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

### Dmitry V. Yakubovich

Universidad Autónoma de Madrid, Spain

## Abstract

Xia proves in that a pure subnormal operator $S$ is completely determined by its selfcommutator $C = S*S – SS*$, restricted to the closure $M$ of its range and the operator $\Lambda = (S*|M)*$. 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.

## Cite this article

Dmitry V. Yakubovich, Subnormal operators of Xia's model and real algebraic curves in $\mathbb C^2$. Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 95–115

DOI 10.4171/RMI/236