Journal of Operator Theory

Volume 61, Issue 2, Spring 2009 pp. 313-330.

Invariant subspaces for the shift on the vector-valued $L^2$ space of an annulus

Authors: Isabelle Chalendar (1), N. Chevrot (2), and J.R. Partington (3)
Author institution: (1) UFR de Mathematiques, Universite Lyon 1, 43 bld. du 11/11/1918, 69622 Villeurbanne Cedex, France
(2) UFR de Mathematiques, Universite Lyon 1, 43 bld. du 11/11/1918, 69622 Villeurbanne Cedex, France
(3) School of Mathematics, University of Leeds, Leeds LS2 9JT, U.K.

Summary: In this paper we study the invariant subspaces of the shift operator acting on the vector-valued $L^2$ space of an annulus, following an approach which originates in the work of Sarason. We obtain a Wiener-type result characterizing the reducing subspaces, and we give a description of all the invariant and doubly-invariant subspaces generated by a single function. We prove that every doubly-invariant subspace contained in the Hardy space of the annulus with values in $\CC^m$ is the orthogonal direct sum of at most $m$ doubly-invariant subspaces, each generated by a single function. As a corollary we prove that a doubly-invariant subspace that is also the graph of an operator is singly generated.

Keywords: Invariant subspace, vector-valued Hardy space, shift operator, multi\-ply-connected domains

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