Spaces of compact operators and their dual spaces (Q5921660)

Let \(L(X,Y)\) be the space of all bounded linear operators from the Banach space \(X\) into the Banach space \(Y\). Let \(w^{\prime}\) denote the dual weak operator topology, \({L}^{w^{\prime}}(X,Y)\) be the space of all \(T\in L(X,Y)\) such that there exists a sequence of compact operators \((T_n)_n\subset K(X,Y)\) satisfying \(T=\lim_nT_n\), and let \(|||T|||=\inf\{\sup_n||T_n||: T_n\in K(X,Y),\;T=\lim_nT_n\}\) (with the limit \(T=\lim_nT_n\) taken in the dual weak operator topology \(w^{\prime}\)). The authors show that \(({L}^{w^{\prime}}(X,Y), |||\cdot |||)\) is a Banach ideal of operators and that the topological dual space \(K(X,Y)^*\) is complemented in \(({L}^{w^{\prime}}(X,Y)), |||\cdot |||)^*\). They also give necessary and sufficient conditions for \(K(X,Y)\) to be reflexive.