Skip to main content
SpringerLink
Log in

Banach Spaces and Linear Operators

  • Chapter
  • First Online:
Nonstandard Analysis for the Working Mathematician

  • 2060 Accesses

Abstract

In this chapter we deal with old and new applications of nonstandard analysis to the theory of Banach spaces and linear operators. In particular we consider the structure theory of Banach spaces, basic operator theory, strongly continuous semigroups of operators, approximation theory of operators and their spectra, and the Fixed Point Property. 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. Non-experts in these field can, however, profit from this chapter by looking at the elementary applications with which we begin every section. Moreover we refer to the book (Siu-Ah Ng, JAMA Nonstandard methods in functional analysis, 2010, [45]) for those, who want to learn functional analysis and simultaneously nonstandard analysis.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. S. Albeverio, J.E. Fenstad, R. Høegh-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. Acta Appl. Math. 64, 33–73 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Ando, U. Haagerup, Ultraproducts of von Neumann algebras. J. Funct. Anal. 266, 6842–6913 (2014)

    Google Scholar 

  4. S.K. Berberian, Approximate proper vectors. Proc. Am. Math. Soc. 13, 111–114 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Baratella, S.A. Ng, Some properties of nonstandard hulls of Banach algebras. Bull. Belg. Math. Soc. Simon Stevin 18, 31–38 (2011)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. S. Buoni, R. Harte, A.W. Wickstead, Upper and lower Fredholm spectra I. Proc. Am. Math. Soc. 66, 309–314 (1977)

    MATH  MathSciNet  Google Scholar 

  8. A. Connes, Noncommutative Geometry (Academic Press, New York, 1994)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. M. Davis, Applied Nonstandard Analysis (Wiley, New York, 1977)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Dunford, J. Schwartz, Linear Operators Part I (Interscience Publishers, New York, 1958)

    Google Scholar 

  14. E.Y. Emel’yanov, U. Kohler, F. Räbiger, M.P.H. Wolff, Stability and almost periodicity of asymptotically dominated semigroups of positive operators. Proc. Am. Math. Soc. 129, 2633–2642 (2001)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  16. E.I. Gordon, A.G. Kusraev, S.S. Kutadeladze, Infinitesimal Analysis (Kluver Academic Publisher, Dordrecht, 2002)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  18. U. Groh, Uniformly ergodic theorems for identity preserving Schwarz maps on \(W^{\star }\)-algebras. J. Oper. Theory 11, 395–402 (1984)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  20. S. Heinrich, C.W. Henson, L.C. Moore, A note on elementary equivalence of \(C(K)\) spaces. J. Symb. Log. 52, 368–373 (1987)

    Google Scholar 

  21. C.W. Henson, L.C. Moore, The nonstandard theory of topological vector spaces. Trans. Am. Math. Soc. 172, 405–435 (1972)

    Article  MathSciNet  Google Scholar 

  22. C.W. Henson, Nonstandard hulls of Banach spaces. Isr. J. Math. 25, 108–144 (1976)

    Article  MATH  MathSciNet  Google Scholar 

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

    Chapter  Google Scholar 

  24. C.W. Henson, J. Iovino, Ultraproducts in analysis, in Analysisand Logic, ed. by C. Finet (e.) et al. Report on three minicourses given at the international conference "Analyse et logique" Mons, Belgium, 25-29 August, Cambridge University Press, London Mathematical Society Lecture Notes Series , vol. 262 (2002), pp. 1-110

    Google Scholar 

  25. E. Hewitt, K.H. Ross, Abstract Harmonic Analysis I (Springer, Berlin, 1963)

    Book  MATH  Google Scholar 

  26. E. Hille, R.S. Phillips, Functional Analysis and Semi-groups, vol. 31 (American Mathematical Society Colloquium Publications, Providence, 1957)

    Google Scholar 

  27. T. Hinokuma, M. Ozawa, Conversion from nonstandard matrix algebras to standard factors of type \(II_{1}\). Ill. Math. J. 37, 1–13 (1993)

    MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  30. G. Janssen, Restricted ultraproducts of finite von Neumann algebras, in Contributions to Non-Standard Analysis. Studies in Logic and the Foundations of Mathematics, vol. 69. ed. by W.A.J. Luxemburg, A. Robinson (North Holland, Amsterdam, 1972), pp. 101–114

