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Noncommutative Trigonometry

  • Karl Gustafson
Part of the Operator Theory: Advances and Applications book series (OT, volume 167)

Abstract

A unified account of a noncommutative operator trigonometry originated in 1966 by this author and its further developments and applications to date will be given within a format of a historical trace. Applications to wavelet and multiscale theories are included. A viewpoint toward possible future enlargement will be fostered.

Keywords

noncommutative operator trigonometry antieigenvalue linear algebra quantum mechanics statistics numerical analysis wavelets multiscale systems iterative methods 

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References

  1. [1]
    K. Gustafson, A Perturbation Lemma, Bull. Amer. Math. Soc. 72 (1966), 334–338.zbMATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    K. Gustafson, Positive Operator Products, Notices Amer. Math. Soc. 14 (1967), Abstract 67T-531, p. 717. See also Abstracts 67T-340, 67T-564, 67T-675.Google Scholar
  3. [3]
    K. Gustafson, A Note on Left Multiplication of Semi-group Generators, Pacific J. Math. 24 (1968a), 463–465.zbMATHMathSciNetGoogle Scholar
  4. [4]
    K. Gustafson, The Angle of an Operator and Positive Operator Products, Bull. Amer. Math. Soc. 74 (1968b), 488–492.zbMATHMathSciNetGoogle Scholar
  5. [5]
    K. Gustafson, Positive (noncommuting) Operator Products and Semigroups, Math. Zeitschrift 105 (1968c), 160–172.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    K. Gustafson, A min-max Theorem, Amer. Math Soc. Notices 15 (1968d), p. 799.Google Scholar
  7. [7]
    K. Gustafson, Doubling Perturbation Sizes and Preservation of Operator Indices in Normed Linear Spaces, Proc. Camb. Phil. Soc. 98 (1969a), 281–294.MathSciNetCrossRefGoogle Scholar
  8. [8]
    K. Gustafson, On the Cosine of Unbounded Operators, Acta Sci. Math. 30 (1969b), 33–34 (with B. Zwahlen).zbMATHMathSciNetGoogle Scholar
  9. [9]
    K. Gustafson, Some Perturbation Theorems for Nonnegative Contraction Semigroups, J. Math. Soc. Japan 21 (1969c), 200–204 (with Ken-iti Sato).zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    K. Gustafson, A Simple Proof of the Toeplitz-Hausdorff Theorem for Linear Operators, Proc. Amer. Math. Soc. 25 (1970), 203–204.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    K. Gustafson, Anti-eigenvalue Inequalities in Operator Theory, Inequalities III (O. Shisha, ed.), Academic Press (1972a), 115–119.Google Scholar
  12. [12]
    K. Gustafson, Multiplicative Perturbation of Semigroup Generators, Pac. J. Math. 41 (1972b), 731–742 (with G. Lumer).zbMATHMathSciNetGoogle Scholar
  13. [13]
    K. Gustafson, Multiplicative Perturbation of Nonlinear m-accretive Operators, J. Funct. Anal. 10 (1972c), 149–158 (with B. Calvert).zbMATHMathSciNetCrossRefGoogle Scholar
  14. [14]
    K. Gustafson, Numerical Range and Accretivity of Operator Products, J. Math. Anal. Applic. 60 (1977), 693–702 (with D. Rao).zbMATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    K. Gustafson, The RKNG (Rellich, Kato, Nagy, Gustafson) Perturbation Theorem for Linear Operators in Hilbert and Banach Space, Acta Sci. Math. 45 (1983), 201–211.zbMATHMathSciNetGoogle Scholar
  16. [16]
    K. Gustafson, Antieigenvalue Bounds, J. Math. Anal. Applic. 143 (1989), 327–340 (with M. Seddighin).zbMATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    K. Gustafson, Antieigenvalues in Analysis, Fourth International Workshop in Analysis and its Applications, Dubrovnik, Yugoslavia, June 1–10, 1990 (C. Stanojevic and O. Hadzic, eds.), Novi Sad, Yugoslavi (1991), 57–69.Google Scholar
  18. [18]
