Let $T:\Bbb{V}\rightarrow\Bbb{V}$ be a linear transformation over an inner-product space. Suppose $TT^* = 7T-12Id_v$. Prove that T is diagonalizable and find all possible eigenvalues.
I managed to show that T is invertible but I don't think it helps. I would appreciate clues on how to solve this.
Hint: Use the equation that $T$ is Hermitian, so that diagonalizability follows from the spectral theorem.
To determine the eigenvalues, use the fact that we can rewrite $$ TT^* = 7T - 12 \operatorname{Id}_V \implies T^2 = 7T - 12 \operatorname{Id}_V. $$
If you want to avoid the spectral theorem: use the fact that $(T - 3 \operatorname{Id})(T - 4\operatorname{Id}) = 0$ to show that every vector $x \in \Bbb V$ can be written as the following sum of eigenvectors: $$ x = (4 \operatorname{Id} - T) x + (T - 3 \operatorname{Id}) x. $$