Algebra i Analiz
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Algebra i Analiz:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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
\Bibitem{DubEst08}
\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
\mathnet{http://mi.mathnet.ru/aa531}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2492360}
\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}
Linking options:
  • https://www.mathnet.ru/eng/aa531
  • https://www.mathnet.ru/eng/aa/v20/i5/p83
  • Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Алгебра и анализ St. Petersburg Mathematical Journal
    Statistics & downloads:
    Abstract page:263
    Full-text PDF :80
    References:40
    First page:14
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024