  We consider the spherically symmetric, asymptotically flat Einstein-Vlasov
system. We find explicit conditions on the initial data, with ADM mass M, such
that the resulting spacetime has the following properties: there is a family of
radially outgoing null geodesics where the area radius r along each geodesic is
bounded by 2M, the timelike lines $r=c\in [0,2M]$ are incomplete, and for r>2M
the metric converges asymptotically to the Schwarzschild metric with mass M.
The initial data that we construct guarantee the formation of a black hole in
the evolution. We also give examples of such initial data with the additional
property that the solutions exist for all $r\geq 0$ and all Schwarzschild time,
i.e., we obtain global existence in Schwarzschild coordinates in situations
where the initial data are not small. Some of our results are also established
for the Einstein equations coupled to a general matter model characterized by
conditions on the matter quantities.