    Google Scholar 

  31. M.A. Khamsi, B. Sims, Ultra-methods in metric fixed point theory, in Handbook of Metric Fixed Point Theory, ed. by W. Kirk, B. Sims (Kluwer Academic Publishers, Dordrecht, 2001), pp. 177–199

    Chapter  Google Scholar 

  32. T. Kato, Perturbation Theory of Operators (Springer, Berlin, 1976)

    Book  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  34. H.E. Lacey, The Isometric Theory of Classical Banach Spaces (Springer, New York, 1974)

    Book  MATH  Google Scholar 

  35. J. Lindenstrauss, H. Rosenthal, The \({\cal {L}}_{p}\) spaces. Isr. J. Math. 7, 325–349 (1969)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  37. W.A.J. Luxemburg, Near–standard compact internal linear operators, in: Developments in nonstandard mathematics, eds. by N. J. Cutland et al. Pitman Res. Notes Math. Ser. vol. 336 (London, 1995) pp. 91–98

    Google Scholar 

  38. A. Martínez-Abejón, An elementary proof of the principle of local reflexivity. Proc. Am. Math. Soc. 127, 1397–1398 (1999)

    Article  MATH  Google Scholar 

  39. B. Maurey, Points fixes des contractions de certains faiblement compact de \(L^{1}\). Seminaire d’analyses fonctionelle 1980–81 (Ecole Polytechnique, Palaiseau, 1981)

    Google Scholar 

  40. P. Meyer-Nieberg, Banach Lattices (Springer, New York, 1991)

    Book  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  42. G.J. Murphy, \({C^\ast }\) -Algebras and Operator Theory (Academic Press, Boston, 1990)

    Google Scholar 

  43. R. Nagel (ed.), One–Parameter Semigroups of Positive Operators. Lecture Notes in Mathematics vol. 1184 (Springer, Berlin, 1986)

    Google Scholar 

  44. E. Nelson, Internal set theory: a new approach to nonstandard analysis. Bull. Am. Math. Soc. 83, 1165–1198 (1977)

    Article  MATH  Google Scholar 

  45. Siu-Ah Ng, Nonstandard methods in Functional Analysis. Lecture and Notes (World Scientific, Singapore, 2010)

    Google Scholar 

  46. S. Prus, Banach spaces which are uniformly noncreasy, in Proceedings of 2nd World Congress of Nonlinear Analysis (Athens 1996). Nonlinear Anal. 30, 2317–2324 (1997)

    Google Scholar 

  47. H.J. Reinhardt, Analysis of Approximation Methods for Differential and Integral Equations (Springer, Berlin, 1985)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    Article  MATH  Google Scholar 

  51. F. Räbiger, M.P.H. Wolff, On the approximation of positive operators and the behaviours of the spectra of the approximants. Integral Equ. Oper. Theory 28, 72–86 (1997)

    Article  MATH  Google Scholar 

  52. F. Räbiger, M.P.H. Wolff, Spectral and asymptotic properties of resolvent-dominated operators. J. Aust. Math. Soc. 68, 181–201 (2000)

    Article  MATH  Google Scholar 

  53. C.E. Rickart, General Theory of Banach Algebras (Van Nostrand, Princeton, 1960)

    MATH  Google Scholar 

  54. A. Robert, Functional analysis and NSA, in Developments in Nonstandard Mathematics, Pitman Research Notes in Mathematics Series, vol. 336, ed. by N.J. Cutland, et al. (Springer, London, 1995), pp. 73–90

    Google Scholar 

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

    MathSciNet  Google Scholar 

  56. S. Sakai, \({C^\ast }\) -Algebras (Springer, Berlin, 1971)

    Google Scholar 

  57. H.H. Schaefer, Banach Lattices and Positive Operators (Springer, Berlin, 1974)

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  59. M. Seidel, B. Silbermann, Banach algebras of operator sequences. Op. Matrices 6, 385–432 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  60. B. Sims, “Ultra”-techniques in Banach Space Theory. Queen’s Papers in Pure and Applied Mathematics , vol. 60 (Queen’s University, Kingston, 1982)

    Google Scholar 

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

    MATH  Google Scholar 

  62. 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, 45–49 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  63. D.G. Tacon, Generalized semi-fredholm transformations. J. Aust. Math. Soc. Ser. A 34, 60–70 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  64. M. Takesaki, Theory of Operator Algebras (Springer, Berlin, 1978)

