×

zbMATH — the first resource for mathematics

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.

MSC:
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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ansari, S.I., Existence of hypercyclic operators on topological vector spaces, J. funct. anal., 148, 384-390, (1997) · Zbl 0898.47019
[2] Atzmon, A., An operator without invariant subspaces on a nuclear Fréchet space, Ann. of math., 117, 669-694, (1983) · Zbl 0553.47002
[3] L. Bernal, On hypercyclic operators in Banach spaces, Proc. Amer. Math. Soc. · Zbl 0911.47020
[4] Bierstedt, K.D.; Meise, R.M.; Summers, W.H., A projective description of weighted inductive limits, Trans. amer. math. soc., 272, 107-160, (1982) · Zbl 0599.46026
[5] Bonet, J.; Lindström, M., Spaces of operators between Fréchet spaces, Math. proc. Cambridge phil. soc., 115, 133-144, (1994) · Zbl 0804.46011
[6] Enflo, P., On the invariant subspace problem for Banach spaces, Acta math., 158, 213-313, (1987) · Zbl 0663.47003
[7] Esterle, J., Countable inductive limits of Fréchet algebras, J. anal. math., 71, 195-204, (1997) · Zbl 0905.46032
[8] Gethner, R.M.; Shapiro, J.H., Universal vectors for operators on spaces of holomorphic functions, Proc. amer. math. soc., 100, 281-288, (1987) · Zbl 0618.30031
[9] Godefroy, G.; Shapiro, J.H., Operators with dense, invariant, cyclic vector manifolds, J. funct. anal., 98, 229-269, (1991) · Zbl 0732.47016
[10] K. G. Große-Erdmann, Universal families and hypercyclic operators, 1998
[11] Grothendieck, A., Topological vector spaces, (1973), Gordon & Breach New York · Zbl 0275.46001
[12] Hilden, H.M.; Wallen, L.J., Some cyclic and non-cyclic vectors of certain operators, Indiana univ. math. J., 24, 557-565, (1974) · Zbl 0274.47004
[13] Kitai, C., Invariant closed sets for linear operators, (1982), Univ. of Toronto
[14] Köthe, G., Topological vector spaces, I,, (1969), Springer-Verlag Berlin/New York
[15] Metafune, G.; Moscatelli, V.B., Dense subspaces with continuous norm in Fréchet spaces, Bull. Polish acad. sci. math., 37, 477-479, (1989) · Zbl 0762.46001
[16] Pérez Carreras, P.; Bonet, J., Barrelled locally convex spaces, North-holland math. stud., (1987), North-Holland Amsterdam
[17] Read, C., A solution to the invariant subspace problem on the spacel1, Bull. London math. soc., 17, 305-317, (1985) · Zbl 0574.47006
[18] Salas, H., Hypercyclic weighted shifts, Trans. amer. math. soc., 347, 993-1004, (1995) · Zbl 0822.47030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.