PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2014 | 12 | 10 | 1433-1446
Tytuł artykułu

Composition results for strongly summing and dominated multilinear operators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this paper we prove some composition results for strongly summing and dominated operators. As an application we give necessary and sufficient conditions for a multilinear tensor product of multilinear operators to be strongly summing or dominated. Moreover, we show the failure of some possible n-linear versions of Grothendieck’s composition theorem in the case n ≥ 2 and give a new example of a 1-dominated, hence strongly 1-summing bilinear operator which is not weakly compact.
Twórcy
autor
Bibliografia
  • [1] Alencar R., Matos M. C., Some classes of multilinear mappings between Banach spaces, Publicaciones del Departamento de Análisis Matemático, sec. 1, no. 12 (1989), Universidad Complutense de Madrid
  • [2] Bernardino A. T., On cotype and a Grothendieck-type theorem for absolutely summing multilinear operators, Quaest. Math., 2011, 34(4), 513–519 http://dx.doi.org/10.2989/16073606.2011.640747
  • [3] Bombal F., Pérez-García D., Villanueva I., Multilinear extensions of Grothendieck’s theorem, Q. J. Math., 2004, 55(4), 441–450 http://dx.doi.org/10.1093/qmath/hah017
  • [4] Botelho G., Weakly compact and absolutely summing polynomials, J. Math. Anal. Appl., 2002, 265(2), 458–462 http://dx.doi.org/10.1006/jmaa.2001.7674
  • [5] Botelho G., Ideals of polynomials generated by weakly compact operators, Note Mat. 2005/2006, 25, 69–102
  • [6] Botelho G., Pellegrino D., Rueda P., A unified Pietsch domination theorem, J. Math. Anal. Appl., 2010, 365(1), 269–276 http://dx.doi.org/10.1016/j.jmaa.2009.10.025
  • [7] Botelho G., Braunss H.-A., Junek H., Pellegrino D., Holomorphy types and ideals of multilinear mappings, Studia Math., 2006, 177, 43–65. http://dx.doi.org/10.4064/sm177-1-4
  • [8] Botelho G., Pellegrino D., When every multilinear mapping is multiple summing, Math. Nachr., 2009, 282(10), 1414–1422 http://dx.doi.org/10.1002/mana.200710112
  • [9] Çalişkan E., Pellegrino D., On the multilinear generalizations of the concept of absolutely summing operators, Rocky Mountain J. Math., 2007, 37(4), 1137–1154 http://dx.doi.org/10.1216/rmjm/1187453101
  • [10] Carando D., Dimant V., On summability of bilinear operators, Math. Nachr. 2003, 259(1), 3–11 http://dx.doi.org/10.1002/mana.200310090
  • [11] Carl B., Defant A., Ramanujan M. S., On tensor stable operator ideals, Mich. Math. J., 1989, 36(1), 63–75 http://dx.doi.org/10.1307/mmj/1029003882
  • [12] Defant A., Floret K., Tensor norms and operator ideals, North-Holland, Math. Studies, 176, 1993
  • [13] Defant A., Popa D., Schwarting U., Coordinatewise multiple summing operators in Banach spaces, Journ. Funct. Anal., 2010, 259(1), 220–242 http://dx.doi.org/10.1016/j.jfa.2010.01.008
  • [14] Diestel J., Jarchow H., Tonge A., Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge University Press, 1995
  • [15] Diestel J., Fourie J. H., Swart J., The metric theory of tensor products. Grothendieck’s résumé revisited, American Mathematical Society, Providence, RI, 2008
  • [16] Dimant V., Strongly p-summing multilinear operators, J. Math. Anal. Appl., 2003, 278(1), 182–193 http://dx.doi.org/10.1016/S0022-247X(02)00710-2
  • [17] Dineen S., Complex analysis in locally convex spaces, North-Holland Mathematics Studies, 57, 1981
  • [18] Dineen S., Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, 1999 http://dx.doi.org/10.1007/978-1-4471-0869-6
  • [19] Dubinsky Ed., Pelczyński A., Rosenthal H. P., On Banach spaces X for which Π2 (ℒ ∞, X) = B (ℒ ∞, X), Studia Math., 1972, 44, 617–648
