The Commutant of Multiplication by z on the Closure of Rational Functions in $L^t(\mu)$

Abstract

For a compact set $K\subset \mathbb C,$ a finite positive Borel measure $\mu$ on $K,$ and $1 \le t < \i,$ let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K, \mu)$ be the closure of $\text{Rat}(K)$ in $L^t(\mu).$ For a bounded Borel subset $\mathcal D\subset \mathbb C,$ let $\area_{\mathcal D}$ denote the area (Lebesgue) measure restricted to $\mathcal D$ and let $H^\i (\mathcal D)$ be the weak-star closed sub-algebra of $L^\i(\area_{\mathcal D})$ spanned by $f,$ bounded and analytic on $\mathbb C\setminus E_f$ for some compact subset $E_f \subset \mathbb C\setminus \mathcal D.$ We show that if $R^t(K, \mu)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\mathcal R \subset K$ whose closure contains the support of $\mu$ and there exists an isometric isomorphism and a weak-star homeomorphism $\rho$ from $R^t(K, \mu) \cap L^\infty(\mu)$ onto $H^\infty(\mathcal R)$ such that $\rho(r) = r$ for all $r\in\text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for $\rtkmu$.