The Dunford–Pettis property, the Gelfand–Phillips property, and L-sets

Property $ \bf{(wL)}$ and the reciprocal Dunford-Pettis property in projective tensor products

“…property (wL)) if and only if any operator T : Y → X * such that T * | X is completely continuous, is weakly compact (resp. weakly precompact) (by Theorem 4.7 of [23]). …”

Some isomorphic properties in $K(X,Y)$ and in projective tensor products

“…The hypothesis that, for every T ∈ L(X, Y * * ), for every weakly null sequence (x n ) ⊂ X, the sequence (T (x n )) is an L-set, is more general than the hypothesis that L(X, Y * * ) = K(X, Y * * ). As we shall prove in the following Proposition, the condition L(X, Y * * ) = K(X, Y * * ) is equivalent to the hypothesis in Ghenciu and Lewis's Theorem [19]. In the following example we consider two Banach spaces X and Y without the Dunford Pettis property since if one of them has the Dunford Pettis property, then the thesis of Theorem 3 follows immediately and it is contained in [19].…”

“…It is also known (see again [15]) that every dual Banach space has the (DPrcp) if and only if it has the weak Radon Nikodym property that is if and only if its predual does not contain a copy of l 1 . In [15] and in [19] the authors studied the lifting of the (DPrcp) from X * , the dual space of a Banach space X, and from a Banach space Y to the space K(X, Y ) of all compact operators from X into Y. In this note we shall furnish a new condition in order that K(X, Y ) has the (DPrcp).…”

Complemented Subspaces of Linear Bounded Operators

“…Apply Lemma 2. (v) Since X has the DPP and X * does not have the Schur property, ℓ 1 ֒→ X (see [8,22]). Then L 1 ֒→ X * (by a result in [26]), hence ℓ 2 ֒→ X * (see [7]).…”

“…A Banach space X has the Schur property if every weakly null sequence in X is norm null. Corollary 3.10 of [22] shows that if X is a separable Banach space, then X has the Schur property if and only if L(X, c 0 ) = CC(X, c 0 ).…”