The spectrum of some compressions of unilateral shifts
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- by S. Dubernet and J. Esterle
- St. Petersburg Math. J. 20 (2009), 737-748
- DOI: https://doi.org/10.1090/S1061-0022-09-01070-X
- Published electronically: July 21, 2009
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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” \[ \sum _{n\ge 0}\frac {\log \| S^n\|+\log \| T^n\|}{ 1+n^2}<+\infty , \] 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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Bibliographic 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
- MR Author ID: 64315
- Email: esterle@math.u-bordeaux1.fr
- Received by editor(s): August 12, 2006
- Published electronically: July 21, 2009
- © Copyright 2009 American Mathematical Society
- Journal: St. Petersburg Math. J. 20 (2009), 737-748
- MSC (2000): Primary 47B37
- DOI: https://doi.org/10.1090/S1061-0022-09-01070-X
- MathSciNet review: 2492360