  We develop a Birman-Schwinger principle for the spherically symmetric,
asymptotically flat Einstein-Vlasov system. It characterizes stability
properties of steady states such as the positive definiteness of an
Antonov-type operator or the existence of exponentially growing modes in terms
of a one-dimensional variational problem for a Hilbert-Schmidt operator. This
requires a refined analysis of the operators arising from linearizing the
system, which uses action-angle type variables. For the latter, a single-well
structure of the effective potential for the particle flow of the steady state
is required. This natural property can be verified for a broad class of
singularity-free steady states. As a particular example for the application of
our Birman-Schwinger principle we consider steady states where a Schwarzschild
black hole is surrounded by a shell of Vlasov matter. We prove the existence of
such steady states and derive linear stability if the mass of the Vlasov shell
is small compared to the mass of the black hole.
