Let $H$ be a nonzero complex [Hilbert space](/page/Hilbert%20Space), let $I_H:H\to H$ denote the identity operator, and let $N \in \mathcal L(H)$ be a bounded normal linear operator, meaning $N^*N = NN^*$. Let $\sigma(N) \subset \mathbb C$ denote the spectrum of $N$ in the unital Banach algebra $\mathcal L(H)$. Then