Aspects of non-commutative function theory

Jim Agler; John E. McCarthy

Concrete Operators (2016)

  • Volume: 3, Issue: 1, page 15-24
  • ISSN: 2299-3282

Abstract

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

How to cite

top

Jim Agler, and John E. McCarthy. "Aspects of non-commutative function theory." Concrete Operators 3.1 (2016): 15-24. <http://eudml.org/doc/277087>.

@article{JimAgler2016,
abstract = {We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.},
author = {Jim Agler, John E. McCarthy},
journal = {Concrete Operators},
keywords = {Noncommutative function; Free holomorphic function; noncommutative functions; several matrix variables},
language = {eng},
number = {1},
pages = {15-24},
title = {Aspects of non-commutative function theory},
url = {http://eudml.org/doc/277087},
volume = {3},
year = {2016},
}

TY - JOUR
AU - Jim Agler
AU - John E. McCarthy
TI - Aspects of non-commutative function theory
JO - Concrete Operators
PY - 2016
VL - 3
IS - 1
SP - 15
EP - 24
AB - We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.
LA - eng
KW - Noncommutative function; Free holomorphic function; noncommutative functions; several matrix variables
UR - http://eudml.org/doc/277087
ER -

References

top
  1. [1] Agler, J., McCarthy, J.E., Non-commutative functional calculus, Journal d’Analyse, to appear. arXiv:1504.07323,  
  2. [2] Agler, J., McCarthy, J.E., Global holomorphic functions in several non-commuting variables, 2015, Canad. J. Math., 67, 2, 241–285,  Zbl1311.32001
  3. [3] Agler, J., McCarthy, J.E., Non-commutative holomorphic functions on operator domains, 2015, European J. Math, 1, 4, 731–745,  Zbl1339.46048
  4. [4] Agler, J., McCarthy, J.E., The implicit function theorem and free algebraic sets, 2016, Trans. Amer. Math. Soc., 368, 5, 3157–3175,  Zbl1339.32004
  5. [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. [6] Ambrozie, C.-G., Timotin, D., A von Neumann type inequality for certain domains in Cn, 2003, Proc. Amer. Math. Soc., 131, 859–869,  Zbl1047.47003
  7. [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,  Zbl1061.47014
  8. [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,  Zbl1110.47066
  9. [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,  Zbl0816.93004
  10. [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,  Zbl1270.14029
  11. [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. [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. [13] Helton, J. William, Klep, Igor, McCullough, Scott, Analytic mappings between noncommutative pencil balls, 2011, J. Math. Anal. Appl., 376, 2, 407–428,  Zbl1210.47039
  14. [14] Helton, J. William, Klep, Igor, McCullough, Scott, Proper analytic free maps, 2011, J. Funct. Anal., 260, 5, 1476–1490,  Zbl1215.47011
  15. [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,  Zbl1334.90103
  16. [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,  Zbl1280.46043
  17. [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,  Zbl1179.47012
  18. [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,  Zbl1260.14011
  19. [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. [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. [21] Kaliuzhnyi-Verbovetskyi, Dmitry S., Vinnikov, Victor, Foundations of free non-commutative function theory, AMS, Providence, 2014,  Zbl1312.46003
  22. [22] McCarthy, J.E., Timoney, R., Nc automorphisms of nc-bounded domains, Proc. Royal Soc. Edinburgh, to appear,  
  23. [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,  Zbl1291.46043
  24. [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. [25] Pascoe, J.E., Tully-Doyle, R., Free Pick functions: representations, asymptotic behavior and matrix monotonicity in several noncommuting variables, arXiv:1309.1791,  
  26. [26] Popescu, Gelu, Free holomorphic functions on the unit ball of B.H/n, 2006, J. Funct. Anal., 241, 1, 268–333,  Zbl1112.47004
  27. [27] Popescu, Gelu, Free holomorphic functions and interpolation, 2008, Math. Ann., 342, 1, 1–30,  Zbl1146.47010
  28. [28] Popescu, Gelu, Free holomorphic automorphisms of the unit ball of B.H/n, 2010, J. Reine Angew. Math., 638, 119–168,  
  29. [29] Popescu, Gelu, Free biholomorphic classification of noncommutative domains, 2011, Int. Math. Res. Not. IMRN, 4, 784–850,  Zbl1235.47011
  30. [30] Putinar, M., Positive polynomials on compact semi-algebraic sets, 1993, 42, 969–984,  Zbl0796.12002
  31. [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. [32] Sylvester, J., Sur l’équations en matrices px = xq, 1884, C.R. Acad. Sci. Paris, 99, 67–71,  Zbl16.0108.03
  33. [33] Taylor, J.L., The analytic functional calculus for several commuting operators, 1970, Acta Math., 125, 1–38,  Zbl0233.47025
  34. [34] Taylor, J.L., A joint spectrum for several commuting operators, 1970, J. Funct. Anal., 6, 172–191,  Zbl0233.47024
  35. [35] Taylor, J.L., A general framework for a multi-operator functional calculus, 1972, 0001-8708, Advances in Math., 9, 183–252,  
  36. [36] Taylor, J.L., Functions of several non-commuting variables, 1973, Bull. Amer. Math. Soc., 79, 1–34,  Zbl0257.46055
  37. [37] Voiculescu, Dan, Free analysis questions. I. Duality transform for the coalgebra of @XWB, 2004, Int. Math. Res. Not., 16, 793–822,  Zbl1084.46053

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.