### Abstract:

A Hilbert space operator T is called power bounded if sup∥Tn∥<∞sup‖Tn‖<∞. 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.

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