I am trying to do the following question from Liggett.
Let $A$ be a linear operator defined on the space of continuous functions $C(E)$ for a compact set $E$. Define $A$ by $A=T-I$ where $T$ is a positive operator and $T 1=1$. Then show $A$ is a Markov generator.
In particular, I am struggling to prove that the maximum condition holds (the hint on the link page 13 should help, but I do not see how)
Fix $f \in C(E)$ and $\eta \in E$ such that $$f(\eta) = \min_{\zeta \in E} f(\zeta). $$ Then
$$g(\zeta) := f(\zeta)-f(\eta) \in C(E)$$
defines a non-negative function and therefore, since $T$ is a positive operator,
$$Tg \geq 0. \tag{1}$$
On the other hand, it follows from the linearity of $T$ and $T1 = 1$ that
$$Tg = Tf - f(\eta). \tag{2}$$
Combining $(1)$ and $(2)$ yields
$$Tf(\zeta) \geq f(\eta) \qquad \text{for all $\zeta \in E$}.$$
Hence,
$$Af(\eta) = Tf(\eta)-f (\eta) \geq f(\eta)-f(\eta) = 0$$
Remark: Note that this is equivalent to $$f \in \mathcal{D}(A), f(\eta) = \max_{\zeta \in E} f(\zeta) \implies Af(\eta) \leq 0.$$ If this condition holds for an operator $A$, we say that $A$ satifies the positive maximum principle.