Spectral properties of generalized Ces\`aro operators in sequence spaces

TL;DR

This paper investigates the spectral properties, linear dynamics, and ergodic behavior of generalized Cesàro operators in various non-normable sequence spaces, extending known results from classical Banach spaces.

Contribution

It extends the analysis of Cesàro operators to non-normable Fréchet and (LB)-spaces, exploring their spectral and dynamical properties beyond classical Banach spaces.

Findings

Spectral properties are characterized in new non-normable spaces.

Cesàro operators exhibit specific dynamical behaviors in these spaces.

The study broadens understanding of operator theory in complex sequence spaces.

Abstract

The generalized Ces\`aro operators $C_t$, for $t\in [0,1]$, were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$, such as $\ell^p$, $c_0$, $c$, $bv_0$, $bv$ and, as recently shown, \cite{CR4}, also in the discrete Ces\`aro spaces $ces(p)$ and their (isomorphic) dual spaces $d_p$. In most cases $C_t$ ($t\not=1$) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of $C_t$, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$. Besides $\mathbb{C}^{\mathbb{N}_0}$ itself, the Fr\'echet spaces considered are $\ell(p+)$, $ces(p+)$ and $d(p+)$, for $1\leq p<\infty$, as well as the (LB)-spaces $\ell(p-)$, $ces(p-)$ and $d(p-)$, for $1<p\leq\infty$.