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Functional Analysis

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Nonstandard Analysis for the Working Mathematician

Part of the book series: Mathematics and Its Applications ((MAIA,volume 510))

Abstract

In this chapter we deal with old and new applications of nonstandard analysis to functional analysis. In particular we consider the structure theory of Banach spaces, basic operator theory, strongly continuous semigroups of operators, and approximation theory of operators and their spectra. To include in this chapter interesting examples of nonstandard functional analysis we must assume that the reader is familiar with the basics of Banach spaces and operator theory. Nonexperts in these field can, however, profit from this chapter by looking at the elementary applications with which we begin every section.

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References

  1. S. Albeverio, J. E. Fenstad, R. Hoegh-Krohn, T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orlando, 1986.

    MATH  Google Scholar 

  2. S. Albeverio, E. Gordon, A. Khrennikov, Finite dimensional approximations of operators in the spaces of functions on locally compact abelian groups, submitted to Acta Applic. Mathem, 1999

    Google Scholar 

  3. S. K. Berberian, Approximate proper vectors, Proc. AMS 13(1962), 111–114.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. R. Bernstein, A. Robinson, Solution of an invariant subspace problem of K. T. mith and P. R. Halmos, Pacific J. Math. 16(1966), 421–431

    MathSciNet  MATH  Google Scholar 

  5. S. Buoni, R. Harte, A. W. Wickstead, Upper and lower Fredholm spectra I, Proc. AMS 66(1977), 309–314.

    MathSciNet  MATH  Google Scholar 

  6. A. Connes, Noncommutative Geometry, Academic Press, New York, 1994.

    MATH  Google Scholar 

  7. J. J. M. Chadwick, A. W. Wickstead, A quotient of ultrapowers of Banach spaces and semi-Fredholm operators, Bull. London Math. Soc. 9(1977), 321–325.

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Dacunha-Castelle, J. L. Krivine, Application des ultraproducts a l’etude des espaces et des algebres de Banach, Studia Math. 41(1995), 315–334.

    MathSciNet  Google Scholar 

  9. M. Davis, Applied Nonstandard Analysis, Wiley, New York, 1977.

    MATH  Google Scholar 

  10. R. Derndinger, Über das Spektrum positiver Generatoren, Mathm. Z. 172(1980), 281–293.

    Article  MathSciNet  MATH  Google Scholar 

  11. N. Dunford, J. Schwartz, Linear Operators Part I, Interscience Publishers, New York, 1958.

    Google Scholar 

  12. E. Y. Emel’yanov, U. Kohler, F. Räbiger, M. P. H Wolff, Stability and almost periodicity of asymptotically dominated semigroups of positive operators, 1998, to appear in Proc. Amer. Math. Soc.

    Google Scholar 

  13. P. Enflo, J. Lindenstrauss, G. Pisier, On the “three space Problem” Math. Scand. 36(1975), 199–210.

    MathSciNet  MATH  Google Scholar 

  14. G. Greiner, Zur Perron-Frobenius Theorie stark stetiger Halbgruppen, Math. Z 177(1981), 401–423.

    Article  MathSciNet  MATH  Google Scholar 

  15. U. Groh, Uniformly ergodic theorems for identity preserving Schwarz maps on W*-algebras, J. Operator Theory 11(1984), 395–402.

    MathSciNet  MATH  Google Scholar 

  16. S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313(1980), 72–104.

    MathSciNet  MATH  Google Scholar 

  17. S. Heinrich, C. W. Henson, L. C. Moore, A note on elementary equivalence of C(K) spaces, J. Symbolic Logic 52(1987), 368–373.

    Article  MathSciNet  MATH  Google Scholar 

  18. C. W. Henson, L. C. Moore, Nonstandard Analysis and the theory of Banach spaces, in: Nonstandard Analysis — Recent Developments, A. E. Hurd (ed.), Springer, Berlin, 1983, 27–112.

