Classification of nearly invariant subspaces of the backward shift

Abstract

Let S ∗ {S^*} denote the backward shift operator on the Hardy space H 2 {H^2} of the unit disk. A subspace M M of H 2 {H^2} is called nearly invariant if S ∗ h {S^*}h is in M M whenever h h belongs to M M and h ( 0 ) = 0 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 {H^1} with the set of inner functions.