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St. Petersburg Mathematical Journal

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The spectrum of some compressions of unilateral shifts


Authors: S. Dubernet and J. Esterle
Original publication: Algebra i Analiz, tom 20 (2008), nomer 5.
Journal: St. Petersburg Math. J. 20 (2009), 737-748
MSC (2000): Primary 47B37
Published electronically: July 21, 2009
MathSciNet review: 2492360
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Abstract: Let $ E$ be a star-shaped Banach space of analytic functions on the open unit disk $ \mathbb{D}$. It is assumed that the unilateral shift $ S : z\to zf$ and the backward shift $ T : f\to \frac{f-f(0)}{ z}$ are bounded on $ E$ and that their spectrum is the closed unit disk.

Let $ M$ be a closed $ z$-invariant subspace of $ E$ such that $ \dim(M/zM)=1$, and let $ g\in M$. The main result of the paper states that if $ g$ has an analytic extension to $ \mathbb{D} \cup D(\zeta,r)$ for some $ r>0$, with $ g(\zeta) \neq 0$, and if $ S$ and $ T$ satisfy the ``nonquasianalytic condition''

$\displaystyle \sum_{n\ge 0}\frac{\log\Vert S^n\Vert+\log \Vert T^n\Vert}{ 1+n^2}<+\infty, $

then $ \zeta$ does not belong to the spectrum of the compression $ S_M : f+M\to zf +M$ of the unilateral shift to the quotient space $ E/M$. This shows in particular that $ \mathrm{Spec}(S_M)=\{1\}$ for some $ z$-invariant subspaces $ M$ of weighted Hardy spaces that were constructed by N. K. Nikol'skiĭ in the 1970s by using the Keldysh method.


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Additional Information

S. Dubernet
Affiliation: Professeur de CPES, Lycée Jacques Feyder, 10, rue Henri Wallon, 93800-Epinay sur Seine, France
Email: sebastien.dubernet@gmail.com

J. Esterle
Affiliation: Université Bordeaux 1, IMB, UMR 5251, 351, Cours de la Libération, 33405-Talence, France
Email: esterle@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S1061-0022-09-01070-X
Keywords: Weighted Hardy space, $z$-invariant subspace, one-sided nonquasianalytic condition
Received by editor(s): August 12, 2006
Published electronically: July 21, 2009
Article copyright: © Copyright 2009 American Mathematical Society