Linearization of holomorphic Lipschitz functions

Abstract

Let $X$ and $Y$ be complex Banach spaces with $B_X$ denoting the open unit ball of $X$. This paper studies various aspects of the holomorphic Lipschitz space $H{L}_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets ${Lip}_0(B_X,Y)$ of Lipschitz mappings and $H^{\infty }(B_X,Y)$ of bounded holomorphic mappings, from $B_X$ to $Y$. Thanks to the Dixmier–Ng theorem, $H{L}_0(B_X,C)$ is indeed a dual space, whose predual $G_0\left(B_X\right)$ shares linearization properties with both the Lipschitz-free space and Dineen–Mujica predual of $H^{\infty }\left(B_X\right)$. We explore the similarities and differences between these spaces, and combine techniques to study the properties of the space of holomorphic Lipschitz functions. In particular, we get that $G_0\left(B_X\right)$ contains a 1-complemented subspace isometric to $X$ and that $G_0\left(X\right)$ has the (metric) approximation property whenever $X$ has it. We also analyze when $G_0\left(B_X\right)$ is a subspace of $G_0\left(B_Y\right)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.