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

Motke Porat1, Victor Vinnikov2
1Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
2Department of Mathematics, Ben-Gurion University of the Negev, Beer Sheva, Israel

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Agler, J., McCarthy, J.E.: Global holomorphic functions in several non-commuting variables. Can. J. Math. 67(2), 241–285 (2015)

Agler, J., McCarthy, J.E.: Pick interpolation for free holomorphic functions. Am. J. Math. 137(6), 1685–1701 (2015)

Agler, J., McCarthy, J.E.: Aspects of non-commutative function theory. Concrete Oper. 3(1), 15–24 (2016)

Alpay, D., Kaliuzhnyi-Verbovetskyi, D.S.: On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series. C.R. Math. 339(8), 533–538 (2004)

Alpay, D., Kaliuzhnyi-Verbovetzkyi, D. S.: Matrix $$J$$-unitary noncommutative rational formal power series. Operator Theory: Advances and Applications 161: pp. 49–113, Birkhäuser-Verlag, Basel (2005)

Amitsur, S.A.: Rational identities and applications to algebra and geometry. J. Algebra 3(3), 304–359 (1966)

Ball, J.A., Groenewald, G., Malakorn, T.: Structured noncommutative multidimensional linear systems. SIAM J. Control. Optim. 44(4), 1474–1528 (2005)

Ball, J.A., Groenewald, G., Malakorn, T.: Conservative structured noncommutative multidimensional linear systems. Operator Theory: Advances and Applications 161: pp. 179–223, Birkhäuser, Basel (2006)

Ball, J.A., Groenewald, G., Malakorn, T.: Bounded real lemma for structured noncommutative multidimensional linear systems and robust control. Multidimension. Syst. Signal Process. 17(2–3), 119–150 (2006)

Ball, J.A., Kaliuzhnyi-Verbovetskyi, D.S.: Conservative dilations of dissipative multidimensional systems: the commutative and non-commutative settings. Multidimension. Syst. Signal Process. 19(1), 79–122 (2008)

Ball, J.A., Vinnikov, V.: Lax–Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting. Memoirs of the American Mathematical Society 178(837) (2005)

Bart, H., Gohberg, I., Kaashoek, M.A.: Minimal factorization of matrix and operator functions. Birkhäuser-Verlag, Basel, Operator Theory, Advances and Applications (1979)

Beck, C.: On formal power series representations for uncertain systems. IEEE Trans. Autom. Control 46(2), 314–319 (2001)

Beck, C.L., Doyle, J., Glover, K.: Model reduction of multidimensional and uncertain systems. IEEE Trans. Autom. Control 41(10), 1466–1477 (1996)

Belinschi, S.T., Mai, T., Speicher, R.: Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. J. für die reine und angewandte Math. (Crelles Journal) 732, 21–53 (2017)

Bergman, G.M.: Skew fields of noncommutative rational functions, after Amitsur. S’eminaire Schützenberger–Lentin–Nivat, Année 16, Paris (1969/70)

Bergman, G.M.: Coproducts and some universal ring constructions. Trans. Am. Math. Soc. 200, 33–88 (1974)

Berstel, J., Reutenauer, C.: Rational Series and their Languages. Monographs on Theoretical Computer Science 12, Springer, Berlin (1988)

Camino, J.F., Helton, J.W., Skelton, R.E., Ye, J.: Matrix inequalities: a symbolic procedure to determine convexity automatically. Integr. Eqn. Oper. Theory 46(4), 399–454 (2003)

Cohn, P.M.: On the embedding of rings in skew fields. Proc. Lond. Math. Soc. 3(1), 511–530 (1961)

Cohn, P.M.: The embedding of firs in skew fields. Proc. Lond. Math. Soc. 3(2), 193–213 (1971)

Cohn, P.M.: Universal skew fields of fractions. Sympos. Math. 8, 135–148 (1972)

Cohn, P.M.: Free Rings and their Relations. London Mathematical Society Monographs 2. Academic Press, London (1971)

Cohn, P.M.: Skew fields. Theory of General Division Rings. Encyclopedia of Mathematics and its Applications 57. Cambridge University Press, Cambridge (1995)

Cohn, P.M.: Free Ideal Rings and Localization in General Rings. New Mathematical Monographs 3. Cambridge University Press, Cambridge (2006)

Cohn, P.M., Reutenauer, C.: A normal form in free fields. Can. J. Math. 46(3), 517–531 (1994)

