Completely continuous subspaces of operator ideals (Q2495250)

Let \(W,X,Y\) and \(Z\) denote Banach spaces and \(K(X,Y)\) the space of compact operators from \(X\) to \(Y\) (\(K(X):=K(X,X)\)). If \(\mathcal U\) is a Banach operator ideal, a subspace \(\mathcal M\) of a component \({\mathcal U}(X,Y)\) is said to be \textit{strongly completely continuous} in \(K(X,Y)\) if, for all compact operators \(R:Y\to W\) and \(S:Z \to X\), the corresponding left and right multiplication operators are compact. This is stronger than the usual definition of a completely continuous subalgebra of \(K(X)\), but equivalent in the presence of a density condition. The authors give several conditions equivalent to strong complete continuity and show that, under certain conditions (for example, when \(X\) is an \(\ell_p\)-direct sum of finite dimensional spaces and \(\mathcal M\) is a sub-algebra of \(K(X)\) that satisfies the density condition), these are also equivalent to (a) \({\mathcal M}^*\) having the Schur property, and (b) \({\mathcal M}\) having the Dunford--Pettis property. These results generalize those of \textit{S.\,W.\,Brown} [J.~Oper.\ Theory 33, 33--42 (1995; Zbl 0838.47032)] and \textit{A.\,Ülger} [J.~Oper.\ Theory 37, 371--378 (1997; Zbl 0894.47033)].