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Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Alejandro Mas, Noel Merchán, Elena de la Rosa

Abstract

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Cesàro operator as follows $$ C_ω(f)(z)=\int_0^1 f(tz)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_ω$ on weighted Hardy spaces of analytic functions $\mathcal{H}_γ$, $γ>0$ and on general weighted Bergman spaces $A^2_μ$.

Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Abstract

Let denote the unit disc in . We define the generalized Cesàro operator as follows where are the reproducing kernels of the Bergman space induced by a radial weight in the unit disc . We study the action of the operator on weighted Hardy spaces of analytic functions , and on general weighted Bergman spaces .
Paper Structure (4 sections, 8 theorems, 55 equations)

This paper contains 4 sections, 8 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Previous results.
  3. Proof of the main results.
  4. Compactness

Key Result

Theorem 1

Let $\omega$ be a radial weight, $\gamma>0$. Then $C_{\omega}: \mathcal{H}_{\gamma} \to \mathcal{H}_{\gamma}$ is bounded if and only if $\omega \in \mathcal{D}$.

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • Lemma A
  • Lemma B
  • Theorem C
  • Lemma 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof