Stability of an exponentially stabilizable system

Let A be the generator of a C
0
semigroup
T(t), t \geq 0
, and denote by
S(t), t \geq 0
, the semigroup generated by
A - K
, where
K
is a bounded linear operator on a Hilbert space. In this note we find necessary and sufficient conditions for the original semigroup
T(t), t \geq 0
, to be exponentially stable, given that the "feedback" semigroup
S(t), t \geq 0
, is exponentially stable. Applications to feedback stabilization via a steady-state Riccati equation will then be made.