Structure of subspaces of the compact operators having the Dunford-Pettis property
A Banach space \(X\) is said to have the Dunford-Pettis property (DPP) if weakly compact operators on \(X\) map weakly compact sets to norm compact sets. It has turned out to be an important notion for both Banach spaces and Banach algebras. \textit{S. W. Brown} [J. Oper. Theory 33, No. 1, 33-42 (1995; Zbl 0838.47032)] and \textit{A. Ülger} [J. Oper. Theory 37, No. 2, 371-378 (1997; Zbl 0894.47033)] have recently characterized those subspaces of \(K(H)\) (the compact operators on a not-necessarily-separable Hilbert space) that have DPP. Here the authors show that the same characterization is valid for subspaces of \(K(\ell_p)\) \(1 < p < \infty\) and for compact operators on finite sums of \(\ell_p\) spaces. They show, further, that these subspaces are precisely those that can be embedded in ({ i.e.} are linearly isomorphic to a subspace of) \(c_0\). The paper also contains a number of interesting corollaries and examples.
scientific article; zbMATH DE number 1384885