Page 511 - Advanced Linear Algebra

The Umbral Calculus 495

Figure 19.1
is the umbral operator for
Let us begin with umbral operators. Suppose that
the associated sequence ²%³ , with delta series ²!³  < . Then

º d ²!³ “ % » ~ º ²!³ “ ~ º! “ % »
% » ~ º ²!³ “ ²%³» ~ [ Á
for all and . Hence, d ² ! ³ ~ ! and the continuity of d implies that

d
!~ ²!³

More generally, for any ²!³  < ,
d
²!³ ~ ² ²!³³ (19.9 )

In words, d is composition by ²!³ .
)
(
From 19.9 , we deduce that d is a vector space isomorphism and that

d ´ ²!³ ²!³µ ~ ² ²!³³ ² ²!³³ ~ d ²!³ d ²!³
Hence, d is an automorphism of the umbral algebra . It is a pleasant fact that
<

this characterizes umbral operators. The first step in the proof of this is the
following, whose proof is left as an exercise.
Theorem 19.18 If ; is an automorphism of the umbral algebra, then ;
preserves order, that is, ²; ²!³³ ~ ² ²!³³ . In particular, is continuous.
;
Theorem 19.19 A linear operator on is an umbral operator if and only if
F

<
its adjoint is an automorphism of the umbral algebra . Moreover, if is an
umbral operator, then