Kernel of Vector-Valued Toeplitz Operators

Abstract

Let S be the shift operator on the Hardy space H ² and let S* be its adjoint. A closed subspace ${\mathcal F}$ of H ² is said to be nearly S*-invariant if every element ${f\in\mathcal F}$ with f(0) = 0 satisfies ${S^*f\in\mathcal F}$. In particular, the kernels of Toeplitz operators are nearly S*-invariant subspaces. Hitt gave the description of these subspaces. They are of the form ${\mathcal F=g (H^2\ominus u H^2)}$ with ${g\in H^2}$ and u inner, u(0) = 0. A very particular fact is that the operator of multiplication by g acts as an isometry on ${H^2\ominus uH^2}$. Sarason obtained a characterization of the functions g which act isometrically on ${H^2\ominus uH^2}$. Hayashi obtained the link between the symbol ${\varphi}$ of a Toeplitz operator and the functions g and u to ensure that a given subspace ${\mathcal F=gK_u}$ is the kernel of ${T_\varphi}$. 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.