Taiwanese Journal of Mathematics

CONVERGENCE RATES FOR ERGODIC THEOREMS OF KIDO-TAKAHASHI TYPE

Sen-Yen Shaw and Yuan-Chuan Li

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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 $y\in R(A)$.

Article information

Source
Taiwanese J. Math., Volume 12, Number 6 (2008), 1543-1559.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405039

Digital Object Identifier
doi:10.11650/twjm/1500405039

Mathematical Reviews number (MathSciNet)
MR2444871

Zentralblatt MATH identifier
1197.47022

Subjects
Primary: 47A35: Ergodic theory [See also 28Dxx, 37Axx] 47A58: Operator approximation theory 41A25: Rate of convergence, degree of approximation 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30] 47D09: Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10]

Keywords
strongly regular net of linear functionals $A$-ergodic net mean ergodic theorem $(C_0)$-semigroup

Citation

Shaw, Sen-Yen; Li, Yuan-Chuan. CONVERGENCE RATES FOR ERGODIC THEOREMS OF KIDO-TAKAHASHI TYPE. Taiwanese J. Math. 12 (2008), no. 6, 1543--1559. doi:10.11650/twjm/1500405039. https://projecteuclid.org/euclid.twjm/1500405039


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