    K. Gustafson, A Note on Total Antieigenvectors, J. Math. Anal. Applic. 178 (1993), 603–611 (with M. Seddighin).zbMATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    K. Gustafson, Operator Trigonometry, Linear and Multilinear Algebra 37 (1994a), 139–159.zbMATHMathSciNetGoogle Scholar
  20. [20]
    K. Gustafson, Antieigenvalues, Lin. Alg. and Applic. 208/209 (1994b), 437–454.MathSciNetCrossRefGoogle Scholar
  21. [21]
    K. Gustafson, Computational Trigonometry, Proc. Colorado Conf. on Iterative Methods, Vol. 1 (1994c), p. 1.zbMATHMathSciNetGoogle Scholar
  22. [22]
    K. Gustafson, Matrix Trigonometry, Lin. Alg. and Applic. 217 (1995), 117–140.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [23]
    K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, Kaigai Publishers, Tokyo, Japan (1996a), 169 pp.Google Scholar
  24. [24]
    K. Gustafson, Trigonometric Interpretation of Iterative Methods, Proc. Conf. Algebraic Multilevel Iteration Methods with Applications, (O. Axelsson, B. Polman, eds.), Nijmegen, Netherlands, June 13–15, (1996b), 23–29.Google Scholar
  25. [25]
    K. Gustafson, Commentary on Topics in the Analytic Theory of Matrices, Section 23, Singular Angles of a Square Matrix, Collected Works of Helmut Wielandt 2 (B. Huppert and H. Schneider, eds.), De Gruyters, Berlin (1996c), 356–367.Google Scholar
  26. [26]
    K. Gustafson, Numerical Range: The Field of Values of Linear Operators and Matrices, Springer, Berlin, (1997a), pp. 205 (with D. Rao).Google Scholar
  27. [27]
    K. Gustafson, Operator Trigonometry of Iterative Methods, Num. Lin Alg. with Applic. 34 (1997b), 333–347.MathSciNetCrossRefGoogle Scholar
  28. [28]
    K. Gustafson, Antieigenvalues, Encyclopaedia of Mathematics, Supplement 1, Kluwer Acad. Publ., Dordrecht, (1997c), 57.Google Scholar
  29. [29]
    K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, World Scientific, Singapore, (1997d), pp. 178.zbMATHGoogle Scholar
  30. [30]
    K. Gustafson, Domain Decomposition, Operator Trigonometry, Robin Condition, Contemporary Mathematics 218 (1998a), 455–560.MathSciNetGoogle Scholar
  31. [31]
    K. Gustafson, Operator Trigonometry of Wavelet Frames, Iterative Methods in Scientific Computation (1998b), 161–166; (J. Wang, M. Allen, B. Chen, T. Mathew, eds.), IMACS Series in Computational and Applied Mathematics 4, New Brunswick, NJ.Google Scholar
  32. [32]
    K. Gustafson, Semigroups and Antieigenvalues, Irreversibility and Causality-Semigroups and Rigged Hilbert Spaces, (A. Bohm, H. Doebner, P. Kielanowski, eds.), Lecture Notes in Physics 504, Springer, Berlin, (1998c), pp. 379–384.Google Scholar
  33. [33]
    K. Gustafson, Operator Trigonometry of Linear Systems, Proc. 8th IFAC Symposium on Large Scale Systems: Theory and Applications, (N. Koussoulas, P Groumpos, eds.), Patras, Greece, July 15–17, (1998d), 950–955. (Also published by Pergamon Press, 1999.)Google Scholar
  34. [34]
    K. Gustafson, Symmetrized Product Definiteness? Comments on Solutions 19-5.1–19-5.5, IMAGE: Bulletin of the International Linear Algebra Society 21 (1998e), 22.Google Scholar
  35. [35]
    K. Gustafson, Antieigenvalues: An Extended Spectral Theory, Generalized Functions, Operator Theory and Dynamical Systems, (I. Antoniou, G. Lumer, eds.), Pitman Research Notes in Mathematics 399, (1998f), 144–149, London.Google Scholar
  36. [36]
    K. Gustafson, Operator Trigonometry of the Model Problem, Num. Lin. Alg. with Applic. 5 (1998g), 377–399.zbMATHMathSciNetCrossRefGoogle Scholar
  37. [37]
    K. Gustafson, The Geometry of Quantum Probabilities, On Quanta, Mind, and Matter: Hans Primas in Context, (H. Atmanspacher, A. Amann, U. Mueller-Herold, eds.), Kluwer, Dordrecht (1999a), 151–164.Google Scholar
  38. [38]
    K. Gustafson, The Geometrical Meaning of the Kantorovich-Wielandt Inequalities, Lin. Alg. and Applic. 296 (1999b), 143–151.zbMATHMathSciNetCrossRefGoogle Scholar
