On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

Abstract

<jats:title>Abstract</jats:title><jats:p>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 <jats:inline-formula id="j_forum-2019-0097_ineq_9999_w2aab3b7e5268b1b6b1aab1c15b1b1Aa"><jats:alternatives><jats:tex-math>{L^{2}(\mathbb{R}^{n})}</jats:tex-math></jats:alternatives></jats:inline-formula> is unitary equivalent to the operator of multiplication by some matrix-valued function (its matrix symbol) in the space <jats:inline-formula id="j_forum-2019-0097_ineq_9998_w2aab3b7e5268b1b6b1aab1c15b1b3Aa"><jats:alternatives><jats:tex-math>{L^{2}(\mathbb{R}^{n};\mathbb{C}^{2^{n}})}</jats:tex-math></jats:alternatives></jats:inline-formula>. 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.</jats:p>

Defining Hausdorff operators on Euclidean spaces

“…In the related topic of the compactness (or noncompactness) of the Hausdorff operators, only first results have appeared, see other Mirotin. [2][3][4] There is much to do in this topic; open problems are also discussed in Liflyand. 7,22…”

“…For example, in the apparently most mysterious case of the Hardy spaces H p , 0< p <1, the results in Liflyand and Miyachi 10 should be extended to the global case, being mostly local. In the related topic of the compactness (or noncompactness) of the Hausdorff operators, only first results have appeared, see other Mirotin 2–4 . There is much to do in this topic; open problems are also discussed in Liflyand 7,22 …”

“…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,…”

“…In older days, number series and Fourier series served as application objects, while in the recent results, power series and Fourier transforms came into play. There are more abstract settings as well; see Mirotin 2–4 . In this work, we are mainly concentrated on the Hausdorff operators in the Euclidean space .…”

Hausdorff operators on Lebesgue spaces with positive definite perturbation matrices are non-Riesz

“…The general form of a Hausdorff operator given by definition 1 (with an arbitrary measure space (Ω, µ) instead of R m ) gives us, for example, the opportunity to consider (in the case Ω = Z m ) discrete Hausdorff operators [7], [8].…”