Spectrum of the Cesàro operator in \(\ell ^{p }\)
For \(1<p<\infty\), the Cesàro operator \(C^{(p)}\) on \(\ell^{p}\) assigns to a sequence \((a_n)_{n=1}^{\infty}\) its sequence of averages \((1/n \sum_{k=1}^{n}a_k)_{n=1}^{\infty}\). By Hardy's inequality, \(C^{(p)}\) is bounded. In this paper, a novel proof of the fact that the spectrum of \(C^{(p)}\) is the closed disc \(D[p]\), which is centered at \(q/2\) and has radius \(q/2\), where \(q\) is the conjugate index of \(p\). The difficult part in proving the equality \(D[p]= \sigma(C^{(p)})\) is the inclusion \(D[p]\supseteq \sigma(C^{(p)})\). For this, one has to see that \((C^{(p)}-\lambda I)^{-1}\) is a bounded operator whenever \(\lambda \notin D[p]\). This is managed in a clever way by dominating the matrix \(E_{\lambda}:=-\lambda^2[(C-\lambda I)^{-1}-D_{\lambda}]\) by a more tractable positive matrix \(G_{\lambda}\). Here, \(D_{\lambda}\) is the diagonal part of \((C-\lambda I)^{-1}\), for which it is not hard to prove the boundedness. The problem is thereby reduced to show that \(G_{\lambda}\) is a bounded operator.
scientific article; zbMATH DE number 6152314