Sous-espaces biinvariants pour certains shifts pondérés

O. El-Fallah; Karim Kellay

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1543-1558
  • ISSN: 0373-0956

Abstract

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We study the biinvariant subspaces for the usual shift on the weighted spaces L ω 2 = { f ∈ L 2 ( 𝕋 ) : ∥ f ∥ ω = ∑ n ∈ ℤ | f ( n ) | ω 2 ( n ) 1 / 2 < + ∞ } , where ω ( n ) = ( 1 + n ) p , n ≥ 0 and ω ( n ) ( 1 + | n | ) p n → - ∞ → + ∞ for some integer p ≥ 1 . We show that the analytic part of all biinvariant subspaces is spectral if ∑ n ≥ 2 1 n log ω ( - n ) diverges, but that this does not hold when ∑ n ≥ 2 1 n log ω ( - n ) converges.

How to cite

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El-Fallah, O., and Kellay, Karim. "Sous-espaces biinvariants pour certains shifts pondérés." Annales de l'institut Fourier 48.5 (1998): 1543-1558. <http://eudml.org/doc/75330>.

@article{El1998,
abstract = {Nous étudions les sous-espaces biinvariants du shift usuel sur les espaces à poids\begin\{\}L^2\_\omega =\Big \lbrace f\in L^2(\{\Bbb T\}):\Vert f\Vert \_\omega = \Big (\sum \_\{n\in \{\Bbb Z\}\} \vert f(n)\vert \omega ^2(n)\Big )^\{1/2\}&lt; +\infty \Big \rbrace ,\end\{\}où $\omega (n)=(1+n)^p, n\ge 0$ et $\{\omega (n)\over (1+\vert n\vert )^p\}\{\mathrel \{\mathop \{\hspace\{0.0pt\}n\rightarrow -\infty \}\limits ^\{\rightarrow \}\}\} +\infty $, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque $\sum _\{n\ge 2\}\{1\over n\log \omega (-n)\}$ diverge, mais que ceci n’est plus valable lorsque $\sum _\{n\ge 2\}\{1\over n\log \omega (-n)\}$ converge.},
author = {El-Fallah, O., Kellay, Karim},
journal = {Annales de l'institut Fourier},
keywords = {biinvariant subspace; hyperfunction; Carleson set; weighted shifts; weight asymmetry},
language = {fre},
number = {5},
pages = {1543-1558},
publisher = {Association des Annales de l'Institut Fourier},
title = {Sous-espaces biinvariants pour certains shifts pondérés},
url = {http://eudml.org/doc/75330},
volume = {48},
year = {1998},
}

TY - JOUR
AU - El-Fallah, O.
AU - Kellay, Karim
TI - Sous-espaces biinvariants pour certains shifts pondérés
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1543
EP - 1558
AB - Nous étudions les sous-espaces biinvariants du shift usuel sur les espaces à poids\begin{}L^2_\omega =\Big \lbrace f\in L^2({\Bbb T}):\Vert f\Vert _\omega = \Big (\sum _{n\in {\Bbb Z}} \vert f(n)\vert \omega ^2(n)\Big )^{1/2}&lt; +\infty \Big \rbrace ,\end{}où $\omega (n)=(1+n)^p, n\ge 0$ et ${\omega (n)\over (1+\vert n\vert )^p}{\mathrel {\mathop {\hspace{0.0pt}n\rightarrow -\infty }\limits ^{\rightarrow }}} +\infty $, pour un certain entier $p\ge 1$. Nous montrons que la trace analytique de tout sous-espace biinvariant est de type spectral, lorsque $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ diverge, mais que ceci n’est plus valable lorsque $\sum _{n\ge 2}{1\over n\log \omega (-n)}$ converge.
LA - fre
KW - biinvariant subspace; hyperfunction; Carleson set; weighted shifts; weight asymmetry
UR - http://eudml.org/doc/75330
ER -

