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Positivity

, Volume 19, Issue 3, pp 475–487 | Cite as

Swartz type results for nuclear and multiple 1-summing bilinear operators on \(c_{0}\left( \mathcal {X}\right) \times c_{0}\left( \mathcal {Y}\right) \)

  • Gabriela Badea
  • Dumitru Popa
Article

Abstract

In this paper we investigate the bilinear versions of Swartz’s theorem. Thus we characterize the nuclear and the multiple 1-summing operators on a cartesian product of \(c_{0}\left( \mathcal {X}\right) \). As applications we give the necessary and sufficient conditions for some natural operators on a cartesian product of \(c_{0}\left( \mathcal {X}\right) \) to be multiple 1-summing and nuclear operators. By an example, we show that the natural bilinear version of Swartz’s theorem is not necessarily true.

Keywords

Multiple 1-summing bilinear operators Nuclear operators  Swartz’s Theorem 

Mathematics Subject Classification

Primary 47B10 47L20 Secondary 46B45 

Notes

Acknowledgments

We thank the referee for his/her carefully reading of the manuscript and useful suggestions which have improved the quality of the presentation.

References

  1. 1.
    Badea, G.: On nuclear and multiple summing bilinear operators on \(c_{0}\times c_{0}\). Quaest Math. 33, 1–9 (2010)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Badea, G., Popa, D.: The summing nature of the multiplication operator from \(l_{p}({\cal X})\) into \(c_{0}({\cal Y})\), submitted to Mediterranean J. Math.Google Scholar
  3. 3.
    Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem. Quart. J. Math. 55, 441–450 (2004)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32(3), 600–622 (1931)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Defant, A., Popa, D., Schwarting, U.: Coordinatewise multiple summing operators in Banach spaces. J. Funct. Anal. 259(1), 220–242 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland Math. Stud. 176 (1993)Google Scholar
  7. 7.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Camb. Stud. Adv. Math. 43, Cambridge University Press (1995)Google Scholar
  8. 8.
    Dineen, S.: Complex analysis in locally convex spaces. North-Holland Math. Stud. 57 (1981)Google Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory, Pure and Applied Mathematics, vol. 7. Interscience, New York (1958)Google Scholar
  10. 10.
    Garling, D.J.H.: Diagonal mappings between sequence spaces. Stud. Math. 51, 129–138 (1974)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Matos, M.C.: Fully absolutely summing and Hilbert-Schmidt multilinear mappings. Collect. Math. 54, 111–136 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Péréz-García, D., Villanueva, I.: Multiple summing operators on \(C(K)\)-spaces. Arkiv för Matematik. 42, 153–171 (2004)CrossRefzbMATHGoogle Scholar
  13. 13.
    Pietsch, A.: Operator ideals, Veb Deutscher Verlag der Wiss., Berlin 1978, North Holland (1980)Google Scholar
  14. 14.
    Popa, D.: Nuclear multilinear operators with respect to a partition. Rend. Circ. Mat. Palermo 61, 307–319 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Popa, D.: 2-summing multiplication operators on \(l_{p}({\cal X})\). Studia Math. 216(1), 77–96 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Popa, D.: \(2\)-summing operators on \(l_{2}({\cal X})\), Accepted for Publication in Operator and Matrices (2014). http://files.ele-math.com/preprints/oam-0897-pre.pdf
  17. 17.
    Popa, D.: Remarks on multiple summing operators on \(C(\Omega )\)-spaces. Positivity 18(1), 29–39 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Ramanujan, M.S., Schock, E.: Operator ideals and spaces of bilinear operators. Linear Multilinear Algebra 18, 307–318 (1985)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Swartz, Ch.: Absolutely summing and dominated operators on spaces of vector valued continuous functions. Trans. Am. Math. Soc. 179, 123–132 (1973)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Faculty of Civil EngineeringOvidius University of ConstantaConstantaRomania
  2. 2.Department of MathematicsOvidius University of ConstantaConstantaRomania

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