LEFT l1-FACTORABLE POLYNOMIALS

CILIA, RAFFAELLA ; GUTIÉRREZ, JOAQUÍN M.

Glasgow mathematical journal, Tome 51 (2009) no. 3, pp. 631-649

Résumé

A polynomial P ∈ (kE, F) is left l1-factorable if there are a polynomial Q ∈ (kE, l1) and an operator L ∈ (l1, F) such that P = L ○ Q. We characterise the Radon–Nikodým property by the left l1-factorisation of polynomials on L1(μ). We study the left l1-factorisation of nuclear, compact and Pietsch integral polynomials. For Pietsch integral polynomials, we introduce the left integral l1-factorisation property, obtaining a second polynomial characterisation of the Radon–Nikodým property and showing that it plays a role somehow comparable, in this setting, to nuclearity of operators. A characterisation of 1-spaces is also given in terms of the left compact l1-factorisation of polynomials.

Détail

DOI : 10.1017/S001708950999005X

Mots-clés : Primary 46G25, secondary 47H60, 46B20, 46B22

Bibliographie

[1] 1.Albiac, F. and Kalton, N. J., Topics in Banach space theory, Graduate Texts in Mathematics, vol. 233 (Springer, New York, 2006). Google Scholar

[2] 2.Alencar, R., Multilinear mappings of nuclear and integral type, Proc. Am. Math. Soc. 94 (1985), 33–38. Google Scholar | DOI

[3] 3.Alencar, R., On reflexivity and basis for (mE), Math. Proc. R. Ir. Acad. 85A (1985), 131–138. Google Scholar

[4] 4.Ansemil, J. M. and Floret, K., The symmetric tensor product of a direct sum of locally convex spaces, Studia Math. 129 (1998), 285–295. Google Scholar

[5] 5.Aron, R. M., Hervés, C. and Valdivia, M., Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 (1983), 189–204. Google Scholar | DOI

[6] 6.Aron, R. M. and Prolla, J. B., Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. Google Scholar

[7] 7.Blasco, F., Complementation in spaces of symmetric tensor products and polynomials, Studia Math. 123 (1997), 165–173. Google Scholar | DOI

[8] 8.Botelho, G. and Pellegrino, D. M., Two new properties of ideals of polynomials and applications, Indag. Math. (N.S.) 16 (2005), 157–169. Google Scholar | DOI

[9] 9.Carando, D., Extendible polynomials on Banach spaces, J. Math. Anal. Appl. 233 (1999), 359–372. Google Scholar | DOI

[10] 10.Carando, D., Dimant, V. and Muro, S., Coherent sequences of Banach ideals of polynomials, to appear in Math. Nachr. Google Scholar

[11] 11.Carando, D. and Lassalle, S., E′ and its relation with vector-valued functions on E, Ark. Math. 42 (2004), 283–300. Google Scholar | DOI

[12] 12.Cilia, R., D'Anna, M. and Gutiérrez, J. M., Polynomial characterization of ∞-spaces, J. Math. Anal. Appl. 275 (2002), 900–912. Google Scholar | DOI

[13] 13.Cilia, R. and Gutiérrez, J. M., Nuclear and integral polynomials, J. Austral. Math. Soc. 76 (2004), 269–280. Google Scholar | DOI

[14] 14.Cilia, R. and Gutiérrez, J. M., Extension and lifting of weakly continuous polynomials, Studia Math. 169 (2005), 229–241. Google Scholar | DOI

[15] 15.Cilia, R. and Gutiérrez, J. M., Integral and S-factorizable multilinear mappings, Proc. R. Soc. Edinb. 136A (2006), 115–137. Google Scholar | DOI

[16] 16.Cilia, R. and Gutiérrez, J. M., Ideals of integral and r-factorable polynomials, to appear in Bol. Soc. Mat. Mexicana. Google Scholar

[17] 17.Defant, A. and Floret, K., Tensor norms and operator ideals, Mathematics Studies, vol. 176 (North-Holland, Amsterdam, 1993). Google Scholar

[18] 18.Diestel, J. and Uhl, J. J. Jr., Vector measures, Mathematical Surveys and Monographs, vol. 15 (American Mathematical Society, Providence RI, 1977). Google Scholar | DOI

[19] 19.Dineen, S., Complex analysis on infinite dimensional spaces, Springer Monographs in Mathematics (Springer, Berlin, 1999). Google Scholar | DOI

[20] 20.Figiel, T., Factorization of compact operators and applications to the approximation problem, Studia Math. 45 (1973), 191–210. Google Scholar | DOI

[21] 21.Floret, K., Natural norms on symmetric tensor products of normed spaces, Note Mat. 17 (1997), 153–188. Google Scholar

[22] 22.Floret, K., Minimal ideals of n-homogeneous polynomials on Banach spaces, Results Math. 39 (2001), 201–217. Google Scholar | DOI

[23] 23.Floret, K., On ideals of n-homogeneous polynomials on Banach spaces, in Topological algebras with applications to differential geometry and mathematical physics (Athens, 1999) (Strantzalos, P. and Fragoulopoulou, M., Editors) (University of Athens, Athens, 2002), 19–38. Google Scholar

[24] 24.Floret, K. and Hunfeld, S., Ultrastability of ideals of homogeneous polynomials and multilinear mappings on Banach spaces, Proc. Am. Math. Soc. 130 (2002), 1425–1435. Google Scholar | DOI

[25] 25.González, M. and Gutiérrez, J. M., Factorization of weakly continuous holomorphic mappings, Studia Math. 118 (1996), 117–133. Google Scholar

[26] 26.Johnson, W. B., Factoring compact operators, Isr. J. Math. 9 (1971), 337–345. Google Scholar | DOI

[27] 27.Lindenstrauss, J. and Pełzyński, A., Absolutely summing operators in -spaces and their applications, Studia Math. 29 (1968), 275–326. Google Scholar | DOI

[28] 28.Lindenstrauss, J. and Rosenthal, H. P., The -spaces, Isr. J. Math. 7 (1969), 325–349. Google Scholar | DOI

[29] 29.Mujica, J., Complex analysis in Banach spaces, Mathematics Studies, vol. 120 (North-Holland, Amsterdam, 1986). Google Scholar

[30] 30.Rudin, W., Principles of mathematical analysis (McGraw-Hill, New York, 1976). Google Scholar

[31] 31.Ryan, R. A., Applications of topological tensor products to infinite dimensional holomorphy, PhD Thesis (Trinity College, Dublin, 1980). Google Scholar

[32] 32.Ryan, R. A., Weakly compact holomorphic mappings on Banach spaces, Pac. J. Math. 131 (1988), 179–190. Google Scholar | DOI

[33] 33.Ryan, R. A., Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics (Springer, London, 2002). Google Scholar | DOI

[34] 34.Villanueva, I., Integral mappings between Banach spaces, J. Math. Anal. Appl. 279 (2003), 56–70. Google Scholar | DOI

[35] 35.Wojtaszczyk, P., Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25 (Cambridge University Press, Cambridge, UK 1991). Google Scholar | DOI