Optimal domain of Volterra operators in Korenblum spaces
Angela A. Albanese, José Bonet, Werner J. Ricker
TL;DR
This work extends the theory of optimal domains to Volterra-type operators on Korenblum growth spaces by introducing and analyzing the optimal domains [V_g,A^{-gamma}] and [V_g,A^{-gamma}_0] for non-constant g in the Bloch class. It provides a precise description in terms of fg' belonging to A^{-(gamma+1)} (or A_0^{-(gamma+1)}) and proves these domains are Banach spaces, often strictly larger than the original A^{-gamma} (and A^{-gamma}_0). The authors determine the full multiplier structure of these optimal domains, showing M([V_g,A^{-gamma}]) = H_infty and analogous results for cross-spaces, as well as classical results for A^{-gamma} to A^{-delta} via A^{-(delta-gamma)}. They also investigate the range behavior, establishing when optimal domains enlarge the original spaces and demonstrating non-closed-range phenomena for Cesàro and Volterra-type operators on weighted and Korenblum spaces, highlighting intricate interactions between domain enlargement and operator range. These findings illuminate the structure of analytic function spaces under Volterra-type extensions and have implications for stability and extension properties of a broad class of operators.
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 $X:=A^{-γ}$, for $γ>0$. Previous studies dealt with the classical Cesàro operator $T:=C$ acting in the Hardy spaces $H^p$, $1\leq p<\infty$, \cite{CR}, \cite{CR1}, in $A^{-γ}$, \cite{ABR-R}, and more recently, generalized Volterra operators $T$ acting in $X:=H^p$, \cite{BDNS}.