Generalized Volterra operators mapping between Banach spaces of analytic functions

TL;DR

This paper characterizes the boundedness and compactness of classical and generalized Volterra operators between Banach spaces of analytic functions, addressing open problems and providing new insights into operator behavior.

Contribution

It provides a comprehensive characterization of boundedness and compactness for classical and generalized Volterra operators on various Banach spaces, extending existing results.

Findings

Characterized boundedness of $T_g$ for standard weights $v_{\alpha}$ with $0 \leq \alpha < 1$

Analyzed boundedness, compactness, and weak compactness of $T_g^{\varphi}$ between Banach spaces of analytic functions

Partially answered an open problem posed by Anderson, Jovovic, and Smith

Abstract

We characterize boundedness and compactness of the classical Volterra operator $T_g \colon H_{v_{\alpha}}^{\infty} \to H^{\infty}$ induced by a univalent function $g$ for standard weights $v_{\alpha}$ with $0 \leq \alpha < 1$, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator $T_g^{\varphi}$ mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.