Representing measures for R( X) and their Green's functions

Abstract

Let X be a compact subset of the complex plane with a nonempty interior, R( X) the uniform closure in C( X) of the rational functions with poles off X, and m a representing measure on ∂X for the functional on R( X) of evaluation at a point a in int X. Let N 2 be the space of functions f in L 2( m) satisfying ∝ f dm = ∝ f K dm = 0 for all h in R( X), and let T be the operator on N 2 of multiplication by z followed by projection onto N 2. The spectral properties of T are investigated and shown to depend in part on the behavior of the so-called Green's function of m. In case m is the harmonic measure on ∂X for a the latter function is the classical Green's function for int X with singularity at a. Special attention is paid to the case where X is the closure of a finitely connected Jordan domain whose boundary curves are analytic. In that context, new proofs are given of Beurling's invariant subspace theorem and of Forelli's theorem on extreme points in the unit ball of the Hardy space H 1( m).