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Alexei Aleksandrov

Publications and source records attributed to Alexei Aleksandrov.

6 recordsLinked to original sources
arXiv2016

Operator Lipschitz functions (English translation)

The purpose of this survey is a comprehensive study of operator Lip\-schitz functions. A continuous function $f$ on the real line ${\Bbb R}$ os called operator Lipschitz if $\|f(A)-f(B)\|\le\operatorname{const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$ . Next, we consider operator Lipschitz functions on closed subsets of the plane and introduce the class of commutator Lipschitz functions on such subsets. An important role for the study of such classes of functions is played by double operator integrals and Schur multipliers.

math.FA↗
arXiv2015

Functions of almost commuting operators and an extension of the Helton-Howe trace formula

Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $φ\mapstoφ(A,B)$ for functions $φ$ in the Besov class $B_{\infty,1}^1({\Bbb R}^2)$. This functional calculus is linear, the operators $φ(A,B)$ and $ψ(A,B)$ almost commute for $φ,\,ψ\in B_{\infty,1}^1({\Bbb R}^2)$, and $φ(A,B)=u(A)v(B)$ whenever $φ(s,t)=u(s)v(t)$. We extend the Helton--Howe trace formula for arbitrary functions in $B_{\infty,1}^1({\Bbb R}^2)$. The main tool is triple operator integrals with integrands in Haagerup-like tensor products of $L^\infty$ spaces.

math.FA↗
arXiv2010

Trace formulae for perturbations of class $\bs{\bS_m}$

We obtain general trace formulae in the case of perturbation of self-adjoint operators by self-adjoint operators of class $\bS_m$, where $m$ is a positive integer. In \cite{PSS} a trace formula for operator Taylor polynomials was obtained. This formula includes the Livshits--Krein trace formula in the case $m=1$ and the Koplienko trace formula in the case $m=2$. We establish most general trace formulae in the case of perturbation of Schatten--von Neumann class $\bS_m$. We also improve the trace formula obtained in \cite{PSS} for operator Taylor polynomials and prove it for arbitrary functions in he Besov space $B_{\be1}^m(\R)$. We consider several other special cases of our general trace formulae. In particular, we establish a trace formula for $m$th order operator differences.

math.FA↗
arXiv2010

Functions of normal operators under perturbations

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this paper we extend those results to the case of functions of normal operators. We show that if a function $f$ belongs to the Hölder class $Ł_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{Ł_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $Ł_ø(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\constø(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $ø$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\inŁ_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.

math.FA↗