English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We show in this note that there exists a function      f ∈  ∩  1 &gt; p &gt; + ∞  L C p [ 0 , 1 ] f \in { \cap _1}_{ &gt; p &gt; + \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 ‖ &gt; + ∞  {\text {su}}{{\text {p}}_{n \in {\mathbf {Z}}}}\left \| {{T^n}} \right \| &gt; + \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 &gt; p &gt; + ∞  ) L_{\mathbf {C}}^p(1 &gt; p &gt; + \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.

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