Narrow operators on lattice-normed spaces

Marat Pliev

Open Mathematics (2011)

  • Volume: 9, Issue: 6, page 1276-1287
  • ISSN: 2391-5455

Abstract

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The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.

How to cite

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Marat Pliev. "Narrow operators on lattice-normed spaces." Open Mathematics 9.6 (2011): 1276-1287. <http://eudml.org/doc/269204>.

@article{MaratPliev2011,
abstract = {The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.},
author = {Marat Pliev},
journal = {Open Mathematics},
keywords = {Narrow operators; GAM-compact operators; Dominated operators; Lattice-normed spaces; Banach lattices; narrow operators; dominated operators; lattice-normed spaces},
language = {eng},
number = {6},
pages = {1276-1287},
title = {Narrow operators on lattice-normed spaces},
url = {http://eudml.org/doc/269204},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Marat Pliev
TI - Narrow operators on lattice-normed spaces
JO - Open Mathematics
PY - 2011
VL - 9
IS - 6
SP - 1276
EP - 1287
AB - The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.
LA - eng
KW - Narrow operators; GAM-compact operators; Dominated operators; Lattice-normed spaces; Banach lattices; narrow operators; dominated operators; lattice-normed spaces
UR - http://eudml.org/doc/269204
ER -

References

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  1. [1] Abramovich Y.A., Aliprantis C.D., An Invitation to Operator Theory, Grad. Stud. Math., 50, American Mathematical Society, Providence, 2002 Zbl1022.47001
  2. [2] Aliprantis C.D., Burkinshaw O., Positive Operators, Springer, Dordrecht, 2006 
  3. [3] Andrews K.T., Representation of compact and weakly compact operators on the space of Bochner integrable functions, Pacific J. Math., 1981, 92(2), 257–267 Zbl0427.47021
  4. [4] Bilik D., Kadets V., Shvidkoy R., Sirotkin G., Werner D., Narrow operators on vector-valued sup-normed spaces, Ilinois J. Math., 2006, 46(2), 421–441 Zbl1030.46014
  5. [5] Bilik D., Kadets V., Shvidkoy R., Werner D., Narrow operators and the Daugavet property for ultraproducts, Positivity, 2005, 9(1), 45–62 http://dx.doi.org/10.1007/s11117-003-9339-9 Zbl1099.46009
  6. [6] Boyko K., Kadets V., Werner D., Narrow operators on Bochner L 1-spaces, Zh. Mat. Fiz. Anal. Geom., 2002, 2(4), 358–371 Zbl1147.46006
  7. [7] Enflo P., Starbird T.W., Subspaces of L 1 containing L 1, Studia Math., 1979, 65(2), 213–225 
  8. [8] Flores J., Ruiz C., Domination by positive narrow operators, Positivity, 2003, 7(4), 303–321 http://dx.doi.org/10.1023/A:1026211909760 Zbl1051.47032
  9. [9] Ghoussoub N., Rosenthal H.P., Martingales, G δ-embeddings and quotients of L 1, Math. Ann., 1983, 264(3), 321–332 http://dx.doi.org/10.1007/BF01459128 Zbl0511.46017
  10. [10] Johnson W.B., Maurey B., Schechtman G., Tzafriri L., Symmetric Structures in Banach Spaces, Mem. Amer. Math. Soc., 19, American Mathematical Society, Providence, 1979 Zbl0421.46023
  11. [11] Kadets V.M., Kadets M.I., Rearrangements of Series in Banach Spaces, Transl. Math. Monogr., 86, American Mathematical Society, Providence, 1991 Zbl0743.46011
  12. [12] 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(4), 43–62 (in Russian) Zbl0881.47017
  13. [13] Kadets V.M., Shvidkoy R.V., Werner D., Narrow operators and rich subspaces of Banach spaces with the Daugavet property, Studia Math., 2001, 147(3), 269–298 http://dx.doi.org/10.4064/sm147-3-5 Zbl0986.46010
  14. [14] Kantorovich L.V., On a class of functional equations, Dokl. Akad. Nauk SSSR, 1936, 4(5), 211–216 
  15. [15] Kusraev A.G., Dominated Operators, Math. Appl., 519, Kluwer, Dordrecht, 2000 
  16. [16] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. I, Ergeb. Math. Grenzgeb., 92, Springer, Berlin-New York, 1977 Zbl0362.46013
  17. [17] Lindenstrauss J., Tzafriri L., Classical Banach Spaces. II, Ergeb. Math. Grenzgeb., 97, Springer, Berlin-New York, 1979 Zbl0403.46022
  18. [18] Maslyuchenko O.V., Mykhaylyuk V.V., Popov M.M., A lattice approach to narrow operators, Positivity, 2009, 13(3), 459–495 http://dx.doi.org/10.1007/s11117-008-2193-z Zbl1183.47033
  19. [19] Megginson R.E., An Introduction to Banach Space Theory, Grad. Texts in Math., 183, Springer, New York, 1998 http://dx.doi.org/10.1007/978-1-4612-0603-3 
  20. [20] Plichko A.M., Popov M.M., Symmetric Function Spaces on Atomless Probability Spaces, Dissertationes Math. (Rozprawy Mat.), 306, Polish Academy of Sciences, Warsaw, 1990 

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