Fernando León-Saavedra

Fixed points and orbits of non-convolution operators

A continuous linear operator T on a Fréchet space F is hypercyclic if there exists a vector f∈F (which is called hypercyclic for T) such that the orbit {Tnf:n∈N} is dense in F. A subset M of a vector space F is spaceable if M∪{0} contains an infinite-dimensional closed vector space. In this paper note we study the orbits of the operators Tλ,bf=f′(λz+b) (λ,b∈C) 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 Tλ,b is hypercyclic on H(C). We characterize also when the set of hypercyclic vectors for Tλ,b is spaceable. The fixed point of the map z→λz+b (in the case λ≠1) plays a central role in the proofs.

Powers of Convex-Cyclic Operators

A bounded operator on a Banach space is convex cyclic if there exists a vector such that the convex hull generated by the orbit is dense in . In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operator such that the power fails to be convex cyclic. Using this result we solve three questions posed by Rezaei (2013).