Characterizations of weakly $\mathcal{K}$-analytic and Va\v{s}\'ak spaces using projectional skeletons and using 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 solves a problem suggested by M. Fabian and V. Montesinos. 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 on a Banach space there exists a common subskeleton $(P_s)_{s \in \Gamma}$, which is in addition indexed by the ranges of the projections $\{P_s: s \in \Gamma\}$.