p}\|R\|,\qquad 1\leqslant p\leqslant2.
\end{equation*}
\notag
$$
Let us now define triple operator integrals in the case of integrands in a Haagerup- like tensor product of the second kind. Definition 5. We say that a bounded measurable function $\Psi$ on ${\mathscr X}\times{\mathscr Y}\times Z$ belongs to the Haagerup-like tensor product of the second kind $L^\infty(E_1)\mathbin{\otimes^{\rm h}}L^\infty(E_2)\mathbin{\otimes_{\rm h}}L^\infty(E_3)$ if $\Psi$ admits a representation Definition 6. Suppose now that Moreover, the following inequality holds:
$$
\begin{equation*}
\|W\|_{\boldsymbol{S}_p}\leqslant \|\Psi\|_{L^\infty\otimes^{\rm h} L^\infty\mathbin{\otimes_{\rm h}}L^\infty}\,\|T\|\,\|R\|_{\boldsymbol{S}_p}.
\end{equation*}
\notag
$$
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. Let $f$ be a function in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$, and let $1\leqslant p\leqslant 2$. Suppose that $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of bounded not necessarily commuting self-adjoint operators such that $A_2-A_1\in\boldsymbol{S}_p$ and $B_2-B_1\in\boldsymbol{S}_p$. Then the divided differences ${\mathfrak D}^{[1]}f$ and ${\mathfrak D}^{[2]}f$ defined by (7.32) belong to the Haagerup-like tensor products $L^\infty(E_1)\mathbin{\otimes_{\rm h}}L^\infty(E_2)\mathbin{\otimes^{\rm h}}L^\infty(E_3)$ and $L^\infty(E_1)\mathbin{\otimes^{\rm h}}L^\infty(E_2)\mathbin{\otimes_{\rm h}}L^\infty(E_3)$ for arbitrary Borel spectral measures $E_1$, $E_2$, and $E_3$; moreover, representation (7.29) holds, in which the first triple operator integral should be understood in the sense of Definition 4, while the second triple operator integral should be understood in the sense of Definition 6. Moreover, the following Lipschitz-type estimate holds 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 operatorsWe 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
$$
\begin{equation*}
f_\sharp(s,t)\overset{\rm def}=(1-\mathrm{i} t)^{-1}f(s,t).
\end{equation*}
\notag
$$
Assume that $f_\sharp\in L^\infty_{E_A}\mathbin{\otimes_{\rm h}}L^\infty_{E_B}$. We define the operator $f(A,B)$ by
$$
\begin{equation*}
f(A,B)\overset{\rm def}= f_\sharp(A,B)(I-\mathrm{i} B)= \biggl(\,\int\!\!\!\int_{\mathbb{R}\times\mathbb{R}}f_\sharp(s,t)\,{\rm d} E_A(s)\, {\rm d} E_B(t)\biggr)(I-\mathrm{i} B).
\end{equation*}
\notag
$$
Then $f(A,B)$ is a densely defined operator whose domain coincides with the domain ${\rm D}(B)$ of $B$. It is not necessarily bounded but the operator $f(A,B)(I-\mathrm{i} B)^{-1}$ is bounded. 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
$$
\begin{equation*}
\|f_\sharp\|_{L^\infty_{E_1}\mathbin{\otimes_{\rm h}}L^\infty_{E_2}}\leqslant \mathrm{const}(1+\sigma)\|f\|_{L^\infty(\mathbb{R}^2)}.
\end{equation*}
\notag
$$
This easily implies that if $f$ is a function in the inhomogeneous Besov class $\textit{Б}_{\infty,1}^1(\mathbb{R}^2)$, then $f_\sharp\in L^\infty_{E_1}\mathbin{\otimes_{\rm h}} L^\infty_{E_2}$ and
$$
\begin{equation*}
\|f_\sharp\|_{L^\infty_{E_1}\mathbin{\otimes_{\rm h}}L^\infty_{E_2}}\leqslant \mathrm{const}\|f\|_{\textit{Б}_{\infty,1}^1}
\end{equation*}
\notag
$$
for arbitrary spectral Borel measures $E_1$ and $E_2$ (see [15] and [16]). Thus, for an arbitrary pair $(A,B)$ of not necessarily bounded self-adjoint operators and for a function $f$ in the inhomogeneous Besov class $\textit{Б}_{\infty,1}^1(\mathbb{R}^2)$, we can define the operator $f(A,B)$ as a densely defined operator. Moreover, the following result holds (see [15] and [16]). Theorem 7.19. Let $f\in\textit{Б}_{\infty,1}^1(\mathbb{R}^2)$, and let $p\in[1,2]$. Suppose that $(A_1,B_1)$ and $(A_2,B_2)$ are pairs of not necessarily bounded and not necessarily commuting self-adjoint operators such that $A_2-A_1\in\boldsymbol{S}_p$ and $B_2-B_1\in\boldsymbol{S}_p$. Then 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 operatorsLet $A$ 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
$$
\begin{equation*}
\varphi(A,B)\overset{\rm def}=\sum_{j,k}a_{jk}A^jB^k.
\end{equation*}
\notag
$$
It is easy to verify that if $\varphi$ are $\psi$ polynomials of two real variables, then $[\varphi(A,B), \psi(A,B)]\in\boldsymbol{S}_1$. 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
$$
\begin{equation}
\operatorname{trace}[\varphi(A,B),\psi(A,B)]= -\mathrm{i}\int_{\mathbb{R}^2}\biggl(\frac{\partial\varphi}{\partial x}\, \frac{\partial\psi}{\partial y}- \frac{\partial\varphi}{\partial y}\, \frac{\partial\psi}{\partial x}\biggr)\,{\rm d} P,
\end{equation}
\tag{7.40}
$$
where $P$ is the real-valued measure with compact support, which is determined by the pair $(A,B)$. It was shown in [89] that the measure $P$ is absolutely continuous with respect to planar Lebesgue measure and
$$
\begin{equation*}
{\rm d} P(x,y)=\frac{1}{2\pi}g(x,y)\,{\rm d} \boldsymbol{m}_2(x,y),
\end{equation*}
\notag
$$
where $g$ is the so-called Pincus principal function introduced in [88]. 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
$$
\begin{equation*}
\int_{\mathbb{R}^2}(1+|t|)(1+|s|)\,{\rm d} |\omega|(s,t)<\infty;
\end{equation*}
\notag
$$
moreover, the Helton–Howe trace formula (7.40) was extended to the case of such functions. 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. Theorem 7.20. Let $A$ and $B$ be self-adjoit operators, and let $Q$ be a bounded linear operator such $[A,Q]\in\boldsymbol{S}_1$ and $[B,Q]\in\boldsymbol{S}_1$. Suppose that $\varphi$ is a function in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$. Then $[\varphi(A,B),Q]\in\boldsymbol{S}_1$, Theorem 7.21. Let $A$ and $B$ be almost commuting self-adjoint operators, and let $\varphi$ and $\psi$ be functions in the Besov $B_{\infty,1}^1(\mathbb{R}^2)$. Then Now we can extend the Helton–Howe trace formula to the case of functions in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$. Theorem 7.22. Let $A$ and $B$ be almost commuting self-adjoint operators and let $\varphi$ and $\psi$ be functions in the Besov class $B_{\infty,1}^1(\mathbb{R}^2)$. Then 
Citation in format AMSBIB
\Bibitem{Pel24} |
|
||||||||||||||||||||||||||||||||||||||||||||||||||||
|
Contact us: |
Terms of Use
|
Registration to the website |
Logotypes |
|