The commutant of mulitplication by \(z\) on the closure of polynomials in \(L^ t(\mu)\)
Let \(\mu\) be a positive finite Borel measure having compact support in the complex plane \(\mathbb{C}\), \(1\leq t<\infty\), \(P^t(\mu)\) the closure of polynomials in \(L^t(\mu)\). A point \(\lambda\in\mathbb{C}\) is called a bounded point evaluation for \(P^t(\mu)\), if there is a positive constant \(M\) such that \(|p(\lambda)|\leq M|p|_t\), for every polynomial \(p\). Let \(G\) be the set of all bounded point evaluations for \(P^t(\mu)\). From a theorem of J. Thomson, there exists an isometric isomorphism \(T:H^\infty(G)\to P^t(\mu)\cap L^\infty(\mu)\), \(T(f)= f\sim\), \(\forall f\in H^\infty(G)\). Let \(\varphi: G\to D\) be the Riemann map, where \(D\) denotes the open unit disk; and \(\psi=\varphi^{-1}\), \(\nu=\mu\circ\varphi\sim^{-1}\). Then \(\nu\) is a finite measure on \(\overline D\). The authors prove mainly that: The measure \(\mu|_{\partial G}\) is absolutely continuous to the harmonic measure \(\omega\) on \(G\), and \(\psi\sim\) is one-to-one a.e. \(\nu\) from a carrier of \(\nu|_{\partial D}\) to a carrier of \(\mu|_{\partial G}\). There exists a Borel set \(E\) of \(\partial D\) such that: \(\nu(\partial D\backslash E)= 0\), \(\psi\sim|_E\) is one-to-one, \(\psi\sim(E)\subseteq\partial G\), \(\mu(\partial G\backslash\psi\sim(E))= 0\). This result sharpens some theorems of J. Thomson and D. Sarason.
scientific article; zbMATH DE number 850623