arXiv subjects

Vladimir Peller

Publications and source records attributed to Vladimir Peller.

40 records · Page 3Linked to original sources

arXiv2010

Functions of perturbed unbounded self-adjoint operators. Operator Bernstein type inequalities

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 the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operator. In this paper we extend results of \cite{AP2} and \cite{AP3} to the case of unbounded self-adjoint operators $A$. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of \cite{AP2}. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers \cite{AP2} and \cite{AP3} were announced in \cite{AP1}.

math.FA↗

arXiv2009

Lipschitz functions of perturbed operators

We prove that if $f$ is a Lipschitz function on $\R$, $A$ and $B$ are self-adjoint operators such that ${\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le{\rm const} (1+j)^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class $\boldsymbol{S}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\in\boldsymbol{S}_Ω$, i.e., $\sum_{j=0}^ns_j(f(A)-f(B))\le\const\log(2+n)$. We also obtain more general results about the behavior of double operator integrals of the form $Q=\iint(f(x)-f(y))(x-y)^{-1}dE_1(x)TdE_2(y)$, where $E_1$ and $E_2$ are spectral measures. We show that if $T\in\boldsymbol{S}_1$, then $Q\in\boldsymbol{S}_Ω$ and if $\rank T=1$, then $Q\in\boldsymbol{S}_{1,\be}$. Finally, if $T$ belongs to the Matsaev ideal $\boldsymbol{S}_ω$, then $Q$ is a compact operator.

math.FA↗

arXiv2004

An extension of the Koplienko-Neidhardt trace formulae

Koplienko [Ko] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class $\bS_2$. A similar formula in the case of unitary operators was obtained by Neidhardt [N]. In this paper we improve their results and obtain sharp conditions under which the Koplienko--Neidhardt trace formulae hold.

math.FA↗

arXiv1995

Approximation by analytic matrix functions. The four block problem

We study the problem of finding a superoptimal solution to the four block problem. Given a bounded block matrix function $\left(\begin{array}{cc}Φ_{11} &Φ_{12}\\Φ_{21}&Φ_{22}\end{array}\right)$ on the unit circle the four block problem is to minimize the $L^\infty$ norm of $\left(\begin{array}{cc} Φ_{11}-F&Φ_{12}\\Φ_{21}&Φ_{22}\end{array}\right)$ over $F\in H^\infty$. Such a minimizing $F$ (an optimal solution) is almost never unique. We consider the problem to find a superoptimal solution which minimizes not only the supremum of the matrix norms but also the suprema of all further singular values. We give a natural condition under which the superoptimal solution is unique.

math.CA↗