Holomorphic functions and polynomial ideals on Banach spaces

Abstract

Given A a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, HbA(E). We prove that, under very natural conditions satisfied by many usual classes of polynomials, the spectrum MbA(E) of this algebra “behaves” like the classical case of Mb(E) (the spectrum of Hb(E), the algebra of bounded type holomorphic functions). More precisely, we prove that MbA(E) can be endowed with a structure of Riemann domain over E and that the extension of each f ∈ HbA(E) to the spectrum is an A-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.