[Représentations réductibles des groupes abéliens]
Nous établissons un critère de réductibilité pour certaines représentations des groupes abéliens. Parmi les applications de ce critère, nous donnons une réponse positive au problème du sous-espace invariant par translation pour les espaces pondérés sur les groupes abéliens localement compacts, lorsque les poids sont pairs et .
A criterion for reducibility of certain representations of abelian groups is established. Among the applications of this criterion, we give a positive answer to the translation invariant subspace problem for weighted spaces on locally compact abelian groups, for even weights and .
Keywords: abelian groups, reducible representations, translation invariant subspaces
Mot clés : groupes abéliens, représentations réductibles, sous-espaces invariants par translation
Atzmon, Aharon 1
@article{AIF_2001__51_5_1407_0,
author = {Atzmon, Aharon},
title = {Reducible representations of abelian groups},
journal = {Annales de l'Institut Fourier},
pages = {1407--1418},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {51},
number = {5},
year = {2001},
doi = {10.5802/aif.1859},
zbl = {0980.43005},
mrnumber = {1860670},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1859/}
}
TY - JOUR AU - Atzmon, Aharon TI - Reducible representations of abelian groups JO - Annales de l'Institut Fourier PY - 2001 SP - 1407 EP - 1418 VL - 51 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1859/ DO - 10.5802/aif.1859 LA - en ID - AIF_2001__51_5_1407_0 ER -
%0 Journal Article %A Atzmon, Aharon %T Reducible representations of abelian groups %J Annales de l'Institut Fourier %D 2001 %P 1407-1418 %V 51 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1859/ %R 10.5802/aif.1859 %G en %F AIF_2001__51_5_1407_0
Atzmon, Aharon. Reducible representations of abelian groups. Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1407-1418. doi : 10.5802/aif.1859. https://aif.centre-mersenne.org/articles/10.5802/aif.1859/
[1] Hyperinvariant subspaces for bilateral weighted shifts, Integral Equations and Operator Theory, Volume 7 (1984), pp. 1-9 | MR | Zbl
[2] The existence of hyperinvariant subspaces, J. Operator Theory, Volume 11 (1984), pp. 3-40 | MR | Zbl
[3] An operator on a Fréchet space with no common invariant subspace with its inverse, J. Funct. Anal., Volume 55 (1984), pp. 68-77 | DOI | MR | Zbl
[4] Irreducible representations of abelian groups, Proceedings, Harmonic Analysis (Lecture Notes in Mathematics), Volume 1359 (1987), pp. 83-92 | MR | Zbl
[5] Entire functions, invariant subspaces and Fourier transforms, Israel Math. Conf. Proc., Volume 11 (1997), pp. 37-52 | MR | Zbl
[6] Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type, J. Funct. Anal., Volume 169 (1999), pp. 164-188 | DOI | MR | Zbl
[7] Addendum to the paper ``Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type", J. Funct. Anal., Volume 175 (2000), pp. 248-249 | DOI | MR | Zbl
[8] The existence of translation invariant subspaces of symmetric self-adjoint sequence spaces on , J. Funct. Anal., Volume 178 (2000), pp. 372-380 | DOI | MR | Zbl
[9] L'analyse harmonique dans les groupes abélien, L'enseignement Mathématique, Volume II (1956) no. 5 | MR | Zbl
[10] Finitely connected domains G, representations of , and invariant subspaces, J. Operator Theory, Volume 6 (1981), pp. 375-405 | MR | Zbl
[11] Smoothness and renorming in Banach spaces, Pitman monographs and surveys, 64, Longman, Harlow, 1993 | Zbl
[12] On the existence of nontrivial or nonstandard translation invariant subspaces of weighted and , 18th Scandinavian Congress of Mathematics (Aarhus, 1980) (Progress in Mathematics), Volume 11 (1981), pp. 226-235 | Zbl
[13] Translation invariant subspaces of weighted and spaces, Math. Scand., Volume 49 (1981), pp. 133-144 | MR | Zbl
[14] Translation invariant subspaces of weighted , Proceeding, Commutative Harmonic Analysis (Contemporary Mathematics), Volume 91 (1987) | Zbl
[15] Entire functions of order with bounds on both axes, Ann. Acad. Sci. Fenn. Math., Volume 22 (1997), pp. 339-348 | MR | Zbl
[16] Singular inner functions and biinvariant subspaces for disymmetric weighted shifts, J. Funct. Anal., Volume 144 (1997), pp. 61-104 | MR | Zbl
[17] Sous-espaces invariant par translations bilatérales de certains espaces de Hilbert de suites quasianalytiquement pondérées, C.R. Acad. Sci. Paris, Série I, Volume 326 (1998), pp. 295-300 | MR | Zbl
[18] Analytic left-invariant subspaces of weighted Hilbert spaces of sequences (J. Operator Theory, to appear) | Zbl
[19] Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences (To appear in Ann. Sci. École Norm. Sup.) | Numdam | Zbl
[20] Hyperinvariant subspaces of bilateral weighted shifts, Indiana Univ. Math. J., Volume 23 (1974), pp. 771-790 | DOI | MR | Zbl
[21] Topological vector spaces I, Springer-Verlag, New York, 1969 | Zbl
[22] On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 967-970 | DOI | MR | Zbl
[23] A solution to the invariant subspace problem in the space , Bull. London Math. Soc., Volume 17 (1985), pp. 305-317 | DOI | MR | Zbl
[24] Weighted shift operators and analytic function theory, Topics in Operator Theory (Amer. Math Soc.), Volume 13 (1974) | Zbl
Cité par Sources :
