A "typical" contraction is unitary

Abstract

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.

On typical properties of Hilbert space operators

“…Proof. Statements 1 and 2 follow from Theorem 5.2 and the results of [29], while 3 is a corollary of [8,Proposition 4.3].…”

“…We obtain the following results. In Section 3, we recall some results obtained by the first author (see [8], [10]) about typical properties in the weak topology. In this topology, a typical contraction is unitary, it has maximal spectrum and empty point spectrum, it can be embedded into a C 0 -semigroup, and typical contractions are not unitarily equivalent.…”

Embedding operators into strongly continuous semigroups

“…Theorem 5.5. (see Eisner [4]) The set of all embeddable contractions on a separable infinitedimensional Hilbert space form a residual set for the weak operator topology.…”

“…To prove this fact, one first shows that unitary operators form a residual set (see Eisner [4,Theorem 3.3]). Then Proposition 4.1 finishes the argument.…”

The space of contractive $C_{0}$-semigroups is a Baire space

“…Various residuality results for the space of contractive C 0 -semigroups over a separable, infinite dimensional Hilbert space (under the topology of uniform weak convergence on compact subsets of R `) as well as for the space of contractions over a separable, infinite dimensional Hilbert space (under the pw-topology) have been investigated in [ES09], [Eis10], [EM10], etc. Residuality is only meaningful (viz.…”