On the convergence rate in the uniform ergodic theorem (Q1915594)

The present paper is a contribution to the classical topic constituted by Kryloff-Bogoliuboff and Yosida-Kakutani theorems on the speed of convergence for Cesàro averages of the powers of Markov and quasi-compact operators, respectively. Let \(X\) be a Banach space with norm \(\delta_X(\cdot)\) and \(B[X,X]\) be the Banach algebra of bounded linear operators from \(X\) to \(X\) with norm \(\pi_B(\cdot)\). The main results are the following two theorems. Theorem 1. Let \(T\in B[X,X]\) be quasi-compact and \(\pi_B(T^n)\leq K\) \((n=0,1,2,\dots)\) for some constant \(K>0\). Then for any \(\alpha>0\) there exists a compact projection \(\Pi_\alpha\in B[X,X]\) such that \[ \pi_B\Biggl[ (A_n^\alpha)^{-1} \sum_{k=0}^n A_{n-k}^{\alpha-1} T^k- \Pi_\alpha\Biggr]= O\biggl( \frac{1}{n^\omega} \biggr), \qquad \omega=\min(\alpha,1), \] where the \(A_k^\alpha\)'s are the \((C,\alpha)\)-coefficients of order \(\alpha\). Theorem 2. Let \(\{T(t): t\geq 0\}\) be a uniformly continuous semigroup in \(B[X,X]\). Suppose that (i) \(\sup_{t\geq 0} \pi_B[T(t)]= M<\infty\), and (ii) \(T(1)\) is quasi-compact. Then for any \(\alpha\geq 1\) there exists a compact projection \(\Pi_\alpha\in B[X,X]\) such that \[ \pi_B\Biggl[ \frac{\alpha}{u^\alpha} \int^u_0 (u-t)^{\alpha-1} T(t)dt- \Pi_\alpha \Biggr]= O\biggl(\frac{1}{u}\biggr). \]