On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract

The main goal of this work is to examine the structure of normal Hausdorff operators on Rⁿ. We show that normal Hausdorff operator in L² (Rⁿ) is unitary equivalent to the operator of multiplication by some matrix-function (its matrix symbol) in the space L²(Rⁿ; C²ⁿ). Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.