Cohn, P.M., Reutenauer, C.: On the construction of the free field. Int. J. Algebra Comput. 9(3–4), 307–323 (1999)

Effros, E.G.: Advances in quantized functional analysis, pp. 906–916. Proceedings of the International Congress of Mathematicians, Berkeley (1986)

Fliess, M.: Sur le plongement de l’algèbre des séries rationnelles non commutatives dans un corps gauche. Comptes Rendus de l’Académie des Sciences Ser. A 271: pp. 926–927 (1970)

Fliess, M.: Matrices de Hankel. J. de Math. Pures et Appl. 53(9), 197–222 (1974)

Fliess, M.: Sur divers produits de series formelles. Bull. Soc. Math. France 102, 181–191 (1974)

Galkowski, K.: Minimal state-space realization for a class of linear, discrete, nD. SISO systems. Int. J. Control 74(13), 1279–1294 (2001)

Gelfand, I., Gelfand, S., Retakh, V., Wilson, R.L.: Quasideterminants. Adv. Math. 193(1), 56–141 (2005)

Gelfand, I., Retakh, V.: Determinants of matrices over noncommutative rings. Funct. Anal. Appl. 25(2), 91–102 (1991)

Helton, J.W.: “Positive’’ noncommutative polynomials are sums of squares. Ann. Math. 156(2), 675–694 (2002)

Helton, J.W.: Manipulating Matrix Inequalities Automatically. Mathematical Systems Theory in Biology, Communications, Computation, and Finance, Springer, New York, pp. 237–256 (2003)

Helton, J.W., Klep, I., McCullough, S.: Analytic mappings between noncommutative pencil balls. J. Math. Anal. Appl. 376(2), 407–428 (2011)

Helton, J.W., Klep, I., McCullough, S.: Proper analytic free maps. J. Funct. Anal. 260(5), 1476–1490 (2011)

Helton, J.W., Klep, I., McCullough, S., Volčič, J.: Noncommutative polynomials describing convex sets. Foundations of Computational Mathematics, pp. 1–37 (2020)

Helton, J.W., Mai, T., Speicher, R.: Applications of realizations (aka linearizations) to free probability. J. Funct. Anal. 274(1), 1–79 (2018)

Helton, J.W., McCullough, S.: Every convex free basic semi-algebraic set has an LMI representation. Ann. Math. 176(2), 979–1013 (2012)

Helton, J.W., McCullough, S.: Free convex sets defined by rational expressions have LMI representations. J. Convex Anal. 21(2), 425–448 (2014)

Helton, J.W., McCullough, S., Vinnikov, V.: Noncommutative convexity arises from linear matrix inequalities. J. Funct. Anal. 240(1), 105–191 (2006)

Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Corrected reprint of the 1991 original, Cambridge University Press (1994)

Hughes, I.: Division rings of fractions for group rings. Commun. Pure Appl. Math. 23(2), 181–188 (1970)

Kaczorek, T.: Two Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, vol. 68. Springer, Berlin (1985)

Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Foundations of Free Noncommutative Function Theory. Mathematical Surveys and Monographs 199, American Mathematical Society, Providence, RI (2014)

Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Realization theory for noncommutative rational functions around a matrix point. Oberwolfach Reports, Vol. 12, Issue 2: pp. 1600–1603 (Report No. 28/2015, https://doi.org/10.4171/OWR/2015/28, Free Probability Theory) (2015)

Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Noncommutative rational functions, their difference-differential calculus and realizations. Multidimension. Syst. Signal Process. 23(1–2), 49–77 (2012)

Kaliuzhnyi-Verbovetskyi, D.S., Vinnikov, V.: Singularities of rational functions and minimal factorizations: the noncommutative and the commutative settings. Linear Algebra Appl. 430(4), 869–889 (2009)

Kalman, R.E., Arbib, M.A., Falb, P.L.: Topics in Mathematical Systems Theory. McGraw Hill, New York (1969)

Kleene, S.C.: Representation of events in nerve nets and finite automata. Automata Studies, Annals of Mathematics Studies 34: pp. 3–41, Princeton University Press, Princeton (1956)

Klep, I., Vinnikov, V., Volčič, J.: Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation. Trans. Am. Math. Soc. 373, 5587–5625 (2020)

Lewin, J.: Fields of fractions for group algebras of free groups. Trans. Am. Math. Soc. 192, 339–346 (1974)

Lichtman, A.I.: On universal fields of fractions for free algebras. J. Algebra 231(2), 652–676 (2000)

