TY - JOUR

T1 - Realizations of Non-commutative Rational Functions Around a Matrix Centre, II

T2 - The Lost-Abbey Conditions

AU - Porat, Motke

AU - Vinnikov, Victor

N1 - Funding Information:
The authors would like to thank Dmitry Kalyuzhnyi-Verbovetskyi, Roland Speicher, and Juri Volčič for their helpful comments and discussions. A special gratitude is due to Igor Klep and Juri Volčič for the details of Example 4.13. The idea of developing realization theory around a matrix point, along the lines presented here, was first explored by the second author at a talk at MFO workshop on free probability theory in 2015 [48]; the second author would like to thank MFO and the workshop organizers for their hospitality. Finally, we would like to thank the anonymous referee for helpful comment.
Funding Information:
The research of both authors was partially supported by the US–Israel Binational Science Foundation (BSF) Grant No. 2010432, Deutsche Forschungsgemeinschaft (DFG) Grant No. SCHW 1723/1-1, and Israel Science Foundation (ISF) Grant No. 2123/17.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2023/3/1

Y1 - 2023/3/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85143353203&partnerID=8YFLogxK

U2 - 10.1007/s00020-022-02718-z

DO - 10.1007/s00020-022-02718-z

M3 - Article

AN - SCOPUS:85143353203

SN - 0378-620X

VL - 95

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

IS - 1

M1 - 1

ER -