Let $H$ be a Hilbert space. An operator $T$ in $L(H)$ is normal if $T T^{*}=T^{*} T$. Suppose that $T$ is normal and that $\sigma(T) \subseteq \mathbb{R}$. Let $U=(T+i I)(T-i I)^{-1}$.

(a) Suppose that $A$ is invertible and $A T=T A$. Show that $A^{-1} T=T A^{-1}$.

(b) Show that $U$ is normal, and that $\sigma(U) \subseteq\{\lambda:|\lambda|=1\}$.

(c) Show that $U^{-1}$ is normal.

(d) Show that $U$ is unitary.

(e) Show that $T$ is Hermitian.

[You may use the fact that, if $S$ is normal, the spectral radius of $S$ is equal to $\|S\| .$ ]