Skip to main content
SpringerLink
Log in

Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials

  • Survey
  • Published:
Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We draw a fundamental compendium of the most valuable results of the theory of summing linear operators and detail those that are not shared by known multilinear and polynomial extensions of absolutely summing linear operators. The lack of such results in the theory of non-linear summing operators justifies the introduction of a class of polynomials and multilinear operators that satisfies at once all related non-linear results. Surprisingly enough, this class, defined by means of a summing inequality, happens to be the well known ideal of composition with a summing operator.

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. Achour, D., Dahia, E., Rueda, P., Sánchez-Pérez, E.A.: Factorization of absolutely continuous polynomials. J. Math. Anal. Appl. 405(1), 259–270 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albiac, F., Kalton, N.: Topics in Banac Space Theory. Springer, Berlin (2005)

    Google Scholar 

  3. Alencar, R., Matos, M.C.: Some classes of multilinear mappings between Banach spaces, Publicaciones del Departamento de Análisis Matemático 12, Universidad Complutense Madrid (1989)

  4. Bombal, F., Pérez-García, D., Villanueva, I.: Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55(4), 441–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Botelho, G., Braunss, H.-A., Junek, H., Pellegrino, D.: Holomorphy types and ideals of multilinear mappings. Studia Math. 177, 43–65 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Botelho, G., Pellegrino, D.: Scalar-valued dominated polynomials on Banach spaces. Proc. Am. Math. Soc. 134, 1743–1751 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Botelho, G., Pellegrino, D.: Absolutely summing polynomials on Banach spaces with unconditional basis. J. Math. Anal. Appl. 321, 50–58 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Botelho, G., Pellegrino, D.: Coincidence situations for absolutely summing non-linear mappings. Port. Math. (N.S.) 64(2), 175–191 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Botelho, G., Pellegrino, D., Rueda, P.: Pietsch’s factorization theorem for dominated polynomials. J. Funct. Anal. 243(1), 257–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365, 269–276 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Botelho, G., Pellegrino, D., Rueda, P.: Dominated polynomials on infinite dimensional spaces. Proc. Am. Math. Soc. 138(1), 209–216 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Botelho, G., Pellegrino, D., Rueda, P.: Cotype and absolutely summing linear operators. Math. Z. 267(1–2), 1–7 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Botelho, G., Pellegrino, D., Rueda, P.: On Pietsch measures for summing operators and dominated polynomials. Linear Multilinear Algebra 62(7), 860–874 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Çalışkan, E., Pellegrino, D.M.: On the multilinear generalizations of the concept of absolutely summing operators. Rocky Mountain J. Math. 37, 1137–1154 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carando, D., Dimant, V.: On summability of bilinear operators. Math. Nachr. 259, 3–11 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Carando, D., Dimant, V., Muro, S.: Coherent sequences of polynomial ideals on Banach spaces. Math. Nachr. 282, 1111–1133 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland Mathematics Studies, 176. North-Holland Publishing Co., Amsterdam (1993)

    Google Scholar 

  19. Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  20. Dimant, V.: Strongly \(p\)-summing multilinear operators. J. Math. Anal. Appl. 278, 182–193 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dineen, S.: Complex analysis on infinite-dimensional spaces. Springer, London (1999)

    Book  MATH  Google Scholar 

  22. Fabian, M., Hájek, P., Montesinos-Santalucía, V., Pelant, J., Zizler, V.: Functional analysis and infinite-dimensional geometry. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 8. Springer, New York (2001)

  23. Floret, K.: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17, 153–188 (1997)

    MathSciNet  MATH  Google Scholar 

  24. Geiss, H.: Ideale multilinearer Abbildungen. Diplomarbeit, Brandenburgische Landeshochschule (1985)

  25. Grothendieck, A.: Résumé de la théorie métrique des produits tensoriels topologiques (French). Bol. Soc. Mat. São Paulo 8, 1–79 (1953)

    MathSciNet  Google Scholar 

  26. Jarchow, H., Palazuelos, C., Pérez-García, D., Villanueva, I.: Hahn-Banach extension of multilinear forms and summability. J. Math. Anal. Appl. 336, 1161–1177 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in \({\cal L}_{p}\) spaces and their applications. Studia Math. 29, 275–326 (1968)

