Completely J-positive linear systems of finite order

Gheondea, Aurelian and Ober, Raimund J. (1999) Completely J-positive linear systems of finite order. Mathematische Nachrichten, 203 (1), 75-101. (doi:10.1002/mana.1999.3212030105).

Record type: Article

Abstract

Completely J-positive linear systems of finite order are introduced as a generalization of completely symmetric linear systems. To any completely J-positive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral representation. It is proved that these systems are asymptotically stable. The observability and reachability operators obey a certain duality rule and the number of negative squares of the Hankel operator is estimated. The Hankel operator is bounded if and only if a certain measure associated with the defining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization with a completely J-symmetric linear space which is reachable, observable and parbalanced. Uniqueness and spectral minimality of the completely J-symmetric realizations are discussed.

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Published date: 1999

Keywords: Asymptotic stability, Completely J-positive linear system of finite order, Defining measure, Definitizable operator, Discrete time linear system, Kreǐn space, Realization theory, Sign symmetry, Spectral minimality