A class of summing operators acting in spaces of operators

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