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\TopicFR{Analyse fonctionnelle, Analyse harmonique}
\TopicEN{Functional analysis, Harmonic analysis}

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\title{Relatively bounded and relatively trace class perturbations}
\alttitle{Perturbations relativement born\'ees et relativement de classe trace}

\author{\firstname{Aleksei} \middlename{B.} \lastname{Aleksandrov}}
\address{St.\ Petersburg State University, Universitetskaya nab., 7/9, 199034 St.\ Petersburg, Russia}
\address{St.\ Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St.\ Petersburg, Russia}
\email{aall54eexx@gmail.com}

\author{\firstname{Vladimir} \middlename{V.} \lastname{Peller}\CDRorcid{0000-0002-7414-7625}}
\address[1]{St.\ Petersburg State University, Universitetskaya nab., 7/9, 199034 St.\ Petersburg, Russia}
\address[2]{St.\ Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St.\ Petersburg, Russia}
\email{peller@math.msu.edu}

\subjclass{47A55}

\keywords{\kwd{Relatively bounded perturbation} \kwd{relatively trace class perturbation} \kwd{relatively operator Lipschitz class} \kwd{trace formula} \kwd{self-adjoint operators} \kwd{double operator integrals}}
\altkeywords{\kwd{Perturbations relativement born\'ees} \kwd{perturbations relativement \`a trace} \kwd{fonctions relativement lipschitziennes op\'eratorielles} \kwd{formules des traces} \kwd{op\'erateurs auto-adjoints} \kwd{int\'egrales d'op\'erateurs doubles}}

\thanks{The research on Sections~2--4 is supported by the Russian Science Foundation (grant number 23-11-00153). The research on Sections~5--6 is supported by a grant of the Government of the Russian Federation for the state support of scientific research, carried out under the supervision of leading scientists, agreement 075-15-2024-631}
\CDRGrant[Russian Science Foundation]{23-11-00153}
\CDRGrant[Russian Federation for the state support of scientific research]{075-15-2024-631}

\begin{abstract}
In this note we study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. We introduce and study the class of relatively operator Lipschitz functions. We obtain a trace formula in the case of relatively trace class perturbations and show that this class of functions is the maximal class of functions for which the trace formula holds. Our method also gives us a new approach to the inequality $\int\lvert\boldsymbol{\xi}(t)\rvert(1+\lvert t\rvert)^{-1}\,\mathrm{d}t&lt;\infty$ for the spectral shift function $\boldsymbol{\xi}$ in the case of relatively trace class perturbations.
\end{abstract}

\begin{altabstract}
Dans cette note nous \'etudions le comportement des fonctions d'op\'erateurs auto-adjoints sous des perturbations relativement dans la classe des op\'erateurs \`a trace et relativement born\'ees. Nous introduisons
et \'etudions la classe de fonctions relativement lipschitziennes op\'eratorielles. Nous obtenons une formule de trace dans le cas de perturbations relativement \`a trace et montrons que cette classe de fonctions est maximale pour la validit\'e de la formule de trace. Notre m\'ethode nous donne une nouvelle approche de l'in\'egalit\'e $\int\lvert\boldsymbol{\xi}(t)\rvert(1+\lvert t\rvert)^{-1}\,\mathrm{d}t&lt;\infty$ pour la fonction $\boldsymbol{\xi}$ de d\'ecalage spectral.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

The purpose of this note is to study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. It turns out that the natural class of functions to be considered in this connection is the class of relatively operator Lipschitz functions, which we are going to define in the introduction. We are going to obtain various characterizations of this class. An important role will be played by double operator integrals.

Let $A$ be a not necessarily bounded self-adjoint operator in Hilbert space. A self-adjoint operator $K$ (also, not necessarily bounded) is called a \emph{relatively bounded perturbation of $A$} if
\begin{equation}\label{otogner}
\norm{Kv}\le\const\parens[\big]{\norm{v}+\norm{Av}}, \quad v\in\Dom(A),
\end{equation}
where $\Dom(A)$ stands for the domain of the operator $A$. It is easy to see that property~\eqref{otogner} is equivalent to the fact that the operator
\begin{equation}\label{C}
C\df K(A+\ri I)^{-1}
\end{equation}
is bounded. We say that a self-adjoint operator $K$ is a \emph{relatively trace class perturbation of $A$} if the operator $C$ defined by~\eqref{C} belongs to the trace class $\bS_1$.

