12 AMREIN BOUTET DE MONVEL-BERTHIER GEORGESCU

those functions having compact support in I . We say that a linear sub-

space F c H is a k-semilocal subspace if it is stable under multiplication

by functions from Ck(I) , i.e. from n € C (I) and f € F it follows that

nf € F. The most important examples of k-semilocal spaces we have in mind

are the Sobolev spaces of sections of direct integrals of Hilbert spaces

continuously embedded in H (see Section 5 ) .

Assume F is a k-semilocal subspace of H . We define F = {f € F | the

support of f is a compact subset of I }, F = {f G L* (I;H) | if

n € C (I) , then nf G F} . Clearly F is a k-semilocal subspace of H and

2

F _ is a linear subspace of Hn = Ln (I;H) which is stable under mul-

loc

r

loc loc

tiplication by functions from C (I) ( we say that F is a k-semilocal

subspace of H. ) . But now we have more . namelJ y F and F. , are stable

lo c * c lo c

la -

under multiplication by any function from C (I) .

We say that a mapping T : F - H is local if , whenever J c I is open,

f1 , f2 G F and f1|J = f2|j , we have (Tf1)|j = (Tf2)|J . If T is (p-linear

(i.e. linear for the structures of complex vector spaces ) , then this is

equivalent to supp Tf c supp f for all f € F . We say that T is k-ultralo-

cal if it is dJ-linear and Tnf = nTf for all n € Ck(I), f G F. Remark

that if T is k-ultralocal . then it is local . TF c F and Tnf = nTf for

c c

all n G Ck(I) and f G F .

Assume that T : F • * H is local . Then there is a unique mapping

T : F- . -» - H. . which is local and coincides with T on F ( if f C f, and

lo c lo c lo c

J c I is open with compact closure in I , choose n € C (I) such that

n(t) = 1 in a neighbourhood of J and define (Tf)|j = (Tnf)|J.). Moreover ,

°° ^ lr

if T is k-ultralocal , then one has Tnf = nTf for each n G C (I) and all

f G F, ^ . From now on we shall denote the extension T by the same let-

loc °

ter T .

If G c H is a linear subspace . we define its k-semilocal closure

c 9 •

to be the smallest k-semilocal subspace of H that contains G . Clearly it

is equal to the linear subspace generated by elements of the form nf with