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

TL;DR

This paper extends previous estimates of finite differences for functions of bounded self-adjoint operators to unbounded cases and establishes Bernstein type inequalities for entire functions and unitary operators, providing new proofs and insights.

Contribution

It introduces operator Bernstein inequalities for unbounded self-adjoint and unitary operators, extending prior finite difference estimates and offering alternative proofs.

Findings

Extended finite difference estimates to unbounded operators

Established Bernstein inequalities for entire functions of exponential type

Provided alternative proofs of previous main results

Abstract

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}.