Let $H$ be a complex [Hilbert space](/page/Hilbert%20Space) with [inner product](/page/Inner%20Product) $(\cdot,\cdot)_H$ and norm $\|\cdot\|_H$, let $\mathcal L(H)$ denote the Banach algebra of bounded linear maps $H\to H$ equipped with the operator norm $\|\cdot\|_{\mathcal L(H)}$, and let $I \in \mathcal L(H)$ be the identity operator. For $T\in\mathcal L(H)$, write $T^*$ for the Hilbert-space adjoint of $T$. Let $U \in \mathcal L(H)$ be unitary, meaning $U^*U=UU^*=I$. Then

\begin{align*} \sigma(U) \subset \{z \in \mathbb C : |z| = 1\}. \end{align*}

Equivalently, if $\lambda \in \mathbb C$ and $|\lambda| \ne 1$, then $U-\lambda I$ is invertible in $\mathcal L(H)$.