    Chapter  Google Scholar 

  19. E. Hewitt, K. H. Ross, Abstract Harmonic Analysis I, Springer, Berlin, 1963.

    MATH  Google Scholar 

  20. E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31, Providence, 1957.

    Google Scholar 

  21. T. Hinokuma, M. Ozawa, Conversion from nonstandard matrix algebras to standard factors of type II 1, Illinois Math. J. 37(1993), 1–13.

    MathSciNet  MATH  Google Scholar 

  22. A. E. Hurd, P. A. Loeb, An Introduction to Nonstandard Real Analysis, Academic Press, Orlando, 1985.

    MATH  Google Scholar 

  23. R. C. James, Characterizations of reflexivity, Studia Math. 23(1963/64), 205–216

    MathSciNet  Google Scholar 

  24. G. Janssen, Restricted ultraproducts of finite von Neumann algebras, in: Contributions to non-standard analysis, W. A. J. Luxemburg and A. Robinson (ed.), Studies in Logic and Found. Math. 69(1972), North Holland, Amsterdam, 101–114.

    Chapter  Google Scholar 

  25. T. Kato, Perturbation theory of operators Springer, Berlin, 1976.

    Book  Google Scholar 

  26. A. Krupa, On various generalizations of the notion of an F-power to the case of unbounded operators, Bull. Pol. Acad. Sci. Math. 38(1990), 159–166.

    MathSciNet  MATH  Google Scholar 

  27. H. E. Lacey, The isometric theory of classical Banach spaces Springer, New York, Heidelberg, 1974.

    Book  MATH  Google Scholar 

  28. W. A. J. Luxemburg, A general theory of monads, in: Applications of Model Theory to Algebra, Analysis, and Probability, W. A. J. Luxemburg (ed.), Holt, Rinehart, and Winston, New York, 1969, 18–86.

    Google Scholar 

  29. —, Near-standard compact internal linear operators, in: Developments in nonstandard mathematics, N. J. Cutland et al. (eds.), Pitman Res. Notes Math. Ser. 336, London, 1995, 91–98

    Google Scholar 

  30. P. Meyer-Nieberg, Banach lattices, Springer, New York, Heidelberg, 1991.

    Book  MATH  Google Scholar 

  31. G. Mittelmeyer, M. P. H. Wolff, Über den Absolutbetrag auf komplexen Vektorverbänden, Math. Z. 137(1974), 87–92.

    Article  MathSciNet  MATH  Google Scholar 

  32. G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, Boston, San Diego, 1990.

    Google Scholar 

  33. R. Nagel (ed.), One-parameter semigroups of positive operators, Lecture Notes in Math. 1184, Springer, Berlin, 1986.

    MATH  Google Scholar 

  34. H. J. Reinhardt, Analysis of approximation methods for differential and integral equations, Springer, Berlin, 1985.

    Book  MATH  Google Scholar 

  35. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.

    Book  MATH  Google Scholar 

  36. F. Räbiger, Wolff, M. P. H., Superstable semigroups of operators, Indag. Mathem., N.S. 6(1995), 481–494.

    Article  MATH  Google Scholar 

  37. F. Räbiger, M. P. H. Wolff, Spectral and asymptotic properties of dominated operators, J. Austral. Math. Soc. (Series A) 63(1997), 16–31.

    Article  MATH  Google Scholar 

  38. F. Räbiger, M. P. H. Wolff, On the approximation of positive operators and the behavious of the spectra of the approximants, Integral equ. oper. theory 28(1997), 72–86.

    Article  MATH  Google Scholar 

  39. F. Räbiger, M. P. H. Wolff, Spectral and asymptotic properties of resolvent-dominated operators, submitted to J. Austral Math. Soc., 1998.

    Google Scholar 

  40. C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, Princeton, 1960.

    MATH  Google Scholar 

  41. A. Robert, Functional analysis and NSA, in: Developments in nonstandard mathematics, N. J. Cutland et al. (eds.), Pitman Res. Notes Math. Ser. 336, London, 1995, 73–90.

