Spectral trajectories and duality

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Eijndhoven, van, S. J. L., & Kruszynski, P. (1986). Spectral trajectories and duality. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8604). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1986

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Memorandum 86-04 October 1986

SPECTRAL TRAJECTORIES AND DUALITY by

S.J.L. van Eijndhoven and P. Kruszynski

Eindhoven University of Technology

Department of Mathematics and Computing Science P.O. Box 513

5600 MB Eindhoven The Netherlands

Department of Mathematics and Computing Science Eindhoven University of Technology

P.O. Box 513, 5600 MB Eindhoven The Netherlands

Sunnnary

Let E be a spectal measure on a ring of sets E , .with values in the set of projection operators in a Hilbert space H. An H-valued set function

~ on L is called a spectral trajectory controlled by the measure E if

for any ~I' ~2 E E E(~1)~(~2)

=

~(6In~2)' In other words ~ is a countably

additive orthogonally scattered measure on E , controlled by E (cf. [7], [9]) •

For a given locally convex space SR originating from a "generating" family R of operators on the Hilbert space H (cf. the authors' papers

[2,3,5]). It is proved that the topological dual Si is isomorphic to the space TR of R-bounded spectral traj ectories controlled by the joint spectral measure of the family R. The present paper contains a study of the duality between the spaces SR and T

R, and of the topological properties of the space T

R•

It extends the results of the authors' papers [2,3,4,5], which are the main references to the subject.

Introduction

Our previous paper [3] contains the construction of a locally convex topological vector space SR (called the initial space), which is

associated with a generating family R of operators acting on a Hilbert space H. In comparison with our former constructions, which were based on a generating family of functions (cf. [2], [5]), the approach in [3] lacks a proper description of the topological dual of the space SR' Presently we prove that the strong dual of SR can be represented as a locally convex topological vector space TR consisting of spectral trajectories (i.e. R-bounded countably additive orthogonally scattered measures) over the joint spectrum A of the family R.

The general idea behind these topics originates from our previous papers [2], [3], [5]. However, the use of orthogonally scattered measures, somewhat in the spirit of the Riesz' representation theorem for spaces of continuous functions, seems to be the most complete and elegant one. Moreover, it may provide an interesting global point of view on certain spaces of generalized functions and the generalized eigenvalue problem for self-adjoint operators. The spectral theory connected with our construction will be studied elsewhere.

The main reference is our paper [3], some technicalities come from [2], [4], and [5]. We use freely the general topics on locally convex

1. Preliminaries

Let L be a family of bounded operators in a Hilbert space H.

(1.1) Definition

A densely defined linear operator L in a Hilbert space H is called L-bounded if for each A E L the operator LA is everywhere defined and

bounded.

The vector space of L-bounded operators is denoted by LB(H).

(1.2) Definition

Let K c LB(H) be a family of operators. The set

c

K

=

{K E LB(H)IVLEK VAEL KLE LB(H), LK E LB(H), LKA

= KLA}

is called the L-commutant of K.

The L-bicommutant of K is defined by KCC

=

(Kc)c.

Now let H be a separable Hilbert space. As in our paper [3) we introduce a generating family of operators R:

(1.3) Definition

Let R c L(H) consist of bounded self-adjoint operators in the Hilbert space H. R is called a generating family of operators if:

1. V A E R :

(positivity and boundedness).

2. V A, B E R : (commutativity).

3. V A, B E R 3 C E R A ~ C and B ~ C (directedness). 4. V A E R 3 B E R 1 A2 ~ B (sub-semigroup property). * "

5. In the W -algebra R , generated by R and the identity I , there exists a sequence {p} N

n nE of mutually orthogonal projections 00 such that E n=l P

= I ,

n and a) V n E N 3 A E R 3 c 1 E Rl, c] > 0 P n ~ c1·A • b) V A E R 3 B E R 3 c 2 E Rl, c2 > 0 V n E N

n2 IIAP II

~

c inf {IIBP yll

I

lip yll = I, Y E H}

n 2 n n 6. Let R** = {L E R"

I

V L'ERcC L'L E

R"}~·

Then V

Q

E R** 3 A E R 3 C 3E Rl, c3 > 0

*

2 Q Q ~ c 3'A •

Let E be the joint resolution of the identity for the generating family R (or simply the joint spectral measure) E can be defined as a projection valued countably additive measure which extends the product measure

X

EA

AER

(cf.

[1]).

