Optimal domain of Volterra operators in Korenblum spaces

TL;DR

This paper investigates the maximal domain spaces for generalized Volterra operators acting on Korenblum spaces, extending the understanding of operator behavior in analytic function spaces.

Contribution

It characterizes the largest Banach spaces where these operators can be extended continuously, generalizing previous results on classical and generalized Volterra operators.

Findings

Identifies the maximal domain spaces for Volterra operators in Korenblum spaces.

Extends known results from Hardy and classical spaces to Korenblum spaces.

Provides a framework for analyzing operator extensions in analytic function spaces.

Abstract

The aim of this article is to study the largest domain space $[T,X]$, whenever it exists, of a given continuous linear operator $T\colon X\to X$, where $X\subseteq H(\mathbb{D})$ is a Banach space of analytic functions on the open unit disc $\mathbb{D}\subseteq \mathbb{C}$. That is, $[T,X]\subseteq H(\mathbb{D})$ is the \textit{largest} Banach space of analytic functions containing $X$ to which $T$ has a continuous, linear, $X$-valued extension $T\colon [T,X]\to X$. The class of operators considered consists of generalized Volterra operators $T$ acting in the Korenblum growth Banach spaces $X:=A^{-\gamma}$, for $\gamma>0$. Previous studies dealt with the classical Ces\`aro operator $T:=C$ acting in the Hardy spaces $H^p$, $1\leq p<\infty$, \cite{CR}, \cite{CR1}, in $A^{-\gamma}$, \cite{ABR-R}, and more recently, generalized Volterra operators $T$ acting in $X:=H^p$, \cite{BDNS}.