Resumen de Subnormal operators of finite type I.: Xia's model and real algebraic curves in C2.
Xia proves in [9] that a pure subnormal operator S is completely determined by its self-commutator C = S*S - SS*, restricted to the closure M of its range and the operator ? = (S*|M)*. In [9], [10], [11] he constructs a model for S that involves this 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,?) 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 ?. We also give a new explicit formula for Xia's mosaic.
