Composition results for strongly summing and dominated multilinear operators

Dumitru Popa

Open Mathematics, Tome 12 (2014), p. 1433-1446 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2014-01-01

Zbl 1308.47063

EUDML-ID : urn:eudml:doc:269356

Bibliographie

[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 | Zbl 1274.46092

[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 | Zbl 1078.46030

[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 | Zbl 1193.46026

[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 | Zbl 1112.46038

[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 | Zbl 1191.46041

[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 | Zbl 1152.46034

[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 | Zbl 1037.46045

[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 | Zbl 0669.47025

[12] Defant A., Floret K., Tensor norms and operator ideals, North-Holland, Math. Studies, 176, 1993 | Zbl 0774.46018

[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 | Zbl 1205.46026

[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 | Zbl 1186.46004

[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 | Zbl 1043.47018

[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 | Zbl 1049.46030

[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 | Zbl 0195.41601

[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 | Zbl 1161.46025

[25] Lindenstrauss J., Pełczynski A., Absolutely summing operators in ℒ p-spaces and their applications, Studia Math. 1968, 29, 257–326 | Zbl 0183.40501

[26] Matos M. C., On multilinear mappings of nuclear type, Rev. Mat. Univ.Complut. Madrid, 1993, 6(1), 61–81 | Zbl 0807.46022

[27] Matos M. C., Absolutely summing holomorphic mappings, An. Acad. Bras. Ciênc., 1996, 68(1), 1–13 | Zbl 0854.46042

[28] Matos M. C., Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 2003, 54(2), 111–136 | Zbl 1078.46031

[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 | Zbl 1274.47001

[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 | Zbl 1248.47024

[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 | Zbl 1080.46030

[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 | Zbl 1080.47047

[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 | Zbl 1105.46030

[35] Pietsch A., Absolut p-summierende Abbildungen in normierten Räumen, Studia Math., 1967, 28, 333–353 | Zbl 0156.37903

[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 | Zbl 0561.47037

[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 | Zbl 1244.46006

[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 | Zbl 1198.47036

[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 | Zbl 1196.47019

[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 | Zbl 1228.46041

[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