inner product space, non negative transformation

Question

I'm new to inner product space and I have a hard time with this question, can someone help me: Let $V$ be a unitary space with finite dimension and let $T:V\to V$ be a non negative linear transformation. Assume that there is $n \in N$ that $T^n=I$. is necessarily $T=I$ ?

Thanks!

Answer 1

$T$ is selfadjoint. Let $ \lambda$ be an eigenvalue of $T$. Then $\lambda^n$ is an eigenvalue of $T^n$. Hence $\lambda^n=1$. Since $T$ is non negative, $ \lambda=1$. Therefore $T-I$ is selfadoint and has only one eigenvalue $0$.

From $||T-I||=r(T-I)=0$, we get $T=I$

($r(T)$ = spectral radius of $T$).