Fine spectra and compactness of generalized Cesàro operators in Banach lattices in ${\mathbb{C}}^{{\mathbb{N}}_0}$

Abstract

The generalized Cesàro operators ${\mathcal{C}}_t$, for $t\in [0,1)$, introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro 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.