Page 267 - Advanced Linear Algebra
P. 267
Structure Theory for Normal Operators 251
and so . Conversely, if all eigenvalues of are nonnegative, then
~ b Ä b Á
and since ~ b b Ä ,
º #Á #»~ º #Á #»~ ) # )
Á
and so is positive. Part 2) is proved similarly.
If is a positive operator, with spectral resolution
~ b Ä b Á
then we may take the positive square root of ,
j j ~ b j Ä
b
where j is the nonnegative square root of . It is clear that
² ³ j ~
and it is not hard to see that j is the only positive operator whose square is .
In other words, every positive operator has a unique positive square root.
Conversely, if has a positive square root, that is, if ~ , for some positive
operator , then is positive. Hence, an operator is positive if and only if it
has a positive square root.
If is positive, then j is self-adjoint and so
² ³ j i ~ j
Conversely, if ~ i for some operator , then is positive, since it is clearly
self-adjoint and
º #Á #»~º i #Á #»~º #Á #»
Thus, is positive if and only if it has the form ~ i for some operator .
(A complex number is nonnegative if and only if has the form ' ~ $ $ for
'
some complex number .)
$
B
Theorem 10.23 Let ²= ³ .
)
1 is positive if and only if it has a positive square root.
2 ) is positive if and only if it has the form ~ i for some operator .
Here is an application of square roots.
Theorem 10.24 If and are positive operators and ~ , then is
positive.
