Let $T$ be a linear transformation in a complex finite dimensional vector space equipped with a positive definite inner product. Suppose that $TT^* = 4T - 3I,$ where $I$ is the identity and $T^*$ is the adjoint of $T$.
Prove that $T$ is positive definite and find all possible eigenvalues of $T.$
I have a feeling that using Tr(T) might come in handy here... but I haven't been able to show either result. Any help is greatly appreciated.