spectum of self adjoint operators

Question

Let $H$ be an Hilbert space and $S = \displaystyle{ \sum_{i=1}^nS_i}$ where $(S_i)_{i = 1}^{n}$ are self adjoint with compact resolvent. Is it true that the spectrum of $S$ is the sum of the spectra of $S_i$?

Answer 1

As Igor mentioned, this fails in finite dimension. Let $$ S_1=\begin{bmatrix}1&0\\0&0\end{bmatrix},\ \ S_2=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}. $$ Then both $S_1,S_2$ have spectrum $\{0,1\}$, but $S_1+S_2$ has spectrum $\{1+1/\sqrt2,1-1/\sqrt2\}$.