Spectral properties of generalized Cesaro operators in sequence spaces

Abstract

[EN] The generalized Cesaro operators C-t, for t is an element of[0, 1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in C(N)0, such as l(p), c(0), c, bv(0), bv and, as recently shown in Curbera et al. (J Math Anal Appl 507:31, 2022) [26], also in the discrete Cesaro spaces ces(p) and their (isomorphic) dual spaces d(p). In most cases C-t (t not equal 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 C(N)0. Besides C(N)0 itself, the Frechet spaces considered are l(p+), ces(p+) and d( p+), for 1 <= p < infinity, as well as the (LB)-spaces l(p-), ces(p-) and d(p-), for 1 < p <= infinity.