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

Abstract

Let $S_E$ be the shift operator on vector-valued Hardy space $H_E^2.$ Beurling-Lax-Halmos Theorem identifies the invariant subspaces of $S_E$ and hence also the invariant subspaces of the backward shift $S_E^{\ast }.$ In this paper, we study the invariant subspaces of $S_E\oplus S_F^{\ast }.$ We establish a one-to-one correspondence between the invariant subspaces of $S_E\oplus S_F^{\ast }$ and a class of invariant subspaces of bilateral shift $B_E\oplus B_F$ which were described by Helson and Lowdenslager. As applications, we express invariant subspaces of $S_E\oplus S_F^{\ast }$ 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, and Timotin where the case with one dimensional $E$ and $F$ was considered.