Suppose V is a unitary space [over C] and T: V -> V is a normal transformation that satisfies T-1=-T. Prove that T is unitary transformation.
I know that T is unitary if and only if it is normal and the absolute value of its eigenvalues is 1. [*2]
The Attempt at a Solution
T-1=-T so T2=-I, now suppose a is an eigenvalue of T so T2v=a2v=-Iv what in turn means [itex]a=\pm 1[/itex].
So from [*2] we can conclude that T is unitary.
Have I missed something?