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

TL;DR

This paper thoroughly investigates the optimal domain spaces for generalized Volterra and Cesàro operators across various Banach spaces of analytic functions, including Hardy and weighted spaces.

Contribution

It identifies conditions under which the optimal domain is larger, equal, or a new space, enhancing understanding of operator domains in analytic function spaces.

Findings

Optimal domain spaces can be larger or equal to initial spaces.

In some cases, the optimal domain is a known Banach space.

The study covers Hardy, Korenblum, and weighted spaces.

Abstract

A thorough investigation is made of the optimal domain space of generalized Volterra operators, Ces\`aro 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.