Doubly power-bounded operators on L^p, 2 ≠ p > 1

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 [Formula presented]∑_k=1 ⁿT^kf (or the averages along the primes or the squares) fail to be a.e. convergent.