Page 265 - Advanced Linear Algebra

Structure Theory for Normal Operators 249

Commutativity
The functional calculus can be applied to the study of the commutativity
properties of operators. Here are two simple examples.

Theorem 10.20 Let be a finite-dimensional complex inner product space.
=

For  Á B = ² ³ , we write © to denote the fact that and commute. Let
and have spectral resolutions
~ b Ä
b
~ b Ä
b
Then
)

B
1 For any ²= ³ ,
© ¯ © for all
2)
© ¯ © , for all Á

) ¹ ¦ - ÁÃÁ ¹ ¦ - are injective functions,
3 If ¢ ¸ Á ÃÁ and ¢¸
then

² ³ © ² ³ ¯ ©

Proof. For 1), if © for all , then © and the converse follows from the

fact that is a polynomial in . Part 2) is similar. For part 3), © clearly

implies ² ³ © ² ³ . For the converse, let $ ~ ¸ Á Ã Á ¹ . Since is
injective, the inverse function c ¢ ² ³ $ ¦ $ is well-defined and

c ² ² ³ ³ ~ . Thus, is a function of ² ³ . Similarly, is a function of ² ³ .

It follows that ² ³ © ² ³ implies © .

Theorem 10.21 Let and be normal operators on a finite-dimensional

complex inner product space . Then and commute if and only if they have

=
the form
~ ² ² Á ³³

~ ² ² Á ³³
where ²%³Á ²%³ and ²%Á&³ are polynomials.

Proof. If and are polynomials in ~ ² Á ³ , then they clearly commute.
For the converse, suppose that ~ and let
~ b Ä
b
and
b
~ b Ä
be the orthogonal spectral resolutions of and .