  39. [39]
    K. Gustafson, Symmetrized Product Definiteness: A Further Comment, IMAGE: Bulletin of the International Linear Algebra Society 22 (1999c), 26.Google Scholar
  40. [40]
    K. Gustafson, A Computational Trigonometry and Related Contributions by Russians Kantorovich, Krein, Kaporin, Computational Technologies 4(No. 3) (1999d), 73–83, (Novosibirsk, Russia).zbMATHMathSciNetGoogle Scholar
  41. [41]
    K. Gustafson, On Geometry of Statistical Efficiency, (1999e), preprint.Google Scholar
  42. [42]
    K. Gustafson, The Trigonometry of Quantum Probabilities, Trends in Contemporary Infinite-Dimensional Analysis and Quantum Probability, (L. Accardi, H. Kuo, N. Obata, K. Saito, S. Si, L. Streit, eds.), Italian Institute of Culture, Kyoto (2000a), 159–173.Google Scholar
  43. [43]
    K. Gustafson, Semigroup Theory and Operator Trigonometry, Semigroups of Operators: Theory and Applications, (A.V. Balakrisnan, ed.), Birkhäuser, Basel (2000b), 131–140.Google Scholar
  44. [44]
    K. Gustafson, Quantum Trigonometry, Infinite-Dimensional Analysis, Quantum Probability, and Related Topics 3 (2000c), 33–52.zbMATHMathSciNetCrossRefGoogle Scholar
  45. [45]
    K. Gustafson, An Extended Operator Trigonometry, Lin. Alg. & Applic. 319 (2000d), 117–135.zbMATHMathSciNetCrossRefGoogle Scholar
  46. [46]
    K. Gustafson, An Unconventional Linear Algebra: Operator Trigonometry, Unconventional Models of Computation, UMC’2K, (I. Antoniou, C. Calude, M. Dinneen, eds.), Springer, London (2001a), 48–67.Google Scholar
  47. [47]
    K. Gustafson, Probability, Geometry, and Irreversibility in Quantum Mechanics, Chaos, Solitons and Fractals 12 (2001b), 2849–2858.zbMATHMathSciNetCrossRefGoogle Scholar
  48. [48]
    K. Gustafson, Operator Trigonometry of Statistics and Econometrics, Lin. Alg. and Applic. 354 (2002a), 151–158.MathSciNetGoogle Scholar
  49. [49]
    K. Gustafson, CP-Violation as Antieigenvector-Breaking, Advances in Chemical Physics 122 (2002b), 239–258.Google Scholar
  50. [50]
    K. Gustafson, Operator Trigonometry of Preconditioning, Domain Decomposition, Sparse Approximate Inverses, Successive Overrelaxation, Minimum Residual Schemes Num. Lin Alg. with Applic. 10 (2003a), 291–315.zbMATHMathSciNetCrossRefGoogle Scholar
  51. [51]
    K. Gustafson, Bell’s Inequalities, Contributions to the XXII Solvay Conference on Physics, (A. Borisov, ed.), Moscow-Izhevsk, ISC, Moscow State University (ISBN: 5-93972-277-6) (2003b), 501–517.Google Scholar
  52. [52]
    K. Gustafson, Bell’s Inequality and the Accardi-Gustafson Inequality, Foundations of Probability and Physics-2, (A. Khrennikov, ed.) Växjo University Press, Sweden (2003c), 207–223.Google Scholar
  53. [53]
    K. Gustafson, Bell’s Inequalities, The Physics of Communication, Proceedings of the XXII Solvay Conference on Physics, (I. Antoniou, V. Sadovnichy, H. Walther, eds.), World Scientific (2003d), 534–554.Google Scholar
  54. [54]
    K. Gustafson, Preconditioning, Inner Products, Normal Degree, 2003 International Conference on Preconditioning Techniques for Large Sparse Matrix Problems in Scientific and Industrial Applications, (E. Ng, Y. Saad, W.P. Tang, eds.), Napa, CA, 27–29, October (2003e), pp. 3.Google Scholar
  55. [55]
    K. Gustafson, An Inner Product Lemma, Num. Lin. Alg. with Applic. 11 (2004a), 649–659.MathSciNetCrossRefzbMATHGoogle Scholar
  56. [56]
    K. Gustafson, Normal Degree, Num. Lin. Alg. with Applic. 11 (2004b), 661–674.MathSciNetCrossRefzbMATHGoogle Scholar
  57. [57]
    K. Gustafson, Interaction Antieigenvalues, J. Math Anal. Applic. 299 (2004c), 174–185.zbMATHMathSciNetCrossRefGoogle Scholar
  58. [58]
    K. Gustafson, The Geometry of Statistical Efficiency, Research Letters Inf. Math. Sci. 8 (2005a), 105–121.Google Scholar
