As above, I'm trying to find all linear operators $L: C([0,1]) \to C([0,1])$ which satisfy the following 2 conditions:
- I) $Lf \, \geq \, 0$ for all non-negative $f\in C([0,1])$.
- II) $Lf = f$ for $f(x)= 1$, $f(x)=x$, and $f(x)=x^2$.
I'm honestly not sure where to start here - I'm struggling to use these conditions to pare down the class of linear operators which could satisfy the conditions significantly. Could anyone help me get a result out of this? Thank you!