Multiplication Operators on Hilbert Spaces

Let S be a subnormal operator on a separable complex Hilbert space H and let μ be the scalar-valued spectral measure for the minimal normal extension N of $S.$ Let R^ (σ(S),μ) be the weak-star closure in L⁽μ) of rational functions with poles off σ(S), the spectrum of $S.$ The multiplier algebra $M(S)$ consists of functions f∈ L^∞(μ) such that f(N) H ⊂ H. The multiplication operator MS,f of f∈ M(S) is defined MS,f = f(N) |H. We show that there exists an {analytic} weak-star closed algebra Mₐ(S) satisfying R^ (σ(S),μ) ⊂ Mₐ(S) ⊂ M(S) such that if f∈ Mₐ(S), then (1) MS,f is invertible iff f is invertible in Mₐ(S) iff f is invertible in $M(S)$ and (2) MS,f is Fredholm iff there exists f₀∈ Mₐ(S) and a polynomial p such that f=pf₀, MS,f₀ is invertible, and p has only zeros in σ (S) σₑ (S), where σₑ (S) denotes the essential spectrum of $S.$ As applications, for f∈ Mₐ(S), we characterize σ(MS,f) and σₑ(MS,f) in terms of some cluster subsets of $f.$ Moreover, we show that if S is an irreducible subnormal operator and f ∈ R^ (σ(S),μ), then MS,f is invertible iff f is invertible in R^ (σ(S),μ). The results answer the spectral mapping theorem questions raised by J. Dudziak in 1984 for subnormal operators.