Functions of continuous Ces\'aro operators

TL;DR

This paper investigates the properties of Cesàro operators, including their holomorphic functions, fractional powers, and logarithms, across various L^2 spaces, providing spectral analysis and concrete examples.

Contribution

It introduces and analyzes the spectral properties of logarithms and fractional powers of Cesàro operators in multiple L^2 spaces, extending their functional calculus.

Findings

Spectral properties of Cesàro operators are characterized.

Logarithms of Cesàro operators are defined and studied.

Examples illustrate the theoretical results.

Abstract

We describe holomorphic functions and fractional powers of Ces\'{a}ro operators in $L^2(\mathbb{R})$, $L^2(\mathbb{R}_+)$, and $L^2[0,1]$. Logarithms of Ces\'{a}ro operators are introduced as well and their spectral properties are studied. Several examples are considered.