Hereditarily hypercylic operators.

*(English)*Zbl 0941.47002The 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.

Reviewer: P.Dyshlovenko (Ul’yanovsk)

##### MSC:

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 |

##### Keywords:

hypercyclicity criterion; continuous linear operators; Fréchet space; weighted shifts; linear chaos; hypercyclic operators with a dense generalized kernel; hypercyclic operators with a dense set of periodic points; chaotic operators
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\textit{J. Bès} and \textit{A. Peris}, J. Funct. Anal. 167, No. 1, 94--112 (1999; Zbl 0941.47002)

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##### References:

[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, Contemporary mathematics, 232, (1999), 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) · JFM 55.0192.07 |

[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 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). |

[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 New York · Zbl 0791.30033 |

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