### Infinitesimal Generator of Mean Ergodic Theorem in Semigroup of Linear Operator

#### Abstract

In this paper, results of Ergodic Theorem were obtained by investigating the behavior of $\omega$-order reversing partial contraction mapping (semigroup of linear operator) as $t$ tends to $+\infty$ of the means $\frac{1}{t}\int_{0}^{t}T(s)ds$ of a $C_0$-Semigroup of contraction $\{T(t);t \geq 0\}$ and it was shown that the infinitesimal generator of a $C_0$-Semigroup of contraction $\{T(t);t \geq 0\}$ is densely defined, linear, closed, reflexive and there exists a continuous bounded function.

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