Let $\phi:[0,1]\to\mathbb R$ contionuous and $A:L_2([0,1])\to L_2([0,1])$ defined by
$(Af)(x)=\phi(x)\int_{0}^{1}\phi(t)f(t)dt$
I already showed that $A=A^*$ and that $A$ is positive, but I would like to know two things:
1) When is A an orthogonal projector?
2) Is there a $\lambda\ge0$, such that $A^2=\lambda A$