Analytic Bounded Point Evaluations for Spaces of Rational Functions

For a compact subset of the complex plane K and a regular Borel measure μ supported on K, Rp(K, μ) denotes the closure in Lp(μ) of thè rational functions with poles off K. This paper examines the existence of analytic bounded point evaluations on Rp(K, μ) and extends the work of Thomson on the existence of analytic bounded point evaluations for the closure in Lp(μ) of the analytic polynomials. Provided Rp(K, μ) is "pure," it is shown that the closure of the set of analytic bounded point evaluations for Rp(K, μ) equals the closure of the interior of the spectrum of the operator multiplication by z on Rp(K, μ). In fact this is derived as a consequence of Thomson′s result for polynomials. The bulk of the paper is devoted to certain interpolation problems for Rp(K, μ), thus producing some information on the structure of Rp(K, μ).