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 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 Cesaro operator Tφ with holomorphic symbol φ and characterize those φ for which Tφ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those φ for which Tφ 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.