A Note on Spectral Mapping Theorems for Subnormal Operators

For a compact subset $K\subset \mathbb C$ and a positive finite Borel measure $μ$ supported on $K,$ let $\text{Rat}(K)$ denote the space of rational functions with poles off $K,$ let $R^\infty (K,μ)$ be the weak-star closure of $\text{Rat}(K)$ in $L^\infty (μ),$ and let $R^2 (K,μ)$ be the closure of $\text{Rat}(K)$ in $L^2(μ).$ We show that there exists a compact subset $K\subset \mathbb C,$ a positive finite Borel measure $μ$ supported on $K,$ and a function $f\in R^\infty (K,μ)$ such that $R^\infty (K,μ)$ has no non-trivial direct $L^\infty$ summands, $f$ is invertible in $R^2 (K,μ)\cap L^\infty(μ),$ and $f$ is not invertible in $R^\infty (K,μ).$ The result answers an open question concerning spectral mapping theorems for subnormal operators raised by J. Dudziak in 1984.