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Algebra i Analiz, 2008, Volume 20, Issue 5, Pages 83–98 (Mi aa531)  

Research Papers

The spectrum of some compressions of unilateral shifts

S. Duberneta, J. Esterleb

a Professeur de CPES, Epinay sur Seine, France
b Université Bordeaux 1, Talence, France
References:
Abstract: Let $E$ be a star-shaped Banach space of analytic functions on the open unit disc $\mathbb D$. We assume that the unilateral shift $S\colon z\to zf$ and the backward shift $T\colon f\to\frac{f-f(0)}{z}$ are bounded on $E$ and that their spectrum is the closed unit disc.
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 shows that if $g$ has an analytic extension to $\mathbb D\cup D(\zeta,r)$ for some $r>0$, with $g(\zeta)\ne 0$, 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 $S_M\colon f+M\to zf+M$ of the unilateral shift to the quotient space $E/M$. This shows in particular that $\operatorname{Spec}(S_M)=\{1\}$ for some $z$-invariant subspaces $M$ of weighted Hardy spaces constructed by N. K. Nikol'skiĭ in the seventies by using the Keldysh method.
Keywords: Unilateral shift, nonquasianalyticty condition, spectrum.
Received: 12.08.2006
English version:
St. Petersburg Mathematical Journal, 2009, Volume 20, Issue 5, Pages 737–748
DOI: https://doi.org/10.1090/S1061-0022-09-01070-X
Bibliographic databases:
Document Type: Article
MSC: 47B37
Language: English
Citation: S. Dubernet, J. Esterle, “The spectrum of some compressions of unilateral shifts”, Algebra i Analiz, 20:5 (2008), 83–98; St. Petersburg Math. J., 20:5 (2009), 737–748
Citation in format AMSBIB
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\by S.~Dubernet, J.~Esterle
\paper The spectrum of some compressions of unilateral shifts
\jour Algebra i Analiz
\yr 2008
\vol 20
\issue 5
\pages 83--98
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\zmath{https://zbmath.org/?q=an:1206.30017}
\elib{https://elibrary.ru/item.asp?id=11682838}
\transl
\jour St. Petersburg Math. J.
\yr 2009
\vol 20
\issue 5
\pages 737--748
\crossref{https://doi.org/10.1090/S1061-0022-09-01070-X}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000270134200004}
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    Алгебра и анализ St. Petersburg Mathematical Journal
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