# On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

## TL;DR

This paper analyzes the structure of normal Hausdorff operators on multidimensional Lebesgue spaces, showing their unitary equivalence to multiplication operators and relating their properties to their matrix symbols.

## Contribution

It establishes that normal Hausdorff operators on ^n are unitarily equivalent to multiplication operators, linking their spectral properties to their matrix symbols.

## Key findings

- Normal Hausdorff operators are unitarily equivalent to multiplication operators.
- The norm and spectrum of these operators are characterized by their matrix symbols.
- Properties of Hausdorff operators are closely related to properties of their symbols.

## Abstract

The main goal of this work is to examine the structure of normal Hausdorff operators on $\mathbb{R}^n$. We show that normal Hausdorff operator in $L^2(\mathbb{R}^n)$ is unitary equivalent to the operator of multiplication by some matrix-function (its matrix symbol) in the space $L^2(\mathbb{R}^n; \mathbb{C}^{2^n}).$ Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.