CONVERGENCE RATES FOR ERGODIC THEOREMS OF KIDO-TAKAHASHI TYPE

Abstract

et $\{T(t);t\ge 0\}$ be a uniformly bounded $(C_0)$-semigroup of operators on a Banach space $X$ with generator $A$ such that all orbits are relatively weakly compact. Let $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$ be two nets of continuous linear functionals on the space $C_b[0,\infty)$ of all bounded continuous functions on $[0,\infty)$. $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$ determine two nets $\{A_\alpha\},\ \{B_\alpha\}$ of operators satisfying $\langle A_\alpha x,x^*\rangle=\phi_\alpha(\langle T(\cdot)x,x^*\rangle)$ and $\langle B_\alpha x,x^*\rangle=\psi_\alpha(\langle T(\cdot)x,x^*\rangle)$ for all $x\in X$ and $x^*\in X^*$. Under suitable conditions on $\{\phi_\alpha\}$ and $\{\psi_\alpha\}$, this paper discusses: 1) the convergence of $\{A_\alpha\}$ and $\{B_\alpha\}$ in operator norm; 2) rates of convergence of $\{A_\alpha x\}$ and $\{A_\alpha y\}$ for each $x\in X$ and $y\in R(A)$.