Composition results for strongly summing and dominated multilinear operators

Dumitru Popa

Open Mathematics (2014)

  • Volume: 12, Issue: 10, page 1433-1446
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Dumitru Popa. "Composition results for strongly summing and dominated multilinear operators." Open Mathematics 12.10 (2014): 1433-1446. <http://eudml.org/doc/269356>.

@article{DumitruPopa2014,
abstract = {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.},
author = {Dumitru Popa},
journal = {Open Mathematics},
keywords = {p-summing; Dominated; Strongly summing; Multilinear operators; strongly summing multilinear operators; dominated multilinear operators},
language = {eng},
number = {10},
pages = {1433-1446},
title = {Composition results for strongly summing and dominated multilinear operators},
url = {http://eudml.org/doc/269356},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Dumitru Popa
TI - Composition results for strongly summing and dominated multilinear operators
JO - Open Mathematics
PY - 2014
VL - 12
IS - 10
SP - 1433
EP - 1446
AB - 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.
LA - eng
KW - p-summing; Dominated; Strongly summing; Multilinear operators; strongly summing multilinear operators; dominated multilinear operators
UR - http://eudml.org/doc/269356
ER -

References

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  1. [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. [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 Zbl1274.46092
  3. [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 Zbl1078.46030
  4. [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. [5] Botelho G., Ideals of polynomials generated by weakly compact operators, Note Mat. 2005/2006, 25, 69–102 
  6. [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 Zbl1193.46026
  7. [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 Zbl1112.46038
  8. [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 Zbl1191.46041
  9. [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 Zbl1152.46034
  10. [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 Zbl1037.46045
  11. [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 Zbl0669.47025
  12. [12] Defant A., Floret K., Tensor norms and operator ideals, North-Holland, Math. Studies, 176, 1993 Zbl0774.46018
  13. [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 Zbl1205.46026
  14. [14] Diestel J., Jarchow H., Tonge A., Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge University Press, 1995 
  15. [15] Diestel J., Fourie J. H., Swart J., The metric theory of tensor products. Grothendieck’s résumé revisited, American Mathematical Society, Providence, RI, 2008 Zbl1186.46004
  16. [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 Zbl1043.47018
  17. [17] Dineen S., Complex analysis in locally convex spaces, North-Holland Mathematics Studies, 57, 1981 
  18. [18] Dineen S., Complex Analysis on Infinite Dimensional Spaces, Springer-Verlag, 1999 http://dx.doi.org/10.1007/978-1-4471-0869-6 
  19. [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. [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 Zbl1049.46030
  21. [21] Geiss S., Ideale multilinearer Abbildungen, Diplomarbeit, 1984 
  22. [22] Grothendieck A., Résume de la théorie metrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paolo 8 (1953/1956), 1–79 
  23. [23] Holub J. R., Tensor product mappings, Math. Ann., 1970, 188, 1–12 http://dx.doi.org/10.1007/BF01435409 Zbl0195.41601
  24. [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 Zbl1161.46025
  25. [25] Lindenstrauss J., Pełczynski A., Absolutely summing operators in ℒ p-spaces and their applications, Studia Math. 1968, 29, 257–326 Zbl0183.40501
  26. [26] Matos M. C., On multilinear mappings of nuclear type, Rev. Mat. Univ.Complut. Madrid, 1993, 6(1), 61–81 Zbl0807.46022
  27. [27] Matos M. C., Absolutely summing holomorphic mappings, An. Acad. Bras. Ciênc., 1996, 68(1), 1–13 Zbl0854.46042
  28. [28] Matos M. C., Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 2003, 54(2), 111–136 Zbl1078.46031
  29. [29] Mujica J., Complex Analysis in Banach Spaces, Dover Publications, 2010 
  30. [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 Zbl1274.47001
  31. [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 Zbl1248.47024
  32. [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 Zbl1080.46030
  33. [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 Zbl1080.47047
  34. [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 Zbl1105.46030
  35. [35] Pietsch A., Absolut p-summierende Abbildungen in normierten Räumen, Studia Math., 1967, 28, 333–353 Zbl0156.37903
  36. [36] Pietsch A., Operator ideals, Veb Deutscher Verlag der Wiss., Berlin, 1978; North Holland, 1980 
  37. [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 Zbl0561.47037
  38. [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. [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 Zbl1244.46006
  40. [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 Zbl1198.47036
  41. [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 Zbl1196.47019
  42. [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 Zbl1228.46041
  43. [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. [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 

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