On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

The main goal of this work is to examine the structure of normal Hausdorff operators on ${\mathbb{R}}^n$. We show that normal Hausdorff operator in $L^2({\mathbb{R}}^n)$ is unitary equivalent to the operator of multiplication by some matrix-function (its matrix symbol) in the space $L^2($\mathbb{R}$^n; $\mathbb{C}$^{2^n}).$ 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.