One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices
A
A
under consideration. This so-called resolvent condition is known to imply, for all
n
≥
1
n\ge 1
, the upper bounds
‖
A
n
‖
≤
e
K
(
N
+
1
)
\|A^n\|\le eK(N+1)
and
‖
A
n
‖
≤
e
K
(
n
+
1
)
\|A^n\|\le eK(n+1)
. Here
‖
⋅
‖
\|\cdot \|
is the spectral norm,
K
K
is the constant occurring in the resolvent condition, and the order of
A
A
is equal to
N
+
1
≥
1
N+1\ge 1
. It is a long-standing problem whether these upper bounds can be sharpened, for all fixed
K
>
1
K>1
, to bounds in which the right-hand members grow much slower than linearly with
N
+
1
N+1
and with
n
+
1
n+1
, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each
ϵ
>
0
\epsilon >0
, there are fixed values
C
>
0
,
K
>
1
C>0, K>1
and a sequence of
(
N
+
1
)
×
(
N
+
1
)
(N+1)\times (N+1)
matrices
A
N
A_N
, satisfying the resolvent condition, such that
‖
(
A
N
)
n
‖
≥
C
(
N
+
1
)
1
−
ϵ
\|(A_N)^n\|\ge C(N+ 1)^{1-\epsilon }
=
C
(
n
+
1
)
1
−
ϵ
=C(n+1)^{1-\epsilon }
for
N
=
n
=
1
,
2
,
3
,
…
N=n=1,2,3,\ldots
. The result proved in this paper is also relevant to matrices
A
A
whose
ϵ
\epsilon
-pseudospectra lie at a distance not exceeding
K
ϵ
K\epsilon
from the unit disk for all
ϵ
>
0
\epsilon >0
.