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arXiv · 1306.1973

Semigroups of Partial Isometries

Alexey I. Popov·Heydar Radjavi
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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 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.

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Alexey I. Popov, Heydar Radjavi. 2013-06-09. Semigroups of Partial Isometries. https://arxiv.org/abs/1306.1973

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