Shift invariant subspaces in the Bloch space

Abstract: We consider weak-star closed invariant subspaces of the shift operator in the classical Bloch space. We prove that any bounded analytic function decomposes into two factors, one which is cyclic and another one generating a proper shift invariant subspace, satisfying a permanence property, which in a certain way is opposite to cyclicity. Singular inner functions play the crucial role in this decomposition. We show in several different ways that the description of shift invariant subspaces generated by inner functions in the Bloch spaces deviates substantially from the corresponding description in the Bergman spaces, provided by the celebrated Korenblum and Roberts Theorem. Furthermore, the relationship between invertibility and cyclicity is also investigated and we provide an invertible function in the Bloch space which is not cyclic therein. Our results answer several open questions stated in the early nineties.