Homomorphisms between Algebras of Holomorphic Functions

Abstract

For two complex Banach spacesXandY, in this paper, we study the generalized spectrumℳb(X,Y)of all nonzero algebra homomorphisms fromℋb(X), the algebra of all bounded type entire functions onX, intoℋb(Y). We endowℳb(X,Y)with a structure of Riemann domain overℒ(X*,Y*)wheneverXis 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 setℳb,∞(X,BY)of all nonzero algebra homomorphisms fromℋb(X)intoℋ∞(BY)of bounded holomorphic functions on the open unit ball ofYandℳ∞(BX,BY)of all nonzero algebra homomorphisms fromℋ∞(BX)intoℋ∞(BY).