Jordan Model Research Articles

The Hyperinvariant Subspace Lattice of a Contraction of Class C. 0

It is shown that if T is a C.0 contraction with finite defect indices, then Hyperlat T is (lattice) generated by those subspaces which are either ker Ai(T) or ran i(T), where ip and J are scalar-valued inner functions. For a bounded linear operator T on a complex Hilbert space H, Hyperlat T denotes the lattice of all hyperinvariant subspaces for T, that is, the lattice of those subspaces which are invariant for all operators commuting with T. Recently, Fillmore, Herrero and Longstaff [1] showed that on a finite-dimensional space H, Hyperlat T is (lattice) generated by those subspaces which are either kerp(T) or ran q(T), where p and q are polynomials. In this note we generalize this to the following THEOREM. Let T be a contraction of class C.0 with finite defect indices acting on a separable Hilbert space. Then Hyperlat T is (lattice) generated by those subspaces which are either ker 4(T) or ran ~(T), where 4 and ( are scalarvalued inner functions. Recall that a contraction T (1 TIJ 6 1) is of class C.0 if T*nx -O0 for all x. The defect indices of T are, by definition, dT = rank(l-T*T)1/2 and dm, = rank(l TT*)1/2. If T is of class iC.0, then dT 6 dr,. For operators T, T' acting on H, H', respectively, T -< T' means that there exists a family of operators {Xa } from H to H' such that (i) for each a, Xax is one-to-one, (ii) Cl Ci VaXaH = H', and (iii) for each a, Xa T = T'X .If T < T' and T' < T, then T, T' are said to be completely injection-similar, and this is denoted by T c' T. For contractions of class C.0 and with dT = m < oo, d,. = n < oo, there has been developed a Jordan model which is, in a certain sense, analogous to the Jordan model for finite matrices. More specifically, if T is such a contraction then it is completely injection-similar to a uniquely determined Jordan operator of the form S((p1) D e * * D S(pk) ED Sn, Received by the editors July 19, 1977 and, in revised form, January 25, 1978. AMS (MOS) subject classifications (1970). Primary 47B99; Secondary 47A15.

The Nagy-Foias operator models, networks, and systems

We present in this paper some of the relationships and applications of the operator model theory of Sz-Nagy and Foias to system and network theory. The three basic objects of the Nagy-Foias theory are: a special Hilbert space called the Nagy-Foias space, an operator on this space called the compressed shift operator, and the characteristic operator function of a contraction operator. It will be shown that these objects fit naturally in the state space realization of linear systems, and in the scattering synthesis of linear passive networks. We conclude the paper with a brief discussion of the Jordan model for a class of operators, and some of its potential applications to systems and networks.

Quasi-similar models for nilpotent operators

Every nilpotent operator on a complex Hilbert space is shown to be quasi-similar to a canonical Jordan model. Further, the para-reflexive operators are characterized generalizing a result of Deddens and Fillmore.