The spectrum of analytic mappings of bounded type (Q1570209)

For a complex Banach space \(E\) and a complex Banach algebra \(F\), let \({\mathcal H}_b(E,F)\) denote the algebra of all the holomorphic mappings from \(E\) to \(F\) which are bounded on bounded sets, endowed with the Fréchet topology of uniform convergence on the bounded sets. The set \({\mathcal M}({\mathcal H}_b(E, F),F)\) of all nonzero continuous algebra homomorphisms from \({\mathcal H}_b(E,F)\) into \(F\) is called the generalized spectrum of \({\mathcal H}_b(E,F)\), and the article under review is devoted to the study of this object. For \(E\) and \(F\) as above, let \(G_{EF}= {\mathcal L}({\mathcal L}(E, F),F)\) with the usual operator norm. This space was presented by \textit{I. Zalduendo} [Trans. Am. Math. Soc. 320, No. 2, 747-763 (1990; Zbl 0709.46027)] as the canonical candidate to extend holomorphically the elements of \({\mathcal H}_b(E,F)\). The authors first establish a relation between the Zalduendo extension and `Nicodemi sequences', cf. \textit{P. Galindo}, \textit{D. García}, \textit{M. Maestre} and \textit{J. Mujica} [Stud. Math. 108, No. 1, 55-76 (1994; Zbl 0852.46004)], and select some elements of \(G_{EF}\) which allow to connect the Zalduendo extension with the Aron-Berner extension and the Arens product. Thus, the Zalduendo extension is described as an iterated limit on a subset of \(G_{EF}\). Next, the authors start the study of the generalized spectrum \({\mathcal M}({\mathcal H}_b(E, F),F)\) by presenting a set on which the Zalduendo extension is multiplicative. Examples show that, in general, the sets \(G_{EF}\) and \({\mathcal M}({\mathcal H}_b(E,F), F)\) do not coincide. The main result, however, is the following: Let \(E\) be a symmetrically regular complex Banach space; i.e., let each continuous symmetric linear mapping from \(E\) to \(E'\) be weakly compact. Let \(F\) be a complex Banach algebra so that each continuous linear mapping from \(E\) to \(F\) is weakly compact. Then \({\mathcal M}({\mathcal H}_b(E, F),F)\) has an analytic structure as a manifold over \(E''\). If \(F\) is, in addition, commutative and not isometrically isomorphic to the complex field, then \({\mathcal M}({\mathcal H}_b(E,F), F)\) is not connected. -- This main result is a vector valued generalization of one due to \textit{R. Aron}, \textit{P. Galindo}, \textit{D. García} and \textit{M. Maestre} [Trans. Am. Math. Soc. 348, No. 2, 543-559 (1996; Zbl 0844.46024)].