LAUSR

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.