  We find characterizations of Va\v{s}\'ak spaces and weakly
$\mathcal{K}$-analytic spaces using the notions of separable projectional
resolution of the identity (SPRI) and of projectional skeleton. This in
particular addresses a recent challenge suggested by M. Fabian and V.
Montesinos in \cite{FM18}. Our method of proof also gives similar
characterizations of WCG spaces and their subspaces (some aspects of which were
known, some are new). Moreover we show that for countably many projectional
skeletons $\{\mathfrak{s}_n: n \in \omega\}$ on a Banach space inducing the
same set, there exists a projectional skeleton on the space (indexed by ranges
of the corresponding projections) which is isomorphic to a subskeleton of each
$\mathfrak{s}_n$, $n \in \omega$.
