Show that $T f(x) = \frac{1}{x^2}\int\limits_0^x t f(t) dt$ is not compact.
Solution 1:
$x^{1/(n-1)}$ is an eigen function corresponding to the eigenvalue $\frac {n-1} {2n-1}$. If $T$ is compact then the only possible limit point for the set of eigen values is $0$. Since $\frac 1 2$ is a limit point in this case it follows that $T$ is not compact.