    MathSciNet  MATH  Google Scholar 

  28. Matos, M.C.: Absolutely summing holomorphic mappings. Anais da Academia Brasileira de Ciências 68, 1–13 (1996)

    MathSciNet  MATH  Google Scholar 

  29. Matos, M.C.: Fully absolutely summing and Hilbert–Schmidt multilinear mappings. Collectanea Math. 54, 111–136 (2003)

    MathSciNet  MATH  Google Scholar 

  30. Matos, M.C.: Nonlinear absolutely summing mappings. Math. Nachr. 258, 71–89 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Meléndez, Y., Tonge, A.: Polynomials and the Pietsch domination theorem. Proc. R. Irish Acad. Sect. A 99, 195–212 (1999)

    Google Scholar 

  32. Montanaro, A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53(12), 122–206 (2012)

    Article  MathSciNet  Google Scholar 

  33. Mujica, J.: Complex analysis in Banach spaces. Dover Publications, Mineola (2010)

    Google Scholar 

  34. Pellegrino, D.: Cotype and absolutely summing homogeneous polynomials in \({\cal L}_{p}\) spaces. Studia Math. 157, 121–131 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pellegrino, D., Ribeiro, J.: On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility. Monatsh. Math. 173(3), 379–415 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Pellegrino, D., Santos, J.: A general Pietsch domination theorem. J. Math. Anal. Appl. 375, 371–374 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Pellegrino, D., Santos, J.: Absolutely summing multilinear operators: a panorama. Quaest. Math. 34(4), 447–478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pellegrino, D., Santos, J.: On summability of nonlinear mappings: a new approach. Math. Z. 270(1–2), 189–196 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Pellegrino, D., Santos, J., Seoane-Sepúlveda, J.B.: Some techniques on nonlinear analysis and applications. Adv. Math. 229, 1235–1265 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  40. Pérez-García, D.: David Comparing different classes of absolutely summing multilinear operators. Arch. Math. (Basel) 85(3), 258–267 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pietsch, A.: Absolut p-summierende Abbildungen in normierten Räumen. (German) Studia Math. 28, 333–353 (1966/1967)

  42. Pietsch, A.: Ideals of multilinear functionals (designs of a theory). Proceedings of the second international conference on operator algebras, ideals, and their applications in theoretical physics (Leipzig, 1983), 185–199, Teubner-Texte Math., 67, Teubner, Leipzig, (1984)

  43. Pisier, G.: Grothendieck’s theorem, past and present. Bull. Am. Math. Soc. (N.S.) 49(2), 237–323 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rueda, P., Sánchez-Pérez, E.A.: Factorization of \(p\)-dominated polynomials through \(L^{p}\)-spaces. Michigan Math. J. 63(2), 345–353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Rueda, P., Sánchez-Pérez, E.A.: Factorization theorems for homogeneous maps on Banach function spaces and approximation of compact operators. Mediterr. J. Math. 12(1), 89–115 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ryan, R.A.: Applications of Topological Tensor Products to Infinite Dimensional Holomorphy, Ph.D. Thesis, Trinity College, Dublin, (1980)

Download references

Acknowledgments

The authors are indebted to M. López Pellicer for his valuable suggestions that helped to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, 58051-900, Brazil

    Daniel Pellegrino

  2. Departamento de Análisis Matemático, Universidad de Valencia, 46100, Burjassot, Valencia, Spain

    Pilar Rueda

  3. Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, Camino de Vera s/n, 46022, Valencia, Spain

    Enrique A. Sánchez-Pérez

Authors
  1. Daniel Pellegrino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pilar Rueda
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Enrique A. Sánchez-Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pilar Rueda.

Additional information

D. Pellegrino acknowledges with thanks the support of CNPq Grant 401735/2013-3—PVE (Linha 2)—Brazil. P. Rueda acknowledges with thanks the support of the Ministerio de Economía y Competitividad (Spain) MTM2011-22417. E. A. Sánchez Pérez acknowledges with thanks the support of the Ministerio de Economía y Competitividad (Spain) MTM2012-36740-C02-02.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pellegrino, D., Rueda, P. & Sánchez-Pérez, E.A. Surveying the spirit of absolute summability on multilinear operators and homogeneous polynomials. RACSAM 110, 285–302 (2016). https://doi.org/10.1007/s13398-015-0224-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13398-015-0224-8

Keywords

Mathematics Subject Classification

Search

Navigation