The Rhaly Operators on K\"othe Spaces

TL;DR

This paper introduces and analyzes the Rhaly operator on K"othe spaces, exploring its properties, integral representations, and ergodic behavior in various function spaces.

Contribution

It provides a comprehensive study of the Rhaly operator on K"othe spaces, including conditions for well-definedness, continuity, compactness, and ergodic properties, with new integral representations.

Findings

Characterization of the Rhaly operator's boundedness and compactness

Integral formulas for the operator on entire and holomorphic function spaces

Results on mean ergodicity and Cesàro boundedness of the operators

Abstract

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.