The asymptotically flat Einstein-Vlasov system is used to describe the evolution of an isolated globular cluster or galaxy in a relativistic setting. We consider this system in Schwarzschild coordinates with a fixed central black hole as well as without singularity. After constructing compactly supported equilibria in these cases, we discuss their stability properties. We derive a sharp criterion for linear stability by developing a Birman-Schwinger type principle originating from quantum mechanics. We then conclude by discussing applications of our criterion, including the linear stability of "small" Vlasov matter shells surrounding a black hole. The talk is based on joint work with Sebastian Günther and Gerhard Rein.