Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Abstract

Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}. (Less)