The Spectrum of Analytic Mappings of Bounded Type

A Banach space E is said to be symmetrically.regular if every continuous symmetric.linear mapping from E to E9 is weakly compact. For a complex Banach space E and a complex Banach algebra F, let Hb E, F.denote the algebra 1 Supported in part by DGESIC Spain.pr. 96-0758. 2 Supported in part by agreement USPrUV. 3 Partially supported by DGESIC Spain.pr. 96-0758 and Grant PR 1997]0186. 4 Supported in part by CNPq, Brazil, Research Grant 300016r82-4. 447 0022-247Xr00 $35.00 Copyright Q 2000 by Academic Press All rights of reproduction in any form reserved. 448 GARC´ıA ET AL. of holomorphic mappings from E to F which are bounded on bounded sets. We endow Hb E, F.with the usual Fr´echet 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..