Linnell, P.A.: Division rings and group von Neumann algebras. Forum Math. 5(5), 561–576 (1993)

Linnell, P.A.: Noncommutative localization in group rings. Non-commutative localization in algebra and topology, vol. 330 of London Mathematical Society Lecture Note Series, pp. 40–59, Cambridge University Press, Cambridge (2006)

Lu, W.M., Zhou, K., Doyle, J.C.: Stabilization of uncertain linear systems: an LFT approach. IEEE Trans. Autom. Control 41(1), 50–65 (1996)

Muhly, P.S., Solel, B.: Progress in noncommutative function theory. Sci. China Math. 54(11), 2275–2294 (2011)

Nemirovskii, A.: Advances in convex optimization: conic programming. Plenary Lecture, International Congress of Mathematicians 1: pp. 413–444, Madrid, Spain (2006)

Nesterov, Y., Nemirovskii, A.: Interior-Point Polynomial Algorithms in Convex Programming. Studies in Applied Mathematics 13, Philadelphia, PA (1994)

Popescu, G.: Free holomorphic functions on the unit ball of $$B(H)^n$$. J. Funct. Anal. 241(1), 268–333 (2006)

Popescu, G.: Free holomorphic automorphisms of the unit ball of $$B(H)^n$$. J. für die reine und angewandte Math. (Crelles Journal) 638, 119–168 (2010)

Porat, M., Vinnikov, V.: Realizations of non-commutative rational functions around a matrix centre, I: Synthesis, minimal realizations and evaluation on stably finite algebras. ArXiv preprint, arXiv:1905.11304 (2019)

Porat, M., Vinnikov, V.: Realizations of Non-commutative Rational Functions Around a Matrix Center. Kronecker-Fliess Theorem and the Free Skew Field. To appear, III, Functional Models (2020)

Reutenauer, C.: Malcev-Neumann series and the free field. Expo. Math. 17(5), 469–478 (1999)

Rowen, L.H.: Polynomial Identities in Ring Theory. Pure and Applied Mathematics 84, Academic Press, New York, London (1980)

Skelton, R.E., Iwasaki, T., Grigoriadis, K.M.: A Unified Algebraic Approach to Linear Control Design. Taylor & Francis (1997)

Schrempf, K.: A standard form in (some) free fields: How to construct minimal linear representations. arXiv preprint, arXiv:1803.10627 (2018)

Schrempf, K.: Free fractions: An invitation to (applied) free fields. arXiv preprint, arXiv:1809.05425 (2018)

Schrempf, K.: Linearizing the word problem in (some) free fields. Int. J. Algebra Comput. 28(7), 1209–1230 (2018)

Schrempf, K.: On the factorization of non-commutative polynomials (in free associative algebras). J. Symb. Comput. 94, 126–148 (2019)

Schützenberger, M.P.: On the definition of a family of automata. Inf. Control 4(2–3), 245–270 (1961)

Schützenberger, M.P.: Certain elementary families of automata. In: Proceedings of the Symposium on Mathematical Theory of Automata (New York, 1962), pp. 139–153, Polytechnic Press of Polytechnic Institute of Brooklyn, Brooklyn, New York (1963)

Speicher, R.: Polynomials in asymptotically free random matrices. Acta Phys. Pol. B 46(9), 1611–1624 (2015)

Speicher, R.: Free Probability Theory. The Oxford Handbook of Random Matrix Theory, pp. 452–470, Oxford University Press, Oxford (2011)

Taylor, J.L.: A general framework for a multi-operator functional calculus. Adv. Math. 9, 183–252 (1972)

Taylor, J.L.: Functions of several noncommuting variables. Bull. Am. Math. Soc. 79(1), 1–34 (1973)

Voiculescu, D.V.: Free analysis questions I: duality transform for the coalgebra of $$\partial _{X:B}$$. Int. Math. Res. Not. 16, 793–822 (2004)

Voiculescu, D.V.: Free analysis questions II: the Grassmannian completion and the series expansion at the origin. J. für die reine und Angew. Math. (Crelles Journal) 645, 155–236 (2010)

Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. A Noncommutative Probability Approach to free Products with Applications to Random Matrices, Operator Algebras and Harmonic Analysis on Free Groups. CRM Monograph Series 1, American Mathematical Society, Providence, RI (1992)

Volčič, J.: On domains of noncommutative rational functions. Linear Algebra Appl. 516, 69–81 (2017)

Volčič, J.: Matrix coefficient realization theory of noncommutative rational functions. J. Algebra 499, 397–437 (2018)