Here the measure

X

EA is defined on AER

cylinders in the Tichonoff compact space

A

= X

supp E

A, where each AER

E

We recall that a cylinder in the product space A is a set of the form:

AA E V., i= 1,

• 1

1

, k},

where each A. E R, and where each V. is a bounded Borel set

1 1

in the support supp EA. of the spectral measure EA. corresponding to

1 1

the operator A .•

1

We introduce the following semi-ring

r

of subsets of A:

(1.4)

r

= {

6. c A

I

6. is Borel and 1

E-measurable, 3AER, 3 c E R , c > 0,

It follows from Def. 1.3 3) and 5) that (1.4) defines a ring of subsets, which, however, is not a a-ring in general.

(1.5) Lennna

For each 6. E E there exists a natural number no such that no

E n=l

P

n

where the projections P are given by Def. 1.3 5).

n

Proof:

Assume the contrary. Let 6. E

r

be such that there exists a subsequence {P } _{of the sequence} {p }

~ kEIN n nEIN

such that for each k

=

1, 2, •.•

p _{E(il) ~ 0,} ~ and 00 L: P E(il)

₌

E(il)

.

k=1 ~

Let A E R be such that

for some number c > 0 •

Then by Def. 1.3 S.b) the inequality lip E(l\) II = 1 ~ c,Up All <

~ ~

< c' _{- 2} IIBII

~

holds for some B E R. This yields a contradiction.

•

(1.6) Definition

Let L be a ring of subsets of A and let E be a projection valued measure on the V-algebra generated by the ring L in A.

A function ~ : L + H is called a spectral trajectory controlled (or

propagated) by E if Vl\,l\'EL

E(l\) ~ (l\')

=

~ (l\nl\').

( ~ is an E-trajectory)

Each spectral trajectory ~ satisfies the following conditions

(1.8)

i.e. ~ is a so called orthogonally scattered measure.

I f {l\ } C L,

for n

=

m, and u l'::, E I, then

nelN n

00

ll( U l'::, )

₌

I 1-1 (l'::, )

nelN n n=1 n

where the series is convergent in norm in H.

Therefore II is a so called countably additive orthogonally scattered measure (c.a.o.s. , cf. P. Masani [7]).

The converse is also true, i.e. each c.a.o.s. measure ~ on the semi-ring L is controlled by some spectral measure Q, the so called spatial measure of ~ (cf. [9]). However, we maintain our terminology originat from the theory of analyticity and trajectory spaces (cf. [2]).

1.9 Definition

Let R be a generating family of operators in a Hilbert space H, E its joint spectral measure and L the ring of subsets of the joint

spectrum A of R, given by (1.4).

An E-spectral trajectory 1-1 : L ~ H is called an R-bounded spectral trajectory if for every A e R the map

is bounded, i.e.

sup IIA 1-1([1.)11 < 0 0 .

l'::,d

1.10 Lemma

For each A E R and each ~ E TR there exists a vector ~A € H such that for each ~ E L

E(~) ~A

=

A ~(~).

Proof

In L we can introduce an order relation ~ ~ ~ ~' iff ~.2. ~ I •

Thus L becomes a directed set.

Consider the net of elements of H: {A ~ (~)} ~El: • It is bounded and hence it has weak cluster points. Let x E H be one of them. We have

II xii :::;; sup IIA~(~)II. Let ~ E L. Then for every E > 0 and z € H ~€L

there exists ~, € L such that ~' ~ ~ and

I

(E(~)zlx - A~(~'»I < E. It follows that

I (zl E(~)x - A~(L~»I < £ •

This holds for arbitrary z E Hand E > 0, hence E(~)x

=

A~(~).

Let rCA) be the right support of the operator A in RI!. Put ~ == r(A)x.

A

It is easy to see that ~A does not depend on the choice of x.

1.11 Corollary

The vector ~A is uniquely defined and r(A)~A == ~A' Moreover the set

is dense in the Hilbert space r(A)H.

Proof

Obviously HA c r(A)H. Let v E r(A)H and v ~ H

A, Then for each ~ E TR

(r(A)v I ~A)

=

O.

In particular it holds for all ~ E TR of the form

s(~)

=

E(~)z , where Z E H.

Since sA = A z (1.12)

we have:

(Ar(Av)lz)

= 0 for all z

E H.

It follows that v

=

O.

We have sup IIA~(l'l)1I

=

1I11AII for all A E R, \l E TR' So we can ME

introduce the seminorms _{IIA on the space TR·}

1.13 Definition

For A E R and ~ E TR put

The locally convex topology on the space TR generated by the seminorms

II IIA wi 11 be denoted by T • •

proJ

It easily follows from Def. 1.3.5 a) that the topology T • is pro] Hausdorff and so we shall always refer to the space TR as to the

locally convex topological vector space endowed with the topology T • • proJ ].14 Lemma Let A, B E R, ~ E T R• Then Proof

For each 8 E I and Z E H we have

( E(8)Z I A ~B - B ~A )

=

= ( z I AB ~(8) - BA ~(8) )

=

0 •

Since {E(8)Z I 8 E I, Z E H} is a dense set in H the result follows •

Now we give a complete characterization of the elements of TR by means of the strong bicommutant RCC of R.

