Hilbert–Schmidt and multiple summing operators

• Autores: Gabriela Badea, Dumitru Popa
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 63, Fasc. 2, 2012, págs. 181-194
• Idioma: inglés
• DOI: 10.1007/s13348-010-0028-2
• Texto completo no disponible (Saber más ...)
• Resumen

  • We characterize the multiple p-summing operators for p ≥ 2, in the situation where the range and some of the factors of the cartesian product are Hilbert spaces. As a consequence, we complete some recent results obtained in this area. Thus, we prove a coincidence result for multiple p-summing operators for p ≥ 2, which has no linear analogue. Also, we give the necessary and sufficient conditions for some concrete bilinear operators to be multiple p-summing, p ≥ 1.

• Referencias bibliográficas
  • Acosta M., García D., Maestre M.: A multilinear Lindenstrauss theorem. J. Funct. Anal. 235, 122–136 (2006)
  • Bohnenblust H.F., Hille E.: On the absolute convergence of Dirichlet series. Ann. Math. 32, 600–622 (1931)
  • Bombal F., Pérez-García D., Villanueva I.: Multilinear extensions of Grothendieck’s theorem. Q. J. Math. 55, 441–450 (2004)
  • Botelho, G., Pellegrino, D.: Coincidences for multiple summing mappings, In: Segundo Enama, 27–28, Joao Pessoa, Brazil (2008)
  • Botelho G., Michels C., Pellegrino D.: Complex interpolation and summability properties of multilinear operators. Revista Matematica Complutense...
  • Botelho G., Pellegrino D.: When every multilinear mapping is multiple summing. Mathematische Nachrichten 282, 1414–1422 (2009)
  • Çaliskan E., Pellegrino D.: On the multilinear generalizations of the concept of absolutely summing operators. Rocky Mt. J. Math. 37, 1137–1154...
  • Defant, A., Floret, K.: Tensor norms and operator ideals, North-Holland (1993)
  • Defant A., García D., Maestre M., Pérez-García D.: Bohr’s strip for vector valued Dirichlet series. Mathematische Annalen 342, 533–555 (2008)
  • Defant A., Pérez-García D.: A tensor norm preserving unconditionality for Lp-spaces. Trans. Am. Math. Soc. 360, 3287–3306 (2008)
  • Defant A., Sevilla-Peris P.: A new multilinear insight on Littlewood’s 4/3-inequality. J. Funct. Anal. 256, 1642–1664 (2009)
  • Defant A., Popa D., Schwarting U.: Coordinatewise multiple summing operators in Banach spaces. J. Funct. Anal. 259(1), 220–242 (2010)
  • Diestel J., Jarchow H., Tonge A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)
  • Dineen S.: Complex Analysis on Infinite Dimensional Spaces. Springer, London (1999)
  • Garling D.J.H.: Diagonal maps between sequence spaces. Stud. Math. 51, 129–138 (1974)
  • Littlewood J.E.: On bounded bilinear forms in an infinite number of variables. Q. J. (Oxford Ser.) 1, 164–174 (1930)
  • Matos M.C.: Fully absolutely summing and Hilbert–Schmidt multilinear mappings. Collect. Math. 54, 111–136 (2003)
  • Matos M.C., Pellegrino D.: Fully summing mappings between Banach spaces. Stud. Math. 178, 47–61 (2007)
  • Mujica J.: Complex Analysis in Banach Spaces. North-Holland, Amsterdam (1986)
  • Pełczyński A.: A characterization of Hilbert–Schmidt operators. Stud. Math. 28, 355–360 (1967)
  • Pellegrino D., Souza M.: Fully summing multilinear and holomorphic mappings into Hilbert spaces. Mathematische Nachrichten 278, 877–887 (2005)
  • Peréz-García D., Villanueva I.: Multiple summing operators on Banach spaces. J. Math. Anal. Appl. 285, 86–96 (2003)
  • Pérez-García D.: The inclusion theorem for multiple summing operators. Stud. Math. 165(3), 275–290 (2004)
  • Pérez-García D., Villanueva I.: Multiple summing operators on C(K) spaces. Ark. för Mat. 42, 153–171 (2004)
  • Peréz-García D.: Comparing different classes of absolutely summing multilinear operators. Archiv der Mathematik (Basel) 85, 258–267 (2005)
  • Pietsch, A.: Operator Ideals, North Holland (1980)
  • Pietsch, A.: Ideals of multilinear functionals (designs of a theory). In: Proceedings of second International Conference Operator Alg, Leipzig,...
  • Pisier, G.: Factorization of linear operators and geometry of Banach spaces. CBMS Regional Conf. Series 60, Amer. Math. Soc. (1986)
  • Popa D.: Reverse inclusions for multiple summing operators. J. Math. Anal. Appl. 350(1), 360–368 (2009)
  • Popa D.: Multilinear variants of Maurey and Pietsch theorems and applications. J. Math. Anal. Appl. 368(1), 157–168 (2010)
  • Ramanujan M.S., Schock E.: Operator ideals and spaces of bilinear operators. Linear Multilinear Algebra 18, 307–318 (1985)
  • Tomczak-Jagermann, N.: Banach-Mazur distances and finite dimensional operator ideals, Pitman Monogr. Surveys Pure Appl. Math., vol. 38, Longman...