On subnormal operators whose spectrum are multiply connected domains (Q554279)

Let \(\Omega\subset \mathbb{C}\) be a connected bounded domain with finitely many holes and boundary which consists of disjoint simple closed rectifiable nontrivial Jordan curves. Let \(\mu\) be a finite Borel-measure on \(\partial \Omega\) and let \(H\) be an \(M_z\) invariant subspace of \(L^2(\mu, \ell^2)\). Set \(T=M_z|_H\). The main result of the paper says that the multiplicity function of a minimal normal extension of \(T\) equals \(-\operatorname{ind}(T-\lambda)~\mu\)-a.e.\ if \(T\) is a pure subnormal operator such that \(\sigma(T)=\overline{\Omega}\). Here, \(\lambda \in \Omega\) is fixed. Moreover, it is shown that there exist \(f_1, \dots, f_n \in H\) such that \(H\) is spanned by functions \(rf_j\), where \(r\) runs through the set of rational functions with poles outside \(\sigma(T)\) and \(j=1, \dots, n\).