  • [20] Floret K., García D., On ideals of polynomials and multilinear mappings between Banach spaces, Arch. Math. (Basel), 2003, 81(3), 300–308 http://dx.doi.org/10.1007/s00013-003-0439-3
  • [21] Geiss S., Ideale multilinearer Abbildungen, Diplomarbeit, 1984
  • [22] Grothendieck A., Résume de la théorie metrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paolo 8 (1953/1956), 1–79
  • [23] Holub J. R., Tensor product mappings, Math. Ann., 1970, 188, 1–12 http://dx.doi.org/10.1007/BF01435409
  • [24] Jarchow H., Palazuelos C., Pérez-García D., Villanueva I., Hahn-Banach extension of multilinear forms and summability, J. Math. Anal. Appl., 2007, 336(2), 1161–1177 http://dx.doi.org/10.1016/j.jmaa.2007.03.057
  • [25] Lindenstrauss J., Pełczynski A., Absolutely summing operators in ℒ p-spaces and their applications, Studia Math. 1968, 29, 257–326
  • [26] Matos M. C., On multilinear mappings of nuclear type, Rev. Mat. Univ.Complut. Madrid, 1993, 6(1), 61–81
  • [27] Matos M. C., Absolutely summing holomorphic mappings, An. Acad. Bras. Ciênc., 1996, 68(1), 1–13
  • [28] Matos M. C., Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 2003, 54(2), 111–136
  • [29] Mujica J., Complex Analysis in Banach Spaces, Dover Publications, 2010
  • [30] Pellegrino D., Santos J., Absolutely summing operators: a panorama, Quaest. Math., 2011, 4, 447–478 http://dx.doi.org/10.2989/16073606.2011.640459
  • [31] Pellegrino D., Santos J., Seoane-Sepúlveda J. B., Some techniques on nonlinear analysis and applications, Adv. Math., 2012, 229(2), 1235–1265 http://dx.doi.org/10.1016/j.aim.2011.09.014
  • [32] Pellegrino D., Souza M., Fully summing multilinear and holomorphic mappings into Hilbert spaces, Math. Nachr., 2005, 278(7–8), 877–887 http://dx.doi.org/10.1002/mana.200310279
  • [33] Pérez-García D., Comparing different classes of absolutely summing multilinear operators, Arch. Math. (Basel), 2005, 85(3), 258–267 http://dx.doi.org/10.1007/s00013-005-1125-4
  • [34] Péréz-García D., Villanueva I., A composition theorem for multiple summing operators, Monatsh. Math. 2005, 146, 257–261 http://dx.doi.org/10.1007/s00605-005-0316-1
  • [35] Pietsch A., Absolut p-summierende Abbildungen in normierten Räumen, Studia Math., 1967, 28, 333–353
  • [36] Pietsch A., Operator ideals, Veb Deutscher Verlag der Wiss., Berlin, 1978; North Holland, 1980
  • [37] 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, 1983, 185–199
  • [38] Pisier G., Factorization of linear operators and geometry of Banach spaces, CBMS Regional Conference Series in Mathematics 60, American Mathematical Society, Providence, RI, 1986
  • [39] Pisier G., Grothendieck’s theorem, past and present, Bull. Amer. Math. Soc., New Ser., 2012, 49(2), 237–323 http://dx.doi.org/10.1090/S0273-0979-2011-01348-9
  • [40] Popa D., Multilinear variants of Maurey and Pietsch theorems and applications, J. Math. Anal. Appl., 2010, 368(1), 157–168 http://dx.doi.org/10.1016/j.jmaa.2010.02.019
  • [41] Popa D., Multilinear variants of Pietsch’s composition theorem, J. Math. Anal. Appl., 2010, 370(2), 415–430 http://dx.doi.org/10.1016/j.jmaa.2010.05.018
  • [42] Popa D., A new distinguishing feature for summing, versus dominated and multiple summing operators, Arch. Math. (Basel), 2011, 96(5), 455–462 http://dx.doi.org/10.1007/s00013-011-0258-x
  • [43] Popa D., Nuclear multilinear operators with respect to a partition, Rend. Circolo Matematico di Palermo, 2012, 61(3), 307–319 http://dx.doi.org/10.1007/s12215-012-0091-5
  • [44] N. Tomczak-Jagermann, Banach-Mazur distances and finite dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 38, Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc., 1989
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-014-0423-0
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.