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, H bA (E). We prove that, under very natural conditions satisfied by many usual classes of polynomials, the spectrum M bA (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 bA (E) can be endowed with a structure of Riemann domain over E and that the extension of each f ∈ H bA (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.Keywords Polynomial ideals · Holomorphic functions · Riemann domains over banach spaces Mathematics Subject Classification (2000) 47H60 · 46G20 · 30H05 · 46M05 D.

Generalized Multiple Summing Multilinear Operators on Banach Spaces

“…The concept of coherence and compatibility was introduced in the literature initially by D. Pellegrino and G. Botelho in [4,8] (with a different terminology) and also was studied by D. Carando, V. Dimant and S. Muro in [13,14,15]. The terms "coherence and compatibility" were coined in [13].…”

A Lindenstrauss theorem for some classes of multilinear mappings

“…For example, the ideals of nuclear, integral, extendible, multiple psumming multilinear forms (1 ≤ p < ∞) are multiplicative (see [18,20,35] for the proof in the polynomial case, the multilinear one being analogous). Then, the Lindenstrauss theorem holds for all these ideals.…”

Coherence and compatibility: a stronger approach

“…The study of coherence and compatibility was motivated by the concepts of ideals of polynomials closed under differentiation and closed for scalar multiplication that were introduced in [4] (see also [3]) as an attempt to identify a set of properties that polynomial ideals are expected to have in order to maintain some harmony between the different levels of homogeneity. In that spirit, Carando, Dimant and Muro (related papers can be seen in [13,14]) introduced a notion of coherent and compatible polynomial ideals with the same aim of filtering good polynomial extensions of given operator ideals. One question that arises at this point is: Should we develop the study of the coherence and compatibility of multilinear mappings and homogenous polynomials independently or should we compare these classes in any sense?…”