2-absolutely summing operators on the space \(C(T,X)\) (Q1961021)

Given a compact Hausdorff space \(T\) and Banach spaces \(X\) and \(Y\), every bounded linear operator \(U:{\mathcal C}(T, X)\to Y\) gives rise to a bounded linear operator \(U^{\#}:{\mathcal C}(T)\to{\mathcal L}(X, Y):\varphi\mapsto (x\mapsto U(\varphi\otimes x))\). The paper provides general examples of operators \(U\) such that \(U^{\#}\) takes its values in the Banach space \(\Pi_2(X, Y)\) of absolutely 2-summing operators from \(X\) into \(Y\) and is itself absolutely 2-summing as an operator \({\mathcal C}(T)\to \Pi_2(X, Y)\). It is also shown that this can happen if \(U\) is not absolutely 2-summing.