标题： Rhaly 算子在 Köthe 空间上 显示英文标题

标题： The Rhaly Operators on Köthe Spaces

摘要： 我们引入并研究了Köthe空间上的Rhaly算子，主要关注理解其定义明确性、连续性和紧性。 我们特别考察了作用于无限型和有限型幂级数空间上的算子。 随后，我们提供了Rhaly算子在整函数空间$H(\mathbb{C})$和单位圆上全纯函数空间$H(\mathbb{D})$上的积分表示。 我们还研究了Rhaly算子的拓扑可化性和幂有界性，这导致了关于它们的平均遍历性、一致平均遍历性和Cesàro有界性的发现。 显示英文摘要

摘要： We introduce and study the Rhaly operator on K\"othe spaces, with a primary focus on understanding its well-definedness, continuity, and compactness. We especially examine operators acting on power series spaces of both infinite and finite type. In the sequel, we provide integral representations for the Rhaly operator on the space of entire functions $H(\mathbb{C})$ and the space of holomorphic functions on the unit disc $H(\mathbb{D})$. We also investigate the topologizability and power boundedness of the Rhaly operators, which leads to findings about their mean ergodicity, uniform mean ergodicity, and Ces\`aro boundedness.