Semigroups of Partial Isometries
Alexey I. Popov, Heydar Radjavi

TL;DR
This paper characterizes self-adjoint semigroups of partial isometries on Hilbert spaces, linking them to inverse semigroups and providing a structure theorem involving weighted composition operators and atomic measure spaces.
Contribution
It establishes a comprehensive structure theorem for self-adjoint semigroups of partial isometries, connecting them to inverse semigroup representations and weighted composition operators.
Findings
Semigroups coincide with faithful inverse semigroup representations
Structure theorem involves weighted composition operators
Atomic measure spaces correspond to zero-unitary matrices
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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