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*. In this paper, we study the invariant subspaces of S-E circle plus S-F*. We establish a one-to-one correspondence between the invariant subspaces of S-E circle plus S-F* and a class of invariant subspaces of bilateral shift BE B F which were described by Helson and Lowdenslager [15]. As applications, we express invariant subspaces of S-E circle plus S-F* 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 Camara and Ross [5], and Timontin [22] where the case with one dimensional E and F was considered. (C) 2022 Elsevier Inc. All rights reserved.
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