We consider parametric families $A_t=A+tK$, $t\in\R$, where $A$ and $K$ are self-adjoint operators and $K$ is a relatively bounded perturbation of $A$ and we obtain a formula for the derivative of the function $t\mapsto f(A_t)-f(A)$ in the strong operator topology in terms of double operator integrals.

This will be used to obtain a trace formula and compute the trace of $f(A+K)-f(A)$ in the case when $K$ is a relatively trace class perturbation of $A$ and $f$ is a relatively operator Lipschitz function. Our results allow us to give a new approach to the inequality $\int\abs{\bs{\xi}(t)}(1+\abs{t})^{-1}\dd t&lt;\be$, where $\bs\xi$ is the spectral shift function that corresponds to the pair $\{A,A+K\}$.

\begin{defi}
A continuous function $f$ on the real line $\R$ is called \emph{relatively operator Lipschitz} if there is a positive number $k$ such that
\begin{equation}\label{rOL}
\norm[\big]{f(B)-f(A)}\le k\norm[\big]{(B-A)(A+\ri I)^{-1}}
\end{equation}
whenever $A$ and $B$ are self-adjoint operators such that $B-A$ is a relatively bounded perturbation of $A$.
\end{defi}
We denote by $\ROL$ the class of relatively operator Lipschitz functions on $\R$.

\begin{lemm}
Let $f$ be a continuous function on\/ $\R$. It is relatively operator Lipschitz if and only if
\[
f(B)-f(A)\in\bS_1
\]
whenever $A$ and $B$ are self-adjoint operators such that $B-A$ is a relatively trace class perturbation of $A$.
\end{lemm}

It is easy to verify that $f$ is a relatively operator Lipschitz function if it is a \emph{resolvent Lipschitz function}, i.e.,
\[
\norm[\big]{f(B)-f(A)}\le\const\norm[\big]{(B+\ri I)^{-1}-(A+\ri I)^{-1}},
\]
for arbitrary self-adjoint operators $A$ and $B$.

On the other hand, it is obvious that if $f$ is a relatively operator Lipschitz function, then it is an \emph{operator Lipschitz function}, i.e.,
\begin{equation}\label{OpLip}
\norm[\big]{f(B)-f(A)}\le\const\norm{A-B}
\end{equation}
for arbitrary self-adjoint operators $A$ and $B$, no matter bounded or unbounded. The class of operator Lipschitz functions on $\R$ is denoted by $\OL$; the norm $\norm{f}_\OL$ of a function $f$ in $\OL$ is, by definition, the smallest possible constant on the right-hand side of~\eqref{OpLip}.

We refer the reader to the survey~\cite{AP} for detailed information about operator Lipschitz functions. It is well known (see~\cite{AP}) that operator Lipschitz functions must be differentiable everywhere on the real line $\R$, and so relatively operator Lipschitz functions must also be differentiable on $\R$.

\section{Characterizations of the class of relatively operator Lipschitz functions}

In this section we give several necessary and sufficient conditions for a function on $\R$ to be relatively operator Lipschitz. We also obtain similar characterizations of the class of resolvent Lipschitz functions. In Section~\ref{doich} we obtain another characterization of the class of relatively operator Lipschitz functions in terms of Schur multipliers. Yet another characterization of the class $\ROL$ (in terms of commutator estimates) will be given in Section~\ref{komu tatory?}.

\begin{defi}
We say that a function $f$ on $\R$ is a multiplier of the space $\OL$ of operator Lipschitz functions if
\[
g\in\OL \quad \Longrightarrow \quad fg\in\OL.
\]
\end{defi}
We denote by $\MOL$ the class of all multipliers of $\OL$.

\begin{theo}\label{x+i}
Let $f$ be a continuous function on\/ $\R$. The following are equivalent:
\begin{enumerate}\alphenumi
\item $f\in\ROL$;
\item the function $x\mapsto(x+\ri)f(x)$ is an operator Lipschitz function;
\item $f\in\MOL$.
\end{enumerate}
\end{theo}

The following theorem provides a similar characterization of the class of resolvent Lipschitz functions.
\begin{theo}\label{resLip}
Let $f$ be a continuous function on\/ $\R$. The following are equivalent:
\begin{enumerate}\alphenumi
\item $f$ is a resolvent Lipschitz function;
\item \label{resLip_b} the function $x\mapsto(x+\ri)^2\parens[\big]{f(x)-c}$ is an operator Lipschitz function for some $c\in\C$;
\item \label{resLip_c} the function $x\mapsto(x+\ri)\parens[\big]{f(x)-c}$ belongs to $\MOL$ for some $c\in\C$.
\end{enumerate}
If $c\in\C$ satisfies~\eqref{resLip_b} or~\eqref{resLip_c}, then $c$ must be equal to $\lim_{\abs{x}\to\be}f(x)$.
\end{theo}

