Advertisement

Positivity

, Volume 23, Issue 4, pp 829–851 | Cite as

Unbounded asymptotic equivalences of operator nets with applications

  • Nazife Erkurşun-Özcan
  • Niyazi Anıl GezerEmail author
Article
  • 74 Downloads

Abstract

Present paper deals with applications of asymptotic equivalence relations on operator nets. These relations are defined via unbounded convergences on vector lattices. Given two convergences \(\mathfrak {c}\) and \(\mathfrak {d}\) on a vector lattice, we study \(\mathfrak {d}\)-asymptotic properties of operator nets formed by \(\mathfrak {c}\)-continuous operators. Asymptotic equivalences are known to be useful and extremely important tools to study infinite behaviors of strongly convergent operator nets and continuous semigroups. After giving a general theory, paper focuses on \(\mathfrak {d}\)-martingale and \(\mathfrak {d}\)-Lotz–Räbiger nets.

Keywords

Unbounded convergence Asymptotic equivalence Operator nets 

Mathematics Subject Classification

47A35 47B65 46B42 54A20 

Notes

Acknowledgements

We thank the referee for his/her valuable comments and suggestions.

References

  1. 1.
    Abramovich, Y., Aliprantis, C.D.: An Invitation to Operator Theory, vol. 50. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beattie, R., Butzmann, H.-P.: Convergence Structures and Applications to Functional Analysis. Kluwer, Dordrech (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Deng, Y., O’Brien, M., Troitsky, V.G.: Unbounded norm convergence in Banach lattices. Positivity 21(3), 963–974 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dabboorasad, Y.A.M., Emel’yanov, E.Yu.: Unbounded convergence in the convergence vector lattices: a survey. Vladikavkaz. Mat. Zh 20(2), 49–56 (2018)Google Scholar
  6. 6.
    Emel’yanov, E.Yu.: On quasi-compactness of operator nets on Banach spaces. Studia Math. 203(2), 163–170 (2011)Google Scholar
  7. 7.
    Emel’yanov, E.Yu.: Asymptotic behaviour of Lotz-Räbiger and martingale nets. Sib. Math. J. 51(5), 810–817 (2010)Google Scholar
  8. 8.
    Emel’yanov, E.Yu.: Non-spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Advances and Applications, vol. 173. Birkhauser Verlag, Basel (2007)Google Scholar
  9. 9.
    Emel’yanov, E.Yu., Erkursun, N.: Asymptotically absorbing nets of positive operators. Sib. Adv. Math. 22(4), 243–260 (2012)Google Scholar
  10. 10.
    Emel’yanov, E.Yu., Erkursun, N.: Generalization of Eberlein’s and Sine’s ergodic theorems to \(LR\)-nets. Vladikavkaz. Mat. Zh 9(3), 22–26 (2007)Google Scholar
  11. 11.
    Emel’yanov, E.Yu., Erkursun, N.: Lotz-Räbiger’s nets of Markov operators in \(L^1\)-spaces. J. Math. Anal. Appl. 371(1), 777–783 (2010)Google Scholar
  12. 12.
    Erkursun-Ozcan, N.: Asymptotic behaviour of operator sequences on KB-spaces. Positivity 22(3), 803–814 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Erkursun-Ozcan, N.: Stability and lower-bound functions of C0-Markov semigroups on KB-spaces. Commun. Fac. Sci. Univ. Ank. Sr. A1 Math. Stat. 67(1), 242–247 (2018)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gao, N.: Unbounded order convergence in dual spaces. J. Math. Anal. Appl. 419, 347–354 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gao, N., Troitsky, V.G., Xanthos, F.: Uo-convergence and its applications to Cesro means in Banach lattices. Israel J. Math. 220(2), 649–689 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gao, N., Xanthos, F.: Unbounded order convergence and application to martingales without probability. J. Math. Anal. Appl. 415, 931–947 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kandić, M., Marabeh, M.A.A., Troitsky, V.G.: Unbounded norm topology in Banach lattices. J. Math. Anal. Appl. 451(1), 259–279 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lotz, H.P., Tauberian theorems for operators on \(L^{\infty }\) and similar spaces. In: Functional Analysis: Surveys and Recent Resuls. III (Paderborn, : North-Holland Math. Stud., 90. Amsterdam), 1984, 117–133 (1983)Google Scholar
  19. 19.
    Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  20. 20.
    Räbiger, F.: Stability and ergodicity of dominated semigroups: II. The strong case. Math. Ann. 297, 103–116 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Troitsky, V.G.: Measures of non-compactness of operators on Banach lattices. Positivity 8(2), 165–178 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Wickstead, A.W.: Weak and unbounded order convergence in Banach lattices. J. Aust. Math. Soc. Ser. A 24, 312–319 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zaanen, A.C.: Riesz Spaces II, vol. 30. North-Holland Mathematical Library, Amsterdam (1983)CrossRefzbMATHGoogle Scholar
  24. 24.
    Zabeti, O.: Unbounded absolute weak convergence in Banach lattices. Positivity 22(3), 803–814 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsHacettepe UniversityAnkaraTurkey
  2. 2.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey

Personalised recommendations