Tame Factorization Property II

Abstract

We investigate the relationship between the tame factorization property, denoted by $\mathfrak{TF}$ and introduced in the companion paper \cite{CDI}, and the DN-$\Omega$ type linear topological invariants of Fr\'echet spaces. Combining the basic properties of $\mathfrak{TF}$ with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fr\'echet 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 Fr\'echet space $X$, a nuclear Fr\'echet space $Y$ satisfying the properties $\underline{DN}$ and $\Omega$, and a power series space of finite type $\Lambda_1(\mathcal{E})$ or infinite type $\Lambda_\infty(\mathcal{E})$. 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 $\Lambda_1(\mathcal{E})$ or $\Lambda_\infty(\mathcal{E})$.