Page 509 - Advanced Linear Algebra

The Umbral Calculus 493

(
)
Hence, 19.8 gives
B
L ; ² ! ³ c % M ~ L ; ² ! ³ c % M
~ ~

~ L ; ´ ² ! ³ µ c % M
~
B
~ L ; ´ ² ! ³ µ c % M
~
which implies the desired result.
Operator Adjoints
(
)
F
If F ¦ ¢ F is a linear operator on , then its operator adjoint d is an
operator on F i < ~ defined by
d ´ ²!³µ ~ ²!³ k
In the symbolism of the umbral calculus, this is

º d ²!³ “ ²%³» ~ º ²!³ “ ²%³»

(We have reduced the number of parentheses used to aid clarity.³

Let us recall the basic properties of the adjoint from Chapter 3.

Theorem 19.16 For  Á B F ³ ,
²
d

1) ²b ³ ~ d b d

)
d

2 ² ³ ~ d for any  d
d
3) ² ³ d ~ d
4 ² ) c d ³ ~ ² d ³ c for any invertible  ² F ³ B
d
¢
²
¢²
Thus, the map B F³ ¦ B <³ that sends F ¦ F to its adjoint ¢ < ¦ <
d
²
is a linear transformation from BF²³ to B<³ . Moreover, since ~ implies
that º ²!³“ ²%³»~ for all ²!³ < and ²%³ F , which in turn implies

that ~ , we deduce that is injective. The next theorem describes the range
of .

Theorem 19.17 A linear operator ; ² ³ is the adjoint of a linear operator
B<
;
B B ² ³ if and only if is continuous.
F
F
B
Proof. First, suppose that ; ~ d for some  ² ³ and let ² ²!³³ ¦ B . If

‚ , then for all   we have
º d ² ! ³ “ % » ~ º ² ! ³ “ % »

and so it is only necessary to take large enough that ² ²!³³ € deg ²% ³ for

all   , whence