Invariance and near invariance for non-cyclic shift semigroups

This paper characterises the subspaces of H² (D) simultaneously invariant under S² and S^2k+1, where S is the unilateral shift, then further identifies the subspaces that are nearly invariant under both (S²) ^* and (S^2k+1) ^* for k 1. More generally, the simultaneously (nearly) invariant subspaces with respect to (Sᵐ) ^* and (S^km+) ^* are characterised for m 3, k 1 and \1, 2, , 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.