A Hausdorff Operator on Lebesgue Space With Commuting Perturbation Matrices Is a Non-Riesz Operator

TL;DR

This paper proves that a generalized Hausdorff operator on Lebesgue spaces, with commuting perturbation matrices, is not a Riesz operator if it is non-zero, highlighting its non-decomposability into quasinilpotent and compact parts.

Contribution

It establishes that such generalized Hausdorff operators are inherently non-Riesz, extending understanding of their spectral properties and operator classifications.

Findings

The operator is not a Riesz operator when non-zero.

It cannot be expressed as a sum of quasinilpotent and compact operators.

The result applies to operators with commuting perturbation matrices.

Abstract

We consider a generalization of Hausdorff operators on Lebesgue spaces and under natural conditions prove that such an operator is not a Riesz operator provided it is non-zero. In particular, it cannot be represented as a sum of a quasinilpotent and compact operators.