A closed linear subspace
ℋp(G) is said to be invariant if zf(z) is in ℳ for all f(z) ∈ℳ. It is said to be fully
invariant if r(z)f(z) is in ℳ for all f ∈ℳ and all rational functions r(z) with poles
in the complement of G. This paper investigates those invariant subspaces of ℋp(G),
for a multiply connected G, which are invariant but not fully invariant. We show that
an invariant subspace ℳ fails to be fully invariant if and only if there is
one bounded component Gi of the complement of G such that the ratio
of any two functions in ℳ has a pseudo-continuation to a meromorphic
function in the Nevanlinna class of Gi. This allows us to give a complete
characterization of those invariant subspaces of ℋp(G) which contain the
constants.
