Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space

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 Cesàro operator $T_φ$ with holomorphic symbol $φ$ and characterize those $φ$ for which $T_φ$ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those $φ$ for which $T_φ$ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol $φ$. When $\mathbb{B}$ is the open unit ball of a finite dimensional complex Banach space $X$, this additional assumption is automatically satisfied.