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2 Minimal realizations in non-commutative de Branges–Rovnyak spaces . . . . . . . . . . . . . . . . . . . 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Matrix-J-unitary Rational Formal Power Series 51 1. Introduction In the present paper we study a non-commutative analogue of rational matrixvalued functions which are J-unitary on the imaginary line or on the unit circle and, as a special case, J-inner ones.

PhD thesis, The Weizmann Institute of Sciences, 1986. G. Kre˘n. ˘ Continuous analogues of propositions for polynomials orthogonal on the unit circle. Dokl. Akad. Nauk. SSSR, 105:637–640, 1955. G. Kre˘n. Topics in differential and integral equations and operator theory, volume 7 of Operator theory: Advances and Applications. Birkhauser ¨ Verlag, Basel, 1983. Edited by I. Gohberg, Translated from the Russian by A. Iacob. G. Kre˘n ˘ and H. Langer. Uber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume Πk .

1 Minimal Givone–Roesser realizations and the Stein equation . . . 2 The associated structured Hermitian matrix . . . . . . . . . . . 3 Minimal matrix-J-unitary factorizations . . . . . . . . . . . . . 4 Matrix-unitary rational formal power series . . . . . . . . . . . . 85 6 Matrix-J-inner rational formal power series . . . . . . . . . . . . . . 1 A multivariable non-commutative analogue of the half-plane case . . . . . . . .

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