  59. [59]
    K. Gustafson, On the Eigenvalues Which Express Antieigenvalues, International J. of Mathematics and Mathematical Sciences 2005:10 (2005b), 1543–1554. (with M. Seddighin).zbMATHMathSciNetGoogle Scholar
  60. [60]
    K. Gustafson, Bell and Zeno, International J. of Theoretical Physics 44 (2005c), 1931–1940.MathSciNetGoogle Scholar
  61. [61]
    M. Krein, Angular Localization of the Spectrum of a Multiplicative Integral in a Hilbert Space, Funct. Anal. Appl. 3 (1969), 89–90.zbMATHMathSciNetCrossRefGoogle Scholar
  62. [62]
    H. Wielandt, Topics in the Analytic Theory of Matrices, University of Wisconsin Lecture Notes, Madison, (1967).Google Scholar
  63. [63]
    E. Asplund and V. Ptak, A Minmax Inequality for Operators and a Related Numerical Range, Acta Math. 126 (1971), 53–62.zbMATHMathSciNetCrossRefGoogle Scholar
  64. [64]
    P. Hess, A Remark on the Cosine of Linear Operators, Acta Sci Math. 32 (1971), 267–269.zbMATHMathSciNetGoogle Scholar
  65. [65]
    C. Davis, Extending the Kantorovich Inequalities to Normal Matrices, Lin. Alg. Appl. 31 (1980), 173–177.zbMATHCrossRefGoogle Scholar
  66. [66]
    B.A. Mirman, Antieigenvalues: Method of Estimation and Calculation, Lin. Alg. Appl. 49 (1983), 247–255.zbMATHMathSciNetCrossRefGoogle Scholar
  67. [67]
    L. Kantorovich, Functional Analysis and Applied Mathematics, Uspekhi Mat. Nauk 3 No. 6 (1948), 89–185.zbMATHGoogle Scholar
  68. [68]
    S. Drury, S. Liu, C.Y. Lu, S. Puntanen, and G.P.H. Styan, Some Comments on Several Matrix Inequalities with Applications to Canonical Correlations: Historical Background and Recent Developments, Sankhya: The Indian J. of Statistics 64, Series A, Pt. 2 (2002), 453–507.MathSciNetGoogle Scholar
  69. [69]
    R. Khatree, On Calculation of Antiegenvalues and Antieigenvectors, J. Interdisciplinary Math 4 (2001), 195–199.Google Scholar
  70. [70]
    R. Khatree, On Generalized Antieigenvalue and Antieigenmatrix of order r, Amer. J. of Mathematical and Management Sciences 22 (2002), 89–98.Google Scholar
  71. [71]
    R. Khatree, Antieigenvalues and Antieigenvectors in Statistics, J. of Statistical Planning and Inference 114 (2003), 131–144.CrossRefGoogle Scholar
  72. [72]
    K.C. Das, M. Das Gupta, K. Paul, Structure of the Antieigenvectors of a Strictly Accretive Operator, International J. of Mathematics and Mathematical Sciences 21 (1998), 761–766.zbMATHCrossRefGoogle Scholar
  73. [73]
    C.R. Rao, Anti-eigen and Anti-singular Values of a Matrix and Applications to Problems in Statistics, Research Letters Inf. Math. Sci. 2005:10 (2005), 53–76.Google Scholar
  74. [74]
    K. Gustafson, R. Hartman, Divergence-free Bases for Finite Element Schemes in Hydrodynamics, SIAM J. Numer. Analysis 20 (1983), 697–721.zbMATHMathSciNetCrossRefGoogle Scholar
  75. [75]
    M. Seddighin, Antieigenvalues and Total Antieigenvalues of Normal Operators, J. Math. Anal. Appl. 274 (2002), 239–254.zbMATHMathSciNetCrossRefGoogle Scholar
  76. [76]
    M. Seddighin, Antieigenvalue Inequalities in Operator Theory, International J. of Mathematics and Mathematical Sciences (2004), 3037–3043.Google Scholar
  77. [77]
    M. Seddighin, On the Joint Antieigenvalues of Operators in Normal Subalgebras, J. Math. Anal. Appl. (2005), to appear.Google Scholar
  78. [78]
    K. Gustafson, Distinguishing Discretization and Discrete Dynamics, with Application to Ecology, Machine Learning, and Atomic Physics, in Structure and Dynamics of Nonlinear Wave Phenomena, (M. Tanaka, ed.), RIMS Kokyuroku 1271, Kyoto, Japan (2002c), 100–111.Google Scholar
  79. [79]
    F. Chaitin-Chatelin, S. Gratton, On the Condition Numbers Associated with the Polar Factorization of a Matrix, Numerical Linear Algebra with Applications 7 (2000), 337–354.zbMATHMathSciNetCrossRefGoogle Scholar
