Boundary values in $R^t(K,\hat{I}¼)$-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 $\hat{I}¼$ supported on $K,$ $R^t(K,\hat{I}¼)$ denotes the closure in $L^t(\hat{I}¼)$ of rational functions with poles off $K.$ The paper examines the boundary values of functions in $R^t(K,\hat{I}¼)$ for certain compact subset $K$ and extends the work of Aleman, Richter, and Sundberg on nontangential limits for the closure in $L^t(\hat{I}¼)$ 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 $R^t(K,\hat{I}¼)$ and provide alternative short proofs of their theorems as special cases.