MAXWELL - DIRAC EQUATIONS 23
Proof. Let (1 —A)1/2 denote the positive self-adjoint square root of the positive self-adjoint
operator (1 —A) 1. Since, according to Lemma 2.3, the norms || ||H and ||(1 A)
n
/
2
- H^
are equivalent, it is enough to prove i)-iv) with || \\H replaced by ||(1 A)
n
/
2
\\H.
Let
/
oo
Xdex, (2.8)
where A i— e\ is the spectral resolution of (1 A)
1
/
2
and let p G
CQ°(M),
where tp(s) = 1
for \s\ 1, p(s) = 0 for \s\ 2 and 0 (p(s) 1 for s G R. We define Tr, r 0, by
/
oo
p(X/T)d(exf), feH,r0.
(2.9)
Since the map r i— y?r, yr(A) = ^(A/r), is C°° from ]0,oo[ to L°°(IR), it follows using
(2.9) that the map r -• TT is C°° from ]0, oo[ to Lh(H, Hoo).
We have for / belonging to the domain of (1
A)z/2,
/
oo
X2n(p(\M)2d\\exf\\2H
(2.10)
/
OO
A2("-i)(^(A/r))2A2'd||eA/||2f
sup (A2("-'(p(A/r))2)||(1 - A ) ( / 2 / | |
H
, n,l0.
\1
If n /, then
sup(A 2 ("- i )(^(A/r)) 2 )l, r 0 ,
A1
which together with (2.10) proves statement i). If n Z, then
sup(A
2
(
ri
-
z
)(^(A/r))
2
)=r
2
(
n
-^ sup
s2^n-lH^(s))2
\1 s^r-1
2
2 ( n - Z )
r
2 ( n - Z )
) T
Q?
which together with (2.10) proves statement ii). We have for / belonging to the domain
of (1 - A)'/
2
and n /,
/
oo
A2n(l-V(A/T))2d||eA/|&
/
OO
(A/T)2»-'(l-V(A/T))2Aad||eA/|&
r2"-') suP52("-')(l - p(s))2\\(l-A)l'2f\\%.
seR
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