Semigroups of partial isometries |
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Authors: | Alexey I. Popov Heydar Radjavi |
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Affiliation: | 1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada
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Abstract: | 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 finitely generated then the underlying measure space is purely atomic, so that the semigroup is represented as “zero-unitary” matrices. The same is true if the semigroup contains a compact operator, in which case it is not even required that the semigroup be self-adjoint. |
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