Finite-dimensional Toeplitz kernels and nearly-invariant subspaces

A systematic analysis of the structure of finite-dimensional nearly-invariant subspaces of the Hardy space on the half-plane of index p (with 1 < p < 1) is made, and a criterion given by which they may be recognised. As a consequence, a new approach to Hitt's theorem on nearly-invariant subspaces is developed. Moreover, an analogue is given of Hayashi's theorem for finite-dimensional Toeplitz kernels; this is used to establish a necessary and suffcient condition for a Toeplitz kernel to be non-trivial and of dimension n, in terms of a factorisation of its symbol, analogous to Nakazi's work for the disc.