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Matrix-J-unitary non-commutative rational formal power series
Functional Analysis · Submitted 22 Jul 2004
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.99 | 0.89 | 0.50 | asserted 1.00 |
| s1 | 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. | background 0.70 | 0.39 | 0.08 | asserted 1.00 |
| s2 | The close relationship between minimal realizations and structured Hermitian solutions $H$ of the Lyapunov or Stein equations is established. | result 0.75 | 0.84 | 0.17 | asserted 1.00 |
| s3 | The results are specialized for the case of \emph{matrix-$J$-inner} rational formal power series. | contribution 0.86 | 0.84 | 0.08 | asserted 0.97 |
| s4 | In this case $H>0$, however the proof of that is more elaborated than in the one-variable case and involves a new technique. | result 0.43 | 0.85 | 0.08 | asserted 1.00 |
| s5 | 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. | result 0.68 | 0.78 | 0.17 | asserted 1.00 |
| s6 | A version of the theory for \emph{matrix-selfadjoint} rational formal power series is also presented. | contribution 0.98 | 0.86 | 0.04 | asserted 1.00 |
| s7 | 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. | contribution 0.98 | 0.86 | 0.14 | asserted 1.00 |
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Original abstract
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.
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