On the description of multidimensional normal Hausdorff operators on Lebesgue spaces

2019/02/20 by Mirotin, A. R.
#46E30 #47B15 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1902.07671

Abstract

The main goal of this work is to examine the structure of normal Hausdorff operators on ℝⁿ. We show that normal Hausdorff operator in L²(ℝⁿ) is unitary equivalent to the operator of multiplication by some matrix-function (its matrix symbol) in the space L²(ℝⁿ; ℂ²ⁿ). 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.

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• The structure of normal Hausdorff operators on Lebesgue spaces
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• On a general concept of a Hausdorff-type operator
• Boundedness of Hausdorff-type operators with two-variable kernels on Lebesgue spaces