Invariant subspaces of the direct sum of forward and backward shifts on vector-valued Hardy spaces

Abstract

Let SE be the shift operator on vector-valued Hardy space HE2. Beurling-Lax-Halmos Theorem identifies the invariant subspaces of SE and hence also the invariant subspaces of the backward shift SE⁎. In this paper, we study the invariant subspaces of SE⊕SF⁎. We establish a one-to-one correspondence between the invariant subspaces of SE⊕SF⁎ and a class of invariant subspaces of bilateral shift BE⊕BF which were described by Helson and Lowdenslager [15]. As applications, we express invariant subspaces of SE⊕SF⁎ as kernels or ranges of mixed Toeplitz operators and Hankel operators with partial isometry-valued symbols. Our approach greatly extends and gives different proofs of the results of Câmara and Ross [5], and Timontin [22] where the case with one dimensional E and F was considered.