Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 403-415.

Nearly invariant subspaces for backwards shifts on vector-valued Hardy spaces

Authors: I. Chalendar (1), N. Chevrot (2), and J.R. Partington (3)
Author institution: (1) Universite de Lyon; Universite Lyon 1; INSA de Lyon; Ecole Centrale de Lyon; CNRS, UMR5208, Institut Camille Jordan; 43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
(2) Universite de Lyon; Universite Lyon 1; INSA de Lyon; Ecole Centrale de Lyon; CNRS, UMR5208, Institut Camille Jordan; 43 bld. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France
(3) School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.

Summary: This paper characterizes the subspaces of a vector-valued Hardy space that are nearly invariant under the backward shift, providing a vectorial generalization of a result of Hitt. Compact perturbations of shifts that are pure isometries are studied, and the unitary operators that are compact perturbations of restricted shifts are described, extending a result of Clark. The classification of nearly invariant subspaces gives information on the simply shift-invariant subspaces of the vector-valued Hardy space of an annulus. Finally, as a generalization of a result of Aleman and Richter it is proved that any such subspace has index bounded by the dimension of the vector space in which the functions take their values.

Keywords: Nearly invariant subspace, invariant subspace, vector-valued Hardy space, shift operator, annulus, Toeplitz operator

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