Suppose $T:E\rar E$ is bounded and $P_t$ a continuous contraction semigroup with generator $L$. $T$ and $P_t$ commut if and only if $T$ and $L$ commute and $T(\dom L)\sbe\dom L$.
Suppose $x\in\dom L$, $T(\dom L)\sbe\dom L$ and $0\leq s\leq t$, then
$$
\ttd sP_{t-s}TP_sx
=P_{t-s}(-L)TP_sx+P_{t-s}TLP_sx
=0
$$
Putting $s=0$ and $s=t$ we get: $P_tTx=TP_tx$. If these conditions are satisfied we obtain by differentiation at $t=0$: $LTx=TLx$.