Lecture Details
Name: James Brennan (University of Kentucky)Speaking: Thu 9th May 12:30 - 12:55Title: The Structure of Certain Spaces Spanned by Rational FunctionsAbstract: Let $\mu$ be a positive compactly supported measure in the complex plane $\mathbb{C}$, and for each $p,1\leq p<\infty,$ let $H^p(\mu)$ be the closed subspace of $L^p(\mu)$ spanned by the polynomials. In 1991 Thomson gave a complete description of its structure, expressing $H^p(\mu)$ as the direct sum of invariant subspaces, all but one of which is irreducible in the sense that it contains no non-trivial characteristic function. Years later, Aleman, Richter and Sundberg gave a more detailed analysis of the invariant subspaces in any irreducible summand. Here we discuss the extent to which those earlier results can be extended to $R^p(\mu)$, the closed subspace of $L^p(\mu)$ spanned by the rational functions having no poles on the support of $\mu$. In particular we shall extend the results mentioned above to sets of infinite connectivity for which the diameters of their complementary components are bounded away from zero. An essential feature of our argument is that the Dirichlet problem is solvable in this context.