Semigroups of Partial Isometries

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.