The commutant of rationally cyclic subnormal operators and rational approximation (Q1378044)

Let \(\mu\) be a measure supported on a compact subset \(K\) of the complex plane and let \(R^q(K,\mu)\) be the closure in \(L^q(\mu)\) of the rational functions with poles off \(K\). If \(\Omega\) denotes the set of analytic point evaluations for \(R^q(K,\mu)\) then by representing the point evaluations by \(L^p\)-functions to each \(f\in R^q(K,\mu)\) there is assigned a function \(\widehat{f}\) analytic on \(\Omega\). The map \(e:R^q(K,\mu)\cap L^\infty(\mu)\to H^\infty(\mu)\) is defined by \(e(f):= \widehat{f}\). If \(e\) is surjective and \(\Omega\) is finitely connected it is shown that \(\mu|\partial \Omega\) is absolutely continuous with respect to harmonic measure of \(\Omega\) -- which fails when \(\Omega\) is infinitely connected -- and each function in \(H^\infty(\Omega)\) has nontangential limits on \(\partial \Omega\) a.e. Necessary and sufficient conditions for \(e\) to be surjective are given.