Homomorphisms and composition operators on algebras of analytic functions of bounded type

Let U and V be convex and balanced open subsets of the Banach spaces X and Y, respectively. In this paper we study the following question: given two Frechet algebras of holomorphic functions of bounded type on U and V, respectively, that are algebra isomorphic, can we deduce that X and Y (or X* and Y*) are isornorphic? We prove that if X* or Y* has the approximation property and H-wu(U) and H-wu(V) are topologically algebra isomorphic, then X* and Y* are isomorphic (the converse being true when U and V are the whole space). We get analogous results for H-b(U) and H-b(V), giving conditions under which an algebra isomorphism between H-b(X) and H-b(Y) is equivalent to an isomorphism between X* and Y*. We also obtain characterizations of different algebra homomorphisms as composition operators, study the structure of the spectrum of the algebras under consideration and show the existence of homomorphisms on H-b(X) with pathological behaviors. (c) 2004 Elsevier Inc. All rights reserved.