Invariant Subspaces for the Backward Shift on Hilbert Spaces of Analytic Functions with Regular Norm

We investigate the structure of invariant subspaces of backward shift operator Lf = (f − f(0))/ζ on a large class of abstract Hilbert spaces of analytic functions on the unit disc where the forward shift operator Mζf = ζf acts as a contraction. Our main results show that under certain regularity conditions on the norm of such a space, the functions in a nontrivial invariant subspace of L have meromorphic pseudocontinuations in the Nevanlinna class of the exterior of the unit disc. We also provide a regularity condition which implies that the subspace itself is contained in the Nevanlinna class of the disc. These results imply that the spectrum of the restriction of L to these subspaces intersects the unit disc in a discrete set and this fact is then applied to prove a general index-one theorem for the forward shift invariant subspaces of the Cauchy dual of the original space. Finally, we give a detailed discussion of the weighted shift operators for which our main results apply.