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

DANIEL CARANDO; DOMINGO GARCÍA; MANUEL MAESTRE

ADVANCES IN MATHEMATICS

Elsevier Science (Academic Press)

Lugar: San Diego; Año: 2005 vol. 197 p. 607 - 629

0001-8708

Let $U$ and $V$ be convex and balanced open subsets of theBanach spaces $X$ and $Y$ respectively. In this paper we study thefollowing question:  Given two Fr{´e}chet algebras of holomorphicfunctions of bounded type on  $U$ and $V$ respectively that arealgebra-isomorphic,  can we deduce that $X$ and $Y$ (or $X^ast$and $Y^ast$) are isomorphic? We prove that if $X^ast$ or$Y^ast$ has the approximation property and $H_{wu}(U)$ and$H_{wu}(V)$ are topologically algebra-isomorphic, then $X^ast$and $Y^ast$ 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 analgebra-isomorphism between $H_{b}(X)$ and $H_{b}(Y)$ isequivalent to an isomorphism between $X^*$ and $Y^*$. We alsoobtain characterizations of different algebra-homomorphisms ascomposition operators, study the structure of the spectrum of thealgebras under consideration and show the existence  ofhomomorphisms on $H_{b}(X)$ with pathological behaviors.