Approximation in the mean by rational functions II

For $1\le t<\mathrm{\infty }$, a compact subset $K\subset \mathbb{C}$, and a finite positive measure $\mu$ supported on $K$, $R^t(K,\mu )$ denotes the closure in $L^t(\mu )$ of rational functions with poles off $K$. Conway and Yang (2019) introduced the concept of non-removable boundary $\mathcal{F}$ and removable set $\mathcal{R}=K\setminus \mathcal{F}$ for $R^t(K,\mu )$. We continue the previous work and obtain structural results for $R^t(K,\mu )$. Assume that $S_{\mu }$, the multiplication by $z$ on $R^t(K,\mu )$, is pure ($R^t(K,\mu )$ does not have $L^t$ summand). Let $H^\infty_{\mathcal R}(A_{\mathcal R})$betheweak$^*$closurein$L^\infty (A_{\mathcal R})$ of the functions that are bounded analytic off compact subsets of $\mathcal{F}$, where $A_{\mathcal{R}}$ denotes the area measure restricted to $\mathcal{R}$. $\mathcal{R}$ is $\gamma$-connected ($\gamma$ denotes analytic capacity) if for any two disjoint open set $G_1$ and $G_2$ with $\mathcal R \subset G_1 \cup G_2 ~\gamma-a.a.$,then$\mathcal R \subset G_1 ~\gamma-a.a.$or$\mathcal R \subset G_2 ~\gamma-a.a.$. We prove: (1) $R^t(K,\mu )$ contains no non-trivial characterization functions if and only if the removable set $\mathcal{R}$ is $\gamma$-connected. (2) There is an isometric isomorphism and a weak${}^{\ast }$ homeomorphism from $R^t(K,\mu )\cap L^{\mathrm{\infty }}(\mu )$ onto $H_{\mathcal{R}}^{\mathrm{\infty }}(A_{\mathcal{R}})$.