Invariance and near invariance for non-cyclic shift semigroups

Abstract

We characterise the subspaces of $H^2(\mathbb{D})$ that are invariant under the semigroups generated by two higher-order shifts $S^m$ and $S^n$. Complete descriptions are obtained for both invariant and nearly invariant subspaces associated with these operators and their adjoints. The approach is based on vector-valued Hardy spaces, the Beurling--Lax theorem, and matrix-valued inner functions. Finally we apply these results to non-cyclic shift semigroups and to Toeplitz operators induced by finite Blaschke products.

15 pages. Some minor corrections and updates