The Alternative Dunford-Pettis Property in Subspaces of Operator Ideals

Strongly alternative Dunford–Pettis subspaces of operator ideals

“…Proof. Since X and Y are reflexive Banach spaces, Theorem 2.2 of [8] shows that all evaluation operators are DP1. So by Theorem 2.1, M is strongly DP1 in K(X, Y ).…”

“…The next Corollary 2.5 proves that for some Banach spaces X and Y, having the Schauder decompositions [7], and some operator ideal between them, the strong DP1 is also a sufficient condition for the DP1 property. The concept of Schauder decomposition, as a generalization of Schauder basis of Banach spaces, provides a good location for introducing the concept of P-property for subspaces of operator spaces, which has an essential role in the results of [8,11].…”

“…Finally, we need to remember the following theorem of [8]. For undefined terminologies about Schauder decomposition, we refer the reader to [7].…”

“…For undefined terminologies about Schauder decomposition, we refer the reader to [7]. Theorem 2.5 (Theorems 2.4 and 2.6 of [8]). Let X and Y have monotone finite dimensional Schauder decompositions such that the decomposition of X is shrinking.…”

“…This is similar to Theorem 1 of [4] which stay that a Banach space X has the DP property if and only if every weakly compact operator T : X → Y is completely continuous. We refer the reader for additional properties of these concepts to [1,2,4,6,8].…”