Holomorphic Functions and polynomial ideals on Banach spaces

Abstract

Given \u a multiplicative sequence of polynomial ideals, we consider the associated algebra of holomorphic functions of bounded type, H_{b\u}(E). We prove that, under very natural conditions verified by many usual classes of polynomials, the spectrum M_{b\u}(E) of this algebra "behaves" like the classical case of $M_{b}(E)$ (the spectrum of $H_b(E)$, the algebra of bounded type holomorphic functions). More precisely, we prove that M_{b\u}(E) can be endowed with a structure of Riemann domain over $E"$ and that the extension of each f\in H_{b\u}(E) to the spectrum is an \u-holomorphic function of bounded type in each connected component. We also prove a Banach-Stone type theorem for these algebras.Comment: 19 page