Invariant subspaces of perturbed backward shift

We represent closed subspaces of the Hardy space that are invariant under finite-rank perturbations of the backward shift. We apply this to classify almost invariant subspaces of the backward shift and represent closed subspaces that are invariant under a more refined version of nearly invariant subspaces of the backward shift. Kernels of certain perturbed Toeplitz operators are examples of the newly introduced nearly invariant subspaces.