Journal of Operator Theory

Volume 45, Issue 2, Spring 2001 pp. 265-301.

Analytic left-invariant subspaces of weighted Hilbert spaces of sequences

Authors: J. Esterle (1), and A. Volberg (2)
Author institution: (1) Laboratoire de Mathematiques Pures, UPRESA 5467, Universite Bordeaux I, 351, cours de la Liberation, F-33405 Talence, France
(2) Department of Applied Mathematics, Michigan State University, East Lansing, MI 48824, USA

Summary: Let

$\omega$ be a weight on

${\bbb Z}$ , and assume that the translation operator

$S:(u_n)_{n\in{\bbb Z}}\rightarrow (u_{n-1})_{n\in{\bbb Z}}$ is bounded on

$\ell^2_\omega({\bbb Z})$ , and that the spectrum of

$S$ equals the unit circle. A closed subspace

$G$ of

$\ell^2_\omega({\bbb Z})$ is said to be left-invariant (respecti vely translation invariant, respectively right-invariant) if

$S^{-1}(G)\subset G$ (respectively

$S(G)=G$ , respectively

$S(G)\subset G)$ and

$G$ is said to be analytic if

$G$ contains a nonzero sequen ce

$(u_n)_{n\in{\bbb Z}}$ such that

$u_n=0$ for

$n<0$ . We show that if the weight

$\omega(n)$ grows sufficiently fast as

$n\rightarrow-\infty$ , then all analytic left-invariant sub spaces of

$\ell^2_\omega({\bbb Z})$ are generated by their intersection with

$\ell^2_\omeg a({\bbb Z}^+) :=\{(u_n)_{n\in{\bbb Z}}\in \ell^2_\omega({\bbb Z})\}: u_n=0$ for

$n<0)$ . Variou s concrete examples of weights

$\omega$ for which this situation occurs are obtained by usi ng sharp estimates of Matsaev-Mogulskii about the rate of growth of quotients of analytic functions in the disc. We also discuss the existence of right-invariant subspaces of

$\ell^2_\omega({\b bb Z}^+)$ having a specific division property needed to obtain analytic translation invariant sub spaces of

$\ell^2_\omega({\bbb Z})$ .

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