A class of summing operators acting in spaces of operators

José Rodríguez¹ ; Enrique A. Sánchez-Pérez²

¹ Universidad de Murcia, Dpto. de Ingeniería y Tecnología de Computadores
² Universitat Politècnica de València, Instituto Universitario de Matemática Pura y Aplicada

Annales Fennici Mathematici, Tome 46 (2021) no. 2, pp. 667-681.

Résumé

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 $1/p+1/q \leq 1$).

Détail

Keywords: Summing operator, dominated operator, ε-product of Banach spaces, strong operator topology, universally measurable function