发表机构
Chouaib Doukkali University; Mathematical Institute of the Serbian Academy of Sciences and Arts（舒艾布杜卡利大学; 塞尔维亚科学与艺术学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在线性动力学中引入广义数值传递性与C*-传递性，证明二者与多种标准动力学概念不同，构造了相关算子并论证存在弱超循环但非C*-传递的算子。
AI 中文摘要
受数值超循环性概念的启发，本文在线性动力学中引入两个新的概念：广义数值传递性和C*-传递性。这两个概念均在巴拿赫代数上的算子框架下提出，不过C*-传递性的概念也适用于希尔伯特-施密特算子空间上的算子。我们证明这些新概念彼此互不相同，且与动力学中的标准概念如超循环性、超循环性、弱超循环性、弱超循环性、数值超循环性也存在差异。特别地，我们给出了数值超循环算子的例子，这些算子既不是C*-传递的，也不是广义数值传递的。此外，我们在可分希尔伯特空间上紧算子构成的C*-代数上构造了非超循环且非弱超循环的C*-传递及广义数值传递算子，还在标准希尔伯特C*-模上构造了C*-传递及广义数值传递算子。最后，我们论证存在弱超循环但非C*-传递的算子。
英文摘要
Motivated by the concept of numerical hypercyclicity, in this paper, we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, C*-transitivity. The first notion is defined for operators on general Banach spaces, whereas the latter two are formulated in the setting of operators on Banach algebras. The concept of C*-transitivity is also applicable to operators on the space of Hilbert-Schmidt operators. We prove that these new notions are mutually different, and we also show that they differ from the standard notions in dynamics. In particular, while topological numerical transitivity implies numerical hypercyclicity, as we argue in the paper, we provide examples of numerically hypercyclic and strongly numerically hypercyclic operators that are not topologically numerically transitive. Also, we construct nonsupercyclic C* transitive and generalized numerically transitive operators on the C*-algebra of compact operators on a separable Hilbert space, as well as C*-transitive nonsupercyclic operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional and separable Hilbert spaces, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. Further, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive. All these results are illustrated with concrete examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, and we prove that topological numerical transitivity and C*-transitivity genuinely differ from all these standard concepts in dynamics.
