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Spectrum of Hausdorff operators on weighted Bergman and Hardy spaces of the upper half-plane

Carlo Bellavita, Georgios Stylogiannis

Abstract

We characterize the spectrum of Hausdorff operators on weighted Bergman and power weighted Hardy spaces of the upper half-plane.

Spectrum of Hausdorff operators on weighted Bergman and Hardy spaces of the upper half-plane

Abstract

We characterize the spectrum of Hausdorff operators on weighted Bergman and power weighted Hardy spaces of the upper half-plane.
Paper Structure (6 sections, 14 theorems, 89 equations)

This paper contains 6 sections, 14 theorems, 89 equations.

Table of Contents

  1. Preliminaries
  2. Spectrum of $H_\phi$ on $L^{p}(\mathbb{R},|\cdot|^{a})$
  3. Proof of Theorem \ref{['Theorem spectrum weighted Hardy']}
  4. Proof of Theorem \ref{['Theorem spectrum Bergman']}
  5. Cesaro Type operators: proof of Corollary \ref{['last corollary']}
  6. Hausdorff operators defined by measures

Key Result

Theorem 1

Let $1\leq p<\infty, a>-1$ and $\phi$ be a measurable function in $[0,\infty)$ which satisfies Then $\sigma(\mathcal{H}_{\phi}, H^{p}_{|\cdot|^{a}}(\mathbb{U}))=\overline{\widehat{k_{a,p}}(\mathbb{R})}$ where $\widehat{k_{a,p}}(\xi)=\int^{\infty}_{0}\phi(t)t^{\frac{a+1}{p}}\frac{dt}{t^{1+i\xi}}$ for $\xi\in\mathbb{R}$.

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • Theorem 7
  • Lemma 8
  • ...and 14 more