We investigate the relationship between the tame factorization property, denoted by TF and introduced in the companion paper CDI, and the DN-Ω type linear topological invariants of Fréchet spaces. Combining the basic properties of TF with known characterizations of tameness and boundedness, we obtain several results identifying the triples of Fréchet spaces that possess TF. We further exhibit examples showing that tame factorization property is a strictly weaker condition than tameness, indeed, we construct triples possessing TF
Nearby in the stack
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
DN
and
Ω
, and a power series space of finite type
Λ1(E)
or infinite type
Λ∞(E)
. We show that requiring such a triple to possess the tame factorization property
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