Exercise 7.C.6

Answers

Proof. From 7.6 (e), we have

{(T^{k})}^{∗} = {(T^{∗})}^{k} = T^{k} .

Therefore $T^{k}$ is self-adjoint.

To prove $T^{k}$ is positive, there are two cases.

If $k$ is odd, we have $k = 2 n + 1$ for some nonnegative integer $n$ . Then, for all $v \in V$ ,

⟨ T^{k} v, v ⟩ = ⟨ T^{n} T T^{n} v, v ⟩ = ⟨ T T^{n} v, T^{n} v ⟩ \geq 0

where the the inequality follows because $T$ is positive.

If $k$ is even, we have $k = 2 n$ for some nonnegative integer $n$ . Then, for all $v \in V$ ,

⟨ T^{k} v, v ⟩ = ⟨ T^{n} T^{n} v, v ⟩ = ⟨ T^{n} v, T^{n} v ⟩ \geq 0 .

Therefore $T^{k}$ is positive. □

celiopassos

2017-10-06 00:00