Weighted Composition Operators from the Minimal Möbius Invariant Space into the Bloch Space

Abstract

Let \({{\varphi}}\) be an analytic self-map of the open unit disk \({{\mathbb{D}}}\) in the complex plane \({{\mathbb{C}, H(\mathbb{D})}}\) the space of complex-valued analytic functions on \({{\mathbb{D}}}\) , and let u be a fixed function in \({{H(\mathbb{D})}}\) . The weighted composition operator is defined by $$(uC_{\varphi}f)(z) = u(z)f({\varphi}(z)), \quad z \in \mathbb{D}, f \in H(\mathbb{D}).$$ In this paper, we study the boundedness and the compactness of the weighted composition operators from the minimal Mobius invariant space into the Bloch space and the little Bloch space.