The Spectrum of Unitary Operators on a Complex Inner Product Space lies on the Unit Circle

Theorem

• Let $V$ be a Complex Inner Product Space
• Let $U\in \mathcal{L}(V)$ be Unitary
• Let $|\cdot |$ be the Complex Number Modulus
• Then, $|\lambda |=1,\forall \lambda \in \sigma (u)$ In other words, every characteristic value of $U$ is a complex number of modulus one.

Proof

1. Let $Ux=\lambda x$ as $U$ is Norm Preserving
2. $||x||=||ux||=||\lambda x||$