  Feedback asymptotic stabilization of control systems is an important topic of
control theory and applications. Broadly speaking, if the system $\dot{x} =
f(x,u)$ is locally asymptotically stabilizable, then there exists a feedback
control $u(x)$ ensuring the convergence to an equilibrium for any trajectory
starting from a point sufficiently close to the equilibrium state. In this
paper, we develop a reasonably natural and general composition operator
approach to stabilizability. To begin with, we provide an extension of the
classical Hautus lemma to the generalized context of composition operators and
show that Brockett's theorem is still necessary for local asymptotic
stabilizability in this generalized framework. Further, we employ a powerful
version of the implicit function theorem--as given by Jittorntrum and
Kumagai--to cover stabilization without differentiability requirements in this
expanded context. Employing the obtained characterizations, we establish
relationships between stabilizability in the conventional sense and in the
generalized composition operator sense. This connection allows us to show that
the stabilizability of a control system is equivalent to the stability of an
associated system. That is, we reduce the question of stabilizability to that
of stability.
