Exercise 6.4.5

Answers

Observe that ${(T - cI)}^{∗} = T^{∗} - \bar{c} I$ and check

(T - cI) {(T - cI)}^{∗} = (T - cI) (T^{∗} - \bar{c} I) = T T^{∗} - \bar{c} T - c T^{∗} + | c |^{2} I

and

{(T - cI)}^{∗} (T - cI) = (T^{∗} - \bar{c} I) (T - cI) = T^{∗} T - \bar{c} T - c T^{∗} + | c |^{2} I .

They are the same because $T T^{∗} = T^{∗} T$ .

jephianlin

2011-06-27 00:00