Boundary values in $R^t(K,μ)$-spaces and invariant subspaces

Abstract

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $μ$ supported on $K,$ $R^t(K, μ)$ denotes the closure in $L^t (μ)$ of rational functions with poles off $K.$ The paper examines the boundary values of functions in $R^t(K, μ)$ for certain compact subset $K$ and extends the work of Aleman, Richter, and Sundberg on nontangential limits for the closure in $L^t (μ)$ of analytic polynomials (Theorem A and Theorem C in \cite{ars}). We show that the Cauchy transform of an annihilating measure has some continuity properties in the sense of capacitary density. This allows us to extend Aleman, Richter, and Sundberg's results for $R^t(K, μ)$ and provide alternative short proofs of their theorems as special cases.