Xia proves in that a pure subnormal operator is completely determined by its selfcommutator , restricted to the closure of its range and the operator . In [9–11] he constructs a model for 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 . He finds all pure subnormals with rank . We will give a complete description of pairs of matrices that correspond to some for the case of the self-commutator of arbitrary finite rank. It is given in terms of a topological property of a certain algebraic curve, associated with and . We also give a new explicit formula for Xia's mosaic.
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Dmitry V. Yakubovich, Subnormal operators of Xia's model and real algebraic curves in . Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 95–115DOI 10.4171/RMI/236