Spectrum of Unitary Operators

Question

Let $T_1$ and $T_2$ be two unitary operators. Is it true that the spectrum of $T_1+T_2$ is contained in the closed disc of radius 2?

Answer 1

Yes, this follows from $\left\|T_{1}+T_{2}\right\| \le \left\|T_{1}\right\| +\left\|T_{2}\right\|$.

To wit, suppose $v$ is an eigenfunction of $T_{1}+T_{2}$ with eigenvalue $\lambda$. Then \begin{align} |\lambda|\left\|v\right\| = \left\|(T_{1}+T_{2})v\right\| \le \left\|(T_{1}+T_{2})\right\| \left\|v\right\| \le (\left\|T_{1}\right\| + \left\|T_{2}\right\| ) \left\|v\right\| = 2\left\|v \right\| \end{align} So $|\lambda| \le 2$.