Convergence of monomial expansions in banach spaces

Abstract

[EN] If E is a Banach sequence space, then each holomorphic function defines a formal power series ¿ ¿ c ¿(f) z ¿. The problem of when such an expansion converges absolutely and actually represents the function goes back to the very beginning of the theory of holomorphic functions on infinite-dimensional spaces. Several very deep results have been given for scalar-valued functions by Ryan, Lempert and Defant, Maestre and Prengel. We go on with this study, looking at monomial expansions of vector-valued holomorphic functions on Banach spaces. Some situations are very different from the scalar-valued case. © 2011 Published by Oxford University Press. All rights reserved.