Homomorphisms between Algebras of Holomorphic Functions

Abstract

For two complex Banach spaces $X$ and $Y$, in this paper, we study the generalized spectrum ${ℳ}_b(X,Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_b(X)$, the algebra of all bounded type entire functions on $X$, into ${\mathscr{H}}_b(Y)$. We endow ${ℳ}_b(X,Y)$ with a structure of Riemann domain over $ℒ(X^{\mathrm{*}},Y^{\mathrm{*}})$ whenever $X$ is symmetrically regular. The size of the fibers is also studied. Following the philosophy of (Aron et al., 1991), this is a step to study the set ${ℳ}_{b,\mathrm{\infty }}(X,B_Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_b(X)$ into ${\mathscr{H}}_{\mathrm{\infty }}(B_Y)$ of bounded holomorphic functions on the open unit ball of $Y$ and ${ℳ}_{\mathrm{\infty }}(B_X,B_Y)$ of all nonzero algebra homomorphisms from ${\mathscr{H}}_{\mathrm{\infty }}(B_X)$ into ${\mathscr{H}}_{\mathrm{\infty }}(B_Y)$.