2013/06/09 by Alexey I. Popov, Heydar Radjavi, Popov, Alexey I. +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA) #math.FA #math.GR #math.OA
paper · pdf · doi:10.48550/arxiv.1306.1973
To appear in Semigroup Forum
arxiv created 2013/06/12 · arxiv updated 2013/06/13
We study self-adjoint semigroups of partial isometries on a Hilbert space. These semigroups coincide precisely with faithful representations of abstract inverse semigroups. Groups of unitary operators are specialized examples of self-adjoint semigroups of partial isometries. We obtain a general structure result showing that every self-adjoint semigroup of partial isometries consists of "generalized weighted composition" operators on a space of square-integrable Hilbert-space valued functions. If the semigroup is irreducible and contains a compact operator then the underlying measure space is purely atomic, so that the semigroup is represented as "zero-unitary" matrices. In this case it is not even required that the semigroup be self-adjoint.