Mathematics of the USSR-Sbornik
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Forthcoming papers
Archive
Impact factor
Guidelines for authors
License agreement
Submit a manuscript

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mat. Sb.:
Year:
Volume:
Issue:
Page:
Find






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


Mathematics of the USSR-Sbornik, 1972, Volume 18, Issue 1, Pages 111–138
DOI: https://doi.org/10.1070/SM1972v018n01ABEH001617
(Mi sm3221)
 

This article is cited in 8 scientific papers (total in 8 papers)

Invariant subspaces of the shift operator in weighted Hilbert space

B. I. Korenblum
References:
Abstract: A complete description is given of the closed ideals of the algebra $H_1^2$ of functions $\widehat x(z)$ which are regular in the circle $U$ ($|z|<1$) and such that $\widehat x'\in H^2$, with the norm
$$ \|\widehat x\|_{H_1^2}=(\|\widehat x\|_{H^2}^2+\|\widehat x'\|_{H^2}^2)^{1/2} $$
and the usual multiplication. This is equivalent to a description of the invariant subspaces of the one-sided shift operator on the weighted Hilbert space of sequences with weights $p_k=1+k^2$ ($k=0,1,\dots$). It is shown that each closed ideal $I$ of the algebra $H_1^2$ has the form $I=\overline I\cap A$, where $\overline I$ is the closure of $I$ in the space $A$ of functions which are regular in $U$ and continuous in $\overline U$ with the uniform norm. Thus the ideals of the algebra $H_1^2$ have a structure similar to the structure of the ideals of the algebra $A$: each ideal $I$ is uniquely determined by an interior function $G$, which is the greatest common divisor of the interior parts of the functions $\widehat x\in I$, and the set $K\subset\partial U$ of the common zeros of the functions $\widehat x\in I$.
Bibliography: 19 titles.
Received: 09.09.1971
Bibliographic databases:
UDC: 513.882+517.5
MSC: Primary 30A78; Secondary 47A15
Language: English
Original paper language: Russian
Citation: B. I. Korenblum, “Invariant subspaces of the shift operator in weighted Hilbert space”, Math. USSR-Sb., 18:1 (1972), 111–138
Citation in format AMSBIB
\Bibitem{Kor72}
\by B.~I.~Korenblum
\paper Invariant subspaces of the shift operator in weighted Hilbert space
\jour Math. USSR-Sb.
\yr 1972
\vol 18
\issue 1
\pages 111--138
\mathnet{http://mi.mathnet.ru//eng/sm3221}
\crossref{https://doi.org/10.1070/SM1972v018n01ABEH001617}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=317073}
\zmath{https://zbmath.org/?q=an:0257.46074}
Linking options:
  • https://www.mathnet.ru/eng/sm3221
  • https://doi.org/10.1070/SM1972v018n01ABEH001617
  • https://www.mathnet.ru/eng/sm/v131/i1/p110
  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024