The Arendt-Batty-Lyubich-Vũ theorem is a sufficient condition for the strong stability of a bounded strongly continuous
semigroup on a Banach space 
. Let 
be such a semigroup,
let 
be its infinitesimal generator, and let
be the adjoint of 
on the dual vector space 
.
If the intersection of the operator spectrum
of 
with the imaginary axis is countable and the point spectrum of 
contains no point on the imaginary
axis, then
 |
for every 
in the Banach space. This pointwise decay of each orbit is
the form of semigroup stability called strong
stability. A standard corollary for a reflexive space
replaces the condition on 
by the requirement that 
have no eigenvalues on the
imaginary axis.
