The structure of normal Hausdorff operators on Lebesgue spaces

TL;DR

This paper studies the structure and properties of normal generalized Hausdorff operators on Lebesgue spaces, introducing a symbol concept and analyzing invertibility, spectrum, and compactness, with examples including Cesàro operators.

Contribution

It introduces a new symbol notion for generalized Hausdorff operators and characterizes their structure and key properties on Lebesgue spaces.

Findings

Characterization of invertibility and spectrum of these operators

Analysis of norm and compactness properties

Examples including Cesàro operators

Abstract

We consider a generalization of Hausdorff operator and introduce the notion of the symbol of such an operator. Using this notion we describe the structure and investigate important properties (such as invertibility, spectrum, norm, and compactness) of normal generalized Hausdorff operators on Lebesgue spaces over $\mathbb{R}^n.$ The examples of Ces\`{a}ro operators are considered.