On order structure and operators in L ∞(μ)

Irina Krasikova; Miguel Martín; Javier Merí; Vladimir Mykhaylyuk; Mikhail Popov

Open Mathematics (2009)

  • Volume: 7, Issue: 4, page 683-693
  • ISSN: 2391-5455

Abstract

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It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

How to cite

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Irina Krasikova, et al. "On order structure and operators in L ∞(μ)." Open Mathematics 7.4 (2009): 683-693. <http://eudml.org/doc/269189>.

@article{IrinaKrasikova2009,
abstract = {It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.},
author = {Irina Krasikova, Miguel Martín, Javier Merí, Vladimir Mykhaylyuk, Mikhail Popov},
journal = {Open Mathematics},
keywords = {Narrow operator; The space L ∞; AM-compact operator; rearrangement invariant Banach space; narrow operator},
language = {eng},
number = {4},
pages = {683-693},
title = {On order structure and operators in L ∞(μ)},
url = {http://eudml.org/doc/269189},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Irina Krasikova
AU - Miguel Martín
AU - Javier Merí
AU - Vladimir Mykhaylyuk
AU - Mikhail Popov
TI - On order structure and operators in L ∞(μ)
JO - Open Mathematics
PY - 2009
VL - 7
IS - 4
SP - 683
EP - 693
AB - It is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.
LA - eng
KW - Narrow operator; The space L ∞; AM-compact operator; rearrangement invariant Banach space; narrow operator
UR - http://eudml.org/doc/269189
ER -

References

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  7. [7] Kadets V.M., Popov M.M., On the Liapunov convexity theorem with applications to sign-embeddings, Ukr. Mat. Zh., 1992, 44(9), 1192–1200 http://dx.doi.org/10.1007/BF01058369[Crossref] Zbl0779.46039
  8. [8] Kadets V.M., Popov M.M., The Daugavet property for narrow operators in rich subspaces of the spaces C[0,1] and L 1[0,1], Algebra i Analiz, 1996, 8, 43–62 (in Russian), English translation: St. Petersburg Math. J., 1997, 8, 571–584 Zbl0881.47017
  9. [9] Krasikova I.V., A note on narrow operators in L ∞, Math. Stud., 2009, 31(1), 102–106 Zbl1199.47153
  10. [10] Lindenstrauss J., Pełczyński A., Absolutely summing operators in ℒp-spaces and their applications, Studia Math, 1968, 29, 275–326 Zbl0183.40501
  11. [11] Lindenstrauss J., Tzafriri L., Classical Banach Spaces, Vol. 1, Sequence spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977 Zbl0362.46013
  12. [12] Lindenstrauss J., Tzafriri L., Classical Banach Spaces, Vol. 2, Function spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1979 Zbl0403.46022
  13. [13] Maslyuchenko O.V, Mykhaylyuk V.V., Popov M.M., A lattice approach to narrow operators, Positivity, 2009, 13, 459–495 http://dx.doi.org/10.1007/s11117-008-2193-z[Crossref] Zbl1183.47033
  14. [14] Plichko A.M., Popov M.M., Symmetric function spaces on atomless probability spaces, Dissertationes Math., 1990, 306, 1–85 

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