Abstract of a paper by J. Rodríguez, E.A. Sánchez-Pérez
This is an announcement for the paper “A class of summing operators acting in spaces of operators” by J. Rodríguez<https://arxiv.org/search/math?searchtype=author&query=Rodr%C3%ADguez%2C+J>, E.A. Sánchez-Pérez<https://arxiv.org/search/math?searchtype=author&query=S%C3%A1nchez-P%C3%A9rez%2C+E+A>. 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 $S: U \to Z$ 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 $$ \Big( \sum_{i=1}^n \|S(T_i)\|_Z^p \Big)^{1/p} \le K \sup_{x^* \in B_{X^*}} \Big(\sum_{i=1}^n \|T_i(x^*)\|_Y^p\Big)^{1/p} $$ 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 $(\ell^s_p,\ell_p)$-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 Kwapień 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 $1/p+1/q \leq 1$). https://arxiv.org/abs/2003.07252
participants (1)
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Bentuo Zheng (bzheng)