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I'm trying to solve the following exercise:

Let $f\in\mathcal{C}([0,1])$ and let $T$ an operator such that $Tf(x)=\int_0^1(x-t)f(t)dt$. I have proved that $T$ is a bounded linear operator and, by means of Ascoli-Arzelà theorem, that it is a compact operator.

Now I need to find its kernel, its rank (showing a basis) and its spectrum. I'm quite stucked, without an idea which could make me start.

Thank you for any suggestion!

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    Hint: $Tf(x)=\left (\int_0^1 f(t) dt \right ) x + \left ( \int_0^1 -tf(t) dt \right )$.2011-09-12
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    Uh! :) Now it's much clearer as far as the rank and the spectrum are concerned. I don't have a complete result about the kernel, though. I found some functions in the kernel (1-periodic trigonometric functions), but I can't determine them all. A hint which does not employ Fourier series? Thank you!!2011-09-12
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    The kernel of T is the intersection of the kernel of those two linear functionals, $\int_0^1f$ and $\int_0^1 t f$. I doubt there's a much better description than that. You certainly aren't going to get an explicit general description of the functions in a codimension 2 subspace of $C[0,1]$.2011-09-12

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