  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.
