Cyclic behavior of holomorphic functions on a Runge region

Abstract

Let Ω⊂CN be a Runge region and let H(Ω) denote the Fréchet space of holomorphic functions on Ω. In this paper, we provide extensions of some earlier results regarding nonscalar continuous linear operators on H(Ω) commuting with each partial differentiation operator ∂∂zk, where 1≤k≤N. Specifically, we demonstrate that all such operators are hypercyclic and share a dense set of common cyclic vectors. Motivated by our results, we introduce a class of finite sets of Fréchet space operators patterned after the partial differentiation operators, called backward multi-shifts, and show that any nonscalar operator in the commutant of such a finite set is supercyclic. Lastly, we apply our supercyclicity result on weighted differentiation operators on H(CN) and also on double sequence spaces.