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Zbl 1170.46009
Ghenciu, Ioana; Lewis, Paul
Dunford--Pettis properties and spaces of operators. (English)
[J] Can. Math. Bull. 52, No. 2, 213-223 (2009). ISSN 0008-4395; ISSN 1496-4287/e

Authors' summary: {\it 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 $X$, $\ell_{1}$ embeds in $X$, or there is a subspace of $X$ that fails to have the Dunford--Pettis property. {\it C.\,Bessaga} and {\it A.\,Pełczyński} [Stud.\ Math.\ 17, 151--164 (1958; Zbl 0084.09805)] showed that if $c_{0}$ embeds in $X^*$, then $\ell_\infty$ embeds in $X^*$. {\it G.\,Emmanuele} and {\it 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.
[Joe Howard (Portales)]

MSC 2000:: *46B03 Isomorphic theory (including renorming) of Banach spaces
46B28 Normed linear spaces of linear operators, etc.

Keywords: Dunford-Pettis property; Dunford-Pettis set; basic sequence; complemented spaces of operators

Citations: Zbl 0084.09805; Zbl 0903.46006

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