A construction of generalized eigenprojections based on geometric measure theory

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Eijndhoven, van, S. J. L. (1985). A construction of generalized eigenprojections based on geometric measure theory. (Memorandum COSOR; Vol. 8509). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1985

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Memorandum 85-09

A CONSTRUCTION OF GENERALIZED EIGENPROJECTIONS

BASED ON GEOMETRIC MEASURE THEORY

by

S.J.L. van Eijndhoven

Eindhoven University of Technology P.O.Box 513, 5600 MB Eindhoven The Netherlands

by

S.J.L. van Eijndhoven

Abstract

Let

M

denote a a-compact locally compact metric space which satisfies certain geometrical conditions. Then for each a-additive projection valued measure

P

on

M

there can be constructed a "canonical" Radon-Nikodym derivative TI: a ~

n ,

a E

M,

with respect to a suitable basic

a

measure p on M. The family (TIa)aEM consists of generalized eigen-projections related to the commutative von Neumann algebra generated by the projections p(~). ~ a Borel set of

M.

L

In this paper

M

denotes a o-compact locally compact (and hence separable) metric space. It follows that any positive Borel measure on

M

is regular (cf. [3J, p. 162). In the monograph [2J, certain geo-metrical conditions on

M

are introduced, which lead to the following result.

O. Theorem

Let ~ denote a positive Borel measure on

M

with the property that bounded Borel sets of

M

have finite ~-measure. and let f denote a Borel function which is ~-integrable on bounded Borel sets. Then there exists a ~-null

set

N

_f such that for all a

E M \ N

_f both ~(B(a,r» > 0, and the limit

"" _f(a)

₌

_{lim ~(B(a,r»} -1

r

J

f d].J

rfO

B(a,r)

""

exists. We have f

=

f ~- almost everywhere.

(B(a,r) denotes the closed ball with radius r and centre a.)

Remark: In the previous theorem, the Borel function f can be replaced by a Borel measure v with the property that bounded Borel sets of

M

have finite v-measure. Then a "canonical" Radon-Nikodym derivative

:~

is ob-tained, which satisfies

dv (a) d].J

=

lim v(B(a,r» r+O ].J(B(a,r» ].J-almost everywhere.

In the sequel we assume that M also satisfies Federer's geometrical conditions. As examples of such spaces M we mention

finite dimensional vector spaces with metric d(x,y)

=

v(x - y) where v is any norm,

- Riemannian manifolds (of class ~ 2) with their usual metric.

Let X denote a separable Hilbert space with inner product (',.) and let there be given a a-additive projection valued set function P on

M.

So for all Borel sets ~ c

M.

p(~) is an orthogonal projection on X.

00

Moreover, if ~ is the disjoint union

u

~., then P(~)

=

I

p(~.). In

J j=l J

00 00 j=l

particular

I

p(~.)

=

0 if U ~. =

M.

j=l J j=l J

Now let R denote a positive bounded linear operator on X with the

property that for each bounded Borel set ~ the positive operator RP(~)R

is trace class. E.g. for R any positive Hilbert-Schmidt operator can be taken.

For each bounded Borel set ~ we define p(~)

=

trace(RP(~)R). In a natural way, p becomes a a-finite positive Borel measure on

M.

Each bounded

Borel set of

M

has a finite p-measure.

We take a fixed orthonormal basis (vk)kEIN in X, and for each k,~ E

m

we define the set function

~ Borel.

The set functions ~k~ are absolutely continuous with respect to p.