1.15 Theorem

Let ~ be a c.a.o.s. measure on the ring L. Then ~ E TR if and only if there exists L E RCC and y E Hsuch that for every 6 E I

~ (6)

=

L E(6) y.

Proof

~ If ~(~)

=

L E(6)y then ~ is an E-spectral trajectory and it is enough to notice that the measure A'~ is bounded on E since

RCC C RB(H).

Let {p } be the sequence of projections given n by Definition 1.3.5, r(A ) P :s:;; c

.

A n n n n the vector y

=

A -] n n r = lIy II • n n Since E nEN r

;to

n the series n E IN and let A E R, n

=

1 , 2,

...

n

for some numbers c >

O.

Thus, n

P _~A is well defined. Put n n L: nEIN 1 -2- < 00 n be such that for each n E IN

y = E nEN r

;to

n converges in H Put L

=

E nEN r

;to

n • Y n n r n and so Y E H. n • r • P n n

We shall prove that L E Rce• Let A E R. Then we have

nLAIl ~ sup n • r IIA P 1\

nd~ n n IIB-Ip 11 ~ c sup r 2 nEN n n n ~ -1 where c

2 > 0 and B E R are chosen as in Def. 1.3.5 b).

Also we have

So eventually we get

Thus we have proved that L E RB(H).

Obviously L E RCC and for each ~ E E E(~)Y E D(L).

(1.]6)

(1.17)

Now we compute L E(~)y for ~ E L.

By virtue of Lemma 1.5 we deal with finite sums only: Let n E N be such that

o Then

=

= n o L n=I n o P E(l;) n

=

E(lI). L n=1 r

;to

n n • r • P • E (~) • • y ::: n n n n • r n o L P E(lI) • A -] • P _{n n n • ~A}

₌

n=l n r

;to

n n 0 n

(

L E(lI) r(A ) P ) ~(1I)

=

~(l;)

n n n=l

r

;to

n

The last equality follows from the identities:

r(A)P

=

P ,and r(A )P ~(1I) = 0

n n n n n

whenever r

=

IIA -I P]JA

=

O.

n n n

n

We conclude that L E(t;)y

=

~(1I) •

(1.]9)

At the end of this section we recall the definition of the space SR constructed in [3].

Let AH denote the Hilbert space consisting of all elements of H of the form Ax, with A E R and x E H.

AH is endowed with the scalar product

(A x I A Y)A

=

(r(A)xl r(A)Y)H'

1.20 Definition

The space SR is the locally convex inductive limit of the family of Hilbert spaces with respect to the family of embeddings

such that for s E BH, A ~ B, s Bx, x E H

1.21 Remark

We can identify SR with the dense linear subspace u AH of the AER

Hilbert space H. Since the embedding SRC; H is continuous SR is a

Hausdorff topological vector space. In the construction of the inductive limit SR condition 6 in Def. 1.3 is not involved.

It has been proved in [3] that the locally convex inductive limit topology is given by the family of seminorms:

(1. 22)

(cf. 5.4 [3]).

Moreover (cf. 5.10 [3]), it has been proved that SR is a bornological, barreled, semireflexive (hence reflexive) locally convex topological vector space. SR is dense in H. If we assume that the family R

fulfil~ also condition 6 of Def. 1.3 then we obtain the completeness of SR'

2 TR as a locally convex topological vector space

In this section we prove that the space TR is endowed with the projective limit topology T • is an inductive limit of Hilbert

proJ

spaces. A corresponding result has been proved for the space T~(A)

introduced in [5], where the generating family of operators was

obtained by applying a generating family of functions ~ to an n-tuple of commuting operators A. Also we present here a nice characterization of the bounded sets in T

R•

This characterization is essential for the proof of the main result of this paper- namely, the duality theorem in Section 3.

To represent TR as an inductive limit we have to assume that the

generating family of operators R fulfills also condition 6 of Def. 1.3. L _{et +} R cc

=

{L ~ _~ RcclL 1.'S _{essentl.a y se -a JOl.nt on R an} • 11 If d' , S d 3 _{£ >} 0

L ~ £ I}. Let L EO R+cC • We define

For ~1' ~2 EO L·H put where fJ (.) = 1 J.l (.)

