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 {em holomorphic Lipschitz space} $mathcal HL_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 $mathcal H^infty(B_X,Y) of bounded holomorphic mappings, from B_X to Y. Thanks to the Dixmier-Ng theorem, mathcal HL_0(B_X, mathbb C)$ is indeed a dual space, whose predual $mathcal G_0(B_X) shares linearization properties with both the Lipschitz-free space and Dineen-Mujica predual of $mathcal H^infty(B_X). 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 mathcal G_0(B_X) contains a 1-complemented subspace isometric to and that mathcal G_0(X) has the (metric) approximation property whenever X has it. We also analyze when mathcal G_0(B_X) is a subspace of mathcal G_0(B_Y), and we obtain an analogue of Godefroy´s characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.