Exercise 18, § 1

Let $\mathfrak{g}$ be a complex Hilbert space, $Y$ and $Z$ continuous Hermitian operators on $\mathfrak{g}$ and $B = [Z, Y]$. Suppose that $Y$ and $B$ are permutable.

(a) Show that $[Z, Y^n] = n B Y^{n-1}$ (argue by induction on $n$).

(b) Let $f$ be a continuously differentiable function of a real variable. Deduce from (a) that $[Z, f(Y)] = B f'(Y)$.

(c) Deduce from (b) that $B = 0$. (Show that, if $B \neq 0$, $f$ can be chosen such that $\|f(Y)\| \leq 1$ and $\|B f'(Y)\|$ is arbitrarily large.)

(d) Let $g$ be the Lie algebra of continuous operators $T$ on $\mathfrak{g}$ such that $T^* = -T$. Deduce from (c) that the conditions $Y \in g, Z \in g, [[Z, Y], Y] = 0$ imply $[Z, Y] = 0$.