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An operator on a Fréchet space with no common invariant subspace with its inverse. (English) Zbl 0552.47006

The main result of this paper is the construction of a Fréchet-Montel space F of formal Laurent series \(\sum^{\infty}_{-\infty}a_ nz^ n\) in which \((z_ n)^{\infty}_{-\infty}\) is a Schauder basis, such that the operators of formal multiplication by z and \(z^{-1}\) are continuous on F and have no common invariant subspace. This answers in the negative the problem of existence of hyperinvariant subspaces for operators on general Fréchet spaces [the problem for Banach spaces remains open]. The strong dual of the above Fréchet space F also yields an example of an abelian complete locally convex algebra, with unit, which does not have any nontrivial closed ideal.
Reviewer: H.Jarchow

MSC:

47A15 Invariant subspaces of linear operators
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46H10 Ideals and subalgebras
46A35 Summability and bases in topological vector spaces
46A45 Sequence spaces (including Köthe sequence spaces)
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References:

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