Multiplication Operators on Hilbert Spaces

Let $S$ be a subnormal operator on a separable complex Hilbert space $\mathcal H$ and let $\mu$ be the scalar-valued spectral measure for the minimal normal extension $N$ of $S.$ Let $R^\infty (\sigma(S),\mu)$ be the weak-star closure in $L^\infty (\mu)$ of rational functions with poles off $\sigma(S),$ the spectrum of $S.$ The multiplier algebra $M(S)$ consists of functions $f\in L^\infty(\mu)$ such that $f(N)\mathcal H \subset \mathcal H.$ The multiplication operator $M_{S,f}$ of $f\in M(S)$ is defined $M_{S,f} = f(N) |_{\mathcal H}.$ We show that for $f\in R^\infty (\sigma(S),\mu),$ (1) $M_{S,f}$ is invertible iff $f$ is invertible in $M(S)$ and (2) $M_{S,f}$ is Fredholm iff there exists $f_0\in R^\infty (\sigma(S),\mu)$ and a polynomial $p$ such that $f=pf_0,$ $f_0$ is invertible in $M(S),$ and $p$ has only zeros in $\sigma (S) \setminus \sigma_e (S),$ where $\sigma_e (S)$ denotes the essential spectrum of $S.$ Consequently, we characterize $\sigma(M_{S,f})$ and $\sigma_e(M_{S,f})$ in terms of some cluster subsets of $f.$ Moreover, we show that if $S$ is an irreducible subnormal operator and $f \in R^\infty (\sigma(S),\mu),$ then $M_{S,f}$ is invertible iff $f$ is invertible in $R^\infty (\sigma(S),\mu).$ The results answer the second open question raised by J. Dudziak in 1984.