Reducibility of operator semigroups and values of vector states

Abstract

Let S be a multiplicative semigroup of bounded linear operators on a complex Hilbert space H, and let Ω be the range of a vector state on S so that Ω = {⟨Sξ, ξ⟩ : S ∈ S} for some fixed unit vector ξ ∈ H. We study the structure of sets Ω of cardinality two coming from irreducible semigroups S. This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for S. This is made possible by a thorough investigation of the structure of maximal families F of unit vectors in H with the property that there exists a fixed constant ρ ∈ C for which ⟨x, y⟩ = ρ for all distinct pairs x and y in F.