The local spectral behavior of completely subnormal operators

Abstract:

For any compact set $X$, let $C(X)$ denote the continuous functions on $X$ and $R(X)$ the functions on $X$ which are uniformly approximable by rational functions with poles off $X$. Let $A$ denote a subnormal operator having no reducing space on which it is normal. It is shown that a necessary and sufficient condition that $X$ be the spectrum of such an operator $A$ is that $R(X \cap \overline D ) \ne C(X \cap \overline D )$ whenever $D$ is an open disk intersecting $X$ in a nonempty set.