There is a theorem in functional analysis, that says that for a selfadjoint compact operator $T:H\rightarrow H$, either $\lVert T\rVert $ or $-\lVert T\rVert$ is an eigenvalue. For finite dimensional $H$ it is easy to construct examples of operators such that $\lVert T \rVert$ and $-\lVert T\rVert$ are eigenvalues.
But I couldn't find yet an example for infinite dimensional spaces of such operators. Could someone provide me with one or give me a reference ?