Let $X$ be a compact set, $C(X)$ the space of continuous function with values in $\mathbb{R}$ with the usual norm (uniform convergence) and an operator $T: C(X) \to C(X)$ with these conditions
1- Nonnegative: For every $ f\in C(X)$ such that $f \geq 0$ we have $Tf \geq 0$.
2- Let $1(x)=1$ for every $x \in X$, then $T1=1$
I can prove with these hypothesis that $T$ is a contraction. Indeed, for every $f \in C(X)$ $$||Tf(x)|| \leq ||f||||T(1)|| = ||f||.$$ Is it correct? Because I have a problem (the model which I'm working can't get this property), I can prove that $||T||=1$: $$1 = \frac{||T1||}{||1||} \leq \sup_{f \in C(X)}\frac{||Tf||}{||f||} = ||T|| \leq 1$$ These calculation are correct? Thanks in advance.