Fine spectra and compactness of generalized Ces\`{a}ro operators in Banach lattices in ${\mathbb C}^{{\mathbb N}_0}$

TL;DR

This paper investigates the spectral properties and compactness of generalized Cesàro operators in a broad class of Banach lattices, providing a comprehensive analysis of their spectra and norm estimates.

Contribution

It offers a complete spectral characterization and norm estimates for generalized Cesàro operators in Banach lattices, extending known results to a wider class of spaces.

Findings

Operators are always compact in the considered Banach lattices.

Full description of point, continuous, and residual spectra is provided.

Operator norms are estimated explicitly.

Abstract

The generalized Ces\`{a}ro operators $\mathcal{C}_t$, for $t\in[0,1)$, introduced in the 1980's by Rhaly, are natural analogues of the classical Ces\`{a}ro averaging operator $\mathcal{C}_1$ and act in various Banach sequence spaces $X\subseteq {\mathbb C}^{{\mathbb N}_0}$. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invariant sequence spaces, various weighted $c_0$ and $\ell^p$ spaces and many others. In such Banach lattices $X$ the operators $\mathcal{C}_t$, for $t\in[0,1)$, are always compact (unlike $\mathcal{C}_1$) and a full description of their point, continuous and residual spectrum is given. Estimates for the operator norm of $\mathcal{C}_t$ are also presented.