Cyclic behavior of holomorphic functions on a Runge region

[摘要] Let Omega subset of C-N be a Runge region and let H(Omega) denote the Frechet space of holomorphic functions on Omega. In this paper, we provide extensions of some earlier results regarding nonscalar continuous linear operators on H(Omega) commuting with each partial differentiation operator partial derivative/partial derivative(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 Frechet 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(C-N) and also on double sequence spaces. (C) 2021 Elsevier Inc. All rights reserved.

[发布日期] 2021-11-01

[关键词] Runge region;Hypercyclic operator;Supercyclic operator;Common cyclicity;Generalized backward shift;Backward multi-shift