Exercise 3, § 5

Let E be a Hilbert space of countable type. The only closed two-sided ideals of $\mathscr{L}(E)$ are $\{0\},\mathscr{L}^c(E)$ and $\mathscr{L}(E)$. (Use Exercise IV, p. 316, Exercise 10; if I is such an ideal, suppose that there exists in I a nonzero self-adjoint element $u$ which is not compact, and show that for every open interval J in $\mathbf{R}$ such that $0\notin J$, the spectral projector of $u$ on J belongs to I.)