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 $\mathcal{L}(X^{\ast },Y)$, the Banach space of all operators from $X^{\ast }$ to $Y$. An operator $S:U\to Z$ is said to be $({\ell }_p^s,{\ell }_p)$-summing (where $1\le p<\infty$) if there is a constant $K\ge 0$ such that $$(\sum _{i=1}^n\Vert S(T_i){\Vert }_Z^p{)}^{1/p}\le K\underset{x^{\ast }\in B_{X^{\ast }}}{\sup }(\sum _{i=1}^n\Vert T_i(x^{\ast }){\Vert }_Y^p{)}^{1/p}$$ 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 $({\ell }_p^s,{\ell }_p)$-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 Kwapień 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 ($1<p,q<\infty$ and $1/p+1/q\le 1$).