Hereditarily hypercylic operators
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 \textit{C. Kitai}, ``Invariant closed sets for linear operators'' (Ph.D. thesis, Univ. of Toronto) (1982); \textit{R. M. Gethner} and \textit{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 \textit{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 \textit{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.
hypercyclic operators with a dense generalized kernel
hypercyclic operators with a dense set of periodic points
scientific article; zbMATH DE number 1346903
scientific article