Matrix-J-unitary non-commutative rational formal power series
| dc.creator | Alpay, D. | |
| dc.creator | Kalyuzhnyi-Verbovetzkii, D. S. | |
| dc.date | 2004-07-22 | |
| dc.date | 2005-04-05 | |
| dc.date.accessioned | 2026-07-07T05:10:35Z | |
| dc.date.available | 2026-07-07T05:10:35Z | |
| dc.description | In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are $J$-unitary on the imaginary line or on the unit circle is extended to the setting of non-commutative rational formal power series. The property of $J$-unitarity holds on $N$-tuples of $n\times n$ skew-Hermitian versus unitary matrices ($n=1,2,...$), and a rational formal power series is called \emph{matrix-$J$-unitary} in this case. The close relationship between minimal realizations and structured Hermitian solutions $H$ of the Lyapunov or Stein equations is established. The results are specialized for the case of \emph{matrix-$J$-inner} rational formal power series. In this case $H>0$, however the proof of that is more elaborated than in the one-variable case and involves a new technique. For the rational \emph{matrix-inner} case, i.e., when $J=I$, the theorem of Ball, Groenewald and Malakorn on unitary realization of a formal power series from the non-commutative Schur--Agler class admits an improvement: its finite-dimensionality and uniqueness up to a unitary similarity is proved. A version of the theory for \emph{matrix-selfadjoint} rational formal power series is also presented. The concept of non-commutative formal reproducing kernel Pontryagin spaces is introduced, and in this framework the backward shift realization of a matrix-$J$-unitary rational formal power series in a finite-dimensional non-commutative de Branges--Rovnyak space is described. | |
| dc.description | To appear in Oper. Theory Adv. Appl | |
| dc.identifier | https://arxiv.org/abs/math/0407387 | |
| dc.identifier | http://arxiv.org/abs/math/0407387 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71973 | |
| dc.subject | Functional Analysis | |
| dc.subject | Complex Variables | |
| dc.subject | 47A48; 13F25; 46C20; 46E22; 93B20; 93D05 | |
| dc.title | Matrix-J-unitary non-commutative rational formal power series | |
| dc.type | text |