Weighted Composition Operators from Banach Spaces of Holomorphic Functions to Weighted-Type Banach Spaces on the Unit Ball in $$\mathbb {C}^n$$

Abstract

Let X be a Banach space of holomorphic functions on the unit ball $$\mathbb {B}_n$$ in $$\mathbb {C}^n$$ whose point-evaluation functionals are bounded. In this work, we characterize the bounded weighted composition operators from X into a weighted-type Banach space $$H^\infty _\mu (\mathbb {B}_n)$$, where the weight $$\mu $$ is an arbitrary positive continuous function on $$\mathbb {B}_n$$. We determine the norm of such operators in terms of the norm of the point-evaluation functionals. Under some restrictions on X, we characterize the compact weighted composition operators mapping X into $$H^\infty _\mu (\mathbb {B}_n)$$. Under an alternative set of conditions, we provide essential norm estimates. We apply our results to the cases when X is the Hardy space $$H^p(\mathbb {B}_n)$$, the weighted Bergman space $$A_\alpha ^p(\mathbb {B}_n)$$ for $$\alpha >-1$$ and $$1\le p<\infty $$, the Bloch space $$\mathcal {B}$$ and the little Bloch space $$\mathcal {B}_0$$. In all these cases we obtain precise formulas of the essential norm.