Remarks on dilations in 𝐿_{𝑝}-spaces

Akcoglu, M. A.; Sucheston, L.

doi:10.1090/S0002-9939-1975-0377558-X

Remarks on dilations in $L_{p}$-spaces
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by M. A. Akcoglu and L. Sucheston PDF

Proc. Amer. Math. Soc. 53 (1975), 80-82 Request permission

Abstract:

Let $(X,\;\mathcal {F},\;\mu )$ be a nonatomic measure space. It is shown that there exists a unitary operator $U$ on ${L_2} = {L_2}(X,\;\mathcal {F},\;\mu )$, a function $f\epsilon {L_2}$, and a nonnull set $A$ in $\mathcal {F}$ such that ${n^{ - 1}}\Sigma _{i = 1}^n{U^i}f$ diverges on $A$.

References

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Positive contractions of

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Analyse harmonique des opérateurs de l’espace de Hilbert

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Real anàlysis