16 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU

(2.4) ||u||2 = [ ||u||2

+

| | Pu||2

+

||Lou|i2]1/2

If F is a subspace of H, and r a , we denote by F(r) the set of

all u in F such that u(t) = 0 for t r . In particular , this defines

the spaces V(r) and P-ioc(r) •

In the following we shall denote by r(v) a function of v E (0,1]

which takes values in (a+1,00) , is decreasing , satisfies r(v) k v and

is such that for t ^ r(v) :

(2.5) |M(t)-l|B(H) * v , |tM»(t)|B(H) * v .

Later on we shall put stronger conditions on the function r(v) .

We define a symmetric sesquilinear form QT with domain V by the fol-

lowing formula : if u,v £ V then

d

Q'(u,v) = (u,Qfv) = i(Pu,Qv) - i(Qu,Pv) ,

2

hence , formally , Qf = i[P,Q]. If h € C (I) , then the sesquilinear form

hQ1 is naturally defined on V by the formula (u,hQfv) = (hu,Qfv) .

Clearly

(u,hQTv) = (hu,Q»v) = i(Phu,Qv)-i(Qhu,Pv) = i(hPu,Qv)-i(Qu,hPv)-(u,htQv) ,

which for real h and any v £ V implies that

(2.6) (v,hQ'+h'Qv) = -2Im(hPv,Qv) = -2Re(hv',Qv) .

A special role in the later developments will be played by the follo-

wing symmetric sesquilinear form Q , a € 3 R :

(2.7) Qa = - aQ - tQ'