Bloch-type spaces and extended Ces\`{a}ro operators in the unit ball of a complex Banach space

TL;DR

This paper extends Bloch-type spaces to the unit ball of complex Banach spaces using radial derivatives and characterizes the boundedness and compactness of extended Cesàro operators between these spaces based on their symbols.

Contribution

It generalizes Bloch-type spaces to infinite-dimensional Banach spaces and provides a complete characterization of Cesàro operators' boundedness and compactness.

Findings

Characterization of symbols for bounded Cesàro operators

Criteria for compactness of Cesàro operators

Extension of Bloch spaces to infinite-dimensional settings

Abstract

Let $\mathbb{B}$ be the unit ball of a complex Banach space $X$. In this paper, we will generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $\mathbb{B}$ by using the radial derivative. Next, we define an extended Ces\`{a}ro operator $T_{\varphi}$ with holomorphic symbol $\varphi$ and characterize those $\varphi$ for which $T_{\varphi}$ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those $\varphi$ for which $T_{\varphi}$ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol $\varphi$. When $\mathbb{B}$ is the open unit ball of a finite dimensional complex Banach space $X$, this additional assumption is automatically satisfied.