Exercise 6.3.11

Answers

We have

0 = ⟨ T^{∗} T (x), x ⟩ = ⟨ T (x), T (x) ⟩ = ∥ T (x) ∥^{2}

for all $x$ and hence $T (x) = 0$ for all $x$ . And the second statement is also true since we may write

T T^{∗} = T^{∗∗} T^{∗} = T_{0}

and get $T^{∗} = T_{0}$ . Since $T = T^{∗∗} = T_{0}^{∗} = T_{0}$ .

jephianlin

2011-06-27 00:00