Invariant subspaces for finite index shifts in Hardy spaces

LetHbe the finite direct sums ofH2(D). In this paper, we give a characteriza-tion of the closed subspaces ofHwhich are invariant under the shift, thus obtaining a concreteBeurling-type theorem for the finite index shift. This characterization presents any such a sub-space as the finite intersection, up to an inner function, of pre-images of a closed shift-invariantsubspace ofH2(D)under “determinantal operators” fromHtoH2(D), that is, continuous linearoperators which intertwine the shifts and appear as determinants of matrices with entries givenby bounded holomorphic functions. With simple algebraic manipulations we provide a directproof that every invariant closed subspace of codimension at least two sits into a non-trivialclosed invariant subspace. As a consequence every contraction with finite defect has a nontrivialclosed invariant subspace.