Positive linear operator on continuous functions ($f\ge 0\Rightarrow Uf\ge 0$)

A linear map $U:C(K,\mathbb{R})\to C(K,\mathbb{R})$ is positive when $f\ge 0$ implies $Uf\ge 0$. This is positivity for the pointwise order, not positivity of a quadratic form. On a compact space, $|Uf|\le U|f|\le \Vert f{\Vert }_{\infty }U1$, so $\Vert U\Vert =\Vert U1{\Vert }_{\infty }$. This makes positive maps especially useful for uniform approximation.

If a linear operator on real continuous functions preserves every zero, evaluate $f-f(x)1$ at $x$. The zero-preserving hypothesis forces $Lf(x)=f(x)L1(x)$, so the operator is multiplication by the continuous function $L1$. Positivity further makes that multiplier nonnegative; linearity and zero preservation alone already give the multiplication representation.