  In this article, we characterize reducing and invariant subspaces of the
space of square integrable functions defined in the unit circle and having
values in some Hardy space with multiplicity.
  We consider subspaces that reduce the bilateral shift and at the same time
are invariant under the unilateral shift acting locally. We also study
subspaces that reduce both operators. The conditions obtained are of the type
of the ones in Helson and Beurling-Lax-Halmos theorems on characterizations of
the invariance for the bilateral and unilateral shift. The motivations for our
study were inspired by recent results on Dynamical Sampling in shift-invariant
spaces.
