MAXWELL - DIRAC EQUATIONS
21
As | V|
p _ 1 / 2
0 | V|
p _ 1 / 2
0
I:EP

E1/2
is an isomorphism and as the representation
(A
- 1
0 A
- 1
0 / )
U1
is a unitary representation on
E1/2,
it follows that V is unitary in
Ep.
This proves the theorem.
To be able to use the implicit functions theorem in the Frechet space E^, we shall
establish the existence of smoothing operators (cf. [25]). As this can be done in a general
context of unitary representations, in the following lemma and theorem, V will be an
arbitrary unitary representation of a Lie group Q on a Hilbert space H and S will be the
corresponding representation on H^ (the space of C°°-vectors) of the Lie algebra g of Q
by differentiation and II is a general basis of g. The only fact which is really used is that
the Laplace operator for the representation V, A = ]Cxen(^x)
2 m
H with domain H^
is essentially self-adjoint (see [34] Theorem 4.4.4.3). Let A denote the closure of A.
Lemma 2.3. The domain of (1 A)
n
/
2
is Hn (the space of
Cn-vectors
ofV), and the
norms || ||H and ||(1 A)
n
/
2
\\H are equivalent, i.e.
Cn'll/lk ||(1
- S )
B / 2
/ I I H
Cn\\f\\Hn, Cn0JeHn,n 0.
Proof. Let II = {Xi,..., Xr}, r dimg. The statement is true for n 0. Let n 1 and
let f eH^. Then
WfWk ii/iik-! + £ I I ^
•••**»/II2 (2-2)
l i i , . . . , i
n
r
= ii/iik_i+ E (-infsXtn...xtlxil...Xinf),
l i i , . . . , i
n
^
as Sx, X G 0, is skew-symmetric on H^. By successive commutations we obtain
(-1)" E Xin-.-XilXil---Xin (2.3)
lii,...,inr
= (-l)» E ( 4 )
2
- ( ^ )
2
+ 4 n - l + - + ^l
l i i , . . . , i
n
r
= (-*)"+ A2n_1 + --- + Al,
where A\ is a (noncommutative) polynomial of degree I in Xj; II, 1 j r. As Sx, is a
skew-symmetric operator on ifoo, we have
\(f,SAlf)\C\\f\\HJ\f\\Hi_q, 0ql, (2.4)
where C depends on A\.
Equality (2.3) and inequality (2.4) give
E
\\Sxtl...xtJ\\2 (f,(-±)nf)
+ Cn\\f\\HJf\\Hni
l i i , . . . , i
n
r
{f,(i-Arf)+cn\\f\\Hjf\\H
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