Hereditarily hypercylic operators. (English) Zbl 0941.47002

The notion of hereditarily hypercyclic operator is introduced and discussed. Hypercyclicity is closely related to linear chaos on infinite dimensional spaces. It is shown that a continuous linear operator \(T\) on a Fréchet space satisfies the hypercyclicity criterion [see C. Kitai, “Invariant closed sets for linear operators” (Ph.D. thesis, Univ. of Toronto) (1982); R. M. Gethner and J. H. Shapiro, Proc. Am. Math. Soc. 100, No. 2, 281-288 (1987; Zbl 0618.30031)] if and only if it is hereditarily hypercyclic, and if and only if the direct sum \(T\oplus T\) is hypercyclic. As a consequence, it is shown that two classes of hypercyclic operators, namely the hypercyclic operators with a dense generalised kernel and hypercyclic operators with a dense set of periodic points [i.e., chaotic operators in the sense of R. L. Devaney, “An introduction to chaotic dynamical systems, 2nd ed.”, Addison-Wesley (1989; Zbl 0695.58002)] must satisfy the hypercyclicity criterion. In connection with the work of H. N. Salas [Trans. Am. Math. Soc. 347, No. 3, 993-1004 (1995; Zbl 0822.47030)] on hypercyclic weighted shifts, a characterisation of those weighted shifts \(T\) that are hereditarily hypercyclic with respect to a given sequence \((n_k)\) of positive integers is provided, as well as conditions under which \(T\) and \(\{T^{n_k}\}_{k\geq 1}\) share the same set of hypercyclic vectors.


47A16 Cyclic vectors, hypercyclic and chaotic operators
47A15 Invariant subspaces of linear operators
47A65 Structure theory of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46A04 Locally convex Fréchet spaces and (DF)-spaces
Full Text: DOI


[1] Ansari, S. I., Hypercyclic and cyclic vectors, J. Funct. Anal., 128, 374-383 (1995) · Zbl 0853.47013
[2] Aron, R.; Bès, J., Hypercyclic differentiation operators, Function Spaces. Function Spaces, Contemporary Mathematics, 232 (1999), Am. Math. Soc: Am. Math. Soc Providence, p. 39-46 · Zbl 0938.47004
[3] Bès, J., Three Problems on Hypercyclic Operators (1998), Kent State University
[4] Birkhoff, G. D., Démonstration d’un théorème élémentaire sur les fonctions entières, C. R. Acad. Sci. Paris, 189, 473-475 (1929)
[5] Bonet, J.; Peris, A., Hypercyclic operators on non-normable Fréchet spaces, J. Funct. Anal., 159, 587-595 (1998) · Zbl 0926.47011
[6] Bourdon, P. S.; Shapiro, J. H., Cyclic phenomena for composition operators, Mem. Amer. Math. Soc., 125 (1997) · Zbl 0996.47032
[7] Chan, K. C.; Shapiro, J. H., The cyclic behaviour of translation operators on Hilbert spaces of entire functions, Indiana Univ. Math. J., 40, 1421-1449 (1991) · Zbl 0771.47015
[8] Devaney, R. L., An Introduction to Chaotic Dynamical Systems (1989), Addison-Wesley: Addison-Wesley Reading · Zbl 0695.58002
[9] 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
[10] Godefroy, G.; Shapiro, J. H., Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., 98, 229-269 (1991) · Zbl 0732.47016
[11] Grosse-Erdmann, K.-G., Universal families and hypercyclic operators, Bull. Amer. Math. Soc., 36, 345-381 (1999) · Zbl 0933.47003
[12] Gulisashvili, A.; MacCluer, C. R., Linear chaos in the unforced quantum harmonic oscillator, J. Dynam. Systems Measure. Control, 118, 337-338 (1996) · Zbl 0870.58057
[13] Herrero, D. A., Hypercyclic operators and chaos, J. Operator Theory, 28, 93-103 (1992) · Zbl 0806.47020
[14] Herzog, G.; Schomoeger, C., On operators \(T\) such that \(f(T)\) is hypercyclic, Studia Math., 108, 209-216 (1994) · Zbl 0818.47011
[15] Kitai, C., Invariant Closed Sets for Linear Operators (1982), Univ. of Toronto
[16] León-Saavedra, F.; Montes-Rodrı́guez, A., Linear structure of hypercyclic vectors, J. Funct. Anal., 148, 524-545 (1997) · Zbl 0999.47009
[17] F. León-Saavedra, and, A. Montes-Rodrı́guez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc. (to appear).; F. León-Saavedra, and, A. Montes-Rodrı́guez, Spectral theory and hypercyclic subspaces, Trans. Amer. Math. Soc. (to appear). · Zbl 0961.47003
[18] Maclane, G. R., Sequences of derivatives and normal families, J. Anal. Math., 2, 72-87 (1952) · Zbl 0049.05603
[19] Montes-Rodrı́guez, A., Banach spaces of hypercyclic vectors, Michigan Math. J., 43, 419-436 (1996) · Zbl 0907.47023
[20] Protopopescu, V.; Azmy, Y. Y., Topological chaos for a class of linear models, Math. Models Methods Appl. Sci., 2, 79-90 (1992) · Zbl 0770.58024
[21] Read, C., The invariant subspace problem for a class of Banach spaces 2: Hypercyclic operators, Israel J. Math., 63, 1-40 (1998) · Zbl 0782.47002
[22] Rolewicz, S., On orbits of elements, Studia Math., 32, 17-22 (1969) · Zbl 0174.44203
[23] Salas, H., A hypercyclic operator whose adjoint is also hypercyclic, Proc. Amer. Math. Soc., 112, 765-770 (1991) · Zbl 0748.47023
[24] Salas, H., Hypercyclic weighted shifts, Trans. Amer. Math. Soc., 347, 993-1004 (1995) · Zbl 0822.47030
[25] Seidel, W. P.; Walsh, J. L., On approximation by Euclidean and non-Euclidean translates of an analytic function, Bull. Amer. Math. Soc., 47, 916-920 (1941) · Zbl 0028.40003
[26] Shapiro, J. H., Composition Operators and Classical Function Theory (1993), Springer-Verlag: Springer-Verlag New York · Zbl 0791.30033
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.