On Banach spaces of normalized Bloch mappings

Abstract

Applying the theory of tensor products of Banach spaces, we study the Banach spaces of normalized Bloch maps from $\mathbb{D}$ (the complex unit open disc) into $X^*$ (the dual of a complex Banach space $X$) that can be represented canonically as the dual of the completion of the tensor product $\mathrm{lin}(\Gamma(\mathbb{D}))\otimes_\alpha X$, where $\mathrm{lin}(\Gamma(\mathbb{D}))$ is the space of $X$-valued Bloch molecules on $\mathbb{D}$ and $\alpha$ is a Bloch cross-norm on $\mathrm{lin}(\Gamma(\mathbb{D}))\otimes X$. We show that the normalized spaces of Bloch maps, $p$-summing Bloch maps and Bloch maps that factor through a Hilbert space admit such a representation. On the converse problem, we characterize when a Banach normalized Bloch space $B(\mathbb{D},X^*)$ is isometrically isomorphic to $(\mathrm{lin}(\Gamma(\mathbb{D}))\widehat{\otimes}_\alpha X)^*$ for some Bloch cross-norm $\alpha$, in terms of the compactness of its unit ball with respect to the weak* Bloch topology.