Proving an operator is positive definite

Question

Let $V$ be a complex vector space and $T$ a linear operator on it. Suppose $T T^\ast=6T-8I$. I need to prove $T$ is positive definite.

Doesn't this just follow from the fact $TT^\ast, I$ are positive definite and closure of positive-definite under sums with positive coefficients?

($TT^\ast$ cannot have eigenvalue zero because $\ker T=\ker TT^\ast$ would then yield a contradiction.)

Answer 1

Let $v\ne0$ be a vector; then $$ 6v^*Tv=v^*TT^*v+8v^*v>0 $$