Kernel of Vector-Valued Toeplitz Operators

Let S be the shift operator on the Hardy space H(2) and let S* be its adjoint. A closed subspace F of H(2) is said to be nearly S*-invariant if every element f is an element of F with f(0) = 0 satisfies S* f is an element of F. In particular, the kernels of Toeplitz operators are nearly S*-invariant subspaces. Hitt gave the description of these subspaces. They are of the form F = g(H(2) circle uH(2)) with g is an element of H(2) and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on H(2) circle minus uH(2). Sarason obtained a characterization of the functions g which act isometrically on H(2) circle minus uH(2). Hayashi obtained the link between the symbol phi of a Toeplitz operator and the functions g and u to ensure that a given subspace F = gK(u) is the kernel of T(phi). Chalendar, Chevrot and Partington studied the nearly S*-invariant subspaces for vector-valued functions. In this paper, we investigate the generalization of Sarason's and Hayashi's results in the vector-valued context.