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!