The Structure of the Closure of the Rational Functions in $L^{q}$($\mu$)$

TL;DR

This paper investigates the structure of the closure of rational functions in L^q spaces over compact sets in the complex plane, providing conditions for a Thomson-type theorem and addressing a major open problem in subnormal operator theory.

Contribution

It establishes necessary and sufficient conditions on the compact set K for a Thomson-type structure theorem for A^q(K,μ), advancing the understanding of rational function closures in L^q spaces.

Findings

Provides a characterization of the structure of A^q(K,μ)

Solves a major open problem in subnormal operator theory

Establishes conditions for a Thomson-type structure theorem

Abstract

Let $K$ be a compact subset in the complex plane and let $A(K)$ be the uniform closure of the functions continuous on $K$ and analytic on $K^{\circ}$. Let $\mu$ be a positive finite measure with its support contained in $K$. For $1 \leq q < \infty$, let $A^{q}(K,\mu)$ denote the closure of $A(K)$ in $L^{q}(\mu)$. The aim of this work is to study the structure of the space $A^{q}(K,\mu)$. We seek a necessary and sufficient condition on $K$ so that a Thomson-type structure theorem for $A^{q}(K,\mu)$ can be established. Our results essentially give perfect solutions to the major open problem in the research filed of theory of subnormal operators and aproximation by analytic functions in the mean .