  We introduce a rigorous framework for defining quasinormal modes on
stationary, asymptotically flat spacetimes as isolated eigenvalues of the
infinitesimal generator of time translations. We consider time functions
corresponding to a foliation of asymptotically hyperboloidal hypersurfaces and
restrict to suitable Hilbert spaces of functions. These functions have finite
Sobolev regularity in bounded regions, but need to be Gevrey-regular at null
infinity. This framework is developed in the context of sub-extremal Kerr
spacetimes, but also gives uniform-in-$\Lambda$ resolvent estimates on Kerr--de
Sitter spacetimes with a small cosmological constant $\Lambda$. As a corollary,
we also construct the meromorphic continuation (in a sector of the complex
plane) of the cut-off resolvent in Kerr that is associated to the standard
Boyer--Lindquist time function. The framework introduced in this paper bridges
different notions of quasinormal modes found in the literature. As further
applications of our methods, we prove stability of quasinormal frequencies in a
sector of the complex plane, with respect to suitably small perturbations and
establish convergence properties for Kerr--de Sitter quasinormal frequencies
when the cosmological constant approaches zero.