=

2 (2. 1 )

It is easy to see that (2.1) defines a scalar product on L·H under which L·H becomes a Hilbert space.

Our proof of the representation of TR as an inductive limit space of the family of Hilbert spaces {L·H I L E R+cc} is a reformulation of

the proof of Theorem ].7 in [2]. It is based on the following facts: By virtue of Theorem 1.]5 the space TR can be identified with the linear meanifold u L'H •

L R cc

E +

Using the usual ordering of essentially self-adjoint operators in R+cC we can introduce the locally convex inductive limit topology lind on TR ' with respect to the canorical embeddings of the Hilbert spaces L·H into TR'

Recall that a set 0 C TR is open in the topology" d

ln if and only if for each L E R+cC the set 0 n L·H is open in the Hilbert space

cc Since for each A E R the operator LA is bounded for every L € R the seminorms on TR ' ).l H- liJl AliA, are continuous in the topology

" d' Thus " d _{In In}

>, . ,

_proJ

To prove that T • > , . d we shall use the following result, proJ l.n

2,2 Proposition

Let 0 C TR be a convex null-neighborhood in the topology lind'

Then there exists A E R and a number 0 > 0, such that the set

Proof

For each L E R+cC the set 0 n L·R is open and convex in L·R. Let emb denote the canonical embedding of H into TR defined by

E 3 ~ ~ emb(x)(~)ER, where x E H. (2.3)

Since and for each n E N the Hilbert space emb(P H) is n a Hilbert subspace of L'R the set

o

n emb(P H) is open in emb(P H).

n n

For every n E N we define:

r

= sup

{p >

°

I emb(P K(O,p» C Onemb(P H)} =

n n n

= sup

{p > Ol(~ E emb(P R),

n sup

IIPnl1(~)

II < p)

'*./'

~

0;

~El:

Now let us define the following operator

00 K = E n=1 2 2- !!.... P r n n

We are going to show that K E R**. We need the following result:

(2.6) Lemma

(2.4)

(2.5)

For every L E RCC there exists L' E RCC such that for all n E N

n2 IILP II

~

C

n _lip inf _yll=1 IIL'P _n yll n

for some number c > o.

Proof

Define the following unbounded operator

L' = L: n=l

Let A E R. Then there exists B E R such that for IN

=

{n E N I IIAP II:tO} IIL'AII ~ sup ndlI o ~ c· sup nEIN o Hence L' E RCC • Obviously n2 IiLP II

=

n Now let ]J E T R• o n inf Ii BP g 1111 B-1 BP LP II

~

lip gH=1 n n n n

n2 IILP 1/ inf lip gil

~

inf IIL'P gil n lip gll=l n lip gll=l n

n n

This proves Lemma 2.6.

cc Then ]J E L'H for some L E R+ •

Let the operator L' E RCC be defined for L as in Lemma 2.6, i.e. for each n E IN

inf IIL'P gil •

lip gll=l n

n

Since the set 0 n L'·H is a neighborhood of 0 in L'·H there exists £ >

a

such that

{v EO L'·H I IIvIl

L, < d c 0 () L'·H • For each nEON we have:

lip ]JilL' = sup IILt

- Ip ]J(1l)1I

~

( inf !!L'P yll)-l sup lip ]J(1l)1I •

n lid n lip yli:] n lid n

n

So if sup lip ]J(ll)!! < s' inf IIL'P gil

llEOL n lip yll:1 n

n

then P ]J EO 0 () L'· H c O.

n

It follows that for each LEOR+ CC. rn ? s· inf IIL'P yll with L' as in 2.6. lip yll=l n

n

2

By virtue of Lemma 2.6 we have for all LEORcC IIKLII ~ 2 sup ~ ilL Pnll ~ nEON n

2

~

2 sup: ( inf IIL'P yll)-t IlL P Ii

~

2 •

nEON lip yll=1 n n

n

Thus K E R**.

By condition 6) of Def. 1.3 there exists A EO R and a number c

3 > 0 Let 0 = -] and V =V_l,A)= {]J EO TR I iI]Jil A < C 3 } . c 3

We will show that V C

o.

cc

Let ]J EO V. Assume that ]J EO L'H for some L EO R • Then

L

n=]

By virtue of (2.7) for each nEON we have

2 1

n sup lip 1-I(ll)1I =: Zr

n sup IIAP \1(ll)1I

llEOL n llEOL n Thus, by (2.4) we have 1 ~-. 2 C 3 • r • n (2.7)

2

n P \l E 0

n

Consider the following decomposition:

00 \l = n o L -2 n=l 2n 2 2n P \l + ( n L 2nI 2 ) llno n=n +1 o 0:> 00 where _{lln =} ( E - ) 1 -1 L P II n=n +1 2n2 n=n +1 n 0 0 0

Using again Lemma 2.6 we can find L' E

00 00 112 IIll = ( L -2 lip ll1l2 2n2 ) L n 0 L' n=n +1 n=n +1 0 0 4 00 ~ 4 n c·

r

_"""4

1 IIP 2 ~ nllllL 0 n=n +1 n 0 00 ~ 4 • c • E IIP 2 n11ilL n=n +1 0 Hence lI]..In II l ' o + 0 as n + (x). o n L'

.

