Spectral properties of generalized Cesàro operators in sequence spaces

Abstract

The generalized Cesàro operators $C_t$, for $t\in [0,1]$, were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in ${\mathbb{C}}^{{\mathbb{N}}_0}$, such as ${\ell }^p$, $c_0$, $c$, $b{v}_0$, $bv$ and, as recently shown, \cite{CR4}, also in the discrete Cesàro spaces $ces(p)$ and their (isomorphic) dual spaces $d_p$. In most cases $C_t$ ($t\ne 1$) 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 $1\le p<\infty$, as well as the (LB)-spaces $\ell (p-)$, $ces(p-)$ and $d(p-)$, for $1<p\le \infty$.

34 pages, the revised manuscript is accepted to appear in RACSAM