On the punctual and local ergodic theorem for nonpositive power bounded operators in 𝐿^{𝑝}_{𝐶}[0,1],1<𝑝<+∞

Abstract

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.