  We prove a generalized Birman-Schwinger principle in the non-self-adjoint
context. In particular, we provide a detailed discussion of geometric and
algebraic multiplicities of eigenvalues of the basic operator of interest
(e.g., a Schr\"odinger operator) and the associated Birman-Schwinger operator,
and additionally offer a careful study of the associated Jordan chains of
generalized eigenvectors of both operators. In the course of our analysis we
also study algebraic and geometric multiplicities of zeros of strongly analytic
operator-valued functions and the associated Jordan chains of generalized
eigenvectors. We also relate algebraic multiplicities to the notion of the
index of analytic operator-valued functions and derive a general
Weinstein-Aronszajn formula for a pair of non-self-adjoint operators.
