Classify non-splitting nearly S ⊕ S* invariant subspaces of H^2_{C^2}

Determine all closed subspaces M of the vector-valued Hardy space H^2_{C^2} that are nearly invariant under the operator S ⊕ S*, where S is the unilateral shift on H^2 and S* is its adjoint, in the sense that (S ⊕ S*)(M ∩ (S ⊕ S*)^* H^2_{C^2}) ⊆ M (as in Definition 1.2). Restrict attention to non-splitting subspaces, i.e., those M that cannot be written as N1 ⊕ N2 with N1 and N2 closed subspaces of H^2.

Background

The paper develops the structure of nearly U_θ-invariant subspaces of K_θ^⊥, with special attention to the case θ(0)=0, employing Hitt’s algorithm. Through a known unitary equivalence, when θ(0)=0 the dual compressed shift U_θ corresponds to S ⊕ S* on the vector-valued Hardy space H^2_{C^2}. This equivalence allows the authors to derive classifications of splitting subspaces (of the form N1 ⊕ N2 with N1, N2 ⊂ H^2) that are nearly S ⊕ S*-invariant (Corollary \ref{mainconsequence}).

Motivated by these results and by prior observations that certain U_θ-invariant subspaces need not decompose into the form M_- ⊕ M_+, the authors pose the problem of characterizing all non-splitting nearly S ⊕ S*-invariant subspaces of H^2_{C^2} in the sense of their general near-invariance definition (Definition 1.2). This seeks a full classification beyond the splitting case already obtained.