Trying to figure out the ingredients for the spectral representation of the following self-adjoint integral operator:
$(Tf)(s)=\int_0^1 (s-t)^2 f(t) dt$
A hint in the exercise was to consider $im T$ and make an ansatz for the possible eigenfunctions.
I tried using a solution like the one in: How to Find the Spectrum of an Integral Operator but it didn't quite work so now I'm stuck.
Hints: Since $Tf(s)=s^{2}\int_0^{1} f(t)dt-2s\int_0^{1} f(t)dt+\int_0^{1} t^{2}f(t)dt$ the range of $T$ is contained in $span \{1,s,s^{2}\}$. Hence $T$ is a compact operator. So all the non-zero points in its spectrum are eigen values.
Consider $Tf(s)=\lambda f(s)$ with $\lambda \neq 0$. Since LHS is smooth it follows that $f$ is also smooth. Differentiate twice to find eigen values and eigen functions. I hope these hints help.