Skip to main content
SpringerLink
Log in

The positive Schur property on positive projective tensor products and spaces of regular multilinear operators

  • Original Research
  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We characterize the positive Schur property in the positive projective tensor products of Banach lattices, we establish the connection with the weak operator topology and we give necessary and sufficient conditions for the space of regular multilinear operators/homogeneous polynomials taking values in a Dedekind complete Banach lattice to have the positive Schur property.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)

    Book  Google Scholar 

  2. Aqzzouz, B., Elbour, A.: Some characterizations of almost Dunford-Pettis operators and applications. Positivity 15, 369–380 (2011)

    Article  MathSciNet  Google Scholar 

  3. Ardakani, H., Moshtaghioun, S.M., Mosadegh, S.M.S.M., Salimi, M.: The strong Gelfand-Phillips property in Banach lattices. Banach J. Math. Anal. 10, 15–26 (2016)

    Article  MathSciNet  Google Scholar 

  4. Baklouti, H., Hajji, M.: Schur operators and domination problem. Positivity 21, 35–48 (2017)

    Article  MathSciNet  Google Scholar 

  5. Botelho, G., Rueda, P.: The Schur property on projective and injective tensor products. Proc. Amer. Math. Soc. 137, 219–225 (2009)

    Article  MathSciNet  Google Scholar 

  6. Bu, Q., Buskes, G.: Polynomials on Banach lattices and positive tensor products. J. Math. Anal. Appl. 388, 845–862 (2012)

    Article  MathSciNet  Google Scholar 

  7. Dineen, S.: Complex Analysis on Infinite Dimensional Spaces. Springer, Berlin (1999)

    Book  Google Scholar 

  8. Fremlin, D.H.: Tensor products of archimedean vector lattices. Amer. J. Math. 94, 778–798 (1972)

    Article  MathSciNet  Google Scholar 

  9. Fremlin, D.H.: Tensor products of Banach lattices. Math. Ann. 211, 87–106 (1974)

    Article  MathSciNet  Google Scholar 

  10. González, M., Gutiérrez, J.: The Dunford-Pettis property on tensor products. Math. Proc. Cambridge Philos. Soc. 131, 185–192 (2001)

    Article  MathSciNet  Google Scholar 

  11. Ji, D., Navoyan, K., Bu, Q.: Complementation in Fremlin vector lattice symmetric tensor products II. Ann. Funct. Anal. 11, 47–61 (2020)

    Article  MathSciNet  Google Scholar 

  12. Kamińska, A., Mastyło, M.: The Schur and (weak) Dunford-Pettis properties in Banach lattices. J. Austral. Math. Soc. 73, 251–278 (2002)

    Article  MathSciNet  Google Scholar 

  13. Lust, F.: Produits tensoriels injectifs d’espaces de Sidon. Colloq. Math. 32, 285–289 (1975)

  14. Meyer-Nieberg, P.: Banach Lattices. Springer-Verlag, Berlin (1991)

    Book  Google Scholar 

  15. Moussa, M., Bouras, K.: About positive weak Dunford-Pettis operators on Banach lattices. J. Math. Anal. Appl. 381, 891–896 (2011)

    Article  MathSciNet  Google Scholar 

  16. Mujica, J.: Complex Analysis in Banach Spaces. Dover Publications, United States (2010)

    Google Scholar 

  17. Retbi, A.: A complement of positive weak almost Dunford-Pettis operators on Banach lattices. Arch. Math. (Brno) 55(1), 1–6 (2019)

    Article  MathSciNet  Google Scholar 

  18. Ryan, R.A.: The Dunford-Pettis property and projective tensor products. Bull. Polish Acad. Sci. Math. 35, 785–792 (1987)

    MathSciNet  MATH  Google Scholar 

  19. J.A. Sánchez, Operators on Banach lattices (Spanish), Ph.D. Thesis, Universidad Complutense de Madrid, Madrid, 1985

  20. Schep, A.R.: Factorization of positive multilinear maps. Illinois J. Math. 28, 579–591 (1984)

    Article  MathSciNet  Google Scholar 

  21. Tradacete, P.: Positive Schur properties in spaces of regular operators. Positivity 19, 305–316 (2015)

    Article  MathSciNet  Google Scholar 

  22. Wnuk, W.: Some remarks on the positive Schur property in spaces of operators. Funct. Approx. Comment. Math. 21, 65–68 (1992)

    MathSciNet  MATH  Google Scholar 

  23. Wnuk, W.: Banach lattices with the weak Dunford-Pettis property. Atti Sem. Mat. Fis. Univ. Modena 42(1), 227–236 (1994)

    MathSciNet  MATH  Google Scholar 

  24. Wnuk, W.: The strong Schur property in Banach lattices. Positivity 13, 435–441 (2009)

    Article  MathSciNet  Google Scholar 

  25. Wnuk, W.: On the dual positive Schur property in Banach lattices. Positivity 17, 759–773 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank Vladimir G. Troitsky, Anthony W. Wickstead and José Lucas P. Luiz for their very important suggestions.

Author information

Authors and Affiliations

  1. Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, MG, Brazil

    Geraldo Botelho & Khazhak Navoyan

  2. Department of Mathematics, University of Mississippi, Oxford, MS, 38677, USA

    Qingying Bu

  3. Department of Mathematics, Harbin University of Science and Technology, Harbin, 150080, China

    Donghai Ji

Authors
  1. Geraldo Botelho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qingying Bu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Donghai Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Khazhak Navoyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geraldo Botelho.

Additional information

Communicated by Christiane Winter-Todorov.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first author is the corresponding author and is supported by CNPq Grant 304262/2018-8 and Fapemig Grant PPM-00450-17. The fourth author is supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Botelho, G., Bu, Q., Ji, D. et al. The positive Schur property on positive projective tensor products and spaces of regular multilinear operators. Monatsh Math 197, 565–578 (2022). https://doi.org/10.1007/s00605-022-01677-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-022-01677-2

Keywords

Mathematics Subject Classification

Search

Navigation