A "typical" contraction is unitary

Abstract

We show that the set of unitary operators on a separable infinite-dimensional Hilbert space is residual (for the weak operator topology) in the set of all contractions. The same holds for unitary operators in the set of all isometries and with respect to the strong operator topology. The continuous versions are discussed as well. These results are applied to the problem of embedding operators into strongly continuous semigroups.