Quasi-martingale Inequalities in Noncommutative Symmetric Spaces

Article
  • 7 Downloads

Abstract

In this paper, we prove Doob’s inequality and Burkholder–Gundy inequalities for quasi-martingales in noncommutative symmetric spaces. We also consider the inclusion relations between \(\widetilde{\mathcal {H}}_E(\mathcal {M})\) and \(\widetilde{h}_{E}(\mathcal {M})\), where \(\widetilde{\mathcal {H}}_E(\mathcal {M})\) and \(\widetilde{h}_{E}(\mathcal {M})\) are the Hardy spaces of quasi-martingales in noncommutative symmetric spaces.

Keywords

Noncommutative symmetric space Quasi-martingale Hardy space 

Mathematics Subject Classification

46L53 46L52 60G42 

References

  1. 1.
    Bekjan, T.N., Chen, Z.: Interpolation and \(\Phi \)-moment inequalities of noncommutative martingales. Probab. Theory Relat. Fields 152, 179–206 (2012)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press Inc., Boston (1988)MATHGoogle Scholar
  3. 3.
    Chilin, V.I., Sukochev, F.A.: Symmetric spaces over semifinite von Neumann algebras. Dokl. Akad. Nauk SSSR 313, 811–815 (1990)MATHGoogle Scholar
  4. 4.
    Dirksen, S.: Noncommutative Boyd interpolation theorems. Trans. Am. Math. Soc. 367, 4079–4110 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dirksen, S., de Pagter, B., Potapov, D., Sukochev, F.: Rosenthal inequalities in noncommutative symmetric spaces. J. Funct. Anal. 261, 2890–2925 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Jiao, Y.: Martingale inequalities in noncommutative symmetric spaces. Arch. Math. (Basel) 98, 87–97 (2012)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Junge, M.: Doob’s inequality for non-commutative martingales. J. Reine Angew. Math. 549, 149–190 (2002)MathSciNetMATHGoogle Scholar
  8. 8.
    Junge, M., Xu, Q.: Non-commutative Burkholder/Rosenthal inequalities. Ann. Probab. 31, 948–995 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Le Merdy, C., Sukochev, F.: Rademacher averages on noncommutative symmetric spaces. J. Funct. Anal. 255, 3329–3355 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. II. Springer, Berlin (1979). Function spacesCrossRefMATHGoogle Scholar
  11. 11.
    Randrianantoanina, N.: Non-commutative martingale transforms. J. Funct. Anal. 194, 181–212 (2002)MathSciNetMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceXinxiang UniversityXinxiangPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsWuhan UniversityWuhanPeople’s Republic of China

Personalised recommendations