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Positivity

, Volume 13, Issue 3, pp 459–495 | Cite as

A lattice approach to narrow operators

  • Olexandr V. Maslyuchenko
  • Volodymyr V. Mykhaylyuk
  • Mykhaylo M. Popov
Article

Abstract.

It is known that if a rearrangement invariant function space E on [0,1] has an unconditional basis then each linear continuous operator on E is a sum of two narrow operators. On the other hand, the sum of two narrow operators in L1 is narrow. To find a general approach to these results, we extend the notion of a narrow operator to the case when the domain space is a vector lattice. Our main result asserts that the set N r (E, F) of all narrow regular operators is a band in the vector lattice L r (E, F) of all regular operators from a non-atomic order continuous Banach lattice E to an order continuous Banach lattice F. The band generated by the disjointness preserving operators is the orthogonal complement to N r (E, F) in L r (E, F). As a consequence we obtain the following generalization of the Kalton-Rosenthal theorem: every regular operator T : EF from a non-atomic Banach lattice E to an order continuous Banach lattice F has a unique representation as T = T D T N where T D is a sum of an order absolutely summable family of disjointness preserving operators and T N is narrow.

Mathematics Subject Classification (2000)

Primary 47B65 secondary 47B38 

Keywords

Narrow operator Vector lattice Disjointness preserving operator 

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References

  1. 1.
    Y.A. Abramovich, C.D. Aliprantis, An invitation to operator theory, Grad. Stud. Math., Amer. Math. Soc., Providence, Rhode Island, 50 (2002).Google Scholar
  2. 2.
    Y.A. Abramovich, A.K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc., Providence, Rhode Island, 143 (679) (2000).Google Scholar
  3. 3.
    C.D. Aliprantis, O. Burkinshaw, Positive operators, Springer, Dordrecht (2006).Google Scholar
  4. 4.
    J. Bourgain, Dunford-Pettis operators on L1 and the radon-Nikodym property, Isr. J. Math., 37(1–2)(1980), 34–47.Google Scholar
  5. 5.
    J. Bourgain, New classes of \(\mathcal {L}_p\) -spaces, Lect. Notes Math., 889 (1981), 1–143.Google Scholar
  6. 6.
    J. Diestel. Geometry of Banach Spaces, Springer, Berlin (1975).Google Scholar
  7. 7.
    P. Enflo, T. Starbird, Subspaces of L1 containing L1, Stud. Math., 65(2) (1979), 203–225.Google Scholar
  8. 8.
    J. Flores, C. Ruiz, Domination by positive narrow operators, Positivity. 7 (2003), 303–321.Google Scholar
  9. 9.
    J. Flores, P. Tradacete, V.G. Troitsky, Invariant subspaces of positive strictly singular operators on Banach lattices, J. Math. Anal. Appl., (to appear).Google Scholar
  10. 10.
    N. Ghoussoub, H.P. Rosenthal, Martingales, Gδ-embeddings and quotients of L1, Math. Anal., 264(3) (1983), 321–332.Google Scholar
  11. 11.
    W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 19(217), (1979).Google Scholar
  12. 12.
    V.M. Kadets, M.I. Kadets, Rearrangements of series in Banach spaces, Trans. Math. Monogr. Am. Math. Soc., 86 (1991).Google Scholar
  13. 13.
    V.M. Kadets, M.M. Popov, Some stability theorems about narrow operators on L1[0,1] and C(K), Matematicheskaya Fizika, Analiz, Geometria 10(1), (2003), 49–60.Google Scholar
  14. 14.
    N.J. Kalton, The endomorphisms of Lp (0 ≤ p ≤ 1), Indiana Univ. Math., J. 27(3) (1978), 353–381.Google Scholar
  15. 15.
    J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Vol. 1, Sequence spaces, Springer, Berlin (1977).Google Scholar
  16. 16.
    J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, Vol. 2, Function spaces, Springer, Berlin (1979).Google Scholar
  17. 17.
    Z. Liu, A decomposition theorem for operators on L1, J. Oper. Theory. 40(1) (1998), 3–34.Google Scholar
  18. 18.
    O.V. Maslyuchenko, V.V. Mykhaylyuk, M.M. Popov, Asymptotic norm and compact operators, Nauk. Visnyk Chernivets’koho Un-tu., 269 (2005), 73–75 (in Ukrainian).Google Scholar
  19. 19.
    O.V. Maslyuchenko, V.V. Mykhaylyuk, M.M. Popov, Theorems on representation of operators on L1 and their generalizations to vector lattices, Ukr. Mat. Zh. 58(1) (2006), 26–35 (in Ukrainian).Google Scholar
  20. 20.
    V.V. Mykhaylyuk, M.M. Popov, Some geometrical aspects of operators acting from L1, Positivity, 10 (2006), 431–466.Google Scholar
  21. 21.
    A.M. Plichko, M.M. Popov, Symmetric function spaces on atomless probability spaces, Dissertationes Math. (Rozprawy Mat.) 306 (1990), 1–85.Google Scholar
  22. 22.
    M.M. Popov, An elementary proof of the non-exastence of non-zero compact operators acting from the space Lp, 0 < p < 1, Mat. Zametki., 47(5) (1990), 154–155 (in Russian).Google Scholar
  23. 23.
    M.M. Popov, An exact Daugavet type inequality for small into isomorphisms in L1. Ark. Math. 90(6) (2008), 537–544.Google Scholar
  24. 24.
    M.M. Popov, B. Randrianantoanina, A pseudo-Daugavet property for narrow projections in Lorentz spaces, Ill. J. Math. 46(4) (2002), 1313–1338.Google Scholar
  25. 25.
    H.P. Rosenthal, Embeddings of L1 in L1, Contemp. Math., 26 (1984), 335–349.Google Scholar
  26. 26.
    H.H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin (1974).Google Scholar
  27. 27.
    R.V. Shvydkoy. Operators and integrals in Banach spaces, Dissertation, Univ. of Missouri-Columbia (2001).Google Scholar
  28. 28.
    R. Sikorski. A theorem on extension of homomorphisms, Ann. Soc. Pol. Math., 21 (1948), 332–335.Google Scholar
  29. 29.
    R. Sikorski, Boolean Algebras. Springer, Berlin (1964).Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Olexandr V. Maslyuchenko
    • 1
  • Volodymyr V. Mykhaylyuk
    • 1
  • Mykhaylo M. Popov
    • 1
  1. 1.Department of MathematicsChernivtsi National UniversityChernivtsiUkraine

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