Resolvent estimates for a function of a linear operator

TL;DR

This paper investigates how the growth rate of the resolvent of a bounded linear operator relates to the growth rate of the resolvent of a function of that operator, providing insights into spectral analysis.

Contribution

It establishes new relations between the resolvent growth of an operator and its analytic functional calculus, enhancing understanding of spectral properties.

Findings

Derived bounds linking resolvent growth of T and f(T)

Provided conditions under which resolvent estimates are comparable

Extended classical results to broader classes of functions

Abstract

Let $T$ be a bounded linear operator on a Banach space and $f$ an analytic function, defined on the spectrum of $T$. We discuss the relations between the rate of growth of the resolvent of $T$ and of $f(T)$.