Exercise 5.B.7

Proof. Suppose $9$ is an eigenvalue of $T^2$. Let $v$ be a corresponding eigenvector. Then $(T^2-9I)v=0$, which implies $(T-3I)(T+3I)v=0$. If $(T+3I)v=0$ then $-3$ is an eigenvalue of $T$. Otherwise, $(T+3I)v$ is an eigenvector of $T$ and $3$ is a corresponding eigenvalue.

Conversely, suppose $\pm 3$ is an eigenvalue of $T$. Let $v$ be a corresponding eigenvector. Then $T^{2}v=T(\pm 3v)={(\pm 3)}^{2}v=9v$, showing that $9$ is an eigenvalue of $T^2$. □