Weighted composition operators from $H^\infty$ to the Bloch space of a bounded homogeneous domain

TL;DR

This paper characterizes bounded and compact weighted composition operators from the Hardy space to the Bloch space on bounded homogeneous domains, providing norm estimates and conditions for compactness, especially in symmetric domains.

Contribution

It offers new characterizations and norm estimates for weighted composition operators between these function spaces on bounded homogeneous domains.

Findings

Characterization of bounded weighted composition operators

Operator norm estimates for these operators

Necessary and sufficient conditions for compactness in specific domains

Abstract

Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if $D$ is a bounded symmetric domain, the bounded multiplication operators from $H^\infty(D)$ to the Bloch space of $D$ are the operators whose symbol is bounded.