  We discuss abstract Birman-Schwinger principles to study spectra of
self-adjoint operators subject to small non-self-adjoint perturbations in a
factorised form. In particular, we extend and in part improve a classical
result by Kato which ensures spectral stability. As an application, we revisit
known results for Schr\"odinger and Dirac operators in Euclidean spaces and
establish new results for Schr\"odinger operators in three-dimensional
hyperbolic space.
