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The State Space Method Generalizations and Applications pp 49–113Cite as

Matrix-J-unitary Non-commutative Rational Formal Power Series

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Part of the book series: Operator Theory: Advances and Applications ((LOLS,volume 161))

Abstract

Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N-tuples of n × n matrices (with n = 1, 2, ⋯) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering.

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 × n skew-Hermitian versus unitary matrices (n = 1, 2, ⋯), and a rational formal power series is called 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 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 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: the existence of a minimal (thus, finite-dimensional) such unitary realization and its uniqueness up to a unitary similarity is proved. A version of the theory for 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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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    D. Alpay

  2. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    D.S. Kalyuzhnyı-Verbovetzkiı

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Editors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva, 84105, Israel

    Daniel Alpay

  2. School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Israel

    Israel Gohberg

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Alpay, D., Kalyuzhnyı-Verbovetzkiı, D. (2005). Matrix-J-unitary Non-commutative Rational Formal Power Series. In: Alpay, D., Gohberg, I. (eds) The State Space Method Generalizations and Applications. Operator Theory: Advances and Applications, vol 161. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7431-4_2

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