Proving that T=T* for a Normal Linear Transformation

[LeifEricson]

Homework Statement

let T:V \to V be a linear transformation which satisfies T^2 = \frac{1}{2} (T + T^*) and is normal. Prove that T=T^*.

Homework Equations

The Attempt at a Solution

I think we should start like this:
Let \mathbf{A}=[T]_B be the matrix representation of T in the orthonormal base B. We look at A as a matrix in the Complex plane. If we prove that for every Eigenvalue, \lambda, it happens that \lambda \in \mathbb{R}, then T = T^* by a theorem and we finished.
Now, be \mathbf{x} an Eigenvector that satisfies \mathbf{Ax} = \lambda \mathbf{x}, then <\mathbf{Ax},\mathbf{x}> = \lambda ||\mathbf{x}||^2 and \lambda = \frac{<\mathbf{Ax},\mathbf{x}>}{||\mathbf{x}||^2}. If only I could show that <\mathbf{Ax},\mathbf{x}> is a real number, I solved it.

 

[Feldoh]

Prove that the transformation is hermitian, once you do it's pretty east to prove that eigenvalues of a hermitian transformation are real.

 

[LeifEricson]

It doesn't help me, because I plan to show that the transformation is Hermitian by the following theorem:
"If T is a normal transformation whose Characteristic polynomial can be completely factored into linear factors over \mathbb{R}, then T is Hermitian".
And then it follows, of course, that T = T*

 

[LeifEricson]

Can someone help me please? I work on it two days. Nothing works. Please. I spent hours.