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Dunford–Pettis properties and spaces of operators. (English) Zbl 1170.46009

Authors’ summary: J. Elton [“Weakly Null Normalized Sequences in Banach Spaces” (Ph. D. dissertation, Yale University) (1979; per bibl.)] used an application of Ramsey theory to show that if X is an infinite-dimensional Banach space, then c 0 embeds in X, 1 embeds in X, or there is a subspace of X that fails to have the Dunford–Pettis property. C. Bessaga and A. Pełczyński [Stud. Math. 17, 151–164 (1958; Zbl 0084.09805)] showed that if c 0 embeds in X * , then embeds in X * . G. Emmanuele and K. John [Czech. Math. J. 47, No. 1, 19–32 (1997; Zbl 0903.46006)] showed that if c 0 embeds in K(X,Y), then K(X,Y) is not complemented in L(X,Y).

In the paper under review, 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 * (X * ,Y) of w * -w continuous operators is also studied.

46B03Isomorphic theory (including renorming) of Banach spaces
46B28Spaces of operators; tensor products; approximation properties