    Google Scholar 

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

    Google Scholar 

  66. V.G. Troitsky, Measures of noncompactness of operators on Banach lattices. Positivity 8, 165–178 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  67. G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden (Teubner, Leipzig, 1976)

    MATH  Google Scholar 

  68. R. Werner, M.P.H. Wolff, Classical mechanics as quantum mechanics with infinitesimal \(\hbar \). Phys. Lett. Ser. A 202, 155–159 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  69. A. Wiśnicki, Towards the fixed point property for superreflexive spaces. Bull. Aust. Math. Soc. 64, 435–444 (2001)

    Article  MATH  Google Scholar 

  70. A. Wiśnicki, On the structure of fixed point sets of nonexpansive mappings, in: Proceedings of the 3rd Polish Symposium on Nonlinear Analysis. Lecture Note in Nonlinear Analysis vol. 3, (2002), pp. 169-174

    Google Scholar 

  71. A. Wiśnicki, The super fixed point property for asymptotically nonexpansive mappings. Fundam. Math. 217, 265–277 (2012)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  73. M.P.H. Wolff, Spectral theory of group representations and their nonstandard hull. Isr. J. Math. 48, 205–224 (1984)

    Article  MATH  Google Scholar 

  74. M.P.H. Wolff, An application of spectral calculus to the problem of saturation in approximation theory. Note di Matematica XII, 291–300 (1992)

    Google Scholar 

  75. M.P.H. Wolff, A nonstandard analysis approach to the theory of quantum meanfield systems, in Advances in Analysis, Probability and Mathematical Physics—Contributions of Nonstandard Analysis, ed. by S. Albeverio, W.A.J. Luxemburg, M.P.H. Wolff (Kluwer Academic Publisher, Dordrecht, 1995), pp. 228–246

    Chapter  Google Scholar 

  76. M.P.H Wolff, An introduction to nonstandard functional analysis, in: Proceedings of Nonstandard Analysis–Theory and Applications eds. by L.O. Arkeryd, N.J. Cutland C.W. Henson (Kluwer Academic Publisher, Dordrecht, 1997), pp. 121–151

    Google Scholar 

  77. M.P.H. Wolff, On the approximation of operators and the convergence of the spectra of the approximants. Op. Theory: Adv. Appl. 13, 279–283 (1998)

    Google Scholar 

  78. M.P.H. Wolff, Discrete approximation of unbounded operators and the convergence of the spectra of the approximants. J. Approx. Theory 113, 229–244 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  79. M.P.H Wolff, Discrete approximation of compact operators and approximation of their spectra, in: Nonstandard Methods and Applications in Mathematics, Lecture Notes in Logic vol. 25 (206) eds. by N.J. Cutland, Mauri Di Nasso, David A. Ross (A.K. Peters Ltd., Wellesley) pp. 224–231

    Google Scholar 

  80. M. Yahdi, Super-ergodic operators. Proc. Am. Math. Soc. 134, 2613–2620 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  81. K. Yosida, Functional Analysis, 2nd edn. (Springer, Berlin, 1968)

    Book  Google Scholar 

Download references

Acknowledgments

I would like to thank Prof. C. W. Henson, University of Illinois at Champaign-Urbana who gave me the important reference to the work of D. Dacunha-Castelles and J. L. Krivine, and who read very carefully an earlier version of Sects. 4.1 and 4.2; Prof. E. Gordon, University of Nishni Novgorod and now Eastern Illinois University, with whom I discussed his own approach to the theory of discrete approximation; Dr. H. Ploss, University at Vienna, for many helpful discussions on the theory of strongly continuous semigroups, some of which prevented me from unsightly errors; and Prof. Dr. Eduard Emel’yanov from the Middle East Technical University at Ankara who carefully read the final version of the first edition eliminating some more misprints and errors.

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076, Tübingen, Germany

    Manfred P. H. Wolff

Authors
  1. Manfred P. H. Wolff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manfred P. H. Wolff .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Illinois, Urbana, Illinois, USA

    Peter A. Loeb

  2. Mathematical Institute, University of Tübingen, Tübingen, Germany

    Manfred P. H. Wolff

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Wolff, M.P.H. (2015). Banach Spaces and Linear Operators. In: Loeb, P., Wolff, M. (eds) Nonstandard Analysis for the Working Mathematician. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-7327-0_4

Download citation

Publish with us

Policies and ethics

Search

Navigation