Homomorphisms between Algebras of Holomorphic Functions

For two complex Banach spaces X and Y, in this paper, we study the generalized spectrum ℳb(X,Y) of all nonzero algebra homomorphisms from ℋb(X), the algebra of all bounded type entire functions on X, into ℋb(Y). We endow ℳb(X,Y) with a structure of Riemann domain over ℒ(X*,Y*) 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 ℳb,∞(X,BY) of all nonzero algebra homomorphisms from ℋb(X) into ℋ∞(BY) of bounded holomorphic functions on the open unit ball of Y and ℳ∞(BX,BY) of all nonzero algebra homomorphisms from ℋ∞(BX) into ℋ∞(BY).