If $T$ injective or $T$ surjective, what is the composition $T^\ast T$? (where $T^\ast$ denotes adjoint of linear map $T$) Let $T:V \to W$ be a linear transformation between inner product spaces. Then $T^\ast: W \to V$ denotes the linear transformation with the property that for every $v \in V$ and $w \in W$, $$\langle T(v),w \rangle = \langle v, T^\ast(w) \rangle.$$ We call $T^\ast$ the adjoint of $T:V \to W$. 
$\cdot$ If $T$ is injective is $T^\ast T$ injective (or possibly a bijection)?
$\cdot$ If $T$ is surjective is $T T^\ast$ surjective (or a bijection)? 
How can we prove this? Any pointers in the right direction appreciated.

start with 
$$T^*\times T=T\times T^*=\det(T) I_n$$
