Eigenvalues and eigenvectors of an integral operator We have the following integral operator
$$
Ku(t)=\int_0^1 G(t,s)\, u(s)\, ds,\,\, u\in L^2[0,1],
$$
where $$G(t,s)=\begin{cases} s(1-t)~ 0\leq s\leq t\leq 1\\ t(1-s)~ 0\leq t\leq s\leq 1\end{cases}$$if the eingenvalues of $K$ are $1/k^2\pi^2$ what can we say about the eigenvalues of $K^2=K\circ K$?
Thank you. 

If $Ku=\lambda u$, then $K^2 u=\lambda^2 u$. Hence the eigenvalues of $K^2$ are
$$
\frac{1}{\pi^4n^4}, \quad n\in\mathbb N,
$$
