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.
KeywordsBanach lattice Band preserving operator Automatic continuity
Mathematics Subject Classification47B38 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.  to our attention. We are also grateful to H. Vogt for making us aware of Ref. .
- 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.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
- 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
- 10.Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, Volume I. North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam (1971)Google Scholar
- 13.Oikhberg, T., Tradacete, P.: Almost disjointness preservers. Can. J. Math. doi: 10.4153/CJM-2016-020-x (in press)
- 15.Vogt, H., Voigt, J.: Bands in \(L_p\)-spaces. Math. Nachr. doi: 10.1002/mana.201600145 (in press)