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

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.