Convergence rates of ergodic limits and approximate solutions (Q1318231)

Let \(X\) be a Banach space, let \(A: D(A)\subset X\to X\) be a closed linear operator, and let \(\{A_ \alpha\}\) and \(\{B_ \alpha\}\) be two nets of obunded linear operators on \(X\) satisfying the following conditions: (1) \(\| A_ \alpha\|\leq M\) for all \(\alpha\); (2) \(R(B_ \alpha)\subset D(A)\) and \(B_ \alpha A\subset AB_ \alpha= I- A_ \alpha\) for all \(\alpha\); (3) \(R(A_ \alpha)\subset D(A)\) for all \(\alpha\), and \(\| AA_ \alpha\|= O(e(\alpha))\) with \(\lim_ \alpha e(\alpha)=0\); (4) \(B^*_ \alpha x^*=\varphi(\alpha)x^*\) for all \(x^*\in R(A)^ \perp\), and \(|\varphi(\alpha)|\to\infty\). By using an abstract mean ergodic theorem stated in the author's previous paper [J. Funct. Anal. 87, No. 2, 428-441 (1989; Zbl 0704.47006)], in the present paper the author characterizes the Favard (or saturation) classes for the two processes \(\{A_ \alpha\}\) and \(\{B_ \alpha\}\). Applications to particular examples, such as integrated semigroups, cosine operator functions, and tensor product semigroups close the paper.