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Optimal domain of Volterra operators in classes of Banach spaces of analytic functions

Angela A. Albanese, José Bonet, Werner J. Ricker

Abstract

A thorough investigation is made of the optimal domain space of generalized Volterra operators, Cesàro operators and other operators when they act in various Banach spaces of analytic functions. Of particular interest is the situation when the operators act in Hardy spaces, Korenblum growth spaces and more general weighted spaces. The optimal domain space may be genuinely larger than the initial domain of the operator, or not. In the former case, the initial space may or may not be dense in the optimal domain space. Sometimes the optimal domain space can be identified with a known Banach space of analytic functions, on other occasions it determines a new space.

Optimal domain of Volterra operators in classes of Banach spaces of analytic functions

Abstract

A thorough investigation is made of the optimal domain space of generalized Volterra operators, Cesàro operators and other operators when they act in various Banach spaces of analytic functions. Of particular interest is the situation when the operators act in Hardy spaces, Korenblum growth spaces and more general weighted spaces. The optimal domain space may be genuinely larger than the initial domain of the operator, or not. In the former case, the initial space may or may not be dense in the optimal domain space. Sometimes the optimal domain space can be identified with a known Banach space of analytic functions, on other occasions it determines a new space.

Paper Structure

This paper contains 7 sections, 30 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Properties of general optimal domain spaces
  3. Optimal domain of Volterra operators in classes of Banach spaces of analytic functions
  4. Weighted Banach spaces of analytic functions
  5. The Hardy spaces $H^p$ for $1\leq p\leq\infty$
  6. Generalized Cesàro operators $C_t$ for $0\leq t\leq 1$
  7. Finite dimensional kernel

Key Result

Proposition 2.1

Let $X$ be a Banach space of analytic functions on $\mathbb{D}$ and $T\in {\mathcal{L}}(H({\mathbb D}))$ be an isomorphism satisfying $T(X)\subseteq X$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (68)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 58 more