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Hypercyclic operators on non-normable Fréchet spaces. (English) Zbl 0926.47011
An operator \(T\) on a locally convex space (lcs) \(E\) is called hypercyclic if \[ \text{Orb} (T,x):=\{x,Tx,T^2x,\ldots\} \] is dense in \(E\) for some \(x\in E\). In this case \(x\) is a hypercyclic vector for \(T.\) Hypercyclic vectors are of importance in connection with invariant subspaces. It is known that an operator lacks closed non-ntrivial invariant subsets iff every non-zero vector is hypercyclic. S. Ansari [J. Funct. Anal. 148, 384-390 (1997; Zbl 0898.47019)] proved that every infinite dimensional separable Banach space admits a hypercyclic operator. A corollary of the main result of this paper asserts that every infinite dimensional separable Frechet space admits a hypercyclic operator. The proof of this result contains a gap in the case of nonnormable Frechet space. The main purpose of this paper is to show that the result claimed by Ansari does hold for arbitrary Frechet spaces. An example of a complete separable lcs (more precisely, an inductive limit of Banach spaces) for which there is no hypercyclic operator defined on it is also given.

47A65 Structure theory of linear operators
46A04 Locally convex Fréchet spaces and (DF)-spaces
47A15 Invariant subspaces of linear operators
47A16 Cyclic vectors, hypercyclic and chaotic operators
Full Text: DOI
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