Invariance and near invariance for non-cyclic shift semigroups

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.