Dense range perturbations of hypercyclic operators
We show that if $(T_n)$ is a hypercyclic sequence of linear operators on a locally convex space and $(S_n)$ is a sequence of linear operators such that the image of each orbit under every linear functional is non-dense then the sequence $(T_n + S_n)$ has dense range. Furthermore, it is proved that if $T,S$ are commuting linear operators in such a way that $T$ is hypercyclic and all orbits under $S$ satisfy the above non-denseness property then $T - S$ has dense range. Corresponding statements for operators and sequences which are hypercyclic in a weaker sense are shown. Our results extend and improve a result on denseness due to C. Kitai.