Weighted Space and Bloch-Type Space on the Unit Ball of an Infinite Dimensional Complex Banach Space

Abstract

Let $${\mathbf {B}}_{{\mathbb {X}}}$$ be the open unit ball of a complex Banach space $${\mathbb {X}}$$ , which may beinfinite dimensional. The weighted composition operator and weighted space defined on $${\mathbf {B}}_{{\mathbb {X}}}$$ are introduced. We obtain the boundedness and compactness of the weightedcomposition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball $${\mathbb {B}}^n$$ to the case of $${\mathbf {B}}_{{\mathbb {X}}}$$ .