Given the infinitesimal generator
A
A
of a
C
0
C_0
-semigroup on the Banach space
X
X
which satisfies the Kreiss resolvent condition, i.e., there exists an
M
>
0
M>0
such that
‖
(
s
I
−
A
)
−
1
‖
≤
M
R
e
(
s
)
\| (sI-A)^{-1}\| \leq \frac {M}{\mathrm {Re}(s)}
for all complex
s
s
with positive real part, we show that for general Banach spaces this condition does not give any information on the growth of the associated
C
0
C_0
-semigroup. For Hilbert spaces the situation is less dramatic. In particular, we show that the semigroup can grow at most like
t
t
. Furthermore, we show that for every
γ
∈
(
0
,
1
)
\gamma \in (0,1)
there exists an infinitesimal generator satisfying the Kreiss resolvent condition, but whose semigroup grows at least like
t
γ
t^\gamma
. As a consequence, we find that for
R
N
{\mathbb R}^N
with the standard Euclidian norm the estimate
‖
exp
(
A
t
)
‖
≤
M
1
min
(
N
,
t
)
\|\exp (At)\| \leq M_1 \min (N,t)
cannot be replaced by a lower power of
N
N
or
t
t
.