A

such that for all k,t E IN and all a E M \ Nl A . { ¢k~(B(a,r»} ~ (0) = 11m . kt r~O p(B(a,r» 1. Lemma Let a E M \ N

1, Then for all k,£ E IN

Proof, Consider the estimation,

00 lim r~O 2 cI>k~(B(a,r}) p(B(a,r)} { cI>kk(B(a,r» ;;;;; lim r~O p(B(a,r» lim

~£2(B(a,r»l

=

r~O

l-p(B(a,r»

f

\ A A

The function L ~kk is Borel, and the functions ~kk are positive. So k=l

for each bounded Borel set ~, we have

( 00 A J

(I

qlkk)d p

=

~ k=1 co

I

cI>kk(~)

=

k=1 p(~) . o

Then Theorem 0 yields a null set N2 ~ Nl such that for all a E M \ N 2,

r

I

1\

J

\=1 qikk) dp B(a,r) 00

I

~kk(a)

= lim ~~~---

=

1 r+O p(B{a,r» k=l 2, Corollary 00 Let a E

M \ N

2, Then

L

I~

(a)1 2 < 00 k

_,

~=1 k£

Proof. Consider the estimation

3, Definition The operators

B

a x ~ X, a E

M,

are defined by B

=

0 a

B

x

=

a x E X, 1 .

Observe that

B

_a is a Hilbert-Schmidt operator for each a E

M.

o

The operators B are related to the set function P in the following way.

4, Lemma Let ~

E M \ N

2, Then we have lim r+O

liB _

RP(B(~,r»RII ~ p(B(~,r» HS with II·II

HS the Hilbert-Schmidt norm,

Proof For all r > 0,

o

2

liB _

RP(B(~ ,r) )R II ~ p(B(~,r»

=

ro

I

_

~kt(B(~.r»1

I

~k~(~) p(B(~.r»

k,~=l

Let £ > O. Take a fixed A E 1N so large that

ro

(*)

I

~kk(~)

< £2/4 . k=A+l

2

Next, take rO > 0 so small that for all r, 0 < r < r

O' and all k,~ E ~ with k, ~ ;;;; A. and also (***)

I

k=A+l ~kk(B(~,r» < £2 p(B(~.r»

Then we obtain the following estimation A A 00 2

I

)I$k~(a)

~k~ (B(a ,r»

I

(

I I

+ 2

L

p (B(a ,r» k=1 R,=1 k=A+1 -R,=1 2 00 00 l<I>k-R,(B(a,r»I 2

L L

(l~k9.,(a)1

~ E + 4 + p(B(a,r»2

>.

k=A+1 9.=1 By (*)

""

00 co

L L

l~kR,(a)12

~

I

A _qikk(a) ~ E2/4

k=A+1 9.,=1 k=A+1 and by (***) 00 2 00 _{!Q)kR, (B(a ,r» I} 00 _{<I>kk(B(a,r»}

L

_I

_I

2 p(B(a,r»2 ~ p (B(a, r» < E k=A+1 R,=1 k=A+1

Thus it follows that

liB _

RP(B(a,r»RI! <

a p(8(a,r» HS

for all r with 0 < r < rO . o

In a natural way, the projection valued set function P can be linked to the function-algebra L (M,p). To show this, let x,y E X. Then the finite

q>

measure ~ is defined by ~ (~) = (P(~)x,y) where 6 is any Borel set.

x,y x,y

We have

J

d~ = (x,y). Clearly, ~ is absolutely continuous with

M x,y x,y

Let f denote a Borel function on M which is bounded on bounded Borel sets. Then we define the operator T

f by

<If

x,y) ₌

_J

f d]J ,

x,Y y E: X .

M

Observe that T

f is a normal operator in X. Since f is bounded on bounded Borel sets we derive for each r > 0, a E M and x E: X,

I (T

f P(B(a,r»x,x)1

( sup If(A)i) (P(B(a,r»x,x) . AEB(a,r)

So RTfP(B(a,r»R is a trace class operator.

5. Lemma

There exists a null set N3 such that for all a E M \ N3

RT fP(B(a ,r»R

lim

~f(a)Ba

- p(B(a,r»

~HS

= 0 . r+O

Proof

Following Lemma 4, we are ready if we can prove that there exists a null set N3 ~ N2 such that for all a E M \ N3

lim

r-l-O

~(a) RP(B(a,r»R

p(B(a,r»

Therefore we estimate as follows

00

L

p(B(a,r»-2₁ k,.R.=l

J

B(a,r) RTf P(B(a ,r» R p(B(a,r»

II

HS

=

o .

