The Fréchet spaces $ces(p+)$, $1

The Banach spaces $ces(p)$, $1<p<\infty$, were intensively studied by G. Bennett and others. The largest solid Banach lattice in $\mathbb{C}^{\mathbb {N}}$ which contains $\ell_p$ and which the Cesàro operator $C \colon\mathbb{C}^{\mathbb {N}}\to\mathbb{C}^{\mathbb {N}}$ maps into $\ell_p$ is $ces(p)$. For each $1\leq p<\infty$, the (positive) operator $C$ also maps the Fréchet space $\ell_{p+}=\cap_{q>p}\ell_q} into itself. It is shown that the largest solid Fréchet lattice in $\mathbb{C}^{\mathbb {N}$ which contains $\ell_{p+} and which $C$ maps into $\ell_{p+} is precisely $ces(p+) :=\cap_{q>p}ces(q)$. Although the spaces $\ell_{p+}$ are well understood, it seems that the spaces $ces(p+)$ have not been considered at all. A detailed study of the Fréchet spaces $ces(p+)$, $1\leq p<\infty$, is undertaken. They are very different to the Fréchet spaces $\ell_{p+}$ which generate them in the above sense. We prove that each $ces(p+)$ is a power series space of finite type and order one, and that all the spaces $ces(p+)$, $1\leq p<\infty$, are isomorphic.