A "typical" contraction is unitary

TL;DR

This paper proves that in certain topologies, the set of unitary operators is large within all contractions on infinite-dimensional Hilbert spaces, and applies these findings to operator embedding problems.

Contribution

It establishes residuality of unitary operators among contractions in specific topologies and explores implications for embedding operators into semigroups.

Findings

Unitary operators form a residual set among contractions in the weak operator topology.

Analogous residuality results hold for isometries in the strong operator topology.

Applications include embedding operators into strongly continuous semigroups.

Abstract

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