Uniformly summing sets of operators on spaces of vector-valued continuous functions (Q2501149)

Let \({\mathcal L}(X,Y)\) be the space of all operators between Banach spaces \(X\) and \(Y\) and \(\Pi_p(X,Y)\) the linear subspace of all \(p\)-summing operators, \(1\leq p<\infty\). A set \({\mathbb M}\i\Pi_p(X,Y)\) is said to be \textit{uniformly \(p\)-summing} if, for every weak \(\ell_p\) sequence \((x_n)\) in \(X\), the series \(\sum_n\|Tx_n\|\) converges uniformly in \(T\in{\mathbb M}\). In this paper, emphasis is on the case where the domain is \(C(\Omega,X)\), the space of \(X\)-valued continuous functions on a compact Hausdorff space \(\Omega\). Put \(C(\Omega)=C(\Omega,{\mathbb R})\). Via \(T(fx)=(T^{\hbox{\#}}f)x\) (\(f\in C(\Omega)\), \(x\in X\)) each operator \(T:C(\Omega,X)\to Y\) gives rise to an operator \(T^{\hbox{\#}}:C(\Omega)\to{\mathcal L}(X,Y)\). Given \({\mathbb M}\i\Pi_1(C(\Omega,X),Y)\), put \({\mathbb M}^{\hbox{\#}}=\{T^{\hbox{\#}}:T\in{\mathbb M}\}\). One of the main results asserts that, regardless of the space \(Y\), for \(X\) not to contain a copy of \(c_0\) it is necessary and sufficient that a subset \({\mathbb M}\) of \(\Pi_1(C(\Omega,X),Y)\) is uniformly \(1\)-summing if and only if \({\mathbb M}^{\hbox{\#}}=\{T^{\hbox{\#}}:T\in{\mathbb M}\}\) is uniformly \(1\)-summing in \(\Pi_1(C(\Omega),\Pi_1(X,Y))\).