Page 120 - Advanced Linear Algebra

104 Advanced Linear Algebra

:
Proof. Clearly : q ; ~ ¸ ¹ , since any functional that annihilates both and
; : must annihilate l ; ~ = : . Hence, the sum b ; is direct. The rest
follows from Theorem 3.14, since

i

= ~ ¸ ¹ ~ ²: q ;³ ~ :b ; ~ :l ;
Alternatively, since ; ~ : b is the identity map, if  = i , then we can
write

b ³ ~ ² k ³ b ² k ³  : l ;
~ k ² ; : ; :
i

and so =~ : l ; .
Operator Adjoints
*

If ²= Á > ³ , then we may define a map d ¢ > ¦ = i by
B
d ² ³ ~ k ~
)
(
for  > * . We will write composition as juxtaposition. Thus, for any #  = ,
´ d ² ³ µ ² # ³ ~ ² # ³
The map d is called the operator adjoint of and can be described by the

phrase “apply first.”

(
Theorem 3.17 Properties of the Operator Adjoint)
)
1 For  B²= Á> ³ and Á  - ,
Á
d
² b ³ ~ d b d

)
B

2 For ²= Á > ³ and ²> Á <³ ,
B
d
² ³ d ~ d
)

3 For any invertible ²= ³ ,
B
² c d ³ ~ ² d ³ c
)
)
Proof. Proof of part 1 is left for the reader. For part 2 , we have for all  < i ,
d
d
² ³ ² ³ ~ ² ³ ~ d ² ³ ~ d ² d ² ³³ ~ ² d ³² ³

)
)
Part 3 follows from part 2 and
d ² c ³ d ~ c ² ³ d ~ d ~
and in the same way, ² c d d ~ . Hence ² c d ~ ² d c .
³
³
³
i
i
i
i
i
i
If ²= Á > ³ , then d B  ( > Á = ) and so d d B ²= Á > ³ . Of course,
B

dd is not equal to . However, in the finite-dimensional case, if we use the

natural maps to identify = ii with and > ii with > , then we can think of dd
=