Abstract. We discuss the two closely related, but different concepts of weak and almost weak stability for the powers of a contraction on a separable Hilbert space. Extending Halmos' and Rohlin's theorems in ergodic theory as a model, we show that the set of all weakly stable contractions is of first category while the set of all almost weakly stable contractions is of second category and is residual. Analogous statements for unitary and isometric operators are also proved.

AMS Subject Classification (1991): 47A35, 37A25

Keyword(s): Power bounded operators, Hilbert space, stability, weak and strong mixing

Received December 15, 2006, and in final form March 28, 2008. (Registered under 6016/2009.)