  The generalized Ces\`aro 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 $\ell^p$,
$c_0$, $c$, $bv_0$, $bv$ and, as recently shown, \cite{CR4}, also in the
discrete Ces\`aro spaces $ces(p)$ and their (isomorphic) dual spaces $d_p$. In
most cases $C_t$ ($t\not=1$) 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
Fr\'echet 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
$1<p\leq\infty$.
