Abstract. If $T$ is a Hilbert space contraction, then $T^*nT^n\mathop\to ^sA$, where $A$ is a nonnegative contraction. The strong limit $A$ is a projection if and only $T=G\oplus V$, where $G$ is a strongly stable contraction and $V$ is an isometry. This article is an expository paper on the class of contractions $T$ for which $A$ is a projection. After surveying such a class, it is shown that it is quite a large class. Indeed, it includes (i) all contractions whose adjoint has property PF, and also (ii) all contractions whose intersection of the continuous spectrum of its completely nonunitary direct summand with the unit circle has Lebesgue measure zero. Some new questions are investigated as well. For instance, is $A$ a projection for every biquasitriangular contraction $T$? If so, then every contraction not in class $\mathcal C_{00}$ has a nontrivial invariant subspace.

DOI: 10.14232/actasm-013-255-6

AMS Subject Classification (1991): 47A45; 47A15

Keyword(s): partially isometric contractions, biquasitriangular operators, invariant subspaces

Received January 17, 2013, and in final version March 11, 2013. (Registered under 5/2013.)