Subnormal operators of finite type. I: Xia's model and real algebraic curves in \(\mathbb{C}^2\) (Q1265420)

It is known that, for a subnormal operator \(S\) on \({\mathcal H}\) (i.e., \(S\) has a normal extension), the closure \([(S^*S- SS^*){\mathcal H}]^{\sim}\) of the range of \(S^*S- SS^*\) is invariant under \(S^*\). Xia proved that a pure subnormal operator \(S\) (i.e., \(S\) has no normal part) is completely determined by \(S^*S- SS^*\) and \((S^*|_{[(S^*S- SS^*){\mathcal H}]^{\sim}})^*\). By constructing the so-called mosaic which is a projection-valued function, analytic outside the spectrum of the minimal normal extension of \(S\), he found all pure subnormals \(S\) with \(\text{rank}(S^*S- SS^*)= 2\). In this paper, the author gives a new explicit formula for Xia's mosaic and a complete description of pairs of matrices \(S^*S- SS^*\) and \((S^*|_{[(S^*S- SS^*){\mathcal H}]^{\sim}})^*\) for the case where the rank of \(S^*S- SS^*\) is arbitrarily finite.