The discrete Cesàro (Banach) sequence spaces ces(r),1<r<∞,\documentclass[12pt]{minimal}
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\begin{document}$$ {{\text {ces}}}(r), 1< r < \infty ,$$\end{document} have been thoroughly investigated for over 45 years. Not so for their dual spaces d(s)≅(ces(r))′,\documentclass[12pt]{minimal}
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\begin{document}$$ d (s) \cong ( {{\text {ces}}}(r))', $$\end{document} with 1s+1r=1,\documentclass[12pt]{minimal}
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\begin{document}$$ \frac{1}{s} + \frac{1}{r} = 1 ,$$\end{document} which are somewhat unwieldy. Our aim is to undertake a further study of the spaces d(s) and of various operators acting between these spaces. It is shown that d(s)⊆d(t)\documentclass[12pt]{minimal}
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\begin{document}$$ d (s) \subseteq d (t)$$\end{document} whenever s≤t,\documentclass[12pt]{minimal}
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\begin{document}$$ s \le t ,$$\end{document} with the inclusion being compact if s<t.\documentclass[12pt]{minimal}
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\begin{document}$$ s< t .$$\end{document} The classical Cesàro operator C is continuous from d(s) into d(t) precisely when s≤t\documentclass[12pt]{minimal}
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\begin{document}$$ s \le t $$\end{document} and compact precisely when s<t.\documentclass[12pt]{minimal}
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\begin{document}$$ s < t .$$\end{document} Moreover, C even maps the larger space ces(s)\documentclass[12pt]{minimal}
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\begin{document}$$ {{\text {ces}}}(s)$$\end{document} continuously into d(s). This is a consequence of the Hardy–Littlewood maximal theorem and the remarkable property, for each 1<s<∞,\documentclass[12pt]{minimal}
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\begin{document}$$ 1< s < \infty ,$$\end{document} that x∈CN\documentclass[12pt]{minimal}
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\begin{document}$$ x \in \mathbb {C}^{\mathbb {N}} $$\end{document} satisfies C(C(|x|))∈d(s)\documentclass[12pt]{minimal}
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\begin{document}$$ C (C (| x| )) \in d (s)$$\end{document} if and only if C(|x|)∈d(s).\documentclass[12pt]{minimal}
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\begin{document}$$ C (| x | ) \in d (s).$$\end{document} These results are used to analyze the spectrum and to determine the norm and the mean ergodicity of C acting in d(s). Similar properties for multiplier operators are also treated.
