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

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.

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

“…In this note we prove the aforementioned conjecture for the case where A(u) is a commuting family of self-adjoint matrices. The result has been announced in [20]. The case of positive or negative definite perturbation matrices was considered in [15].…”