RCC such that ~ (2.8) (2.9)

The first term in the decomposition (2.9)·belongs to O. The second term for sufficiently large n also belongs to O. Thus 11 is a convex

o

combination of elements of the convex set 0, and therefore II E O. We proved that V C O.

This ends the proof of Proposition 2.2 •

•

Taking into account Proposition 2.2 and the previously stated fact that To d > T 0 we can formulate the following result.

2.10 Theorem

The locally convex topological vector space TR with topology T

proj can be identified with the locally convex inductive limit space

u L·B with topology T. d •

~n

LER+cC

Now we prove another useful result:

2.11 Theorem

Let R be a generating family of operators fulfilling the conditions 1-6 of Def. 1.3 • Let TR be the space of spectral trajectories. Then a set B C TR is bounded in the topology T • if and only if there

pro]

exists L E R cc such that B C L·B and B is bounded in the Bilbert

+ space L·B.

Proof

The idea of the proof is based on the proof of Theorem 2.3 in [5] and the above formulated result representing TR as the inductive limit

u L-B _ Therefore we present here only an outline of it.

L R cc

E +

cc

If a set B c TR is contained in a space L-B for some L E R+ then

it is bounded in TR whenever it is bounded in L-B because the embedding LoBe TR is continuous_

On the other hand assume now that B is bounded in TR' Then for each n E N the number

(2.12)

is well defined if A is chosen as in Def. 1.3.5 a). We define in H an unbounded operator L

=

E n=1 n' s .p n n Obviously SR C D(L). cc Using Def. 1.3.5 b) we can prove that L E RB(H), and hence L E R •

For \.I E B put

00 x = L

- -

.

A-Ip \.I_A \.I n=l n's n n s

;to

n

with A as in 2.12 and \.I

A given by Lemma 1.10. We have the easy estimation

00

2 I

IIx II ~ L - < 00

]l n=1 n2

Therefore x E H and the set B

=

{x I \.I E B} LS uniformly bounded

\.I 0 \.I

in H. One can see that B

=

L·B o

3. Duality

In this section we discuss the duality between the spaces SR and T R• We shall do it on two levels. Under the assumption of conditions

1-5 of Definition 1.3 we prove an algebraic identification of the space TR with the (strong) dual Si of the space SR' and similarly an algebraic identification of the space SR with the (strong) dual TR' of the space T

R•

However, to prove the topological identification of the spaces SR and TR with the strong duals of TR and SR' respectively, we have to assume also condition 6 of Def. 1.3 •

We define the following pairing between spaces SR and T R•

3.1 Definition

Let ~ E TR and s E SR' with s

= A x for some A

E R and x E H. We

define the number

where ~A is given by Lemma 1.10.

3.2 Remark

The numbers < ~, s > are well defined, i.e. they do not depend on the decomposition s

= A x, AER,

XEH.

Proof

Let s

=

A x = A'x' A, A' € R ; x, x' € H.

By the definition the vectors ~A and ~A' are weak cluster points of the nets {E(A)~A}AE~ and {E(A)~A'}A€~ , respectively.

For each A E ~ we have

Thus it follows that

3.3 Proposition The function

1

1+ <~, s > € C

1S a non-degenerate sesquilinear form.

Proof

•

The antilinearity of the form < ~, s > in the first argument ~~TR

follows directly from the definition of the vectors ~A' A E R, Lemma I. 10.

