Let $T : H \rightarrow H$ be a compact linear operator on a Hilbert space which is NOT necessarily self-adjoint. How would you prove that there exists $v \in H$, $\lvert\lvert v \rvert\rvert = 1$, such that $\langle Tv, Tv \rangle = \lvert\lvert T \rvert \rvert^2$ ?
I don't know how I should take on this problem. Of course, I can use the compacity assumption right at the beginning. But then, I'm lost.