Exercise 9.B.8

Proof. Define

for each $j=1,\dots ,n$. By Exercise 4 in section 6B, the list $\frac{1}{\sqrt{2\pi }},f_1,e_1,\dots ,f_n,e_n$ is an orthonormal basis of $V$. Note that $D\frac{1}{\sqrt{2\pi }}=0$, $D{f}_j=j{e}_j$ and $D{e}_j=-j{f}_j$. Hence, the matrix of $D$ with respect to this basis has the desired form, where the first block is the $1$-by-$1$ matrix $\left(\begin{array}{c}\hfill 0\hfill \end{array}\right)$ and others are of the form

for some $j\in \{1,\dots ,n\}$. □