A class of summing operators acting in spaces of operators

Abstract

Let \(X\), \(Y\) and \(Z\) be Banach spaces and let \(U\) be a subspace of \(\mathcal{L}(X^*,Y)\), the Banach space of all operators from \(X^*\) to \(Y\). An operator \(S\colon U \to Z\) is said to be \((\ell^s_p,\ell_p)\)-summing (where \(1\leq p <\infty\)) if there is a constant \(K\geq 0\) such that \(\left( \sum_{i=1}^n \|S(T_i)\|_Z^p \right)^{1/p}\le K\sup_{x^* \in B_{X^*}} \left(\sum_{i=1}^n \|T_i(x^*)\|_Y^p\right)^{1/p}\) for every \(n\in\mathbf{N}\) and all \(T_1,\dots,T_n \in U\). In this paper we study this class of operators, introduced by Blasco and Signes as a natural generalization of the \((p,Y)\)-summing operators of Kislyakov. On the one hand, we discuss Pietsch-type domination results for \((\ell^s_p,\ell_p)\)-summing operators. In this direction, we provide a negative answer to a question raised by Blasco and Signes, and we also give new insight on a result by Botelho and Santos. On the other hand, we extend to this setting the classical theorem of Kwapien characterizing those operators which factor as \(S_1\circ S_2\), where \(S_2\) is absolutely \(p\)-summing and \(S_1^*\) is absolutely \(q\)-summing (\(1<p,q<\infty\) and \(1/p+1/q \leq 1\)).