# Reducibility of operator semigroups and values of vector states

Reducibility of operator semigroups and values of vector states Let $$\mathcal S$$ S be a multiplicative semigroup of bounded linear operators on a complex Hilbert space $$\mathcal H$$ H , and let $$\Omega$$ Ω be the range of a vector state on $$\mathcal S$$ S so that $$\Omega = \{ \langle S \xi , \xi \rangle \,{:}\,S \in \mathcal S\}$$ Ω = { ⟨ S ξ , ξ ⟩ : S ∈ S } for some fixed unit vector $$\xi \in \mathcal H$$ ξ ∈ H . We study the structure of sets $$\Omega$$ Ω of cardinality two coming from irreducible semigroups $$\mathcal S$$ S . This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for $$\mathcal S$$ S . This is made possible by a thorough investigation of the structure of maximal families $$\mathcal F$$ F of unit vectors in $$\mathcal H$$ H with the property that there exists a fixed constant $$\rho \in \mathbb C$$ ρ ∈ C for which $$\langle x, y \rangle = \rho$$ ⟨ x , y ⟩ = ρ for all distinct pairs x and y in $$\mathcal F$$ F . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Semigroup Forum Springer Journals

# Reducibility of operator semigroups and values of vector states

, Volume 95 (1) – Apr 26, 2017
33 pages

/lp/springer_journal/reducibility-of-operator-semigroups-and-values-of-vector-states-9fhiiEWL0T
Publisher
Springer US
Subject
Mathematics; Algebra
ISSN
0037-1912
eISSN
1432-2137
D.O.I.
10.1007/s00233-017-9872-7
Publisher site
See Article on Publisher Site

### Abstract

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

### Journal

Semigroup ForumSpringer Journals

Published: Apr 26, 2017

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