Spectral Radius Inequalities for Functions of Operators Defined by Power Series

TL;DR

This paper establishes inequalities relating the spectral radius of functions of operators, constructed via power series, to their norms, with special attention to commuting operators on complex Hilbert spaces.

Contribution

It introduces new spectral radius inequalities for functions of operators defined by power series, including cases involving commuting operators.

Findings

Spectral radius inequalities for operator functions derived from power series.

Results applicable to bounded linear operators on complex Hilbert spaces.

Analysis of the case involving two commuting operators.

Abstract

By the help of power series f we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f. Utilising these functions we prove some inequalities for the spectral radius of the bounded linear operator f(T) on a complex Hilbert space and some functions of its norm. The case of two commuting operators is also investigated.