Interpolating sequences in the spectrum of $H^\infty $ I

Abstract

Listen

Translate article

We show that a sequence of trivial points in M(Hc?) is interpolating if and only if it is discrete. This answers a question of K. Izuchi. We also give a sufficient topological condition for a sequence of nontrivial points to be interpolating. Let H' be the uniform algebra of all bounded analytic functions in the open unit disk 1D3. Its spectrum, or maximal ideal space, is the space M(H?) of all nonzero multiplicative linear functionals on H' endowed with the weak-*-topology. As usual, we identify a function f in H' with its Gelfand transform, f, defined by f(m) = m(f) for m E M(H'). Moreover, we look upon DI as a subset of M(H?). A sequence (xn)nEN of points in M(H??) is said to be interpolating if for any bounded sequence (Wn)nEN of complex numbers there exists a function f E H' so that f(xn) = Wn for every n. Carleson's well known interpolation theorem describes the interpolating sequences in the unit disk (see the book [4] of Garnett for an extensive presentation of this beautiful theory). In fact, a sequence (an) in 11D is interpolating if and only if (C) iEnfII a > 6> 0. j:j:Ak 1 jak It is the aim of this paper to continue the study of the interpolating sequences in the whole spectrum of H'. For previous results in this direction we refer the reader to [1], [2], [5], [6], [7], [9], [10], [11], [14], [15]. To proceed, we need to present a few definitions. Let x and y be two points in M(H?). We define the pseudohyperbolic distance of x to y by p(x,y) = sup{If(x)I: f E H', ljfj ?o 1, f (y) = O}. Then for a, b E D we have p(a, b) = 1 b It is well known that the relation defined on M(H?) by x y X# p(x) y) < I defines an equivalence relation on M(H?). The equivalence class containing a point m E M(H?) is called the Gleason part of m and is denoted by P(m). If the Received by the editors March 6, 1998 and, in revised form, July 13, 1998. 1991 Mathematics Subject Classification. Primary 46J15. ?2000 American Mathematical Society