The Spectrum of Analytic Mappings of Bounded Type

AbstractA Banach space E is said to be (symmetrically) regular if every continuous (symmetric) linear mapping from E to E′ is weakly compact. For a complex Banach space E and a complex Banach algebra F, let Hb(E,F) denote the algebra of holomorphic mappings from E to F which are bounded on bounded sets. We endow Hb(E,F) with the usual Fréchet topology. M(Hb(E,F),F) denotes the set of all non-null continuous homomorphisms from Hb(E,F) to F. A subset of GEF on which the extension of Zalduendo is multiplicative is presented and it is shown that, in general, the sets GEF and M(Hb(E,F),F) do not coincide. We prove that if E is symmetrically regular and every continuous linear mapping from E to F is weakly compact then there exists an analytic structure on M(Hb(E,F),F)