Invariance and near invariance for non-cyclic shift semigroups

This paper characterises the subspaces of H²( D) simultaneously invariant under S² and S²ᵏ⁺¹, where S is the unilateral shift, then further identifies the subspaces that are nearly invariant under both (S²)^* and (S²ᵏ⁺¹)^* for k≥ 1. More generally, the simultaneously (nearly) invariant subspaces with respect to (Sᵐ)^* and (Skm+γ)^* are characterised for m≥ 3, k≥ 1 and γ∈ \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.