Completely J-positive linear systems of finite order

Completely J-positive linear systems of finite order are introduced as a ge neralization of completely symmetric linear systems. To any completely J-po sitive linear system of finite order there is associated a defining measure with respect to which the transfer function has a certain integral represe ntation. It is proved that these systems are asymptotically stable. The obs ervability and reachability operators obey a certain duality rule and the n umber of negative squares of the Hankel operator is estimated. The Hankel o perator is bounded if and only if a certain measure associated with the def ining measure is of Carleson type. We prove that a real symmetric operator valued function which is analytic outside the unit disk has a realization w ith a completely J-symmetric linear space which is reachable, observable an d parbalanced. Uniqueness and spectral minimality of the completely J-symme tric realizations are discussed.