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 uN( A i ) .  Since the A j  factors commute, we can move A i  to the rightmost position so that L( u )= ji A j A i ( u )=O .  Hence, uN( A i )  Þ uN( 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 Aj of L annihilates u, then L annihilates u.