To prove the linearity of < ~, s > in the second argument s E SR'

let us take s = A x

=

_{A1X 1} + A_ZX₂ with A~ AI' A2 E R, x, xl' x

and ~ E T_R• Then for every

n

E L (~(n) I A x)H

=

=

(E(n)~A I x)H

=

I

Again remembering that the vectors ~A' ~A and ~A are weak cluster

1 2

points of the corresponding nets we obtain the result by "2E-argument". To prove that the form < , > is non-degenerate assume the contrary: suppose there exists v E T

R, v ;E 0 such that < v, s >

=

0 for all s E SR' Then we have for each x E H, each A E R and each nEE :

It follows that vA

=

0 so v

=

O. A contradiction. Now let So E SR be such that for all ~ E TR

Considering the trajectories of the form

we get:

~ (n)

=

E(n)s o

DE(A)s

0

2

=

0 for all A E

r.

o Thus s

=

0 o < ~,s >

=

O. o

•

Now let SR' denote the (strong) topological dual of the inductive limit space SR ' and let TR' be the (strong) topological dual of the space TR endowed with the topology T ••

3.4 Theorem

Let R be a generating family of operators fulfilling conditions 1-5 of Def. 1.3 • Then the following algebraic dualities hold:

Proof

We shall prove the theorem in four steps showing the existence of four anti-linear injections:

I a l SR -+ T ' R I I a 2 T ' R -+ S R III 6_{1 TR -+ S ' R} IV 6₂ : s ' _R -+T _R

and the identities:

a)

o

a

=

idT '

a

0

a

l = ids 2 2 R R 6_t 0 ₆ 2

=

ids I 6 2 0 61

=

idT

,

R R I. The injection a l is defined by

where for each ~ € TR

*

and TR is the algebraic dual of the space T R • By proposition 3.3 the map a

1 is well defined and anti-linear.

It is easy to see that for each T •

pro] -continuous

II. The injection a

2 is constructed as follows:

Let ~ E T ' • Then there exists A E R and a constant c > 0 such that R

On the pre-Hilbert space with the norm II IIH

a funtional ~(~A)

=

~(~).

~ is linear and bounded hence it can be extended onto the Hilbert space r(A)H which contains HA as a dense subset. So there exists x E r(A)H such that for each ~ E TR

~(~) ~(~A)

=

(x I ~A)H

=

< ~, s > where s

=

A x € SR' It is easy to show that the thus defined s is unique. Hence the map TR' 3 ~ ~ a2(~) = s € SR is a well defined anti-linear injection.

One can easily check that the identities

It means that the vector spaces SR and TR' can be identified by means of the anti-linear invertible one-to-one mappings a) and a

2•

III. Now we shall construct the injection

T _R ~ S ' _R

Let ~ E TR and s E SR • Define

It follows from Proposition 3.3 that

BI

is a well defined anti-linear

map from TR into the algebraic dual SR* of the space SR • Let us consider the linear functional

(3.6)

with A E R. We have

Hence the functional (3.6) is continuous. It is sufficient for the continuity of the functional Bl(~)(·) on the inductive limit space SR

(cf. [3]). Hence al(~) E SR' •

IV. Now we construct the injection

S t ~ T .

By virtue of (1.4) for each 8 E ~ the Hilbert space E(~)H can be isometrically embedded into the Hilbert space AH for some A e R. Let £ E SRI. We can restrict £ to the space E(8)H, for every 8 E ~.

£I_E(8)H is continuous on E(8)H, so there exists a unique vector

~(8) e E(8)H, such that for every x E H

t( E(8)x )

=

(~(8) I E(8)x)H Thus we obtain an H-valued set function:

which is obviously an E-spectral trajectory. We will show that

~ e T

R, 1.e. ~ is R-bounded. Let A e R. Then

sup IIA~(8)1I ,s;;; sup sup I (H8)IE(8)Ax)H1

= sup sup

1£(E(~)Ax)I,s;;;

8eI 8eI Uxll') 8eI IIx U,s;;; 1

Hence ~ e T

R• Now we put

(3.7)

Since for each 8 e~, A E R, and x E H t(Ax)

Thus we have defined an anti-linear

52

map from SR' into T_R•

A straightforward computation yields:

5}

0

52

= id

sR, and

B2 0 B)

= id

T • Thus we have established the one-to-one anti linear R

correspondence between the vector spaces TR and SR'.

3.8 Remark

To simplify the notation we shall denote a

and B = B}

In the following part of this section we shall consider the duality between the spaces SR and TR from the topological point of view. Having in mind the characterization of bounded sets in TR given in Section 2 we recall also the following result:

3.9 Proposition (cf. [3], Lemma 5.8)

Let the family R fulfill the conditions 1-6 of Def. 1.3 • Then a set B C SR is bounded if and only if there exists A E R such that B is a bounded subset of the Hilbert space AH.

3. 10 Theorem

Let the family R fulfill the conditions 1-6 of Def. 1.3 • Then the maps a and Bare antiisomorphisms between the space SR and the strong dual T I

R of the space TR ' and, respectively, between the space TR

and the strong dual SR' of the space SR • Thus SR is homeomorphic with TR'

and TR is homeomorphic with SRI

Proof

Consider at first the map a:SR+T R' •

Let B] C TR be a convex bounded set. Then Bl C LiH for some

by the seminorms SR 3 s >+ IILslI

H ,where L E R CC

(cf. [3], Theorem 5.4).

