Volterra type operators between Bloch type spaces and weighted Banach spaces

TL;DR

This paper characterizes the boundedness and compactness of Volterra type operators between Bloch type spaces and weighted Banach spaces with general weights, extending previous results to more general settings.

Contribution

It provides complete characterizations of boundedness and compactness conditions for Volterra operators between Bloch type and weighted Banach spaces with general weights.

Findings

Complete criteria for boundedness of $T_g$ and $S_g$

Complete criteria for compactness of $T_g$ and $S_g$

Extension of previous results to more general weights

Abstract

When the weight $\mu$ is more general than normal, the complete characterizations in terms of the symbol $g$ and weights for the conditions of the boundedness and compactness of $T_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ and $S_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and $S_g$ between Bloch type spaces $\mathcal{B}^\infty_\nu$ and weighted Banach spaces $H^{\infty}_\nu$ with more general weights, which generalize their works.