Sry this the second question from the following article, I am asking in this week.
At page 6 (126), 3th line, of the following article.
THE HEAT EQUATION WITH A SINGULAR POTENTIAL
the authors say by positivity preserving property of the semigroup $\{e^{t \Delta}\}$, we have:
$$ e^{\delta (\Delta+ V_n)}v_0 = \lim_{m \to \infty} (e^{\delta \Delta/m } e^{(\delta/m) V_n})^m v_0 \leq e^{\delta \lambda} e^{\delta \Delta} v_0$$
Here $\delta >0$, $V_n$ is a non-negative bounded function, $\| V_n\|_{\infty}\leq \lambda$ and $v_0$ is an initial value.
I can't understand this inequality. Why this limiting procedure is necessary and where the positivity preserving property is used?