A weak-star rational approximation problem connected with subnormal operators

Abstract:

Let $\mu$ be a positive Borel measure on a compact subset $K$ of the complex plane. Denote the weak-star closure in ${L^\infty }\left ( \mu \right )$ of $R\left ( K \right )$ by ${R^\infty }\left ( {K,\mu } \right )$. Given $f \in {R^\infty }\left ( {K,\mu } \right )$, denote the weak-star closure in ${L^\infty }\left ( \mu \right )$ of the algebra generated by ${R^\infty }\left ( {K,\mu } \right )$ and the complex conjugate of $f$ by ${A^\infty }\left ( {f,\mu } \right )$. This paper determines the structure of ${A^\infty }\left ( {f,\mu } \right )$. As a consequence, a solution is obtained to a problem concerned with minimal normal extensions of functions of a subnormal operator.