Nontangential Limits in P (μ)-spaces and the Index of Invariant Subspaces

Let μ be a finite positive measure on the closed disk D in the complex plane, let 1 ≤ t <∞, and let P t(μ) denote the closure of the analytic polynomials in Lt(μ). We suppose that D is the set of analytic bounded point evaluations for P t(μ), and that P t(μ) contains no nontrivial characteristic functions. It is then known that the restriction of μ to ∂D must be of the form h|dz|. We prove that every function f ∈ P t(μ) has nontangential limits at h|dz|-almost every point of ∂D, and the resulting boundary function agrees with f as an element of Lt(h|dz|). Our proof combines methods from James E. Thomson’s proof of the existence of bounded point evaluations for P t(μ) whenever P t(μ) 6= Lt(μ) 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 ν in the complex plane we are able to describe locations of bounded point evaluations for P t(ν) 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 < ∞ dim M/zM = 1 for every nonzero invariant subspace M of P t(μ) if and only if h 6= 0. We also investigate the boundary behaviour of the functions in P t(μ) near the points z ∈ ∂D where h(z) = 0. In particular, for 1 < t < ∞ we show that there are interpolating sequences for P t(μ) that accumulate nontangentially almost everywhere on {z : h(z) = 0}. Work of the first author was supported by the Royal Swedish Academy, work of the second and third author was supported by the National Science Foundation, grants DMS-0070451 and DMS-0245384