Aspects of non-commutative function theory

Jim Agler ; John E. McCarthy

Concrete Operators, Tome 3 (2016), p. 15-24 / Harvested from The Polish Digital Mathematics Library

Résumé

We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.

Détail

Publié le : 2016-01-01

Zbl 1337.32017

EUDML-ID : urn:eudml:doc:277087

Bibliographie

[1] Agler, J., McCarthy, J.E., Non-commutative functional calculus, Journal d’Analyse, to appear. arXiv:1504.07323,

[2] Agler, J., McCarthy, J.E., Global holomorphic functions in several non-commuting variables, 2015, Canad. J. Math., 67, 2, 241–285, | Zbl 1311.32001

[3] Agler, J., McCarthy, J.E., Non-commutative holomorphic functions on operator domains, 2015, European J. Math, 1, 4, 731–745, | Zbl 1339.46048

[4] Agler, J., McCarthy, J.E., The implicit function theorem and free algebraic sets, 2016, Trans. Amer. Math. Soc., 368, 5, 3157–3175, | Zbl 1339.32004

[5] Alpay, D., Kalyuzhnyi-Verbovetzkii, D. S., Matrix-J-unitary non-commutative rational formal power series, 2006, The state space method generalizations and applications, Oper. Theory Adv. Appl., 161, Birkhäuser, Basel, 49–113,

[6] Ambrozie, C.-G., Timotin, D., A von Neumann type inequality for certain domains in Cn, 2003, Proc. Amer. Math. Soc., 131, 859–869, | Zbl 1047.47003

[7] Ball, J.A., Bolotnikov, V., Realization and interpolation for Schur-Agler class functions on domains with matrix polynomial defining function in Cn, 2004, J. Funct. Anal., 213, 45–87, | Zbl 1061.47014

[8] Ball, Joseph A., Groenewald, Gilbert, Malakorn, Tanit, Conservative structured noncommutative multidimensional linear systems, 2006, bookThe state space method generalizations and applications, Oper. Theory Adv. Appl., 161, Birkhäuser, Basel, 179–223, | Zbl 1110.47066

[9] Boyd, Stephen, El Ghaoui, Laurent, Feron, Eric, Balakrishnan, Venkataramanan, Linear matrix inequalities in system and control theory, SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994, 15, 0-89871-334-X, http://dx.doi.org/10.1137/1.9781611970777, | Zbl 0816.93004

[10] Cimpric, Jakob, Helton, J. William, McCullough, Scott, Nelson, Christopher, A noncommutative real nullstellensatz corresponds to a noncommutative real ideal: algorithms, 2013, 0024-6115, Proc. Lond. Math. Soc. (3), 106, 5, 1060–1086, http://dx.doi.org.libproxy.wustl.edu/10.1112/plms/pds060, | Zbl 1270.14029

[11] Dineen, Seán, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999, 1-85233-158-5, http://dx.doi.org/10.1007/978-1-4471-0869-6,

[12] Helton, J. William, “Positive” noncommutative polynomials are sums of squares, 2002, 0003-486X, Ann. of Math. (2), 156, 2, 675–694, http://dx.doi.org/10.2307/3597203,

[13] Helton, J. William, Klep, Igor, McCullough, Scott, Analytic mappings between noncommutative pencil balls, 2011, J. Math. Anal. Appl., 376, 2, 407–428, | Zbl 1210.47039

[14] Helton, J. William, Klep, Igor, McCullough, Scott, Proper analytic free maps, 2011, J. Funct. Anal., 260, 5, 1476–1490, | Zbl 1215.47011

[15] Helton, J. William, Klep, Igor, McCullough, Scott, Convexity and semidefinite programming in dimension-free matrix unknowns, 2012, bookHandbook on semidefinite, conic and polynomial optimization, Internat. Ser. Oper. Res. Management Sci., 166, Springer, New York, 377–405, http://dx.doi.org/10.1007/978-1-4614-0769-0_13, | Zbl 1334.90103

[16] Helton, J. William, Klep, Igor, McCullough, Scott, Free analysis, convexity and LMI domains, 2012, bookOperator theory: Advances and applications, vol. 41, Springer, Basel, 195–219, | Zbl 1280.46043

