Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Question

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle.
Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy maximum principle if $Lu\geq 0$ implies $u\geq 0$ in $[0,1]$.
Now my question is whether $L$ is a positive definite operator or not. Is there any special structure of self-adjoint positive definite differential operators which also satisfies maximum principle?