AI 中文总结
该研究为弗雷歇空间三元组引入驯服分解性质，通过算子半范数估计完成刻画，针对克塞空间三元组给出显式条件，还证明其与拟对角算子乘积构成的算子族的驯服性等价。
AI 中文摘要
我们为弗雷歇空间的三元组引入了\emph{驯服分解性质}：三元组$(E,G,F)$具有驯服分解性质，记为$(E,G,F) \in \mathfrak{TF}$，当且仅当存在非递减函数$S:\mathbb{N}\to\mathbb{N}$，使得从$E$到$F$的每个通过$G$分解的算子都是$S$-驯服的。我们通过算子半范数估计给出了$\mathfrak{TF}$的完整刻画，将该刻画专门应用于其中一个或多个空间为克塞空间的三元组，给出了克塞矩阵的显式条件，并描述了$\mathfrak{TF}$在克塞空间的投射张量积下的表现。最后，我们证明了克塞空间的三元组具有驯服分解性质当且仅当由两个拟对角算子的乘积构成的算子族是驯服的。
英文摘要
We introduce the \emph{tame factorization property} for triples of Fréchet spaces: a triple $(E,G,F)$ has the tame factorization property, denoted by $(E,G,F) \in \mathfrak{TF}$, if there exists a nondecreasing function $S:\mathbb{N}\to\mathbb{N}$ such that every operator from $E$ to $F$ factoring through $G$ is $S$-tame. We give a complete characterization of $\mathfrak{TF}$ in terms of operator seminorm estimates. We specialize this characterization to triples in which one or more of the spaces are Köthe spaces, putting explicit conditions on the Köthe matrices, and we describe how $\mathfrak{TF}$ behaves under projective tensor products of Köthe spaces. Finally, we show that a triple of Köthe spaces has the tame factorization property if and only if the family of operators that factor as products of two quasi-diagonal operators is tame.
Comments26 pages
