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Operators with hypercyclic Cesaro means

Fernando León-Saavedra — 2002

Studia Mathematica

An operator T on a Banach space ℬ is said to be hypercyclic if there exists a vector x such that the orbit T ⁿ x n ≥ 1 is dense in ℬ. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ℬ. If the arithmetic means of the orbit of x are dense in ℬ then the operator T is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector...

Hypercyclic sequences of operators

Fernando León-Saavedra;  Vladimír Müller — 2006

Studia Mathematica

A sequence (Tₙ) of bounded linear operators between Banach spaces X,Y is said to be hypercyclic if there exists a vector x ∈ X such that the orbit Tₙx is dense in Y. The paper gives a survey of various conditions that imply the hypercyclicity of (Tₙ) and studies relations among them. The particular case of X = Y and mutually commuting operators Tₙ is analyzed. This includes the most interesting cases (Tⁿ) and (λₙTⁿ) where T is a fixed operator and λₙ are complex numbers. We also study when a sequence...

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