Nearly invariant subspaces for operators in Hilbert spaces

For a shift operator T with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly T−1T−1 invariant subspaces in Hilbert space in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product B, we give a description of the nearly T−1BTB−1 invariant subspaces for the operator TBTB of multiplication by B in a scale of Dirichlet-type spaces.