Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

Abstract

In this paper, we prove that for −1/2 ≤ β ≤ 0, suppose M is an invariant subspaces of the Hardy–Sobolev spaces H β 2(D) for T z β , then M ⊖ zM is a generating wandering subspace of M, that is, $$M = {\left[ {M \ominus zM} \right]_{T_z^\beta }}$$ . Moreover, any non-trivial invariant subspace M of H β 2(D) is also generated by the quasi-wandering subspace P M T z β M⊥, that is, $$M = {\left[ {{P_M}T_z^\beta {M^ \bot }} \right]_{T_z^\beta }}$$ .