Page 495 - Advanced Linear Algebra

The Umbral Calculus 479

Here are the operator versions of the functionals in Example 19.2.

Example 19.3
)
1 The operator ! satisfies
B
c
% ~ ! ! % ~ 45 % ~ ²% b ³
!
~ ~
and so
!
²%³ ~ ²% b ³
for all  F . Thus ! is a translation operator .
!
)
2 The forward difference operator is the delta operator c , where
)
!
² c ²%³ ~ ²% b ³ c ² ³
)
!
3 The Abel operator is the delta operator e , where
!
! ! e ² % ³ ~ Z ² % b ³
)
4 The invertible operator ² c !³ c satisfies
B
c
² c !³ ²%³ ~ ²% b "³ c" "

)
!
5 The operator ² c )°! is easily seen to satisfy
!
c %b
²%³ ~ ²"³ "
! %
F
We have seen that all linear functionals on have the form ²!³ , for  < .
However, not all linear operators on have this form. To see this, observe that
F
deg ´ ²!³ ²%³µ  deg ²%³
but the linear operator F ¦ F defined by ² ²%³³ ~ % ²%³ does not have
¢
this property.

Let us characterize the linear operators of the form ²!³ . First, we need a lemma.
;
Lemma 19.5 If is a linear operator on and 7 ; ² ! ³ ~ ² ! ³ ; for some delta
series ²!³ , then deg ²; ²%³³  deg ² ²%³³ .
Proof. For any ‚ ,

deg²;% ³ c ~ deg² ²!³;% ³ ~ deg²; ²!³% ³
and so