Kernels of vector-valued Toeplitz operators

Let $S$ be the shift operator on the Hardy space $H^2$ and let $S^{\ast }$ be its adjoint. A closed subspace $\text{FF}$ of $H^2$ is said to be nearly $S^{\ast }$-invariant if every element $f\in \text{FF}$ with $f(0)=0$ satisfies $S^{\ast }f\in \text{FF}$. In particular, the kernels of Toeplitz operators are nearly $S^{\ast }$-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 $H^2\ominus u{H}^2$. Sarason obtained a characterization of the functions $g$ which act isometrically on $H^2\ominus u{H}^2$. Hayashi obtained the link between the symbol $\text{phii}$ of a Toeplitz operator and the functions $g$ and $u$ to ensure that a given subspace $\text{FF}=g{K}_u$ is the kernel of $T_{\text{phii}}$. Chalendar, Chevrot and Partington studied the nearly $S^{\ast }$-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.