Derivative and factorization of holomorphic functions

Abstract

Let E be a complex Banach space and denote by H b (E) the space of all holomorphic functions f:E→C of bounded type, that is, bounded on bounded sets. It is known that f∈H b (E) admits a factorization of the form f=g∘S , with S a compact linear operator and g a holomorphic function of bounded type if and only if the derivative df:E→E ∗ takes bounded sets of E into relatively compact sets of E ∗ . In the weakly compact case, a similar result was obtained by R. M. Aron and P. Galindo [Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 181–192;]. In the paper under review, the authors extend these results to closed injective operator ideals and the associated families of bounded sets. Moreover, this main result is applied to give examples of factorization through operators belonging to important closed injective operator ideals.