Skip to main content

Topics on the Approximation Problem of Almost Isometric Operators by Isometric Operators

  • Chapter

Part of the book series: Mathematics and Its Applications ((MAIA,volume 356))

Abstract

Let E, E 1 be two Banach spaces and 0 ≤ ε < 1. We say a mapping \( T \in \mathbb{B}(E,E_1 ) \) is “ε—isometric” or “ε—almost isometric” if

$$ (1 - \varepsilon )\left\| x \right\| \leqslant \left\| {Tx} \right\| \leqslant (1 + \varepsilon )\left\| x \right\| $$

for all xE. We say T is “isometric“ if we can take ε = 0.

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

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Michael, E. and Pelczynski, A., Separable spaces which admit approximations, Israel J. Math. 4 (1966), 189–198.

    Article  MathSciNet  MATH  Google Scholar 

  2. Huang Senzhong, Renorm and IAP, Chinese Ann. of Math. 9A (1988), 488–497.

    Google Scholar 

  3. Luo Yaohu, Some notes About ε—isometric operators, J. Shanxi Univ., 1 (1984), 24–30.

    Google Scholar 

  4. Li Shengjia, There is no isometric operator in \({\Bbb B}(C[0,\;1],L^p [0,\;1])\)----, 31–36.

    Google Scholar 

  5. Wang Sichun, The ε—isometric operators in \({\Bbb B}(l_n^\infty \to L(X,\mu ))\) and \({\Bbb B}(l_n^1 \to L(X,\mu ))\)(to appear).

    Google Scholar 

  6. Schechtman, G., Fine imbedding of finite dimensional subspaces of L p , 1 ≤ p < 2, into l 1 Proc. A.M.S. 94 (1985), 617–623.

    Google Scholar 

  7. Wang Risheng, Almost isometric imbeddings of two dimensional real (B)-spaces into l 1 (to appear).

    Google Scholar 

  8. Silvio Machado, Functional analysis, Holomorphy, and Approximation Theory, Lecture Notes in Math., 843, Springer-Verlag, 1981.

    Google Scholar 

  9. Kothe,G. Topological Vector Spaces II, Springer-Verlag, Berlin, 1979.

    Google Scholar 

  10. Ding Guanggui, Isometric and Almost Isometric Operators of \({\Bbb B}\) (L 1), Acta. Math. Sinica, 1 (1985), 126–140.

    Google Scholar 

  11. Schaefer, H. H., Banach Lattices and Positive Operators, Springer-Verlag, Berlin, 1974.

    MATH  Google Scholar 

  12. Huang Senzhong, On Operators that are almost isometric on the positive cones of L p—spaces, 1 < p < ∞, Proc. A. M. S., 106 (1989), 1039–1047.

    MATH  Google Scholar 

  13. Alspach, D. E., Small into isomorphism on \(L^p \) spaces, Illinois J. Math., 27 (1983), 300 – 314.

    MathSciNet  MATH  Google Scholar 

  14. Huang Senzhong, Small into isomorphism on L p spaces, 0 < p ≤ 1, (to appear).

    Google Scholar 

  15. Lacey, H. E., Isometric Theory of Classical Banach Spaces, Springer-Verlag Berlin, 1974.

    MATH  Google Scholar 

  16. Benyamini, Y., Small into isomorphisms between spaces of Continue functions, Proc. A. M. S., 33 (1981), 479–485.

    Article  MathSciNet  Google Scholar 

  17. Benyamini, Y., Small into-isomorphisms between spaces of Continuous functions II, Trans. A. M. S., 277 (1983), 825–833.

    Article  MathSciNet  MATH  Google Scholar 

  18. Ding Guanggui, The Approximation problem of almost isometric operators by isometric operators, Acta. Math. Sinica, 8 (1988), 361–372.

    MATH  Google Scholar 

  19. Ding Guanggui, Small into isomorphisms on \(L_\infty\) space, (to appear in “Acta Math. Sinica”).

    Google Scholar 

  20. Dunford, N. & Schwatz, J. T. Linear Operators I, New York, 1958.

    MATH  Google Scholar 

  21. Xiang Guangping, Approximation of some operators in \({\Bbb B}\) C, C), Acta. Sci. Nat. Univ. Nankai, 2 (1985), 27–33.

    Google Scholar 

  22. Xiao Yuanhui, The isometric approximation of positive operators from C(K) into C(S), Acta Sci. Nat. Univ. Nankai, 3 (1990), 16–22.

    Google Scholar 

  23. Wang Yaoting, A counterexample of approximating an almost isometric operator by an isometric operator in \({\Bbb B}(l^1 \to l^\infty )\) , J. of Shanxi Univ., 1 (1984), 19–23.

    Google Scholar 

  24. Huang Senzhong, Constructing operators which cannot be isometrically approximated, Acta. Math. Sci., 6: 2. (1986), 195–200.

    MathSciNet  MATH  Google Scholar 

  25. Wang Risheng, The problem of isometric approximation on the space\({\Bbb B}\)(l1 (Γ), C(Ω)), Chinese Sci. Bull., 10 (1990), 975–978.

    Google Scholar 

  26. Ding Guanggui, On Almost isometries From L 1 (µ) into L (v)or Cb(∆), Acta. Math. Sci., 12: 3 (1992), 308 – 311.

    MATH  Google Scholar 

  27. Wang Risheng. Isometric Approximations From Uniformly Smooth Spaces into l (Γ) Type spaces, Acta. Math. Sci., 9: 1 (1989),27–32.

    MathSciNet  MATH  Google Scholar 

  28. Liu Fei, The isometric approximation problem from uniformly smooth separable spaces to L (µ), (to appear).

    Google Scholar 

  29. Diestel, J. & Uhl, J. J. Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, Phode Island, 1977.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1996 Kluwer Academic Publishers

About this chapter

Cite this chapter

Ding, G. (1996). Topics on the Approximation Problem of Almost Isometric Operators by Isometric Operators. In: Li, B., Wang, S., Yan, S., Yang, CC. (eds) Functional Analysis in China. Mathematics and Its Applications, vol 356. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0185-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-0185-8_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6567-2

  • Online ISBN: 978-94-009-0185-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics