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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.

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
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