Exercise 9.A.12

Proof. By Exercise 3 in section 8D, the minimal polynomial of $T^2+bT+cI$ is of the form $z^m$ for some positive integer $m$. Let the $p$ be the polynomial defined by

Then $p$ has no real roots (because $z^2+bz+c$ does not) and $p(T_{}$

C) = 0$.Thus8.46and8.49implythat$T_C$hasnorealeigenvalues.Now9.10impliesthat$T$hasnoeigenvalues.\square$