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

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