A Note on Spectral Mapping Theorems for Subnormal Operators

Abstract

For a compact subset $K\subset \mathbb{C}$ and a positive finite Borel measure $\hat{I}¼$ supported on $K,$ let $\text{Rat}(K)$ denote the space of rational functions with poles off $K,$ let $R^{\infty }(K,\hat{I}¼)$ be the weak-star closure of $\text{Rat}(K)$ in $L^{\infty }(\hat{I}¼),$ and let $R^2(K,\hat{I}¼)$ be the closure of $\text{Rat}(K)$ in $L^2(\hat{I}¼).$ We show that there exists a compact subset $K\subset \mathbb{C},$ a positive finite Borel measure $\hat{I}¼$ supported on $K,$ and a function $f\in R^{\infty }(K,\hat{I}¼)$ such that $R^{\infty }(K,\hat{I}¼)$ has no non-trivial direct $L^{\infty }$ summands, $f$ is invertible in $R^2(K,\hat{I}¼)\cap L^{\infty }(\hat{I}¼),$ and $f$ is not invertible in $R^{\infty }(K,\hat{I}¼).$ The result answers an open question concerning spectral mapping theorems for subnormal operators raised by J. Dudziak in 1984.

arXiv admin note: text overlap with arXiv:2301.06305