    Google Scholar 

  42. B. N. Sadovskii, Limit-compact and condensing operators, Uspehi Math. Nauk 27(1972), 81–146.

    MathSciNet  Google Scholar 

  43. S. Sakai, C*-Algebras and W*-Algebras, Springer, Berlin, 1971.

    Book  Google Scholar 

  44. H. H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974.

    Book  MATH  Google Scholar 

  45. A. Schepp, M. P. H. Wolff, Semicompact operators, Indag. Mathem. New Ser. 1(1990), 115–125.

    Article  Google Scholar 

  46. B. Sims, “Ultra”-techniques in Banach Space Theory, Queen’s papers in pure and applied mathematics 60, Queen’s Univ., Kingston, Ontario, 1982.

    MATH  Google Scholar 

  47. K. D. Stroyan, W. A. J. Luxemburg, Introduction to the Theory of Infinitesimals, Academic Press, New York, 1976.

    MATH  Google Scholar 

  48. Yeneng Sun, A Banach space in which a ball is contained in the range of some countable additive measure is superreflexive, Canad. Math. Bull. 33(1990), 45–49.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Takesaki, Theory of Operator Algebras, Springer, Berlin, 1978.

    Google Scholar 

  50. L. N. Trefethen, Pseudospectra of matrices, in: Numerical Analysis, Proc. of the 14th Dundee Conference 1991, D. F. Griffiths (ed), Pitman, London, 1993, 234–264

    Google Scholar 

  51. G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig, 1976.

    MATH  Google Scholar 

  52. R. Werner, M. P. H. Wolff, Classical mechanics as quantum mechanics with infinitesimal h, Physics Letters, Ser. A 202(1995), 155–159.

    Article  MathSciNet  MATH  Google Scholar 

  53. M. P. H Wolff, R. Honegger, On the algebra of fluctuation operators of a quantum meanfield system, Quantum Probability Rel. Top. IX(1994), 401–410.

    Google Scholar 

  54. M. P. H Wolff, R. Honegger, Spectral theory of group representations and their nonstandard hull, IsraelJ. Math. 48(1984), 205–224

    Article  MATH  Google Scholar 

  55. —, An application of spectral calculus to the problem of saturation in approximation theory, Note di Matematica XII(1992), 291–300.

    MathSciNet  Google Scholar 

  56. —, A nonstandard analysis approach to the theory of quantum meanfield systems, in: Advances in Analysis, Probability and Mathematical Physics — Contributions of Nonstandard Analysis, S. Albeverio, W. A. J. Luxemburg and M. P. H. Wolff (eds.), Kluwer Acad. Publ, Dordrecht, 1995, 228–246.

    Google Scholar 

  57. —, An introduction to nonstandard functional analysis, in: Nonstandard Analysis-Theory and Applications, L. O. Arkeryd, N. J. Cutland and C. W. Henson (eds.), Kluwer Acad. Publ. Dordrecht, 1997, 121–151.

    Google Scholar 

  58. —, On the approximation of operators and the convergence of the spectra of the approximants, Operator Theory: Adv. and Apl. 13(1998), 279–283.

    MathSciNet  Google Scholar 

  59. —, Approximation of unbounded operators and the convergence of the spectra of the approximants, 1999, submitted to IMA J. Num. Anal.

    Google Scholar 

  60. K. Yosida, Functional Analysis, Springer, Berlin, 2nd. ed., 1968.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematische Fakultät der Universität Tübingen, Auf der Morgenstelle 10, D-72076, Tübingen, Germany

    Manfred P. H. Wolff

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  1. Manfred P. H. Wolff
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Editors and Affiliations

  1. Department of Mathematics, University of Illinois at Champaign-Urbana, Champaign-Urbana, Illinois, USA

    Peter A. Loeb

  2. Faculty of Mathematics, Universität Tübingen, Tübingen, Germany

    Manfred Wolff

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© 2000 Springer Science+Business Media Dordrecht

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Wolff, M.P.H. (2000). Functional Analysis. In: Loeb, P.A., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Mathematics and Its Applications, vol 510. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4168-0_4

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  • DOI: https://doi.org/10.1007/978-94-011-4168-0_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-0-7923-6341-5

  • Online ISBN: 978-94-011-4168-0

  • eBook Packages: Springer Book Archive

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