Realizations of Non-commutative Rational Functions Around a Matrix Centre, II: The Lost-Abbey Conditions

In a previous paper the authors generalized classical results on minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational fun...

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Published in:	Integral equations and operator theory Vol. 95; no. 1; p. 1
Main Authors:	Porat, Motke, Vinnikov, Victor
Format:	Journal Article
Language:	English
Published:	Cham Springer International Publishing 01.03.2023 Springer Nature B.V
Subjects:	Algebra Analysis Calculus Codes Domains Mathematical analysis Mathematics Mathematics and Statistics Observability (systems) Rational functions Regularity System theory Variables
ISSN:	0378-620X, 1420-8989
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Abstract	In a previous paper the authors generalized classical results on minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational function is contained in the invertibility set of the corresponding pencil of any minimal realization of the function. In this paper we prove an equality between the domain of a nc rational function and the domain of any of its minimal realizations. As for evaluations over stably finite algebras, we show that the domain of the realization w.r.t any such algebra coincides with the so called matrix domain of the function w.r.t the algebra. As a corollary we show that the domain of regularity and the stable extended domain coincide. In contrary to both the classical case and the scalar case—where every matrix coefficients which satisfy the controllability and observability conditions can appear in a minimal realization of a nc rational function—the matrix coefficients in our case have to satisfy certain equations, called linearized lost-abbey conditions, which are related to Taylor–Taylor expansions in nc function theory.
AbstractList	In a previous paper the authors generalized classical results on minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini realizations which are centred at an arbitrary matrix point. In particular, it was proved that the domain of regularity of a nc rational function is contained in the invertibility set of the corresponding pencil of any minimal realization of the function. In this paper we prove an equality between the domain of a nc rational function and the domain of any of its minimal realizations. As for evaluations over stably finite algebras, we show that the domain of the realization w.r.t any such algebra coincides with the so called matrix domain of the function w.r.t the algebra. As a corollary we show that the domain of regularity and the stable extended domain coincide. In contrary to both the classical case and the scalar case—where every matrix coefficients which satisfy the controllability and observability conditions can appear in a minimal realization of a nc rational function—the matrix coefficients in our case have to satisfy certain equations, called linearized lost-abbey conditions, which are related to Taylor–Taylor expansions in nc function theory.
ArticleNumber	1
Author	Porat, Motke Vinnikov, Victor
Author_xml	– sequence: 1 givenname: Motke surname: Porat fullname: Porat, Motke organization: Department of Mathematics, Ben-Gurion University of the Negev – sequence: 2 givenname: Victor orcidid: 0000-0003-2965-1408 surname: Vinnikov fullname: Vinnikov, Victor email: vinnikov@math.bgu.ac.il organization: Department of Mathematics, Ben-Gurion University of the Negev
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Snippet	In a previous paper the authors generalized classical results on minimal realizations of non-commutative (nc) rational functions, using nc Fornasini–Marchesini...
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SubjectTerms	Algebra Analysis Calculus Codes Domains Mathematical analysis Mathematics Mathematics and Statistics Observability (systems) Rational functions Regularity System theory Variables
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Title	Realizations of Non-commutative Rational Functions Around a Matrix Centre, II: The Lost-Abbey Conditions
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