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A note on the concept of factorable strongly p-summing operators

  • Dumitru Popa
Original Paper
  • 69 Downloads

Abstract

It is known that an n-homogeneous polynomial is factorable strongly p-summing if and only if its linearization is absolutely p-summing. We prove that a similar result holds for n-linear operators and, as a consequence, we present simplified proofs of recent results from Pellegrino et al. (Rev R Acad Cienc Exactas Fís Nat Ser A Mat RACSAM 110(1):285–302, 2016). Using this connection and the well-known results of the classical linear theory of absolutely p-summing operators we prove other properties for this class of multilinear operators.

Keywords

Absolutely summing operators Strongly summing multilinear mappings Nuclear operators Projective tensor product 

Mathematics Subject Classification

Primary 46A32 46E30 46B07 Secondary 47B10 

Notes

Acknowledgments

We thank the two reviewers for their useful and constructive suggestions and remarks which improve the first version of this paper.

References

  1. 1.
    Botelho, G., Pellegrino, D., Rueda, P.: On composition ideals of multilinear mappings and homogeneous polynomials. Publ. Res. Inst. Math. Sci. 43(4), 1139–1155 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Carando, D., Dimant, V.: On summability of bilinear operators. Math. Nachr. 259(1), 3–11 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Defant, A., Floret, K.: Tensor Norms and Operator Ideals. North-Holland Mathematics Studies, vol. 176. North-Holland, Amsterdam (1993)Google Scholar
  4. 4.
    Diestel, J., Jarchow, H., Tonge, A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, vol. 43. Cambridge University Press, Cambridge (1995)Google Scholar
  5. 5.
    Dimant, V.: Strongly p-summing multilinear operators. J. Math. Anal. Appl. 278(1), 182–193 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Grothendieck, A.: Résume de la théorie metrique des produits tensoriels topologiques. Bol. Soc. Mat. São Paolo 8, 1–79 (1953/1956)Google Scholar
  7. 7.
    Pérez-García, D.: Comparing different classes of absolutely summing multilinear operators. Arch. Math. 85(3), 258–267 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pietsch, A.: Operator Ideals. Veb Deutscher Verlag der Wiss., Berlin (1978) [North Holland (1980)]Google Scholar
  9. 9.
    Pietsch, A.: Ideals of multilinear functionals. In: Proceedings of the Second International Conference on Operator Algebras, Ideals and Their Applications in Theoretical Physics, Teubner-Texte, Leipzig, pp. 185–199 (1983)Google Scholar
  10. 10.
    Pietsch, A.: Eigenvalues and s-Numbers. Akad. Verlagsgesellschaft Geest & Portig K-G, Leipzig (1987)zbMATHGoogle Scholar
  11. 11.
    Pisier, G.: Factorization of linear operators and geometry of Banach spaces. Reg. Conf. Ser. Math. 60, X (1986)Google Scholar
  12. 12.
    Pellegrino, D., Rueda, P., Sánchez-Pérez, E.A.: Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 110(1), 285–302 (2016)Google Scholar
  13. 13.
    Popa, D.: Nuclear multilinear operators with respect to a partition. Rend. Circ. Mat. Palermo (2) 61(3), 307–319 (2012)Google Scholar
  14. 14.
    Popa, D.: Composition results for strongly summing and dominated multilinear operators. Cent. Eur. J. Math. 12(10), 1433–1446 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Popa, D.: The splitting property for \((p,S)\)-summing operators. Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Mat. RACSAM (2016). doi: 10.1007/s13398-016-0284-4
  16. 16.
    Tomczak-Jagermann, N.: Banach–Mazur Distances and Finite Dimensional Operator Ideals. Pitman Monographs, vol. 38. Longman Scientific & Technical, Harlow (1989)Google Scholar

Copyright information

© Springer-Verlag Italia 2016

Authors and Affiliations

  1. 1.Department of MathematicsOvidius University of ConstantaConstantaRomania

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