Kernels 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 \FF of $H^2$ is said to be nearly $S^*$-invariant if every element f\in\FF with $f(0)=0$ satisfies S^*f\in\FF. In particular, the kernels of Toeplitz operators are nearly $S^*$-invariant subspaces. Hitt gave the description of these subspaces. They are of the form \FF=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 \phii of a Toeplitz operator and the functions $g$ and $u$ to ensure that a given subspace \FF=gK_u is the kernel of T_\phii. 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.Comment: 20 pages, accepted by Integral Equations and Operator Theor