\section{Double operator integrals}

As in the case of operator Lipschitz functions (see~\cite{AP}), an important role to study the class of relatively operator Lipschitz functions is played by double operator integrals. Double operator integrals appeared first in~\cite{DK}; later Birman and Solomyak developed in~\cite{BS1,BS4,BS3} a rigorous mathematical theory of double operator integrals.

Double operator integrals are expressions of the form
\begin{equation}\label{dvopi}
\iint\Phi(x,y)\dd E_1(x)Q\dd E_2(y).
\end{equation}
Here $E_1$ and $E_2$ are spectral measures on Hilbert space, $\Phi$ is a bounded measurable function and $Q$ is a bounded linear operator on Hilbert space.

The starting point of Birman and Solomyak in~\cite{BS1} was to define double operator integrals of the form~\eqref{dvopi} in the case when $Q$ is a Hilbert--Schmidt operator, i.e., $Q\in\bS_2$. In this case $\iint\Phi\dd E_1Q\dd E_2$ also belongs to $\bS_2$ and
\[
\norm*{\iint\Phi\dd E_1Q\dd E_2}_{\bS_2}\le\norm{\Phi}_{L^\be}\norm{Q}_{\bS_2}.
\]
If $\Phi$ has the property
\[
Q\in\bS_1 \quad \Longrightarrow \quad \iint\Phi\dd E_1Q\dd E_2\in\bS_1,
\]
we say that $\Phi$ is a \emph{Schur multiplier} with respect to the spectral measures $E_1$ and $E_2$. In this case we can define by duality the double operator integral~\eqref{dvopi} for an arbitrary bounded linear operator $Q$, see~\cite{BS1,AP} for more detail.

We use the notation $\fM(\R^2)$ for the class of functions on $\R^2$ that are Schur multipliers with respect to arbitrary Borel spectral measures.

\section{A characterization of the class of relatively operator Lipschitz functions in terms of Schur multipliers}\label{doich}

Recall that for a differentiable function $f$ on $\R$, the divided difference $\dg f$ is defined on $\R^2$ by
\[
(\dg f)(x,y)\df
\begin{cases}
\dfrac{f(x)-f(y)}{x-y},	&amp; \text{if} \ x\ne y, \\
f'(x),					&amp; \text{if} \ x=y.
\end{cases}
\]
Put
\[
(\dg f)_\mathrm{x}(x,y)\df(x+\ri)(\dg f)(x,y),\quad(x,y)\in\R^2.
\]

The following result describes the class $\ROL$ in terms of Schur multipliers.
\begin{theo}
Let $f$ be a continuous function on $\R$. Then $f$ is relatively operator Lipschitz if and only if it is differentiable on\/ $\R$ and $(\dg f)_\mathrm{x}\in\fM(\R^2)$.
\end{theo}

Note that the part ``if'' was established in~\cite[Theorem~4.6]{BS4}.

Now we are in a position to obtain a useful formula for $f(B)-f(A)$ in the case when $B-A$ is a relatively bounded perturbation of $A$ and $f$ is a relatively operator Lipschitz function on $\R$.

\begin{theo}
Let $f$ be a relatively operator Lipschitz function on $\R$. If $A$ and $B$ are self-adjoint operators such that $B-A$ is a relatively bounded perturbation of $A$, then
\[
f(B)-f(A) = \iint_{\R\times\R}(\dg f)(x,y)(y+\ri)\dd E_{B}(x)(B-A)(A+\ri I)^{-1}\dd E_A(y).
\]
\end{theo}

\section{A characterization of the class of relatively operator Lipschitz functions in terms of commutators}\label{komu tatory?}

In this section we are going to obtain a characterization of the class of relatively operator Lipschitz functions in terms of commutator and quasi-commutator estimates. The following result means, in particular, that a function is relatively operator Lipschitz if and only if it is \emph{relatively commutator Lipschitz}, i.e.\ satisfies statement~\eqref{theo_ROL_b} of the following theorem.

