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Journal of Operator Theory

Volume 75, Issue 1, Winter 2016  pp. 75-90.

Finite-dimensional Toeplitz kernels and nearly-invariant subspaces

Authors:  M.C. Camara (1) and J.R. Partington (2)
Author institution: (1) Center for Mathematical Analysis, Geometry and Dynamical Systems, Mathematics Department, Instituto Superior T\'ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(2) School of Mathematics, University of Leeds, Leeds LS2~9JT, U.K.

Summary: A systematic analysis of the structure of finite-dimensional nearly-invariant subspaces of the Hardy space on the half-plane of index $p$ (with $p$ larger than $1$) 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 sufficient 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.

Keywords: Toeplitz operator, Toeplitz kernel, model space, nearly-invariant subspace, inner-outer factorization, Riemann-Hilbert problem

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