Subnormal operators of Xia's model and real algebraic curves in ${\mathbb{C}}^2$

Abstract

Xia proves in [9] that a pure subnormal operator $S$ is completely determined by its self-commutator $C=S^{\ast }S–S{S}^{\ast }$, restricted to the closure $M$ of its range and the operator $\Lambda =(S^{\ast }|M{)}^{\ast }$. In [9–11] he constructs a model for $S$ that involves these two operators and the so-called mosaic which is a projection-valued function, analytic outside the spectrum of the minimal normal extension of $S$. He finds all pure subnormals $S$ with rank $C=2$. We will give a complete description of pairs of matrices $(C,\Lambda )$ that correspond to some $S$ for the case of the self-commutator $C$ of arbitrary finite rank. It is given in terms of a topological property of a certain algebraic curve, associated with $C$ and $\Lambda$. We also give a new explicit formula for Xia's mosaic.