Solid Extensions of the Cesàro Operator on ℓp and c0

We introduce and study the largest Banach lattice (for the coordinate-wise order) which is a solid subspace of CN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{C}^\mathbb{N}}$$\end{document} and to which the classical Cesàro operator C:ℓp→ℓp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}\colon\ell^p \to \ell^p}$$\end{document} (a positive operator) can be continuously extended while still maintaining its values in ℓp. Properties of this optimal Banach lattice [C,ℓp]s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\mathcal{C}, \ell^p]_s}$$\end{document} are presented. In addition, all continuous convolution operators of [C,ℓp]s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${[\mathcal{C}, \ell^p]_s}$$\end{document} into itself are identified and the spectrum of C:[C,ℓp]s→[C,ℓp]s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}\colon[\mathcal{C}, \ell^p]_s \to[\mathcal{C}, \ell^p]_s}$$\end{document} is determined. A similar investigation is undertaken for the Cesàro operator C:c0→c0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{C}\colon c_0\to c_0}$$\end{document}.