I want to prove that the operator $$ A: L^2[0,1] \to L^2[0,1], \quad A(u)(s) = \int_0^1 |t-s| u(t) dt $$ is not positive, i.e. $\langle Au, u \rangle \geq 0$ does not hold for every $u \in L^2[0,1]$.
I could not find a counterexample but I think it is possible to prove that this operator has a negative eigenvalue which would imply my claim.
Any idea how to construct a counterexample for the claim itself or maybe an eigenfunction corresponding to a negative eigenvalue?