Let $L: C([0,1]) \rightarrow C([0,1])$ be an operator satisfying the conditions

(i) $L f \geqslant 0$ for any $f \in C([0,1])$ with $f \geqslant 0$,

(ii) $L(a f+b g)=a L f+b L g$ for any $f, g \in C([0,1])$ and $a, b \in \mathbf{R}$ and

(iii) $Z_{f} \subseteq Z_{L f}$ for any $f \in C([0,1])$, where $Z_{f}$ denotes the set of zeros of $f$.

Prove that there exists a function $h \in C([0,1])$ with $h \geqslant 0$ such that $L f=h f$ for every $f \in C([0,1])$.