Reducible representations of abelian groups

Atzmon, Aharon

Annales de l'Institut Fourier, Tome 51 (2001) no. 5, pp. 1407-1418.

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 $L^p$ pondérés sur les groupes abéliens localement compacts, lorsque les poids sont pairs et $1<p<\infty$.

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 $L^p$ spaces on locally compact abelian groups, for even weights and $1<p<\infty$.

Classification : 43A65, 43A15, 47A15, 47B37
Keywords: abelian groups, reducible representations, translation invariant subspaces
Mot clés : groupes abéliens, représentations réductibles, sous-espaces invariants par translation

[1] C. Apostol Hyperinvariant subspaces for bilateral weighted shifts, Integral Equations and Operator Theory, Volume 7 (1984), pp. 1-9 | MR | Zbl

[2] A. Atzmon The existence of hyperinvariant subspaces, J. Operator Theory, Volume 11 (1984), pp. 3-40 | MR | Zbl

[3] A. Atzmon 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] A. Atzmon Irreducible representations of abelian groups, Proceedings, Harmonic Analysis (Lecture Notes in Mathematics), Volume 1359 (1987), pp. 83-92 | MR | Zbl

[5] A. Atzmon Entire functions, invariant subspaces and Fourier transforms, Israel Math. Conf. Proc., Volume 11 (1997), pp. 37-52 | MR | Zbl

[6] A. Atzmon; M. Sodin Completely indecomposable operators and a uniqueness theorem of Cartwright-Levinson type, J. Funct. Anal., Volume 169 (1999), pp. 164-188 | DOI | MR | Zbl

[7] A. Atzmon; M. Sodin 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] A. Atzmon The existence of translation invariant subspaces of symmetric self-adjoint sequence spaces on $\mathbb{Z}$, J. Funct. Anal., Volume 178 (2000), pp. 372-380 | DOI | MR | Zbl

[9] J. Bracconier L'analyse harmonique dans les groupes abélien, L'enseignement Mathématique, Volume II (1956) no. 5 | MR | Zbl

[10] B. Chevreau; C.M. Pearcy; A.L. Shields Finitely connected domains G, representations of $H^{\infty }\left(G\right)$, and invariant subspaces, J. Operator Theory, Volume 6 (1981), pp. 375-405 | MR | Zbl

[11] R. Deville; G. Godefroy; V. Zizler Smoothness and renorming in Banach spaces, Pitman monographs and surveys, 64, Longman, Harlow, 1993 | Zbl

[12] Y. Domar On the existence of nontrivial or nonstandard translation invariant subspaces of weighted ${\ell }^p$ and $L^p$, 18th Scandinavian Congress of Mathematics (Aarhus, 1980) (Progress in Mathematics), Volume 11 (1981), pp. 226-235 | Zbl

[13] Y. Domar Translation invariant subspaces of weighted ${\ell }^p$ and $L^p$ spaces, Math. Scand., Volume 49 (1981), pp. 133-144 | MR | Zbl

[14] Y. Domar Translation invariant subspaces of weighted $L^p$, Proceeding, Commutative Harmonic Analysis (Contemporary Mathematics), Volume 91 (1987) | Zbl

[15] Y. Domar Entire functions of order $\le 1$ with bounds on both axes, Ann. Acad. Sci. Fenn. Math., Volume 22 (1997), pp. 339-348 | MR | Zbl

[16] J. Esterlé Singular inner functions and biinvariant subspaces for disymmetric weighted shifts, J. Funct. Anal., Volume 144 (1997), pp. 61-104 | MR | Zbl

[17] J. Esterlé; A. Volberg 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] J. Esterlé; A. Volberg Analytic left-invariant subspaces of weighted Hilbert spaces of sequences (J. Operator Theory, to appear) | Zbl

[19] J. Esterlé; A. Volberg Asymptotically holomorphic functions and translation invariant subspaces of weighted Hilbert spaces of sequences (To appear in Ann. Sci. École Norm. Sup.) | Numdam | Zbl

[20] R. Gellar; D. Herrero Hyperinvariant subspaces of bilateral weighted shifts, Indiana Univ. Math. J., Volume 23 (1974), pp. 771-790 | DOI | MR | Zbl

[21] G. Köthe Topological vector spaces I, Springer-Verlag, New York, 1969 | Zbl

[22] J. Lindenstrauss On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc., Volume 72 (1966), pp. 967-970 | DOI | MR | Zbl

[23] C.J. Read A solution to the invariant subspace problem in the space ${\ell }_1$, Bull. London Math. Soc., Volume 17 (1985), pp. 305-317 | DOI | MR | Zbl

[24] A.L. Shields Weighted shift operators and analytic function theory, Topics in Operator Theory (Amer. Math Soc.), Volume 13 (1974) | Zbl