The structure of normal Hausdorff operators on Lebesgue spaces

We consider a generalization of Hausdorff operator and introduce the notion of the symbol of such an operator. Using this notion we describe the structure and investigate important properties (such as invertibility, spectrum, norm, and compactness) of normal generalized Hausdorff operators on Lebesgue spaces over ${\mathbb{R}}^n\mathrm{.}$ The examples of Ces`{a}ro operators are considered.