\begin{theo}
Let $f\colon\R\to\C$ be a differentiable function. The following statements are equivalent:
\begin{enumerate}\alphenumi
\item $f$ is a relatively operator Lipschitz function;
\item \label{theo_ROL_b}
\[
\norm[\big]{f(A)R-Rf(A)}\le\const\norm[\big]{(AR-RA)(A+\ri I)^{-1}}
\]
for any bounded operator $R$ and any self-adjoint operator $A$;
\item
\[
\norm[\big]{f(B)R-Rf(A)}\le\const\norm[\big]{(BR-RA)(A+\ri I)^{-1}}
\]
for any bounded operator $R$ and any self-adjoint operators $A$ and $B$.
\end{enumerate}
\end{theo}

\section{Differentiation in a parameter}

To obtain a trace formula for $f(B)-f(A)$ in the case when $K\df B-A$ is a self-adjoint relatively trace class perturbation of $A$, we are going to consider a parametric family $A_t\df A+tK$, $t\in\R$, and differentiate the function
\[
t\mapsto f(A_t)-f(A)
\]
in the parameter $t$ in the strong operator topology.

\begin{theo}
Let $f$ be a relatively operator Lipschitz function. Suppose that $K$ is a relatively bounded perturbation of a self-adjoint operator $A$. Then the function $t\mapsto f(A_t)-f(A)$ is differentiable on\/ $\R$ in the strong operator topology and
\begin{equation}\label{proizvodnaya}
\restr{\frac\dd {\dd s}\parens[\big]{f(A_s)-f(A_t)}}{s=t} = \iint_{\R\times\R}\!\frac{f(x)-f(y)}{x-y}(y+\ri)\dd E_{A_t}(x)K(A_t+\ri I)^{-1}\dd E_{A_t}(y)
\end{equation} for any $t\in\R$.
\end{theo}

\section{The trace formula}

For a pair of self-adjoint operators $\{A,B\}$ with trace class difference the very important Lifshits--Krein trace formula~\cite{L,Kr} gives the following expression for the trace of the difference $f(A)-f(B)$:
\[
\trace\parens[\big]{f(A)-f(B)}=\int_\R f'(t)\bs{\xi}(t)\dd t
\]
for nice functions $f$, where $\bs{\xi}$ is the so-called \emph{spectral shift function} for the pair $\{A,B\}$. It is real-valued and belongs to $L^1(\R)$. It was established in~\cite{Pe} that the maximal class of functions $f$ for which the Lifshits--Krein trace formula holds for arbitrary pairs of self-adjoint operators with trace class difference coincides with the class of operator Lipschitz functions $\OL$.

To obtain a trace formula in the case of relatively operator Lipschitz functions and relatively trace class perturbations, we are going to use an idea of Birman and Solomyak~\cite{BS3}. Namely, we are going to use formula~\eqref{proizvodnaya} to compute the derivative of the function $t\mapsto f(A_t)-f(A)$ in the strong operator topology. Then we compute the trace of the derivative and use a version of the Newton--Leibnitz formula to compute the trace of $f(B)-f(A)$. We also describe the maximal class of functions for which the trace formula holds in the case of arbitrary relatively trace class perturbations.

\begin{theo}\label{spectral'no sdvinem}
Let $A$ be a self-adjoint operator and let $K$ be a relatively trace class perturbation of~$A$. Then there exists a unique measurable real-valued function $\bs\xi$ on $\R$ satisfying
\begin{equation}\label{svesom}
\int_\R\frac{\abs{\bs{\xi}(t)}}{1+\abs{t}}\dd t&lt;\be
\end{equation}
and such that for an arbitrary relatively operator Lipschitz function $f$ on $\R$ the following trace formula holds
\begin{equation}\label{nasledil}
\trace\parens[\big]{f(A+K)-f(A)}=\int_\R f'(t)\bs{\xi}(t)\dd t.
\end{equation}
\end{theo}

The function $\bs\xi$ satisfying~\eqref{nasledil} and~\eqref{svesom} is called the \emph{spectral shift function for the pair $\{A,A+K\}$}.

The fact that the spectral shift function in the case of a relatively trace class perturbation must satisfy~\eqref{svesom} was established earlier in~\cite{CS}.

\begin{theo}\label{maksimalka}
Let $f$ be a differentiable function on\/ $\R$ such that trace formula \eqref{nasledil} holds for arbitrary self-adjoint operators $A$ and $K$ such that $K$ is a relatively trace class perturbation of $A$. Then $f\in\ROL$.
\end{theo}

The details of this note can be found in~\cite{AParXiV}.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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