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

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.