Invariance and near invariance for non-cyclic shift semigroups

This paper characterises the subspaces of $H^2(\mathbb{D})$ simultaneously invariant under $S^2$ and $S^{2k+1}$, where $S$ is the unilateral shift, then further identifies the subspaces that are nearly invariant under both $(S^2{)}^{\ast }$ and $(S^{2k+1}{)}^{\ast }$ for $k\ge 1$. More generally, the simultaneously (nearly) invariant subspaces with respect to $(S^m{)}^{\ast }$ and $(S^{km+\gamma }{)}^{\ast }$ are characterised for $m\ge 3$, $k\ge 1$ and $\gamma \in \{1,2,\cdots ,m-1\},$ which leads to a description of (nearly) invariant subspaces with respect to higher order shifts. Finally, the corresponding results for Toeplitz operators induced by a Blaschke product are presented. Techniques used include a refinement of Hitt’s algorithm, the Beurling–Lax theorem, and matrices of analytic functions.