Power bounded operators, similarity problems and the Blum Hanson property

A Hilbert space operator T is called power bounded if $\sup \|T^n\|<\infty$. Clearly every operator similar to a contraction is power bounded. Problems concerning relations between power bounded operators and operators similar to a contraction have a long history. We give a survey of such results and discuss also the closely related Blum-Hanson property, which comes from the ergodic theory.