Hausdorff operators on Lebesgue spaces with positive definite perturbation matrices are non-Riesz

TL;DR

This paper proves that generalized Hausdorff operators with positive definite, permutable perturbation matrices are not Riesz operators on Lebesgue spaces, provided they are non-zero, highlighting a specific spectral property.

Contribution

It establishes a new non-Riesz property for a class of generalized Hausdorff operators with positive definite, permutable matrices on Lebesgue spaces.

Findings

Such operators are not Riesz if non-zero

Positive definite, permutable matrices influence spectral properties

The result applies to a broad class of Lebesgue space operators

Abstract

We consider generalized Hausdorff operators with positive definite and permutable perturbation matrices on Lebesgue spaces and prove that such operators are not Riesz operators provided they are non-zero.