Dunford–Pettis Properties and Spaces of Operators

Ghenciu, Ioana ; Lewis, Paul

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 213-223

Résumé

J. Elton used an application of Ramsey theory to show that if $X$ is an infinite dimensional Banach space, then ${{c}_{0}}$ embeds in $X$ , ${{\ell }_{1}}$ embeds in $X$ , or there is a subspace of $X$ that fails to have the Dunford–Pettis property. Bessaga and Pelczynski showed that if ${{c}_{0}}$ embeds in ${{X}^{*}}$ , then ${{\ell }_{\infty }}$ embeds in ${{X}^{*}}.$ Emmanuele and John showed that if ${{c}_{0}}$ embeds in $K\left( X,\,Y \right)$ , then $K\left( X,\,Y \right)$ is not complemented in $L\left( X,\,Y \right)$ . Classical results from Schauder basis theory are used in a study of Dunford–Pettis sets and strong Dunford–Pettis sets to extend each of the preceding theorems. The space ${{L}_{{{w}^{*}}}}\left( {{X}^{*}},\,Y \right)$ of ${{w}^{*}}\,-\,w$ continuous operators is also studied.

Détail

DOI : 10.4153/CMB-2009-024-5

Mots-clés : 46B20, 46B28, Dunford–Pettis property, Dunford–Pettis set, basic sequence, complemented spaces of operators

Bibliographie

[1] [1] Andrews, K., Dunford–Pettis sets in the space of Bochner integrable functions.Math. Ann. 241(1979), no. 1, 35–41. Google Scholar

[2] [2] Bator, E. M., Remarks on completely continuous operators. Bull. Polish Acad. Sci.Math. 37(1989), no. 7–12, 409–413. Google Scholar

[3] [3] Bator, E. and Lewis, P., Complemented spaces of operators. Bull. Polish Acad. Sci. Math. 50(2002), no. 4, 413–416. Google Scholar

[4] [4] Bator, E., Lewis, P., and Ochoa, J., Evaluation maps, restriction maps, and compactness. Colloq. Math. 78(1998), no. 1, 1–17. Google Scholar

[5] [5] Bessaga, C. and Pelczynski, A., On bases and unconditional convergence of series in Banach spaces. Studia Math. 17(1958), 151–174. Google Scholar

[6] [6] Bilyeu, R. and Lewis, P., Vector measures and weakly compact operators on continuous function spaces: A survey. In: Measure Theory and Its Applications, Proceedings of the 1980 Conference. Northern Illinois University, Department of Mathematical Sciences, DeKalb, IL, 1981, pp. 165–172. Google Scholar

[7] [7] Brooks, J. and Lewis, P., Linear operators and vector measures. Trans. Amer. Math. Soc. 192(1974), 139–162. Google Scholar

[8] [8] Diestel, J., A survey of results related to the Dunford–Pettis property. Contemp. Math. 2(1980), 15–60. Google Scholar

[9] [9] Diestel, J., Sequences and Series in Banach Spaces. Graduate Texts in Mathematics 92, Springer-Verlag, New York, 1984. Google Scholar

[10] [10] Diestel, J. and Uhl, J. J. Jr., Vector Measures. Mathematical Surveys 15, American Mathematical Society, Providence, RI, 1977. Google Scholar

[11] [11] Elton, J., Weakly Null Normalized Sequences in Banach Spaces. Ph.D. dissertation, Yale University, 1979. Google Scholar

[12] [12] Emmanuele, G., A dual characterization of Banach spaces not containing ℓ 1 . Bull. Polish Acad. Sci. Math. 34(1986), no. 3–4, 155–160. . Google Scholar

[13] [13] Emmanuele, G., Remarks on the uncomplemented subspace W(E, F). J. Funct. Anal. 99(1991), 125–130. Google Scholar

[14] [14] Emmanuele, G., A remark on the containment of c in spaces of compact operators. Math. Proc. Cambridge Philos. Soc. 111(1992), no. 2, 331–335. Google Scholar

[15] [15] Emmanuele, G., Banach spaces in which Dunford–Pettis sets are relatively compact. Arch. Math. Basel 58(1992), no. 5, 477–485. Google Scholar

[16] [16] Emmanuele, G. and John, K., Uncomplementability of spaces of compact operators in larger spaces of operators, Czechoslovak Math. J. 47(122)(1997), no. 1, 19–32. Google Scholar

[17] [17] Feder, M., On the nonexistence of a projection onto the space of compact operators. Canad. Math. Bull. 25(1982), no. 1, 78–81. Google Scholar

[18] [18] Ghenciu, I. and Lewis, P., Tensor products and Dunford–Pettis sets. Math. Proc. Cambridge Philos. Soc. 139(2005), no. 2, 361–369. Google Scholar

[19] [19] Grothendieck, A., Critères de compacité dans les espaces fonctionnels généraux. Amer. J. Math 74(1952), 168–186. Google Scholar

[20] [20] James, R. C., Separable conjugate spaces. Pacific J. Math. 10(1960), 563–571. Google Scholar

[21] [21] John, K., On the uncomplemented subspace K(X, Y). Czechoslovak Math. J. 42(117)(1992), no. 1, 167–173. Google Scholar

[22] [22] Kalton, N., Spaces of compact operators. Math. Ann. 208(1974), 267–278. Google Scholar

[23] [23] Lewis, P., Spaces of operators and c . Studia Math. 145(2001), no. 3, 213–218. Google Scholar

[24] [24] Lewis, P., Dunford–Pettis sets. Proc. Amer. Math. Soc. 129(2001), 3297–3302. Google Scholar

[25] [25] Rosenthal, H., On relatively disjoint families of measures with some applications to Banach space theory. Studia Math. 37(1970), 13–36. Google Scholar

[26] [26] Rosenthal, H., Point-wise compact subsets of the first Baire class. Amer. J. Math. 99(1977), no. 1, 362–378. Google Scholar

[27] [27] Ryan, R. A., The Dunford–Pettis property and projective tensor products. Bull. Polish Acad. Sci. Math. 35(1987), no. 11–12, 785–792. Google Scholar