Is Every Nontrivial Hausdorff Operator on Lebesgue Space a Non-Riesz Operator?

The chapter investigates the nature of Hausdorff operators on Lebesgue spaces, beginning with an introduction to their one-dimensional and multidimensional forms. It discusses the regularity properties of these operators, as established by Hardy and extended in modern theory. The main focus is on the conjecture that nontrivial Hausdorff operators are non-Riesz, particularly when the family of matrices involved is commutative and positive definite. The author presents a proof of this conjecture, employing lemmas on the boundedness and adjoints of Hausdorff operators. The result has implications for the compactness and spectral properties of these operators, making the chapter a valuable contribution to the field of operator theory.