Consider the inequality:

= sup

l<fl,S>!

=

flEB)

~ sup sup I (AL)E(~)Y YEK(O,r) ~d

sup l(fl_A I x)H1 ~ flEE)

where B) C L] • B(O,r) , and where B(O,r) is a ball in H of radius r.

Thus the map a : SR ~ TR' is continuous in the strong dual topology in TR' •

-1

To prove the continuity of the map

a

the following relation:

cc ,

For any L E R ,~TR' we have:

T ' ~ S let us consider

R R

II L a -) (IP) II = II L A x II = sup I ( Y I L A x )H I

=

s~p

I IP(fl) I

lIyn~l flEL -B(O,l)

where IP(fl) = (x I flA)H with x E H, A E R (cf. Theorem 3.4).

.

* (

S1nce the set L ·B 0,1) is bounded in TR the above relation proves

h . . f - )

t e cont1nu1ty 0 a with respect to the strong dual topology in TR' and the inductive limit topology in SR'

Thus we proved that the spaces SR and TR' are topologically isomorphic by means of the map

a.

Now let us consider the map

6 :

TR ~ SRI •

Let B2 C SR be a bounded set in the inductive limit topology in SR'

Thus B2 C A H for some A E Rand

We have for any fl E TR

sup "siiA ~ c < 00 for some c > O. sEB

sup

I

13(11) (s)

I

SEB

2

Hence the map 13 is continuous with respect to the strong dual topology in S ' and the

R topology T pro] • in TR •

-I ,

The inverse map

a

SR ~ TR is also continuous thanks to the following relations:

For any £ E SRI and A E R we have

II 13-1(£) "A

=

sup" A

B-I(£)(~)H

= sup

~E~ ~E~

sup xEH

HxlI~l

= sup sup

1£(E(f:.)Ax) I ~ 11£

I

AH" = sup I £(s) I

~EL. IIxll~l IIxII~1

s=Ax

Since the set { s E SR I s=Ax, IIxll~1 } is bounded in the inductive

limit topology in SR the map B -I is continuous with respect to the strong dual topology in SRI and the topology T • in T

R. pro]

This proves that the spaces SR' and TR are topologically isomorphic •

•

Considering all results of the sections 2 and 3 we have the following final statement:

3.11 Theorem (cf. [ 3], [5], [ 6], [1 0] )

Let a generating family of operators R fulfill the conditions 1-6 of Def. 1.3 and let SR and TR be the initial space and spectral trajectories space endowed with the inductive limit topology and

the topology T . , respectively. Then both spaces are the topological proJ

representations of the strong dual of each other.

Hence, the spaces SR and TR are complete, barreled, bornological, Mackey, reflexive locally convex topological vector spaces.

4. An example

In our previous papers we considered a particular generating family of operators, namely the family ~(A) constructed by means of a generating family ~ of Borel functions on the real line and a self-adjoint operator A in a Hilbert space H. (cf. [2], [5]). Let us recall shortly this construction:

4. 1 Def ini don

A family

~

of real valued functions on R] is called a generating family of functions if it satisfies the following conditions:

A I. Each ~ E ~ is a nonnegative Borel function bounded by I with inverse

~-1

bounded on bounded Borel subsets of the carrier of ~ :

~

= {

A E R 1 I

~

(A) ;t O} •

The family ~ is directed with respect to the usual ordering of real functions.

A II. For each ~ E ~ there exists ~ E ~ such that

~~ ~ ~

.

A III. For each ~ E ~ and each number 0 > 0 there exists Xo E 4l and a number c > 0 such that

A IV. For each ~ E ~ there exists ~ E ~ and a positive number c such that

A

v.

Let

~*

= {flf is a Borel function on RI

and V(,j)E~ sup) If(A)W(A)1 < 00 } •

AER The set

<j)** = { xix

is a Borel function on Rand I Vfd* sup] IX(A)f(A) I < <Xl }

AER

has the property:

For every X E

~**

there exists W E

~

and a number c > 0 such that

IX(A)I < c· wO) A € R • 1

There is no difficulty in defining a generating family of functions n

on R n ~ I. However, the one-dimensional case is illuminating to a satisfactory extent.

Now, let A be a self-adjoint operator 1n a separable Hilbert space H, with the spectral measure E. Then

A

=

II

A dE (A) , R

and for every finite Borel function f the operator

f(A)

=

It

f(A) dE(A) R

is a well defined normal operator in H.

