Subnormal operators and quadrature domains
A bounded linear operator \(S\) on a Hilbert space \({\mathcal H}\) is called subnormal if it has a normal extension. The self-commutator of \(S\) is \([S^*,S]= S^*S- SS^*\). The operator \(S\) is called cyclic if there exists a vector \(x\in {\mathcal H}\) such that \(\{p(S)x:p\) is a polynomial\} is dense in \({\mathcal H}\), and is called rationally cyclic if there exists \(x\) so that \(\{r(S)x:r\) is a rational function with poles off \(\sigma(S)\}\) is dense in \({\mathcal H}\) where \(\sigma(S)\) denotes the spectrum of \(S\). An operator is called irreducible if it commutes with no non-trivial projection. It is known, by Olin, Thomson and Trent, the necessary and sufficient condition that an irreducible cyclic subnormal operator \(S\) has a finite rank self-commutator. The main result in this paper is a generalization of this result in the case where \(S\) is rationally cyclic.
scientific article; zbMATH DE number 1013823
scientific article