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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!

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    Are you viewing $C([0,1])$ as a real vector space with pointwise operations? (I assume $C([0,1])$ is the set of continuous functions $[0,1]\to\mathbb{R}$)2011-01-08
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    Do you require your operators to be continuous? If so, with respect to what topology?2011-01-08
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    @Chris and @Arturo: Positivity and $L1 = 1$ imply $\|L\| = 1$ in the $\sup$-norm because $L(\|f\|\cdot 1 \pm f) \geq 0$, hence $\|Lf\| \leq \|f\|$.2011-01-08
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    Apologies, I tried to correct it but it appears someone beat me to it! By an operator i just mean a mapping between the 2 spaces - no continuity necessary, just linearity in this case. And yes, C([0,1]) is the real vector space of continuous functions onto R as you stated.2011-01-08
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    Why do you want to do this?2011-01-08
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    It was given as an additional (optional) exercise in a mathematics course I'm studying, so it certainly isn't the end of the world if I don't do it but I feel like I should be able to make some progress on it, as it appears to me like it shouldn't be as challenging as I seem to be finding it.2011-01-08
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    Could you provide some more background/context? I don't know if this helps: The dual space of $C([0,1])$ is the space of signed Borel measures and the adjoint $L^{\ast}$ must preserve probability measures on $[0,1]$ (this follows from $L1 = 1$ and $L \geq 0$). In particular, there is a (weak$^{\ast}$-)continuous function $[0,1] \to M_{1}([0,1]), s \mapsto \mu_{s} = L^\ast(\delta_s)$, so that $(Lf)(s) = \int_{[0,1]} f\,d\mu_{s}$. The further requirements $Lx = x$ and $L(x^2) = x^2$ should give some restrictions on the $\mu_s$.2011-01-08
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    What sort of context? I'm studying a mathematics course at university (I'd prefer not to name), and this is a broad covering of a number of topics in analysis - this in particular was loosely related to a proof of the Weierstrass Approximation theorem, but i don't believe any of the proof is useful for this additional exercise which was given to the student. The course isn't at all measure theoretic, this was the start of a brief section on polynomial approximations and operators (specifically focusing on the Bernstein polynomials), so i expect there is a non-measure theory solution too.2011-01-08
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    @Stephen: Thanks! Bernstein & Weierstrass was a very good hint for me. I've finally remembered the argument. You don't need any measure-theory stuff, I hope my hints below help.2011-01-08
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    @AD: Apologies, I'm new to this - I'll know for next time. Theo, ill start working on your hints, thank you!2011-01-08
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    @Stephen: Have fun! Just post a comment to my answer in case you're stuck (then I will be notified). In case of desperation I'll give you full details, but not just now, ok?2011-01-08

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