Positivity

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

Almost band preservers

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Abstract

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.

Keywords

Banach lattice Band preserving operator Automatic continuity 

Mathematics Subject Classification

47B38 46B42 

Notes

Acknowledgements

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].

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

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

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