# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2014oct29.2067

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

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