Classification of nearly invariant subspaces of the backward shift

Abstract:

Let ${S^*}$ denote the backward shift operator on the Hardy space ${H^2}$ of the unit disk. A subspace $M$ of ${H^2}$ is called nearly invariant if ${S^*}h$ is in $M$ whenever $h$ belongs to $M$ and $h(0) = 0$. In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of ${H^1}$ with the set of inner functions.