A class of summing operators acting in spaces of operators

Let $X$, $Y$ and $Z$ be Banach spaces and let $U$ be a subspace of $\mathcal{L}(X^{\ast },Y)$, the Banach space of all operators from $X^{\ast }$ to $Y$. An operator $S:U\to Z$ is said to be $({\mathrm{\ell }}_p^s,{\mathrm{\ell }}_p)$-summing (where $1\leq p <\infty$)ifthereisaconstant$K\geq 0$suchthat$$ \Big( \sum_{i=1}^n |S(T_i)|Z^p \Big)^{1/p} \le K \sup{x^* \in B_{X^}} \Big(\sum_{i=1}^n |T_i(x^)|_Y^p\Big)^{1/p} $$ for every $n\in \mathbb{N}$ and every $T_1,\dots ,T_n\in U$. 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 one hand, we discuss Pietsch-type domination results for $({\mathrm{\ell }}_p^s,{\mathrm{\ell }}_p)$-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 Kwapie'{n} characterizing those operators which factor as $S_1\circ S_2$, where $S_2$ is absolutely $p$-summing and $S_1^{\ast }$ is absolutely $q$-summing ($1<p,q<\mathrm{\infty }$ and $1\mathrm{/}p+1\mathrm{/}q\le 1$).