I found the following statement without proof:
Let us given a self-adjoint Operator $T\colon L^2 \to L^2$ which has n eigenvalues $ \lambda_n \leq \dots \leq\lambda_{n-1} < \lambda_1 =1$ counted according to theire multiplicity. In additon, the spectrum $\sigma(T)$ of T satisfies $ \sigma(T) \subset [a,b] \cup \{ \lambda_n,\dots,\lambda_1 \} $ where $a,b\in(-1,1)$ and $b < \lambda_n$.
Then, $\lambda_n + \dots + \lambda_1 = max \{ \sum_{i=1}^n <T\phi_i,\phi_j> \mid (\phi_1,\dots,\phi_n)$ is a orthonormal system $\}$.
I would like to know the proof. Does anyone know where I can find it? I have been searching for Rayleigh principle but wasnt able to find any proof.