, Volume 21, Issue 4, pp 1393–1423 | Cite as

Almost band preservers



We study the stability of band preserving operators on Banach lattices. To this end the notion of \(\varepsilon \)-band preserving mapping is introduced. It is shown that, under quite general assumptions, a \(\varepsilon \)-band preserving operator is in fact a small perturbation of a band preserving one. However, a counterexample can be produced in some circumstances. Some results on automatic continuity of \(\varepsilon \)-band preserving maps are also obtained.


Banach lattice Band preserving operator Automatic continuity 

Mathematics Subject Classification

47B38 46B42 



TO partially supported by Simons Foundation travel Award 210060. PT partially supported by the Spanish Government Grants MTM2016-76808-P, MTM2016-75196-P, and Grupo UCM 910346. The authors wish to express their gratitude to the anonymous referee for many helpful suggestions and in particular for bringing Ref. [16] to our attention. We are also grateful to H. Vogt for making us aware of Ref. [15].


  1. 1.
    Abramovich, Y.A., Veksler, A.I., Koldunov, A.V.: On operators preserving disjointness. Soviet Math. Dokl. 20(5), 1089–1093 (1979)MATHGoogle Scholar
  2. 2.
    Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. In: Graduate Studies in Mathematics, vol. 50. American Mathematical Society, Providence, RI (2002)Google Scholar
  3. 3.
    Abramovich, Y.A., Kitover, A.K.: Inverses of disjointness preserving operators. Memoirs of the American Mathematical Society, vol. 143, no. 679. American Mathematical Society, Providence, RI (2000)Google Scholar
  4. 4.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)CrossRefMATHGoogle Scholar
  5. 5.
    Diestel, J., Fourie, J., Swart, J.: The Metric Theory of Tensor Products. Grothendieck’s résumé revisited. American Mathematical Society, Providence, RI (2008)CrossRefMATHGoogle Scholar
  6. 6.
    Huijsmans, C.B.: Disjointness preserving operators on Banach lattices. In: Operator Theory in Function Spaces and Banach Lattices. Operator Theory: Advances and Applications, vol. 75, pp. 173–189. Birkhäuser, Basel (1995)Google Scholar
  7. 7.
    Huijsmans, C.B., de Pagter, B.: Ideal theory in f-algebras. Trans. Am. Math. Soc. 269(1), 225–245 (1982)MathSciNetMATHGoogle Scholar
  8. 8.
    Kakutani, S.: Concrete representation of abstract \(L\)-spaces and the mean ergodic theorem. Ann. Math. 42, 523–537 (1941)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces II. Springer, Berlin (1979)CrossRefMATHGoogle Scholar
  10. 10.
    Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, Volume I. North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1971)Google Scholar
  11. 11.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefMATHGoogle Scholar
  12. 12.
    Nakano, H.: Teilweise geordnete algebra. Jpn. J. Math. 17, 425–511 (1941)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Oikhberg, T., Tradacete, P.: Almost disjointness preservers. Can. J. Math. doi: 10.4153/CJM-2016-020-x (in press)
  14. 14.
    Peralta, A.: Orthogonal forms and orthogonality preservers revisited. Linear Multilinear Algebra 65(2), 361–374 (2017)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Vogt, H., Voigt, J.: Bands in \(L_p\)-spaces. Math. Nachr. doi: 10.1002/mana.201600145 (in press)
  16. 16.
    Voigt, J.: The projection onto the center of operators in a Banach lattice. Math. Z. 199(1), 115–117 (1988)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Zaanen, A.C.: Examples of orthomorphisms. J. Approx. Theory 13, 192–204 (1975)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of IllinoisUrbanaUSA
  2. 2.Mathematics DepartmentUniversidad Carlos III de MadridLeganésSpain

Personalised recommendations