Reducibility of operator semigroups and values of vector states

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 = { ⟨ S ξ , 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...

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Bibliographic Details
Published in:Semigroup forum 2017-08, Vol.95 (1), p.126-158
Main Authors: Marcoux, L. W., Radjavi, H., Yahaghi, B. R.
Format: Article
Language:English
Subjects:
Algebra
Group theory
Hilbert space
Linear operators
Mathematics
Mathematics and Statistics
Research Article
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Summary: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 = { ⟨ S ξ , 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 .
ISSN:0037-1912
1432-2137
DOI:10.1007/s00233-017-9872-7