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Hausdorff Operators on de Branges Spaces and Paley-Wiener spaces

A. R. Mirotin

Abstract

For a class of de Branges spaces containing polynomials, sufficient and necessary conditions are given for the boundedness and compactness of the Hausdorff operators under consideration. For the Paly-Wiener spaces we reduce the study of our Hausdorff operators to classical integral ones. The operators that appeared are Carleman and therefore closeble in $L^2(\mathbb{R})$. We obtain also conditions for boundedness, compactness and nuclearity of our operators in the Paley-Wiener space as well as the conditions for their belonging to the Hilbert-Schmidt class.

Hausdorff Operators on de Branges Spaces and Paley-Wiener spaces

Abstract

For a class of de Branges spaces containing polynomials, sufficient and necessary conditions are given for the boundedness and compactness of the Hausdorff operators under consideration. For the Paly-Wiener spaces we reduce the study of our Hausdorff operators to classical integral ones. The operators that appeared are Carleman and therefore closeble in . We obtain also conditions for boundedness, compactness and nuclearity of our operators in the Paley-Wiener space as well as the conditions for their belonging to the Hilbert-Schmidt class.

Paper Structure

This paper contains 6 sections, 21 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The general case of de Branges spaces
  4. Action between two arbitrary spaces
  5. The case of spaces with polynomials
  6. Hausdorff operator in the Paley-Wiener space

Key Result

Theorem 3.1

Let the condition condition holds for functions $E_2$ and $E_1$ of Hermite-Biehler class, let $\Phi(u)=0$ for $\mu$-a.e. $u\in (0,\delta)$ ($\delta>0$), and $\Phi(u)v(u)\sqrt{u}$ be $\mu$-integrable on $[\delta,\infty)$. Then $\mathbb{H}_{\Phi,\mu}$ is bounded as an operator between $\mathcal{H}(E_1 $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (43)

  • Example 1
  • Remark 1
  • Theorem 3.1
  • proof
  • Remark 2
  • Remark 3
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 33 more