Doubly commuting mixed invariant subspaces in the polydisc

Abstract

We obtain a complete characterization for doubly commuting mixed invariant subspaces of the Hardy space over the unit polydisc. We say a closed subspace Q of H2(Dn) is mixed invariant if Mzj(Q)⊆Q for 1≤j≤k and Mzj⁎(Q)⊆Q, k+1≤j≤n for some integer k∈{1,2,…,n−1}. We prove that a mixed invariant subspace Q of H2(Dn) is doubly commuting if and only ifQ=ΘH2(Dk)⊗Qθ1⊗⋯⊗Qθn−k, where Θ∈H∞(Dk) is some inner function and Qθj is either a Jordan block H2(D)⊖θjH2(D) for some inner function θj or the Hardy space H2(D). Furthermore, an explicit representation for the commutant of an n-tuple of doubly commuting shifts as well as a representation for the commutant of a doubly commuting tuple of shifts and co-shifts are obtained. Finally, we discuss some concrete examples of mixed invariant subspaces.