Some theorems on bounded holomorphic functions

E. L. Stout
1964 Bulletin of the American Mathematical Society  
The purpose of this note is the announcement of some results in the theory of bounded holomorphic functions on finite open Riemann surfaces. The proofs are too long to be included ; they will be published elsewhere. Special cases of some of the results of this note are to be found in [4] . We consider a fixed finite open Riemann surface R. We thus assume that R is contained as an open set in some compact Riemann surface Ro and that RQ\R consists of finitely many topological bordered surfaces.
more » ... ordered surfaces. We assume further that dR, the boundary of R with respect to Ro, consists of the m analytic simple closed curves Ti, • • • , T m . Denote by H^j^R] the algebra of all functions holomorphic and bounded on R. Given the norm \\'\\u defined by ||/IU«sup{|/(f)|:r€*}, £foo[i£] is a Banach algebra. We shall denote by SDÎ the maximal ideal space of Jïoo[i?]î 5DÎ will be regarded as the space of all nonzero complex homomorphisms together with the weak* topology. There is a natural embedding of R in SDÎ given by f-^ where 0. Then there exist g u • • • , g n eH" As is known [3, p. 163], this theorem is equivalent to the assertion that R is dense in 2ft in the sense that the set of all homomorphisms of form <fo is dense in 9K. It is this latter fact that our proof establishes. We first obtain the result for the case that R is an annulus. The proof then proceeds along the general lines of the proof of Theorem B of [4] . A different proof of Theorem 1 has been given in [l]. Making use of Theorem 1 in the form of the density of R in 5DÎ and of certain of the constructions in the proof, we are able to obtain some
doi:10.1090/s0002-9904-1964-11124-1 fatcat:btmwzd6dsrfe5pi4u3xbxq3jy4