00 -1

r

p(B(a,r» (

J

₁ f(a) - f(A) ₁2 dp(A» p(B(a,r» -1 \' L <Pk (B(a,r»

B(a,r) k,.R.=l

-1 ( ~ p(B(a,r»

J

B(a,r)

Now there exists a null set N3 ~ N2 such that the latter expression tends to zero as r -I- 0 for all a

E M \ N

3·

II.

In the second part of this paper we employ the above auxilliary results in the announced construction of generalized eigenprojections.

We consider the triple of Hilbert spaces -1

R(X) c X c R (X) .

-1 -1

(u,w)l = (R u, R w) , u,w E R(X) ,

-1

and R (X) is the completion of X with respect to the inner product

'.

.)

\ , -1'

(x,y)_1

=

(Rx, Ry) .

The spaces R(X) and R-1(X) are in duality through the pairing <-,.> ,

<w,G> = (R -1 w, RG),

6. Definition

For each a

E M,

we define the operator n

a -1

Rn

w

= B

R w, a a Cf. Definition 3. w E R(X) . R(x) Observe that

n

a R(x) -+ R-1 (x) is continuous. 7. Theorem

I. For all a E M \ N2 and for all w E R(X)

lim riO ~n w - P(B(a,r» w~ a p(B(a,r» -1 = 0 . -1 -+ R (X) by

II. Let f :

M

-+ ~ be a Borel function which is bounded on bounded Borel

sets. Then there exists a null set N

f ~ N2 such that for all a E M \ Nf and all w E R(X)

lim riO

~f(a) n w -

T

P(B(a,r» w~

Proof

The proof of I follows from Lemma 4 and the inequality

lin w _ P(B(a,r»w II ~

a p(B(a,r» -1- IIRn R _ RP(B(a, r) )R II a p(B(a,r» HS IIR- 1 II w .

The proof of I I follows from Lemma 5 and the inequality

T fP(B(a,r»w IIf(a) n w -a p(B(a,r» 8. Corollary -1 -1

Let the operator RTfR be closable in X. Then T

f is closable as an

-1 -1

-operator from R (X) into R (X). For its closure T

f we have f(a)n w a with w E R(X) and a E M \

N

f• o

The results stated in Theorem 7 and Corollary 8 indicate that the mappings

n : R(X) -+ R-1(x) give rise to ("candidate") generalized eigenspaces

a

naR(X) for the commutative von Neumann algebra {Tflf E Loo<M,p)}.

Finally, we explain in which way the operators n , a E M, can be seen a

9. Lemma

Let w E R(X). Then in weak sense

So for all v E R(X) ,

(v,w) ₌

_f

M

<v,TI w>dp(a) a

Proof. Let ~ be a bounded Borel set. For all v E R(X),

co

I

l$k~(a)(R-1v,V~)(Vk'

R-1w)

I

~

k, £=1

and hence by Fubini's theorem

r

J

<v,TIaw>dP(a)

=

!J.

=

(p(~)v,w) .

Since

M

can be written as the disjoint union of bounded Borel sets it

follows that

J

<v,TIaw>dP(a)

=

(v,w) .

M

Remark: If R is Hilbert-Schmidt, the integral

J

strong sense. So in addition we have

I

~RITawijdP(a)

< 00 •

M

M

IT wdp(a) exists in

a

In IT R(~) we define the sesquilinear form (-,-) by

a a

(F,G)

a <v,

where F

=

IT v, G

=

IT w. (F,G) does not depend on the choice of v and w.

a a a

It can be shown easily that (-,-) is a well-defined non-degenerate

ses-a

quilinear form in IT R(X). By X we denote the completion of IT R(X) with

a a a

respect to this sequilinear form.

