Adjoint of the Toeplitz operator with the singular inner function

Let ψδ1 be the singular inner function associated with the unit point mass at z=1. We shall give an infinite dimensional closed subspace M of kerTψδ1⁎, the kernel of the adjoint of the Toeplitz operator Tψδ1, satisfying that for every nonzero v in M, the smallest backward shift invariant subspace of H2 containing v coincides with kerTψδ1⁎.