We show in this note that there exists a function
f
∈
∩
1
>
p
>
+
∞
L
C
p
[
0
,
1
]
f \in { \cap _1}_{ > p > + \infty }L_{\mathbf {C}}^p[0,1]
and for each
p
p
an isomorphism
T
:
L
C
p
→
L
C
p
T:L_{\mathbf {C}}^p \to L_{\mathbf {C}}^p
such that
su
p
n
∈
Z
‖
T
n
‖
>
+
∞
{\text {su}}{{\text {p}}_{n \in {\mathbf {Z}}}}\left \| {{T^n}} \right \| > + \infty
and
T
T
does not satisfy the punctual ergodic theorem. We give also an example of a one-parameter semigroup
(
T
t
,
t
⩾
0
)
({T_t},t \geqslant 0)
of power bounded operators in each
L
C
p
(
1
>
p
>
+
∞
)
L_{\mathbf {C}}^p(1 > p > + \infty )
for which the assertion of the local ergodic theorem
(
(
1
/
t
)
∫
0
t
T
s
f
d
s
((1/t)\smallint _0^t{T_s}fds
converge almost everywhere as
t
→
0
+
t \to {0_ + }
for all
f
∈
L
p
f \in {L^p}
fails to be true.