Page 512 - Advanced Linear Algebra

496 Advanced Linear Algebra

d
²!³ ~ ² ²!³³

for all ²!³  < . In particular, d ²!³ ~ . !
is an automorphism
Proof. We have already shown that the adjoint of
(
)
satisfying 19.9 . For the converse, suppose that d is an automorphism of .
<
Since d is surjective, there is a unique series ²!³ for which d ²!³ ~ . !
Moreover, Theorem 19.18 implies that ²!³ is a delta series. Thus,

[ Á ~º! “ % » ~º d ²!³ “% »~º ²!³ “ % »
which shows that is the associated sequence for ² ! ³ and hence that is an
%
umbral operator.
Theorem 19.19 allows us to fill in one of the boxes on the right side of Figure
19.1. Let us see how we might use Theorem 19.19 to advantage in the study of
associated sequences.
We have seen that the isomorphism ª d maps the set of umbral operators

K
<
F
< ~
on onto the set aut²³ of automorphisms of < F i . But aut²³ is a group
under composition. So if
¢% ¦ ²%³ and ¢% ¦ ²%³
are umbral operators, then since
² k d d k d
³
~
<
is an automorphism of , it follows that the composition k is an umbral
operator. In fact, since
d
² k ³ ² ²!³³ ~ d k d ² ²!³³ ~ d ²!³ ~ !
we deduce that k ~ k . Also, since
k ~ ~ ! ~
k
we have c ~ .

Thus, the set of umbral operators is a group under composition with
K
k ~ k
and
c ~

Let us see how this plays out with respect to associated sequences. If the