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.