Abstract
A continuous linear operator T on a Frechet space F is hypercyclic if there exists a vector (which is called hypercyclic for T) such that the orbit is dense in F. A subset M of a vector space F is spaceable if contains an infinite-dimensional closed vector space. In this paper note we study the orbits of the operators ( ) defined on the space of entire functions and introduced by Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004). We complete the results in Aron and Markose (J. Korean Math. Soc. 41(1):65-76, 2004), characterizing when is hypercyclic on . We characterize also when the set of hypercyclic vectors for is spaceable. The fixed point of the map (in the case ) plays a central role in the proofs.
Highlights
1 Introduction Let us denote by F a complex infinite dimensional Fréchet space
A continuous linear operator T defined on F is said to be hypercyclic if there exists a vector f ∈ F such that the orbit ({Tnf : n ∈ N}) is dense in F
A subset M of a vector space F is said to be spaceable if M ∪ { } contains an infinite-dimensional closed vector space
Summary
Let us denote by F a complex infinite dimensional Fréchet space. A continuous linear operator T defined on F is said to be hypercyclic if there exists a vector f ∈ F (called hypercyclic vector for T) such that the orbit ({Tnf : n ∈ N}) is dense in F. A subset M of a vector space F is said to be spaceable if M ∪ { } contains an infinite-dimensional closed vector space. This result together with the results in [ ] and [ ] shows the following characterization: Tλ,b is hypercyclic on H(C) if and only if |λ| ≥ .
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