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

Keywords: Invariant subspaces, weighted shifts, quotients of analytic functions

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