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  • ONDŘEJ F. K. KALENDA (a1) and JIŘÍ SPURNÝ (a2)

We show that the dual to any subspace of $c_{0}({\rm\Gamma})$ ( ${\rm\Gamma}$ is an arbitrary index set) has the strongest possible quantitative version of the Schur property. Further, we establish a relationship between the quantitative Schur property and quantitative versions of the Dunford–Pettis property. Finally, we apply these results to show, in particular, that any subspace of the space of compact operators on $\ell _{p}$ ( $1 ) with the Dunford–Pettis property automatically satisfies both its quantitative versions.

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[1]Bourgain, J. and Delbaen, F., ‘A class of special L spaces’, Acta Math. 145(3–4) (1980), 155176.
[2]Brown, S. W., ‘Weak sequential convergence in the dual of an algebra of compact operators’, J. Operator Theory 33(1) (1995), 3342.
[3]Cembranos, P., ‘The hereditary Dunford–Pettis property on C (K, E)’, Illinois J. Math. 31(3) (1987), 365373.
[4]Diestel, J., ‘A survey of results related to the Dunford–Pettis property’, in: Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces (Univ. North Carolina, Chapel Hill, NC, 1979 (Providence, RI, 1980), Contemporary Mathematics, 2 (American Mathematical Society, Providence, RI) 1560.
[5]Freeman, D., Odell, E. and Schlumprecht, T., ‘The universality of 1 as a dual space’, Math. Ann. 351(1) (2011), 149186.
[6]Hagler, J., ‘A counterexample to several questions about Banach spaces’, Studia Math. 60(3) (1977), 289308.
[7]Haydon, R., ‘Subspaces of the Bourgain–Delbaen space’, Studia Math. 139(3) (2000), 275293.
[8]Kačena, M., Kalenda, O. F. and Spurný, J., ‘Quantitative Dunford–Pettis property’, Adv. Math. 234 (2013), 488527.
[9]Kalenda, O. F. K., Pfitzner, H. and Spurný, J., ‘On quantification of weak sequential completeness’, J. Funct. Anal. 260(10) (2011), 29862996.
[10]Kalenda, O. F. K. and Spurný, J., ‘On a difference between quantitative weak sequential completeness and the quantitative Schur property’, Proc. Amer. Math. Soc. 140(10) (2012), 34353444.
[11]Kalton, N. J. and Werner, D., ‘Property (M), M-ideals, and almost isometric structure of Banach spaces’, J. reine angew. Math. 461 (1995), 137178.
[12]Pełczyński, A., ‘On Banach spaces containing L 1(𝜇)’, Studia Math. 30 (1968), 231246.
[13]Pełczyński, A. and Szlenk, W., ‘An example of a non-shrinking basis’, Rev. Roumaine Math. Pures Appl. 10 (1965), 961966.
[14]Saksman, E. and Tylli, H.-O., ‘Structure of subspaces of the compact operators having the Dunford–Pettis property’, Math. Z. 232 (1999), 411425.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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