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Keywords:
Dunford-Pettis property of order $p$; $p$-convergent operator; complemented spaces of operators
Summary:
Equivalent formulations of the Dunford-Pettis property of order $p$ (${DPP}_p$), $1<p<\infty$, are studied. Let $L(X,Y)$, $W(X,Y)$, $K(X,Y)$, $U(X,Y)$, and $C_p(X,Y)$ denote respectively the sets of all bounded linear, weakly compact, compact, unconditionally converging, and $p$-convergent operators from $X$ to $Y$. Classical results of Kalton are used to study the complementability of the spaces $W(X,Y)$ and $K(X,Y)$ in the space $C_p(X,Y)$, and of $C_p(X,Y)$ in $U(X,Y)$ and $L(X,Y)$.
References:
[1] Bahreini M., Bator E., Ghenciu I.: Complemented subspaces of linear bounded operators. Canad. Math. Bull. 55 (2012), no. 3, 449–461. DOI 10.4153/CMB-2011-097-2 | MR 2957262
[2] Bourgain J., Diestel J.: Limited operators and strict cosingularity. Math. Nachr. 119 (1984), 55–58. DOI 10.1002/mana.19841190105 | MR 0774176 | Zbl 0601.47019
[3] Castillo J. M. F., Sanchez F.: Dunford-Pettis like properties of continuous vector function spaces. Rev. Mat. Univ. Complut. Madrid 6 (1993), no. 1, 43–59. MR 1245024
[4] Diestel J.: A survey of results related to the Dunford-Pettis property. Proc. of Conf. on Integration, Topology, and Geometry in Linear Spaces, Univ. North Carolina, Chapel Hill, 1979, Contemp. Math. 2 Amer. Math. Soc., Providence, 1980, pp. 15–60. MR 0621850 | Zbl 0571.46013
[5] Diestel J.: Sequences and Series in Banach Spaces. Graduate Texts in Mathematics, 92, Springer, New York, 1984. MR 0737004
[6] Diestel J., Jarchow H., Tonge A.: Absolutely Summing Operators. Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, Cambridge, 1995. MR 1342297 | Zbl 1139.47021
[7] Diestel J., Uhl J. J. Jr.: Vector Measures. Mathematical Surveys, 15, American Mathematical Society, Providence, 1977. MR 0453964 | Zbl 0521.46035
[8] Drewnowski L., Emmanuele G.: On Banach spaces with the Gelfand-Phillips property II. Rend. Circ. Mat. Palermo (2) 38 (1989), no. 3, 377–391. DOI 10.1007/BF02850021 | MR 1053378 | Zbl 0689.46004
[9] Emmanuele G.: Remarks on the uncomplemented subspace $W(E,F)$. J. Funct. Anal. 99 (1991), no. 1, 125–130. DOI 10.1016/0022-1236(91)90055-A | MR 1120917
[10] Emmanuele G.: A remark on the containment of $c_{0}$ in spaces of compact operators. Math. Proc. Cambridge Philos. Soc. 111 (1992), no. 2, 331–335. DOI 10.1017/S0305004100075435 | MR 1142753
[11] Emmanuele G., John K.: Uncomplementability of spaces of compact operators in larger spaces of operators. Czechoslovak Math. J. 47(122) (1997), no. 1, 19–32. DOI 10.1023/A:1022483919972 | MR 1435603 | Zbl 0903.46006
[12] Feder M.: On subspaces of spaces with an unconditional basis and spaces of operators. Illinois J. Math. 24 (1980), no. 2, 196–205. MR 0575060
[13] Fourie J. H., Zeekoei E. D.: On weak-star $p$-convergent operators. Quaest. Math. 40 (2017), no. 5, 563–579. DOI 10.2989/16073606.2017.1301591 | MR 3691468
[14] Ghenciu I.: The $p$-Gelfand Phillips property in spaces of operators and Dunford-Pettis like sets. available at arXiv:1803.00351v1 [math.FA] (2018), 16 pages. MR 2283818
[15] Ghenciu I., Lewis P.: The Dunford-Pettis property, the Gelfand-Phillips property, and $L$-sets. Colloq. Math. 106 (2006), no. 2, 311–324. DOI 10.4064/cm106-2-11 | MR 2283818
[16] Grothendieck A.: Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$. Canadian J. Math. 5 (1953), 129–173 (French). DOI 10.4153/CJM-1953-017-4 | MR 0058866 | Zbl 0050.10902
[17] John K.: On the uncomplemented subspace $K(X,Y)$. Czechoslovak Math. J. 42(117) (1992), no. 1, 167–173. MR 1152178
[18] Kalton N. J.: Spaces of compact operators. Math. Ann. 208 (1974), 267–278. DOI 10.1007/BF01432152 | MR 0341154 | Zbl 0266.47038
[19] Lohman R. H.: A note on Banach spaces containing $l_1$. Canad. Math. Bull. 19 (1976), no. 3, 365–367. DOI 10.4153/CMB-1976-056-x | MR 0430748
[20] Megginson R. E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, 183, Springer, New York, 1998. DOI 10.1007/978-1-4612-0603-3 | MR 1650235 | Zbl 0910.46008
[21] Pełczyński A.: Banach spaces on which every unconditionally converging operator is weakly compact. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 10 (1962), 641–648. MR 0149295 | Zbl 0107.32504
[22] Salimi M., Moshtaghiun S. M.: The Gelfand-Phillips property in closed subspaces of some operator spaces. Banach J. Math. Anal. 5 (2011), no. 2, 84–92. DOI 10.15352/bjma/1313363004 | MR 2792501

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