Spectral and ergodic properties of the operator $B(r,s)$ over power series spaces $\Lambda_\infty(\alpha)$ of infinite type and their duals

TL;DR

This paper thoroughly analyzes the spectral and ergodic characteristics of the operator $B(r,s)$ on certain infinite-dimensional power series spaces, providing a complete spectrum description and conditions for boundedness and ergodicity.

Contribution

It offers a comprehensive spectral characterization and criteria for power boundedness and ergodicity of $B(r,s)$ on $ ext{Lambda}_ ext{infty}( ext{alpha})$ spaces and their duals, advancing understanding of such operators.

Findings

Complete spectrum characterization of $B(r,s)$

Necessary and sufficient conditions for power boundedness

Criteria for (uniform) mean ergodicity

Abstract

In this paper, we investigate the spectral and ergodic properties of the linear operator $B(r,s)$ acting on power series spaces $\Lambda_\infty(\alpha)$ of infinite type and on their strong duals. Precisely, we provide a complete characterization of its fine spectrum and establish necessary and sufficient conditions for the operator to be power bounded and (uniformly) mean ergodic.