On the Structure of Normal Hausdorff Operators on Lebesgue Spaces

Abstract

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 R n . The examples of Cesàro operators are considered. 1 2 1

On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

“…As was mentioned above, the problem of compactness of nontrivial Hausdorff operators was posed in [14]. In our case the answer to this question is negative (the case of positive definite matrices was considered in [18], [20]). Corollary 6.…”

“…The symbol was first introduced in [20] for the case of positive definite A(u) (in the simplest one-dimensional case the symbol was in fact considered also in [5, Theorem 2.1]). As we shall see properties of a Hausdorff operator are closely related to the properties of its matrix symbol.…”

“…The survey articles [15], [6] contain main results on Hausdorff operators and bibliography up to 2014. For more resent results see, e. g., [16], [5], [20], and [18]. The last paper is devoted to generalizations of Hausdorff operators to locally compact groups.…”

“…It should be noted that the case of L p spaces is more complicated for the lack of Plancherel Theorem, cf. [20], [19].…”

Defining Hausdorff operators on Euclidean spaces

“…There are more abstract settings as well; see Mirotin. [2][3][4] In this work, we are mainly concentrated on the Hausdorff operators in the Euclidean space R n . The most general form in which the Hausdorff operators have been considered in the Euclidean setting until recently is ( )(x) = ( Φ )(x) = ( Φ,A )(x) = ∫ R n Φ(u) (xA(u)) du,…”