论文标题
一类求和操作员在运营商的空间
A class of summing operators acting in spaces of operators
论文作者
论文摘要
令$ x $，$ y $和$ z $为Banach Space，让$ U $为$ \ Mathcal {l}（x^*，Y）$的子空间，所有运营商的Banach空间从$ x^*$到$ y $。运算符$ s：u \ to z $据说为$（\ ell^s_p，\ ell_p）$ - 总和（其中$ 1 \ leq p <\ infty $）如果有常数$ k \ geq 0 $，则 \ big（\ sum_ {i = 1}^n \ | s（t_i）\ | _z^p \ big）^{1/p} \ le k \ sup_ {x^*\ in B_ {X^*}} \ big（\ sum_ {i = 1}^n \ | t_i（x^*）\ | _y^p \ big）在本文中，我们研究了由布拉斯科（Blasco）引入的这类运营商，并作为$（p，y）$的自然概括 - 基斯利科夫（Kislyakov）的运营商。一方面，我们讨论了$（\ ell^s_p，\ ell_p）$ - 总和运算符的Pietsch-Type支配结果。在这个方向上，我们对Blasco和Signes提出的问题提供了负面答案，并且我们还对Botelho和Santos的结果提供了新的见解。另一方面，我们扩展到此设置Kwapień的经典定理，这些经营者的特征是$ s_1 \ circs s_2 $，其中$ s_2 $绝对是$ p $ -s_1^*$ as $ s_1^*$是绝对$ q $ -summing（$ q $ -summing）
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 $1/p+1/q \leq 1$).
