### 8.8.1. Theorem 8.7.

Let *L*
be a constant coefficient operator that can be factorized as a product of
constant coefficient operators

L=
∏
j=1
k
A
j

Then the solution space of
L(
y
)=0
contains the solution space of each
A
j
(
y
)=0
. in orther words,

N(
A
j
)⫅N(
L
)
∀ j=1,⋯,k
(8.11)

#### Proof

Let
u∈N(
A
i
)
. Since the
A
j
factors commute, we can move
A
i
to the rightmost position so that
L(
u
)=
∏
j≠i
A
j
A
i
(
u
)=O
. Hence,
u∈N(
A
i
)
Þ
u∈N(
L
)
∀ j
. QED.

### Comment

If
L(
u
)=O
,
then *L* is said to **annihilate** *u*. Thus, Theorem 8.7 can be
rephrased as follows. If a factor *A*_{j} of *L* annihilates *u*, then *L* annihilates *u*.