Let $\left(T(t)\right)_{t\geq0}$ be a $C_0$-semigroup on a Banach lattice $E$ such that $T(t)$ converges strongly to a positive operator $S$ as $t \to \infty.$ Then $$T(t)S=S \text{ for all }t\geq0.$$
My attempt: I see why this is true is $S$ is a projection but I'm unable to prove it for the general case.
Any hints are appreciated.
This follows since $$\|T(s)Sf-Sf\|=\lim_{t\to\infty}\|T(s)T(t)f-T(t)f\|=\lim_{t\to\infty}\|T(t+s)f-T(t)f\|=0.$$