Spectra of weighted (LB)-algebras of entire functions on Banach spaces

Given a decreasing family of continuous weights on a Banach space X, we consider the weighted inductive limits of spaces of entire functions V H ( X ) and V H 0 ( X ) . Motivated by recent research by D. Carando and P. Sevilla-Peris on weighted Fréchet algebras of entire functions on Banach spaces, we determine conditions on the family of weights to ensure that the corresponding weighted space is an algebra or has polynomial Schauder decompositions. We study Hörmander algebras of entire functions defined on a Banach space and we give a description of them in terms of sequence spaces. We also focus on algebra homomorphisms between these spaces and obtain a Banach–Stone type theorem for a particular decreasing family of weights. Finally, we study the spectra of these weighted algebras, endowing them with an analytic structure.