The Commutant of Multiplication by z on the Closure of Rational Functions in $L^t(\hat{I}¼)$

Abstract

For a compact set $K\subset \mathbb{C},$ a finite positive Borel measure $\hat{I}¼$ on $K,$ and $1\le t<\ddot{A}\pm ,$ let $\text{Rat}(K)$ be the set of rational functions with poles off $K$ and let $R^t(K,\hat{I}¼)$ be the closure of $\text{Rat}(K)$ in $L^t(\hat{I}¼).$ 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 $H^{\ddot{A}}\pm (\mathcal{D})$ be the weak-star closed sub-algebra of $L^ı(\area_{\mathcal D})$ 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 $R^t(K,\hat{I}¼)$ contains no non-trivial direct $L^t$ summands, then there exists a Borel subset $\mathcal{R}\subset K$ whose closure contains the support of $\hat{I}¼$ and there exists an isometric isomorphism and a weak-star homeomorphism $\ddot{I}$ from $R^t(K,\hat{I}¼)\cap L^{\infty }(\hat{I}¼)$ onto $H^{\infty }(\mathcal{R})$ such that $\ddot{I}(r)=r$ for all $r\in \text{Rat}(K).$ Consequently, we obtain some structural decomposition theorems for $\rtkmu$.

arXiv admin note: text overlap with arXiv:2212.10811