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Un opérateur sans sous-espace invariant: Simplification de l’exemple de P. Enflo. (French) Zbl 0571.47002

Most decomposition theories for linear operators require the existence of invariant subspaces before one can begin. This existence problem has attracted much attention during the past several decades with little success until recently.
In [Ann. Math. 117, 669-694 (1983; Zbl 0553.47002)] A. Atzmon exhibited a continuous operator on a nuclear Fréchet space with no proper (closed) invariant subspace. In [Bull. London Math. Soc. 16, 337- 401 (1984)] C. J. Read constructed an example of a bounded linear operator on a nonreflexive Banach space with no proper (closed) invariant subspace.
Almost a decade ago P. Enflo had claimed that he had constructed such an example but his proof was extremely complex and is yet to be published. In this note the author, using the ideas and techniques of P. Enflo, provides another example of a bounded linear operator on a Banach space without a proper (closed) invariant subspace. The argument is complicated and combinatorial in nature.
The existence problem for the case of Hilbert spaces remains, however, unresolved.
Reviewer: R.G.Douglas

MSC:

47A15 Invariant subspaces of linear operators

Citations:

Zbl 0553.47002
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References:

[1] B. BEAUZAMY:Un opérateur avec un ensemble non-dénombrable, dense, de vecteurs hypercycliques, Séminaire d’Analyse Fonctionnelle, Université de Paris VII, 1982/83.
[2] B. BEAUZAMY-P. ENFLO:Estimations de produits de polynômes, A paraître au Journal of Number Theory.
[3] P. ENFLO:On the invariant subspace problem in Banach spaces, A paraître à Acta Mathematica.
[4] C. READ:Solution to the Invariant Subspace Problem, Bulletin London Math. Society, July 1984. · Zbl 0566.47003
[5] C. READ:An operator on l 1 without invariant subspaces. Preliminary version, communicated by A.M. Davie.
[6] A. Atzmon:An operator without invariant subspaces on a nuclear Frechet Space. Annals of Math, (1) 117 (1983), n. 3, 669-694. · Zbl 0553.47002
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