Exercise 7.C.4

Answers

Proof. We have ${(T^{∗} T)}^{∗} = T^{∗} {(T^{∗})}^{∗} = T^{∗} T$ , which implies that $T^{∗} T$ is self-adjoint and, for all $v \in V$ ,

⟨ T^{∗} Tv, v ⟩ = ⟨ Tv, Tv ⟩ \geq 0 .

Therefore, $T^{∗} T$ is positive.

It’s basically the same thing for $T T^{∗}$ . □

celiopassos

2017-10-06 00:00