The Gearhart-Prüss-Huang theorem characterizes exponential stability, a form of semigroup stability,
for a strongly continuous semigroup
on a Hilbert space. If 
is such a bounded semigroup
with infinitesimal generator 
, then it has exponential
stability iff
 |
Here 
is the imaginary axis, 
is the identity operator,
and 
is the resolvent set of 
. The supremum is a uniform bound
on the corresponding resolvent operators.
The theorem is a central frequency-domain test for semigroup
stability.
-Semigroups." Trans. Amer. Math. Soc. 284,
847-857, 1984.