JournalsrmiVol. 14, No. 1pp. 95–115

Subnormal operators of Xia's model and real algebraic curves in C2\mathbb C^2

  • Dmitry V. Yakubovich

    Universidad Autónoma de Madrid, Spain
Subnormal operators of Xia's model and real algebraic curves in $\mathbb C^2$ cover
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Abstract

Xia proves in  that a pure subnormal operator SS is completely determined by its selfcommutator C=SSSSC = S*S – SS*, restricted to the closure MM of its range and the operator Λ=(SM)\Lambda = (S*|M)*. In [9–11] he constructs a model for SS 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 SS. He finds all pure subnormals SS with rank C=2C=2. We will give a complete description of pairs of matrices (C,Λ)(C, \Lambda) that correspond to some SS for the case of the self-commutator CC of arbitrary finite rank . It is given in terms of a topological property of a certain algebraic curve, associated with CC and Λ\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 C2\mathbb C^2. Rev. Mat. Iberoam. 14 (1998), no. 1, pp. 95–115

DOI 10.4171/RMI/236