Saturation for Cesàro means of higher order (Q2869928)

In this paper, the author investigates the Cesàro means of higher order of a bounded operator \(T\) on a Banach space. He proves a general ergodic theorem for the Cesàro averages of higher order. The Cesàro means of order \(p\) is given by NEWLINE\[NEWLINEM_n^{(p)}(T)=\frac{p}{(n+1)\cdots(n+p)}\sum_{j=0}^{n}\frac{(j+p-1)!}{j!}M_j^{(p-1)}(T),NEWLINE\]NEWLINE \(M_0^{(p)}(T)=I\), \(M_n^{(0)}(T)=T^n\) and \(M_n^{(1)}(T)=\frac1{n+1}\sum_{j=0}^{n}T^j.\) In this context, the author further establishes the following.NEWLINENEWLINETheorem. Let \(T\) a bounded operator on a Banach space \(X\) such that \(\|M_n^{(p)}(T)\|=O(1)\) and \(\|M_n^{(p-1)}(T)x\|=o(n)\) as \(n \to +\infty\), for any \(x \in X\) and some \(p \geq 1\). Then, the following statements hold:NEWLINENEWLINE(1) \(\|M_n^{(p+1)}(T)x-Px\| = o\left(\frac1{n}\right)\) as \(n \to +\infty\) if and only if \(x\) is a element of the null space of \(I-T\) (\(x \in \mathcal{N}(I-T)\)), where \(P\) is the ergodic projector associated to \(T\), that is, the bounded projection on the closed subspace \(X_0=\overline{\mathcal{R}(T)}\oplus \mathcal{N}(T)\), \(\mathcal{R}(T)\) being the rank of \(T\).NEWLINENEWLINE(2) \(\|M_n^{(p+1)}(T)x-Px\|=o\left(\frac1{n}\right)\) as \(n \to +\infty\) if and only ifNEWLINENEWLINE(a) \(x \in \widetilde{\mathcal{R}(I-T|X_0)}_{X_0}\oplus \mathcal{N}(I-T)\) or, equivalently,NEWLINENEWLINE(b) \(x \in \widetilde{\mathcal{R}(I-T)}_{X}\oplus \mathcal{N}(I-T)\), where \(\widetilde{\mathcal{R}(I-T|X_0)}_{X_0}\) and \(\widetilde{\mathcal{R}(I-T)}_{X}\) are, respectively, the completion of \(\mathcal{R}(I-T|X_0)\) relative to \(X_0\) and \(\mathcal{R}(I-T)\) to \(X\). Notice that the completion of the subspace \(Y\) relative to \(X\) is the set of \(x \in X\) for which there exists a sequence \((y_n) \subset Y\) with \(\|y_n\|=O(1)\) and \(\|y_n-x\|=o(1)\) as \(n \to +\infty\).