  The object of this paper is to prove a version of the
Beurling-Helson-Lowdenslager invariant subspace theorem for operators on
certain Banach spaces of functions on a multiply connected domain in the
complex plane. The norms for these spaces are either the usual Lebesgue and
Hardy space norms or certain continuous gauge norms. In the Hardy space case
the expected corollaries include the characterization of the cyclic vectors as
the outer functions in this context, a demonstration that the set of analytic
multiplication operators is maximal abelian and reflexive, and a determination
of the closed operators that commute with all analytic multiplication
operators.
