Spectra of Weighted Banach Algebras of Holomorphic Functions

We consider weighted algebras of holomorphic functions on a Banach space. We determine conditions on a family of weights that assure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study the spectra of weighted algebras and endow them with an analytic structure. We also deal with composition operators and algebra homomorphisms, in particular to investigate how their induced mappings act on the analytic structure of the spectrum. Moreover, a Banach-Stone type question is addressed. Introduction This work deals with weighted spaces of holomorphic functions on a Banach space. If X is a (finite or infinite dimensional) complex Banach space and U ⊆ X open and balanced, by a weight we understand any continuous, bounded function v : U −→ [0,∞[. Weighted spaces of holomorphic functions defined by countable families of weights were deeply studied by Bierstedt, Bonet and Galbis in [5] for open subsets of C (see also [6],[9], [10],[11],[13]). Garćıa, Maestre and Rueda defined and studied in [21] analogous spaces of functions defined on Banach spaces. We recall the definition of the weighted space HV (U) = {f : U → C holomorphic : ‖f‖v = sup x∈U v(x)|f(x)| < ∞ all v ∈ V }. We endow HV (U) with the Fréchet topology τV defined by the seminorms (‖ ‖v)v∈V . Since the family V is countable, we can (and will throughout the article) assume it to be increasing. 2000 Mathematics Subject Classification. 46G25, 46A45.