On the punctual and local ergodic theorem for nonpositive power bounded operators in $L^ p_ {\textbf {C}}[0,1],\;1<p<+\infty$

Abstract:

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