AI 中文总结
本文研究驯顺分解性质$\mathfrak{TF}$与Fréchet空间线性拓扑不变量的关系，刻画了具备$\mathfrak{TF}$的空间三元组，证明该性质严格弱于驯顺性，并利用其刻画了特定三元组中空间的拓扑不变量。
AI 中文摘要
我们研究了在配套论文\textit{\cite{CDI}}中提出的、记为$\mathfrak{TF}$的驯顺分解性质（tame factorization property）与Fréchet空间的DN-$\Omega$型线性拓扑不变量之间的关系。结合$\mathfrak{TF}$的基本性质与已知的驯顺性和有界性刻画，我们得到了若干识别具备$\mathfrak{TF}$的Fréchet空间三元组的结果。我们进一步给出例子表明，驯顺分解性质是严格弱于驯顺性的条件：实际上，我们构造出了具备$\mathfrak{TF}$、但其任意两个分量组成的配对都不具备驯顺性的三元组。随后我们研究了由任意Fréchet空间$X$、满足$\underline{DN}$和$\Omega$性质的核Fréchet空间$Y$，以及有限型幂级数空间$\Lambda_1(\mathcal{E})$或无限型幂级数空间$\Lambda_\infty(\mathcal{E})$构成的三元组。我们证明，要求这类三元组具备驯顺分解性质$\mathfrak{TF}$可以刻画$X$的相应线性拓扑不变量；在部分情形下该结论无需对$Y$施加任何限制即可成立，而在另一些情形中则要求$Y$的近似直径维数与$\Lambda_1(\mathcal{E})$或$\Lambda_\infty(\mathcal{E})$的近似直径维数一致。
英文摘要
We investigate the relationship between the tame factorization property, denoted by $\mathfrak{TF}$ and introduced in the companion paper \cite{CDI}, and the DN-$Ω$ type linear topological invariants of Fréchet spaces. Combining the basic properties of $\mathfrak{TF}$ with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess $\mathfrak{TF}$. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing $\mathfrak{TF}$ none of whose individual pairs are tame. We then investigate triples consisting of an arbitrary Fréchet space $X$, a nuclear Fréchet space $Y$ satisfying the properties $\underline{DN}$ and $Ω$, and a power series space of finite type $Λ_1(\mathcal{E})$ or infinite type $Λ_\infty(\mathcal{E})$. We show that requiring such a triple to possess the tame factorization property $\mathfrak{TF}$ characterizes the corresponding linear topological invariants of $X$; in some cases this holds without any restriction on $Y$, while in others it requires the coincidence of the approximate diametral dimension of $Y$ with that of $Λ_1(\mathcal{E})$ or $Λ_\infty(\mathcal{E})$.
Comments22 pages