In particular the family of bounded self-adjoint operators

<p(A)

= { W(A)

I W E ~ }

It is very easy to check that the family q, (A) fulfills the conditions imposed on generating families of operators (with a slight modifi-cation) given in Def. 1.3 •

Thus let R = q,(A) •

For this particular family of operators the spaces SR and TR are constructed in our previous papers [2] and [5]. However, it has been done without an explicit application of spectral trajectories. Hence, we give here an alternative description of these spaces. The definition of the inductive limit space SR remains unchanged. Instead, the space of spectral trajectories needs a more specific description.

Since the joint spectral measure for the family R=~(A) is simply E, we have:

4.2 Proposition

Each element of the topological dual space SR' of the space SR can be represented as an R-bounded spectral trajectory ~ € TR in the following form:

~(~)

=

f

f(A)dE(A)Y ,

~

where ~ is a bounded Borel set, Y E H, and f is a Borel function

• =1= =1= ( ) f cc ( f . 2)

~n ~ • The set ~ A plays the role 0 R c . Sect~on •

Proof

~ (~)

=

L E(~)y ,

with L E RCc• By Theorem 4.1 [3] we can write:

L

=

II

f(A) dE(A) , R

The result easily follows from Theorem 3.4 .

•

The duality between the spaces S¢(A)

= SR and T¢(A)

= TR ' defined

in Def. 3.1 is given now as follows:

< ~, f(A)~(A)(yl dE(A)x) ,

,

~(~)

=

I

f(A) dE(A)Y , with Y E H, ~

f E

¢*

and s E SR' s

=

II

~ (A) dE(A)x, with x E H, ~ E ¢. R

4.4 Remark

Every (not necessarily R-bounded) spectral trajectory can be represented in the form:

~ (~)

=

I

g(A) dE(A)Y , ~

with Y E H, and where g is a Borel function, bounded on bounded

Borel sets.

Let A be the spectrum of the abelian C -algebra C

*

*

(~(A»

generated by the family ~(A).

Let for a suitable Radon measure v $

R =

f

R(A) dV(A) , A

be the decomposition of the Hilbert space H into a direct integral of Hilbert spaces which diagonalizes all elements of the C*-algebra

*

C

(HA».

Thus we can represent every ~ E L (cf. 1.4) by an open-closed subset of A, denoted for simplicity also by ~.

We have then: $

E(~)

=

f

.x~ (A)IA dV(A) , A

where X~ is the characteristic function of the set ~ and IA is the identity operator in the Hilbert space H(A).

Now every E-spectral trajectory can be represented in the form: $

~ (~) =

f

X~ (A) f(A)Y(A) dV(A) , A

*

where f E ¢

f)

and Y E

f

H(A) dv(A). A

The connection with the operator A is preserved by the choice of the measure v.

References

1. Yu.M. Berezanskii, "A projective spectral theorem", (in Russian) Vspekhi Mat. Nauk, 39 No.4 (238), (1984) 1-52.

2. S.J.L. van Eijndhoven, J. de Graaf, and P. Kruszynski, "Dual systems of inductive-projective limits of Hilbert spaces originating from self-adjoint operators", Proceedings of KNAW, September, A 88 (3)

(1985) 277-297.

--3. S.J.L. van Eijndhoven, and P. Kruszynski, "GB*-algebras associated with inductive limits of Hilbert spaces", Studia Mathematica, 85, 2

(1985) to appear.

--4. S.J.L. van Eijndhoven, and P. Kruszynski. "Some trivial remarks on orthogonally scattered measures and related Gel' fand triples", Memo-randum 1984-11, December 1984, Eindhoven University of Technology, the Netherlands

5. S. J. L. van Eijndhoven, and P. Kruszynski. "On Gel' fand triples originating from algebras of unbounded operators", EUT-Report, 84-WSK-02, Eindhoven University of Technology, 1984.

6. P. Kruszynski, "Algebras of extendible unbounded operators", EUT-Re-port, 84-WSK-04, Eindhoven University of Technology, 1984.

7. P. Masani, "Orthogonally scattered measures", Adv. in Math. vol 2 (1968) 61-11 7 .

8. P. Masani, "Remarks on eigenpackets of self-adjoint operators",

Hilbert Space Operators and Operator Algebras, Proceedings, Tihany 1970, North Holland, Amsterdam 1972, pp. 415-441.

9. P. Masani. "Dilations as propagators of Hilbertian varieties", SIAM J. Math. Anal., ~, No 3 (1978) 414-456.

10. H.H. Schaefer, Topological Vector Spaces. The MacMillan Co., New York 1966.