Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro operators Cₜ, for t∈ [0,1], were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in CN₀, such as ᵖ, c₀, c, bv₀, $bv$ and, as recently shown, {CR4}, also in the discrete Cesàro spaces $ces(p)$ and their (isomorphic) dual spaces dₚ. In most cases Cₜ (t=1) is compact and its spectra and point spectrum, together with the corresponding eigenspaces, are known. We study these properties of Cₜ, as well as their linear dynamics and mean ergodicity, when they act in certain non-normable sequence spaces contained in CN₀. Besides CN₀ itself, the Fréchet spaces considered are (p+), $ces(p+)$ and $d(p+)$, for 1≤ p<∞, as well as the (LB)-spaces (p-), $ces(p-)$ and $d(p-)$, for $1