Representing completely continuous operators through weakly compact operators (Q2812000)

Let \(L, K, W\) and \(V\) denote the operator ideals of bounded linear, compact, weakly compact, and completely continuous operators. Let \(X\) and \(Y\) be Banach spaces. An operator \(T \in L(X,Y)\) is weakly \(\infty\)-compact if \(T(B_X)\) is a relatively weakly \(\infty\)-compact subset of \(Y\), that is, if \(T(B_X)\) is contained in \(\{ \sum_{k=1}^\infty a_n x_n : (a_n) \in B_{\ell_1}\}\) where \((x_n)\) form some weakly null sequence. The weakly \(\infty\)-compact operators, \(W_\infty\), is a surjective operator ideal.NEWLINENEWLINEThe right-hand quotient \(A \circ B^{-1}\) of two operator ideals \(A\) and \(B\) consists of all operators \(T \in L(X,Y)\) such that \(TS \in A(Z,Y)\) whenever \(S \in B(Z,X)\) for some Banach space \(Z\). It is well known that \(V = K \circ W^{-1}\). The authors show that \(V = W_\infty \circ W^{-1}\). This provides an alternative proof of a theorem of \textit{P. N. Dowling} et al. [J. Funct. Anal. 263, No.~5, 1378--1381 (2012; Zbl 1255.46004)]: Every weakly compact subset of a Banach space \(X\) is contained in the closed convex hull of a weakly null sequence if and only if \(X\) has the Schur property.NEWLINENEWLINEIt is also shown that \(K \subset W_\infty \subset W\) and \(K \subset W_\infty \subset V\) where the inclusions are strict. Furthermore, for \(T \in L(X,Y)\) it is shown that \(T \in V(X,Y)\) if and only if \(T\) takes relatively weakly compact subsets of \(X\) into relatively weakly \(\infty\)-compact subsets of \(Y\).