Math. Nachr. 236 (2002), 109 – 118 Compact and Weakly Compact Homomorphisms on Fr´ echet Algebras of Holomorphic Functions By Pablo Galindo *) of Valencia, Lilian Lourenc ¸o of S˜ao Paulo and Luiza A. Moraes of Rio de Janeiro (Received April 7, 2000; revised version September 29, 2000; accepted October 26, 2000) Abstract. We study homomorphisms between Fr´ echet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize, up to the approximation property of the underlying Banach space, the weakly compact composition operators on H b (V ),V an absolutely convex open set. 1. Introduction The aim of this paper is to discuss the compact and weakly compact homomor- phisms between Fr´ echet algebras of holomorphic functions of bounded type defined on a complex Banach space. We characterize in terms of the defining map, up to the approximation property of the underlying Banach space, the weak compactness of the composition operators between these algebras and find an example of a weakly compact noncompact composition operator. We show that any pointwise bounded homomorphism between a Fr´ echet algebra and H b (V ) actually ranges continuously in H ∞ (V ) and, in particular, is rank one when V is the whole space. Furthermore for V the open unit ball, whenever a (weakly) compact composition operator is defined by a uniformly continuous mapping, it is also (weakly) compact as operator ranging in H ∞ (V ); we exhibit an example showing that this is no true without the uniform continuity assumption. In some way this article may be considered as a sequel to work by R. Aron, M. Lindstr¨ om, R. Ryan and the first named author in [3], [9] and [10] by whose techniques it is strongly inspired and where the topic is treated for the Banach algebras of bounded holomorphic functions in the ball or their subalge- bras of uniformly continuous functions. The characterization was earlier treated by 2000 Mathematics Subject Classification. Primary: 46E25; Secondary: 46G20. Keywords and phrases. Holomorphic functions of bounded type, composition operator, weakly compact homomorphism. *) Correspondingauthor/Pablo.Galindo@uv.es. c WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0025-584X/02/23603-0109 $ 17.50+.50/0