Exercise 20, § 2

Let E be a complex Hilbert space of countable type and let $u\in \mathscr{L}(E)$ be a hermitian endomorphism. Let $\varepsilon$ be a strictly positive real number. There exist an orthonormal basis B of E, a diagonal hermitian endomorphism $v\in \mathscr{D}_B(E)$ and a compact endomorphism $w\in \mathscr{L}^c(E)$ such that $u=v+w$ and $\|w\|< \varepsilon$. (Consider first the case where $u$ admits a cyclic vector, and then apply the preceding exercise.)