Characterizations of weakly $\mathcal{K}$-analytic and Va\v{s}\'ak spaces using projectional skeletons and separable PRI

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 $\mathfrak{s}_n$, $n \in \omega$.