Linearization of holomorphic Lipschitz functions

Let X$X$ and Y$Y$ be complex Banach spaces with BX$B_X$ denoting the open unit ball of X$X$. This paper studies various aspects of the holomorphic Lipschitz space HL0(BX,Y)$\mathcal {H}L_0(B_X,Y)$, endowed with the Lipschitz norm. This space consists of the functions in the intersection of the sets Lip0(BX,Y)$\operatorname{Lip}_0(B_X,Y)$ of Lipschitz mappings and H∞(BX,Y)$\mathcal {H}^\infty (B_X,Y)$ of bounded holomorphic mappings, from BX$B_X$ to Y$Y$. Thanks to the Dixmier–Ng theorem, HL0(BX,C)$\mathcal {H}L_0(B_X, \mathbb {C})$ is indeed a dual space, whose predual G0(BX)$\mathcal {G}_0(B_X)$ shares linearization properties with both the Lipschitz‐free space and Dineen–Mujica predual of H∞(BX)$\mathcal {H}^\infty (B_X)$. 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 G0(BX)$\mathcal {G}_0(B_X)$ contains a 1‐complemented subspace isometric to X$X$ and that G0(X)$\mathcal {G}_0(X)$ has the (metric) approximation property whenever X$X$ has it. We also analyze when G0(BX)$\mathcal {G}_0(B_X)$ is a subspace of G0(BY)$\mathcal {G}_0(B_Y)$, and we obtain an analog of Godefroy's characterization of functionals with a unique norm preserving extension in the holomorphic Lipschitz context.