Exercise 6.7.25

Answers

1.

T^{∗} T

is invertible. Let

U = {(T^{∗} T)}^{- 1} T^{∗}

. Check that

TUT = T {(T^{∗} T)}^{- 1} T^{∗} T = T

and

UTU = {(T^{∗} T)}^{- 1} T^{∗} T {(T^{∗} T)}^{- 1} T^{∗} = U .

Also, both $TU = T {(T^{∗} T)}^{- 1} T^{∗}$ and $UT = {(T^{∗} T)}^{- 1} T^{∗} T = I$ are self-adjoint. Apply Exercise 6.7.21 and get the result.

2.

By Exercise 6.3.13

T T^{∗}

is invertible. Let

U = T^{∗} {(T T^{∗})}^{- 1}

. Check that

TUT = T T^{∗} {(T T^{∗})}^{- 1} T = T

and

UTU = T^{∗} {(T T^{∗})}^{- 1} T T^{∗} {(T T^{∗})}^{- 1} = U .

Also, both $TU = T T^{∗} {(T T^{∗})}^{- 1} = I$ and $UT = T^{∗} {(T T^{∗})}^{- 1} T$ are self-adjoint. Apply Exercise 6.7.21 and get the result.

jephianlin

2011-06-27 00:00