Spectral properties of generalized Cesàro operators in sequence spaces

The generalized Cesàro 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 $\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 $C_t$ ($t$ 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 $\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 $1\leq p < \infty$, as well as the (LB)-spaces $\ell(p-)$, $ces(p-)$ and $d(p-)$, for $1<\leq\infty$.