Poisson’s Equation for Abel-Ergodic Theorems

The functional equation given by \(y=(I-T)x\) $y=(I-T)x$ is called Poisson’s equation. For a (weakly) mean ergodic operator T, this equation can be solved for a given y if and only if \(x_n:=\dfrac{1}{n}\sum \limits _{k=1}^n \sum \limits _{j=0}^{k-1}T^{j}y\) $x_n:={\displaystyle \frac{1}{n}}\sum _{k=1}^n\sum _{j=0}^{k-1}{T}^{j}y$ (weakly) converges. In this paper, we use Abel summability of \((T^n)\) $(T^n)$ in order to solve the equation \(y=(I-T)x\) $y=(I-T)x$ , where T is a bounded linear operator on a Banach space X.