10. Theorem

I. The Hilbert space X with inner product (-,-) is a Hilbert subspace

a a

of R-1(X). IT maps R(X) continuous y 1nto I ' X .

a a

II. Let f be a Borel function which is bounded on bounded Borel sets.

-1

-Suppose the operator

T

f is closable in R (X) with closure

T

f . Then there exists a null set

N

f such that for each a EM \

N

f and all G E X we have

Proof.

I. Let G E n R(X), G n w. We estimate as follows

a a IIRGl12

=

<R n w, nw> 2 ;;; a a 2 n R2n w>i n w>! ;;; _{<R n w, <w,} a a a a

;;; _{IIRn R Iii IIRn w II lin wll}

a a a a

I t follows that

-1 Hence X can be seen as a subspace of R (X).

a

II. By Corollary 8, there exists a null set N

f such that for all a E M \ Nf and for all wE R(X)

f(a)n w a

-1 Let a E M \ N

f . Since Xa <:;.r R (X) and naR(X) is dense in Xa it follows

that for all G ( X_a' G ( Dom(i_f) and if G

=

f(a)G

a 0

11. Corollary

Let n+ X + R-1(X) denote the adjoint of n .

a a a + Then n n n . a a a Proof Let w,v E R(X). We have <w,n v>

=

(n w,n v) a : a a a + = <w, n n v> a a o

Let , (uk)kEIN denote an orthonormal basis in X which is contained in R(X). For each a E M, the sequence (TIauk)kEIN is total in Xa' 80 the spaces X

a ' a E M, establish a measurable field of Hilbert spaces. Its field structure

8 is defined by

~ E 8 ~ the functions a ~ (~(a), TI uk) are Borel functions.

a a

i

So the direct integral

H

~ ~

X.dP(.) is well-defined.

(For the general theory of direct integrals, see [1], p. 161-172.) The vector fields a ~ TIa~' a

E

M, k E IN, give rise to an orthonormal system (~k)kEIN in

H.

(We recall that the elements of

H

are equivalence classes of square integrable vector fields.) We define the isometry

U :

X +

H

by

00

Then for all x,y

E X

we have

(x,y) =

J

d~X,y

M

x EX.

J

«Ux)(a), (Uy)(a)}a dp(a) .

M

It follows that for all x,y E X and all f E Loo(M,p)

=

f

M

fd~

x.y

J

f(a)«Ux)(a), (Uy}(a»a dp(a) •

and hence we can write

12. Lemma

(jI

UTfx

f

f(a)(Ux)(a)dp(a)

M

The operator

U

x

~

H

is unitary.

Proof. We show that the set

U({T

f Uk!k

E

IN, f f Loo(M,p)}) is total in

H.

Let ~ be a square integrable vector field such that for all f f L (M,p)

00

and all k E IN

(

J f(a) (~(a), nauk)adp(a) .

M

Since f E L_oo(11,p) is arbitrary taken, (.p(a) , na uk) vanishes except on

00

a set Nk of measure zero. Taking N == U N this yields ~(a) = 0 on M \ N,

k==l k

and hence

I

~~(a)~:

dp(a) == 0 . M

Now the mappings n , a f M, can be seen as generalized projections as

a

follows: Let w E R(X). The vector field a I~ n w is a representant of

a

the class

Uw.

These representants a ~ n w, w f R(X), are canonical.

a

o

Indeed, there exists a null set

N

(=

N

2) such that for all w f R(X), and

lim

~TI

W - p(B(a,r»-l r.J-O a r

J

B(a,r)

o .

(Cf. Theorem 7.)

So the family (TIa)aEM selects a canonical representant out of each class

Uw, w E R(X). In this sense, each TI "projects" R(X) densely into X •

a a

References

1. Federer, H., Geometric measure theory.

Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, 1969, Berlin.

2 . Dixmier, J., Von Neumann Algebras.

North-Holland Mathematical Library, Vol. 27, 1981, Amsterdam.

3. Weir, A.J., General integration and measure. Cambridge University Press, 1974, Cambridge.

Eindhoven University of Technology P.O. Box 513

5600 MB Eindhoven The Netherlands.