22
NONLINEAR REPRESENTATION AND SPACES
It follows from the last inequality and (2.2) that
\\f\\ln (/, (1 - A)"/) + Cn\\f\\Hn \\f\\Hni + \\ffHni (2.5)
(/,(l- A)"/) + 2^11/11^11/1^.
Assuming that we have proved that
||/||Hi C,||(l - A ) ; / 2 / I I H , / e ffoo.O J n - 1,
which is true for I = 0, we get by induction using (2.5) that
\\ffHn | | ( 1 - Ar/2/H2H + 2C„||/||
H
J|(l - A)^f\\H
for some Cn. This inequality implies
||/||Hn (||(1 - A)" / 2 /l& + C2||(l - A ) ^ f \ \ % ) ^ 2 + C„||(l - A ) V / l ^ ,
which by induction proves that (redefining Cn)
l l / l l H a | | ( l - A r / 2 / | |
H )
feH^n^O. (2.6)
To prove the second inequality in the theorem for / G #oo we observe that
(1—A)n
= SA' ,
where A'2n is a noncommutative polynomial of degree 2n in Xj G II, 1 j r. It follows
now as in (2.4) that
||(i -
A)"/2/!!2,
= (/,(i - A)-/) = (f,s^J) cl\\f\\%n,
for some Cn.
This inequality and (2.6) give
C ^ l l / l k 11(1 - A r / 2 / | |
H
Cn\\f\\Hn, f e ffoo.n 0. (2.7)
As (1 A)
n
/
2
is a maximal closed operator, it follows from (2.7) that the domain of
(1 A) n / 2 is Hn. By continuity (2.7) is then true for / G Hn, which proves the lemma.
We can now easily prove the existence of smoothing operators for the sequence Hi,
i 0, by using the spectral decomposition of (1 A) 1 / 2 .
Theorem 2.4. We denote by L^H^H^) the space of linear continuous mappings from
H to HQQ endowed with the topology of uniform convergence on bounded sets. There exists
a C°° one-parameter family TT G
L^H^HQO), T
0, such that if f G Hi, I 0, then
i)l|rr/||H„C„,i||/||
i
v nl,
ii)||r
r
/||
H B
C
n
,
J
T n -'||/||
H i
, nl,
iii)||(l-r
T
)/||
i f B
C
n
,,r"-'||/||
H |
, nl,
iv) ||-£rT/||Hn cn,i r"-'-1!!/!!^, n 0,1 o.
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