  It is well known that subspaces of the Hardy space over the unit disk which
are invariant under the backward shift occur as the image of an observability
operator associated with a discrete-time linear system with stable
state-dynamics, as well as the functional-model space for a Hilbert space
contraction operator, while forward shift-invariant subspaces have a
representation in terms of an inner function. We discuss several variants of
these statements in the context of weighted Bergman spaces on the unit disk.
