Analytic structures for subnormal operators (Q913210)

Let \(\mu\) be a compactly supported finite Borel measure on the complex plane \({\mathbb{C}}\). The paper is concerned with the problem of finding a measure \(\mu\), mutually absolutely continuous with respect to \(\mu\), so that \(P^ 2(\mu)\), the closure of the polynomials in \(L^ 2(\mu)\), has an analytic structure, which roughly means that the analytic extensions, via bounded point evaluation, of bounded functions in \(P^ 2(\mu)\) are exactly \(H^{\infty}(U)\), the bounded analytic functions on some open subset U of \({\mathbb{C}}\). Through the study of some structure theorems for hypo-Dirichlet algebras, the author is able to show that if \(P^{\infty}(\mu)\), the weak-star closure of the polynomials in \(L^{\infty}(\mu)\), has no \(L^{\infty}\)-summand, then such a measure \(\mu\) exists for which \(P^ 2(\mu)\) has an analytic structure on the interior of the set of weak-star continuous point evaluations for \(P^{\infty}(\mu)\). As a corollary, this result yields a more direct proof of a characterization, originally due to R. F. Olin and J. E. Thomson, of reductive normal operators: a normal operator is reductive if and only if no part of it has nonvanishing Fredholm index. The author also defines the notion of ultrareflexivitiy: T is an ultrareflexive operator if Alg T\(=Alg Lat T=\{T\}'\), where Alg T, Alg Lat T and \(\{T\}'\) denote the weak-operator closure of polynomials in T, the algebra of operators leaving invariant every invariant subspace of T and the commutant of T, respectively. It is shown that an ultrareflexive subnormal operator must be star-cyclic, but not necessarily cyclic. (Reviewer's remark: The conjecture on p. 267 that the minimal normal extension of an ultrareflexive subnormal operator is always cyclic has since been refuted by an example constructed by W. R. Wogen.)