Let
{
T
(
t
)
:
t
⩾
0
}
\{ T(t):t \geqslant 0\}
be a
C
0
{C_0}
-semigroup on a Banach space
X
X
with generator
A
A
, and let
x
∈
X
x \in X
. If
σ
(
A
)
∩
i
R
\sigma (A) \cap i{\mathbf {R}}
is empty and
t
↦
T
(
t
)
x
t \mapsto T(t)x
is uniformly continuous, then
|
|
T
(
t
)
x
|
|
→
0
||T(t)x|| \to 0
as
t
→
∞
t \to \infty
. If the semigroup is sun-reflexive,
σ
(
A
)
∩
i
R
\sigma (A) \cap i{\mathbf {R}}
is countable,
P
σ
(
A
)
∩
i
R
P\sigma (A) \cap i{\mathbf {R}}
is empty, and
t
↦
T
(
t
)
x
t \mapsto T(t)x
is uniformly weakly continuous, then
T
(
t
)
x
→
0
T(t)x \to 0
weakly as
t
→
∞
t \to \infty
. Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.