References

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  1. [A1] A. ATZMON, Operators which are annihilated by analytic functions and invariant subspaces, Acta Math., 144 (1980), 27-63. Zbl0449.47007MR81c:47007
  2. [A2] A. ATZMON, On the existence of hyperinvariants subspaces, J. Op. Theory, 11 (1984), 3-40. Zbl0583.47009MR85k:47005
  3. [B] V.I. BURENKOV, On the approximation of functions in Sobolev spaces by functions of compact support in an arbitrary open set, Soviet Math. Dokl., 13 (1972), 60-64. Zbl0253.46066MR52 #14964
  4. [C] L. CARLESON, Sets of uniqueness of functions regular in the unit circle, Acta Math., 87 (1952), 325-345. Zbl0046.30005MR14,261a
  5. [D] P.L. DUREN, Theory of Hp Spaces, Pure and Applied Math., Academic Press, New York-London, 1970. Zbl0215.20203MR42 #3552
  6. [Dy] E.M. DYNKIN, Free interpolation set for Hölder classes, Mat. Sbornik, 109 (1979), 107-128. Zbl0407.30024
  7. [EZR] J. ESTERLE, M. ZARRABI and M. RAJOELINA, On contractions with spectrum contained in the Cantor set, Math. Proc. Camb. Phil. Soc., 177 (1995), 339-343. Zbl0830.47001MR95h:47021
  8. [E1] J. ESTERLE, Distributions on Kronecker sets, strong forms of uniqueness, and closed ideals of A+, J. für reine ang. Math., 450 (1994), 43-82. Zbl0791.46026
  9. [E2] J. ESTERLE, Uniqueness, strong forms of uniqueness and negative powers of contractions, Banach Center Publications, 30 (1994), 1-19. 
  10. [E3] J. ESTERLE, Closed ideals in certain Beurling algebras and synthesis of hyperdistributions, Studia Math., 120 (1996), 113-153. Zbl0864.46025MR97i:46093
  11. [E4] J. ESTERLE, Singular inner functions and biinvariant subspaces for dissymetric weighted shifts, J. Func. Analysis, 144 (1997), 64-104. Zbl0938.47003MR98d:47021
  12. [K] J.P. KAHANE, Séries de Fourier absolument convergentes, Ergebnisse der Maths, 50, Berlin-Heidelberg-New York, Springer, 1970. Zbl0195.07602MR43 #801
  13. [Ke] K. KELLAY, Contractions et hyperdistributions à spectre de Carleson, J. London. Math. Soc., to appear. Zbl0945.47029
  14. [Kh] L. KHANIN, Spectral synthesis of ideals in algebras of functions having generalized derivatives, Russian Math. Surveys, 39 (1984), 167-168. Zbl0567.46025MR85d:46072
  15. [Ko1] B.I. KORENBLUM, Invariant subspace of the shift operator in weighted Hilbert space, Mat. Sbornik, 89 (1972), 112-138. Zbl0257.46074
  16. [Ko2] B.I. KORENBLUM, Invariant subspaces for a shift operator in some weighted Hilbert sequence spaces, Soviet Math. Dokl., 13 (1972), 272-275. Zbl0249.47023MR47 #5621
  17. [Ko3] B.I. KORENBLUM, Functions holomorphic in a disc and smooth in its closure, Soviet Math. Dokl., 12 (1971), 1312-1315. Zbl0235.30036
  18. [N] N.K. NIKOLSKII, Lectures on the shift operator IV, J. Soviet Math., 16 (1981), 1118-1139. Zbl0458.47028
  19. [S] F.A. SHAMOYAN, Closed ideals in algebras of functions analytic in the disc and smooth up to its boundary, Mat. Sbornik, 79 (1994), 425-445. Zbl0840.30030MR94m:46087
  20. [TW] B.A. TAYLOR and D.L. WILLIAMS, Ideals in rings of analytic functions with smooth boundary values, Can. J. Math., 22 (1970), 1266-1283. Zbl0204.44302MR42 #7905
  21. [Z] M. ZARRABI, Contractions à spectre dénombrable et propriétés d'unicité des fermés dénombrables du cercle, Ann. Inst. Fourier, 43-1 (1993), 251-263. Zbl0766.47002MR94b:47048

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