Operator Norms and Essential Norms of Weighted Composition Operators from Analytic Function Spaces into Zygmund-type Spaces

Abstract

In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk into Zygmund-type spaces. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little Zygmund-type space are compact and characterize such operators. We apply our results to the cases when X is the Hardy space $$H^p$$ for $$1\le p\le \infty $$ and the weighted Bergman space $$A_\alpha ^p$$ for $$\alpha >-1$$ and $$1\le p<\infty $$ .