  We consider different characterizations of Triebel--Lizorkin type spaces of
analytic functions on the unit disc. Even though our results appear in the
folklore, detailed descriptions are hard to find, and in fact we are unable to
discuss the full range of parameters. Without additional effort we work with
vector-valued analytic functions, and also consider a generalized scale of
function spaces, including for example so-called $Q$-spaces. The primary aim of
this note is to generalize, and clarify, a remarkable result by Cohn and
Verbitsky, on factorization of Triebel--Lizorkin spaces. Their result remains
valid for functions taking values in an arbitrary Banach space, provided that
the vector-valuedness "sits in the right factor". On the other hand, if we
impose vector-valuedness on the "wrong" factor, then the factorization fails
even for separable Hilbert spaces.
