Spectral properties of generalized Cesàro operators in sequence spaces

Abstract

The generalized Cesàro operators C_t, for tin [0,1], were first investigated in the 1980s. They act continuously in many classical Banach sequence spaces contained in {{mathbb {C}}}^{{{mathbb {N}}}_0}, such as ell ^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 Cesàro spaces ces(p) and their (isomorphic) dual spaces d_p. In most cases C_t (tnot =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 {{mathbb {C}}}^{{{mathbb {N}}}_0}. Besides {{mathbb {C}}}^{{{mathbb {N}}}_0} itself, the Fréchet spaces considered are ell (p+), ces(p+) and d(p+), for 1le p<infty , as well as the (LB)-spaces ell (p-), ces(p-) and d(p-), for 1<ple infty .