On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract

Abstract Hausdorff operators originated from some classical summation methods. Now this is an active research field. In the present article, a spectral representation for multidimensional normal Hausdorff operator is given. We show that normal Hausdorff operator in L 2 ⁢ ( ℝ n ) {L^{2}(\mathbb{R}^{n})} is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space L 2 ⁢ ( ℝ n ; ℂ 2 n ) {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.