Quasi-martingale Inequalities in Noncommutative Symmetric Spaces

  • Congbian MaEmail author
  • Youliang Hou


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.


Noncommutative symmetric space Quasi-martingale Hardy space 

Mathematics Subject Classification

46L53 46L52 60G42 


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