Disjoint Ces$\grave{a}$ro-hypercyclic operators
Qing Wang, Yonglu Shu
TL;DR
This work studies disjoint Cesàro-hypercyclicity of operator tuples on separable infinite-dimensional Banach spaces, extending Cesàro-hypercyclicity to multiple operators. It introduces a disjoint Cesàro-Hypercyclicity Criterion with dense subspaces and associated transfer maps, and proves two complementary approaches (constructing a d-CH vector and a Blow-Up/Collapse argument) to certify disjoint Cesàro-hypercyclicity. As an application, it fully characterizes the weight sequences that yield disjoint Cesàro-hypercyclicity for both unilateral and bilateral weighted shifts, providing explicit growth and ratio conditions. The results connect disjointness in the Cesàro sense with topological transitivity and Blow-Up/Collapse behavior, enriching the theory of linear dynamics and weighted-shift operations.
Abstract
In this paper, we investigate the properties of disjoint Ces$\grave{a}$ro-hypercyclic operators. First, the definition of disjoint Ces$\grave{a}$ro-hypercyclic operators is provided, and disjoint Ces$\grave{a}$ro-Hypercyclicity Criterion is proposed. Later, two methods are used to prove that operators satisfying this criterion possess disjoint Ces$\grave{a}$ro-hypercyclicity. Finally, this paper further investigates weighted shift operators and provides detailed characterizations of the weight sequences for disjoint Ces$\grave{a}$ro-hypercyclic unilateral and bilateral weighted shift operators on sequence spaces.