Page 516 - Advanced Linear Algebra

500 Advanced Linear Algebra

Theorem 19.21 Let be a surjective derivation on the umbral algebra . Then
C
<
c
f
C ~ for any constant  < and ²C ²!³³ ~ ² ²!³³ c , if ² ²!³³ ‚ . In
particular, is continuous.
C
Proof. We begin by noting that

C ~ C ~ C b C ~ C
C
and so C ~ C ~ for all constants  < . Since is surjective, there must
exist an ²!³  < for which
C ²!³ ~
Writing ²!³ ~ b ! ²!³ , we have

~ C´ b ! ²!³µ ~ ²C!³ ²!³ b !C ²!³

which implies that ²C!³ ~ . Finally, if ² ²!³³ ~ ‚ , then ²!³ ~ ! ²!³ ,

where ² ²!³³ ~ and so

´C ²!³µ ~ ´C! ²!³µ ~ ´! C ²!³ b ! c ²!³C!µ ~ c

Theorem 19.22 A linear operator on is an umbral shift if and only if its

F
adjoint is a surjective derivation of the umbral algebra . Moreover, if is an
<

d
umbral shift, then ~C is derivation with respect to ²!³ , that is,

d
²!³ ~ ²!³ c

for all ‚ . In particular, d ²!³ ~ .

Proof. We have already seen that d is derivation with respect to ²!³ . For the

converse, suppose that d is a surjective derivation. Theorem 19.21 implies that
there is a delta functional ²!³ such that d ²!³ ~ . If ²%³ is the associated

sequence for ²!³ , then

º ²!³ “ ²%³» ~ º d ²!³ “ ²%³»

~ º ²!³ c d ²!³ “ ²%³»

~º ²!³ c “ ²%³»

~² b 1) [ b Á

~ º ²!³ “ b ²%³»
Hence, b ²%³ , that is, ~ is the umbral shift for ²%³ .
²%³ ~

We have seen that the fact that the set of all automorphisms on is a group
<
under composition shows that the set of all associated sequences is a group
under umbral composition. The set of all surjective derivations on does not
<
form a group. However, we do have the chain rule for derivations[