Sequences in the Maximal Ideal Space of 77°° Sheldon Axler and Pamela Gorkin

This paper studies the behavior of sequences in the maximal ideal space of the algebra of bounded analytic functions on an arbitrary domain. The main result states that for any such sequence, either the sequence has an interpolating subsequence or infinitely many elements of the sequence lie in the same Gleason part. Introduction Fix a positive integer N and fix a nonempty open subset Q. of C . The Banach algebra of bounded analytic functions on Q is denoted by 77°°(Í2) ; its maximal ideal space is denoted by M(H°°(Çl)). This paper studies the behavior of sequences in M(H°°(Q)). The maximal ideal space M(H°°(Çl)) consists of the multiplicative linear functionals from 77°°(Q) onto the complex plane C. For <p, t g M(H°°(Çl)), the pseudohyperbolic distance between <p and t , denoted dn(tp, t) , is defined by dQ{tp, r) = sup{|t(/)| : / G 77°°(n), H/ll^ < 1, and <p(f) = 0} ; here WfW^ is the usual supremum norm defined by ||/||oo = sup{|/(z)|:zGn}. As is well known, da is a metric on M(H°°(Çl)) with the property that any two open balls of radius 1 are either equal or disjoint (see, for example, Sections 1 and 2 of [1]). The open balls of radius 1 are called the Gleason parts of M(H°°(Çl)). Specifically, if tp G M(H°°(Q)), then the Gleason part of <p , denoted G(<p), is defined by G(<p) = {t G M(H°°(Q.)): dn(<p,T) < 1}. A sequence (<pn)TM=x c M(H°°(Q)) is called an interpolating sequence if for every bounded sequence of complex numbers (A„)JJ1, , there exists / G 77°°(Q) such that <P„(f) = ^n for every « G Z+; here Z+ denotes the set of positive integers. Received by the editors March 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J15; Secondary 30H05.