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Concrete Operators

Ed. by Chalendar, Isabelle / Partington, Jonathan

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Mathematical Citation Quotient (MCQ) 2016: 0.38


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2299-3282
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Aspects of non-commutative function theory

Jim Agler / John E. McCarthy
Published Online: 2016-04-11 | DOI: https://doi.org/10.1515/conop-2016-0003

Abstract

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

Keywords: Noncommutative function; Free holomorphic function

References

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

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

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

  • [4] Agler, J., McCarthy, J.E., The implicit function theorem and free algebraic sets, 2016, Trans. Amer. Math. Soc., 368, 5, 3157–3175, Web of ScienceGoogle Scholar

  • [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, Google Scholar

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

  • [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, Google Scholar

  • [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, Google Scholar

  • [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, Google Scholar

  • [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, CrossrefGoogle Scholar

  • [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, Google Scholar

  • [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, CrossrefGoogle Scholar

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

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

  • [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, CrossrefGoogle Scholar

  • [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, Google Scholar

  • [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, CrossrefGoogle Scholar

  • [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, Web of ScienceGoogle Scholar

  • [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, CrossrefGoogle Scholar

  • [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, CrossrefGoogle Scholar

  • [21] Kaliuzhnyi-Verbovetskyi, Dmitry S., Vinnikov, Victor, Foundations of free non-commutative function theory, AMS, Providence, 2014, Google Scholar

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

  • [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, CrossrefGoogle Scholar

  • [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, CrossrefGoogle Scholar

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

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

  • [27] Popescu, Gelu, Free holomorphic functions and interpolation, 2008, Math. Ann., 342, 1, 1–30, Web of ScienceGoogle Scholar

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

  • [29] Popescu, Gelu, Free biholomorphic classification of noncommutative domains, 2011, Int. Math. Res. Not. IMRN, 4, 784–850, Google Scholar

  • [30] Putinar, M., Positive polynomials on compact semi-algebraic sets, 1993, 42, 969–984, Google Scholar

  • [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, CrossrefGoogle Scholar

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

  • [33] Taylor, J.L., The analytic functional calculus for several commuting operators, 1970, Acta Math., 125, 1–38, Google Scholar

  • [34] Taylor, J.L., A joint spectrum for several commuting operators, 1970, J. Funct. Anal., 6, 172–191, Google Scholar

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

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

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

About the article

Received: 2015-10-30

Accepted: 2016-02-10

Published Online: 2016-04-11


Citation Information: Concrete Operators, ISSN (Online) 2299-3282, DOI: https://doi.org/10.1515/conop-2016-0003.

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© 2016 Agler and McCarthy. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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