More specifically, there is an orthonormal basis of $X$ consisting of common eigenvectors.
So far, I've approached proving this by using the spectral theorem for compact, self-adjoint operators. I know that $S$ and $T$ can separately be diagonalized. I've also seen a hint that suggests considering the compact operator $S+ iT$, but this operator isn't self-adjoint, so the spectral theorem won't apply.
Any suggestions on how to proceed with this proof would be appreciated.