Theorem 2.4.4 ([Theorem 4.25, Bau14]).label Let $L$ be an essentially self-adjoint diffusion operator with symmetrising measure $\mu$ and heat semigroup $\bracs{\bp_t|t \ge 0}$, then for any $f \in L^{2}(\mu)$ with $f \ge 0$ (a.e.), $\bp_{t}f \ge 0$.
Proof. By the Dominated Convergence Theorem and the Spectral Theorem, for any $f \in L^{2}(\mu)$,
\[\bp_{t}f = \limv{n}\paren{1 - \frac{t}{n}L}^{-n}f\]
Thus it is sufficient to show that $(1 - \lambda L)^{-1}$ satisfies the positivity criterion for all $\lambda > 0$.
Let $g \in L^{2}(\mu)$ with $g \ge 0$ (a.e.) and $f = R_{\lambda} g$, then
\[\norm{f}_{\lambda}^{2} = \dpn{f, g}{L^2(\mu)}\le \dpn{|f|, g}{L^2(\mu)}= \dpn{|f|, f}{\lambda}\le \norm{f}_{\lambda}^{2}\]
so $f = |f|$, and $R_{\lambda} g \ge 0$.$\square$