Reducing subspaces of compressed analytic Toeplitz operators

Let B be an inner function on the unit disc and let A be any function in H-infinity. Denote by P-B the orthogonal projection of H-2 onto H-2 - BH2. Necessary and sufficient conditions for the adjoint PBTA* of the compression PBTA of the analytic Toeplitz operator T-A : F --> AF on H-2 to H-2 - BN2 to have nontrivial reducing subspaces are given for those operators PBTA* whose eigenvectors span all of H-2 - BN2 and whose eigenspaces are at most two-dimensional.