2023/04/14 by Aron, Richard, Dimant, Verónica, García-Lirola, Luis C. +1 · 3 citations
#46B28 (Secondary) #46E15 #46E50 (Primary) 46B20 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2304.07149
Let X and Y be complex Banach spaces with BX denoting the open unit ball of X. This paper studies various aspects of the \em holomorphic Lipschitz space \mathcal HL0(BX,Y), endowed with the Lipschitz norm. This space is the intersection of the spaces, Lip0(BX,Y) of Lipschitz mappings and \mathcal H^∞(BX,Y) of bounded holomorphic mappings, from BX to Y. Thanks to the Dixmier-Ng theorem, \mathcal HL0(BX, \mathbb C) is indeed a dual space, whose predual \mathcal G0(BX) shares linearization properties with both the Lipschitz-free space and Dineen-Mujica predual of \mathcal H^∞(BX). 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 G0(BX) contains a 1-complemented subspace isometric to X and that \mathcal G0(X) has the (metric) approximation property whenever X has it. We also analyze when \mathcal G0(BX) is a subspace of \mathcal G0(BY), and we obtain an analogous to Godefroy's characterization of functionals with a unique norm preserving extension to the holomorphic Lipschitz context.