On norm attaining Bloch maps

Abstract

Given a complex Banach space X, let $\hat{B}(D,X)$ denote the space of all normalized Bloch maps from the open complex unit disc $D$ into X. We prove that the set of all maps in $\hat{B}(D,X)$ which attain their Bloch norms is norm dense in $\hat{B}(D,X)$. Our approach is based on a previous study of the extremal structure of the unit closed ball of $G\left(D\right)$ (the Bloch-free Banach space over $D$). We prove that normalized Bloch atoms of $D$ are precisely the only extreme points of that ball and, in fact, they are strongly exposed points. Moreover, we characterize the surjective linear isometries of $G\left(D\right)$ involved the Möbius transformations of $D$.