Related papers: Functions of perturbed normal operators

Let $(A_1,\cdots,A_n)$ and $(B_1,\cdots,B_n)$ be $n$-tuples of commuting self-adjoint operators on Hilbert space. For functions $f$ on $\R^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in…

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$,…

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz…

We start with the Birman--Solomyak approach to define double operator integrals and consider applications in estimating operator differences $f(A)-f(B)$ for self-adjoint operators $A$ and $B$. We present the Birman--Solomyak approach to the…

Suppose that f is a Lipschitz function on the real numbers with Lipschitz constant smaller or equal to 1. Let A be a bounded self-adjoint operator on a Hilbert space H. Let 1<p<infinity and suppose that x in B(H) is an operator such that…

The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, H\"older type estimates, Schatten--von Neumann estimates are…

The paper studies the problem, for which continuous functions $f$ on the real line ${\Bbb R}$, the difference of the functions $f(B)-f(A)$ of self-adjoint operators $A$ and $B$ with trace class difference must also be of trace class. The…

We consider the difference $f(H_1)-f(H_0)$ for self-adjoint operators $H_0$ and $H_1$ acting in a Hilbert space. We establish a new class of estimates for the operator norm and the Schatten class norms of this difference. Our estimates…

In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with…

The purpose of this paper is to obtain an integral representation for the difference $f(L_1)-f(L_2)$ of functions of maximal dissipative operators. This representation in terms of double operator integrals will allow us to establish…

We consider the difference $f(H_1)-f(H_0)$, where $H_0=-\Delta$ and $H_1=-\Delta+V$ are the free and the perturbed Schr\"odinger operators in $L^2(\mathbb R^d)$, and $V$ is a real-valued short range potential. We give a sharp sufficient…

We consider the Schr\"{o}dinger operator $\mathcal{L}=-\Delta+V$ on $\mathbb R^d$, $d\geq3$, where the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $RH_s$ for some $s\geq d/2$. A real-valued function $f\in…

Let $X$, $Y$ be Banach spaces and let $\mathcal{L}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. We develop the theory of double operator integrals on $\mathcal{L}(X,Y)$ and apply this theory to obtain commutator estimates…

Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n…

We describe the maximal class of functions $f$ on the real line, for which the Lifshitz--Krein trace formula $\operatorname{trace}(f(A)-f(B))=\int_{\Bbb R} f'(s)\boldsymbol{\xi}(s)\,ds$ holds for arbitrary self-adjoint operators $A$ and $B$…

This paper resolves a number of conjectures in the perturbation theory of linear operators. Namely, we prove that every Lipschitz function is operator Lipschitz in the Schatten-von Neumann ideals $S^\alpha$, $1 < \alpha < \infty$. The…

Let $\theta \in(0,1)$ and $(\mathcal{M},\tau)$ be a semifinite von Neumann algebra. We consider the function spaces introduced by Sobolev (denoted by $S_{d,\theta}$), showing that there exists a constant $d>0 $ depending on $p$, $0<p\le…

This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of…

Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but…

Consider the Schr\"odinger operator $\mathcal{L}=-\Delta+V$ in $\mathbb{R}^n, n\ge 3,$ where $V$ is a nonnegative potential satisfying a reverse H\"older condition of the type \begin{equation*} \left( \frac{1}{|B|}\int_B…