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We show the existence of a doubly power-bounded T on L p , 1 < p < ∞ , p ≠ 2 , such that T is spectral of scalar type (hence polynomially bounded), T is not similar to a Lamperti operator (hence is not similar to an isometry), none of the powers of T is similar to a Lamperti operator, none of the powers is similar to a positive operator, and for some f ∈ L p the averages 1 n ∑ k = 1 n T k f (or the averages along the primes or the squares) fail to be a.e. convergent. [ABSTRACT FROM AUTHOR]

We show that if a σ -finite infinite measure space ( Ω , μ ) is quasi-non-atomic, then the Dunford–Schwartz pointwise ergodic theorem holds for f ∈ L 1 ( Ω ) + L ∞ ( Ω ) if and only if μ { | f | ≥ λ } < ∞ for all λ > 0 . [ABSTRACT FROM AUTHOR]