[17] Helton, J. William, Klep, Igor, McCullough, Scott, Slinglend, Nick, Noncommutative ball maps, 2009, 0022-1236, J. Funct. Anal., 257, 1, 47–87, http://dx.doi.org.libproxy.wustl.edu/10.1016/j.jfa.2009.03.008, | Zbl 1179.47012

[18] Helton, J. William, McCullough, Scott, Every convex free basic semi-algebraic set has an LMI representation, 2012, Ann. of Math. (2), 176, 2, 979–1013, | Zbl 1260.14011

[19] Helton, J. William, McCullough, Scott, Putinar, Mihai, Vinnikov, Victor, Convex matrix inequalities versus linear matrix inequalities, 2009, 0018-9286, IEEE Trans. Automat. Control, 54, 5, 952–964, http://dx.doi.org/10.1109/TAC.2009.2017087,

[20] Helton, J. William, McCullough, Scott A., A Positivstellensatz for non-commutative polynomials, 2004, 0002-9947, Trans. Amer. Math. Soc., 356, 9, 3721–3737 (electronic), http://dx.doi.org.libproxy.wustl.edu/10.1090/S0002-9947-04-03433-6,

[21] Kaliuzhnyi-Verbovetskyi, Dmitry S., Vinnikov, Victor, Foundations of free non-commutative function theory, AMS, Providence, 2014, | Zbl 1312.46003

[22] McCarthy, J.E., Timoney, R., Nc automorphisms of nc-bounded domains, Proc. Royal Soc. Edinburgh, to appear,

[23] Muhly, Paul S., Solel, Baruch, Tensorial function theory: from Berezin transforms to Taylor’s Taylor series and back, 2013, 0378- 620X, Integral Equations Operator Theory, 76, 4, 463–508, http://dx.doi.org.libproxy.wustl.edu/10.1007/s00020-013-2062-4, | Zbl 1291.46043

[24] Pascoe, J. E., The inverse function theorem and the Jacobian conjecture for free analysis, 2014, 0025-5874, Math. Z., 278, 3-4, 987–994, http://dx.doi.org/10.1007/s00209-014-1342-2,

[25] Pascoe, J.E., Tully-Doyle, R., Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables, arXiv:1309.1791,

[26] Popescu, Gelu, Free holomorphic functions on the unit ball of B.H/n, 2006, J. Funct. Anal., 241, 1, 268–333, | Zbl 1112.47004

[27] Popescu, Gelu, Free holomorphic functions and interpolation, 2008, Math. Ann., 342, 1, 1–30, | Zbl 1146.47010

[28] Popescu, Gelu, Free holomorphic automorphisms of the unit ball of B.H/n, 2010, J. Reine Angew. Math., 638, 119–168,

[29] Popescu, Gelu, Free biholomorphic classification of noncommutative domains, 2011, Int. Math. Res. Not. IMRN, 4, 784–850, | Zbl 1235.47011

[30] Putinar, M., Positive polynomials on compact semi-algebraic sets, 1993, 42, 969–984, | Zbl 0796.12002

[31] Schmüdgen, Konrad, The K-moment problem for compact semi-algebraic sets, 1991, 0025-5831, Math. Ann., 289, 2, 203–206, http://dx.doi.org/10.1007/BF01446568,

[32] Sylvester, J., Sur l’équations en matrices px = xq, 1884, C.R. Acad. Sci. Paris, 99, 67–71, | Zbl 16.0108.03

[33] Taylor, J.L., The analytic functional calculus for several commuting operators, 1970, Acta Math., 125, 1–38, | Zbl 0233.47025

[34] Taylor, J.L., A joint spectrum for several commuting operators, 1970, J. Funct. Anal., 6, 172–191, | Zbl 0233.47024

[35] Taylor, J.L., A general framework for a multi-operator functional calculus, 1972, 0001-8708, Advances in Math., 9, 183–252,

[36] Taylor, J.L., Functions of several non-commuting variables, 1973, Bull. Amer. Math. Soc., 79, 1–34, | Zbl 0257.46055

[37] Voiculescu, Dan, Free analysis questions. I. Duality transform for the coalgebra of @XWB, 2004, Int. Math. Res. Not., 16, 793–822, | Zbl 1084.46053