  80. [80]
    R. Varga, Matrix Iterative Analysis, Prentice Hall, NJ, (1962).Google Scholar
  81. [81]
    K. Gustafson, B. Misra, Canonical Commutation Relations of Quantum Mechanics and Stochastic Regularity, Letters in Math. Phys. 1 (1976), 275–280.zbMATHCrossRefMathSciNetGoogle Scholar
  82. [82]
    K. Gustafson, I. Antoniou, Wavelets and Kolmogorov Systems, http://www.auth.gr/chi/PROJECTSWaveletsKolmog.html, (2004).Google Scholar
  83. [83]
    X. Dai, D.R. Larson, Wandering Vectors for Unitary Systems and Orthogonal Wavelets, Amer. Math. Soc. Memoirs 640, Providence, RI, (1998).Google Scholar
  84. [84]
    B. Sz. Nagy, C. Foias, Harmonic Analysis of Operators in Hilbert Space, North Holland, Amsterdam, (1970).Google Scholar
  85. [85]
    I. Antoniou, K. Gustafson, Haar Wavelets and Differential Equations, Differential Equations 34 (1998), 829–832.zbMATHMathSciNetGoogle Scholar
  86. [86]
    K. Gustafson, Wavelets as Stochastic Processes, Workshop on Wavelets and Wavelet-based Technologies, (M. Kobayashi, S. Sakakibara, M. Yamada, eds.), IBM-Japan, Tokyo, October 29–30, (1998h), 40–43.Google Scholar
  87. [87]
    I. Antoniou, K. Gustafson, Wavelets and Stochastic Processes, Mathematics and Computers in Simulation 49 (1999), 81–104.zbMATHMathSciNetCrossRefGoogle Scholar
  88. [88]
    I. Antoniou, K. Gustafson, The Time Operator of Wavelets, Chaos, Solitons and Fractals 11 (2000), 443–452.MathSciNetCrossRefzbMATHGoogle Scholar
  89. [89]
    J.M. Jauch, Foundations of Quantum Mechanics, Addison-Wesley, Reading, MA, (1968).zbMATHGoogle Scholar
  90. [90]
    K. Gustafson, The Geometrical Meaning of the Absolute Condition Number of the Hermitian Polar Factor of a Matrix, (2004d), preprint.Google Scholar
  91. [91]
    L.F. Richardson, The Approximate Arithmetical Solution by Finite Differences of Physical Problems Involving Differential Equations with an Application to the Stresses in a Masonry Dam, Phil. Trans. Roy. Soc. London A242 (1910), 307–357.Google Scholar
  92. [92]
    M. Griebel, P. Oswald, On the Abstract Theory of Additive and Multiplicative Schwarz Algorithms, Numerische Mathematik 70 (1995), 163–180.zbMATHMathSciNetCrossRefGoogle Scholar
  93. [93]
    P. Kopp, Martingales and Stochastic Integrals, Cambridge University Press, Cambridge (1984).zbMATHGoogle Scholar
  94. [94]
    R. Dudley, Real Analysis and Probability, Chapman & Hall, New York (1989).Google Scholar
  95. [95]
    K. Gustafson, Continued Fractions, Wavelet Time Operators, and Inverse Problems, Rocky Mountain J. Math 33 (2003f), 661–668.zbMATHMathSciNetGoogle Scholar
  96. [96]
    N. Levan, C. Kubrusly, A Wavelet “time-shift-detail” Decomposition, Math. Comput. Simulation 63 (2003), 73–78.zbMATHMathSciNetCrossRefGoogle Scholar
  97. [97]
    N. Levan, C. Kubrusly, Time-Shifts Generalized Multiresolution Analysis over Dyadic-scaling Reducing Subspaces, International J. Wavelets, Multiresolution and Information Processing 2 (2004), 237–248.zbMATHMathSciNetCrossRefGoogle Scholar
  98. [98]
    B. Misra, G. Sudarshan, The Zeno’s Paradox in Quantum Theory, J. of Mathematical Physics 18 (1977), 756–763.MathSciNetCrossRefGoogle Scholar
  99. [99]
    K. Gustafson, The Quantum Zeno Paradox and the Counter Problem, Foundations of Probability and Physics-2 (A. Khrennikov, ed.), Växjo University Press, Sweden (2003g), 225–236.Google Scholar
  100. [100]
    K. Gustafson, Reversibility and Regularity (2004), preprint, to appear, International J. of Theoretical Physics (2006).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Karl Gustafson
    • 1
  1. 1.Department of MathematicsUniversity of ColoradoBoulderUSA

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