On Gelfand triples originating from algebras of unbounded operators

operators

Citation for published version (APA):

Eijndhoven, van, S. J. L., & Kruszynski, P. (1984). On Gelfand triples originating from algebras of unbounded

operators. (EUT-Report; Vol. 84-WSK-02). Technische Hogeschool Eindhoven.

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Published: 01/01/1984

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE EN INFORMATICA

EINDHOVEN uNIVERSITY O~ TECHNoLOGY

THE NETHERLANDS

DEPARTMENT OF MATHEMATICS AND COMPUTING SCIENCE

On Gelfand triples originating from algebras of

unbounded operators

by

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

EUT Report 84-WSK-02 ISSN 0167-9708

UNBOUNDED OPERATORS

by

S.J.L. van Eijnhoven and P. KrUszynski*)

*)On leave from the Department of Mathematical Methods in Physics, University of Warsaw, Poland.

SummMY

Il1tJtodu~on 1.

_.

The..

.6

pac.e..

S

A

~c

)

2.

The..

.6

pac.e

T A

~ ( )

3.

The..

p~ng

06

s~CA)

and

T~(A)'

thein duaiity

4.

CoWnUOU.6 ,une..aJL mapp,Ln9.6

n

Jtom

S

A

"erl-to

S

CA

<P₁C 1) 4>2 2)

and

n

Jtom

_T

_A)

"enio

T

A

~1( 1 <!>2( 2)

5.

A

.6UJtve..y

06

Attan'.6

~heoJty

on GB*-alge..bJta.6

6.

The.. aiqe..bJta.6

4> (A) C

and

<!> (A) cc

7.

A

6unmonat c.alc.td.u6 60Jt the atgebJta

<!>

CA)

cc

8.

lUu6.tJraUoY/..6

1 9 23 34 41 47

52

61 80 87

triple S¢(A) c X c family ¢ there are

T¢(A) where S¢(A)

=

U ¢E¢ certain GB*-algebras of

¢(A)

(X). Related to the unbounded operators. They extend the cornrnutant ¢(A) I and bicornrnutant ¢(A)" in B(X) in a natural

way. Further, some of them give rise to a description of the inductive limit topology for S¢(A) in terms of explicit seminorms.

- 1

-IYLtJtoductioYl.

A Gelfand triple consists of three ordered spaces

SeX c

T. In this

triple X is a Hilbert space,

S

is a dense subspace of X which carries

a locally convex topology, and T is a locally convex topological vector

space which is in duality with

S.

Many mathematicians have investigated

this kind of space triplets. We mention Gelfand and Shilov who in fact introduced the notion of Gelfand triple in their illuminating work on

generalized functions ([G5J), and, also, Grossman and Antoine ([AGJ)

who generalized the ideas on countable Hilbert spaces towards the so

called scaled Hilbert spaces. In fact, any distribution theory is based on

some Gelfand triple.

In this paper we continue the investigations which are initiated in [GJ and [ElJ. In those papers two functional analytic frames are created

for two different types of distribution theories. In [GJ, the test~

space is introduced as an inductive limit and the distribution space as a Frechet space. In [ElJ the test space is a Frechet space and the distribution space an inductive limit. In order to clarify the intentions of the present paper we give a rather detailed outline of the theory [GJ.

In [GJ, the Gelfand triple

has been introduced and thoroughly investigated. Here

A

denotes a

positive self-adjoint operator in a separable Hilbert space X. The

set

SX,A

is the analyticity domain of the operator

A,

i.e.

U e-tA(X)

It is natural to endow SX,A with the inductive limit topology 0

ind

-tA tA

generated by the spaces e (X) with norm lie .II X.

The space TX,A consists of all mappings from the open interval

(0,00)

-TA

into X with the property F (t + T) = e F(t), t'L> O. The elements of TX,A are called the trajectories. The topology T

proj for TX,A is the locally convex topology generated by the seminorms

F>+ IIF(t)lI, _F

_E:

_{Tx,A '} t > 0 .

It has been proved that Sx,A and TX,A are in duality.

In [G], B(R) denotes the set of all everywhere finite Borel functions f on R which satisfy

sup If (;\) e

-t>..

I

A>O

< 00

B (R) denotes the positive part of B(R) •

+

Following the spectral theorem we can define the operator f(A) for each f

E:

B(R)

The operator f(A) is unbounded and normal.

For each f E B+(JR) the mapping Sf : SX,A 1+ JR+ defined by

IIf(A)wll , w E S

_X,

A

is a seminorm on SX,Ao The seminorms Sf' f ( B+(JR) , generate a locally convex topology G . for S In

[G]

i t has been proved that a is

pro] X,A" ind

('quivalent to () .. But S . also equals the intersection

pro]

X,A

n

D(f(A»

fEB

(R) +

3

-Thus i t can be seen as a projective limit (D(f(A)) denotes the maximal

domain of the unbounded operator f(A). See also [LTJ).

Correspondingly, for each trajectory F E TX,A there are 9 E

B+(E)

and x E X such that F(t) = g(A)e-tA

x,

t >

O.

So we have

U g(A)tX)

gEB+(JR)

where g(A) (X) denotes the set of all trajectories F

g,x

where

x

E X. With the inner product

-tA

t -+ 9(A) e x ,

g(A) (X) becomes a Hilbert space. The set B (JR) is a lattice. So i t

+

makes sense to introduce the inductive limit topology T. d for In

T A

=

U g(A) (X). From the results in this paper i t follows that

X, 9 B+(JR)

T . is equivalent to T

pro) indO

Since the theories [G] and [EJ are a kind of reverse copies of each other we give only a short summary of the latter. In [EJ, the Gelfand triple

T(X,A) c XC o(X,A)

has been studied. Here T(X,A) D (e), Le.00

A

T(X

,A)

seminorms

The space T(X,A) is a projective limit with topology induced by the

tA

wt+ lie wll

_xt

t > O. We observe that T (X,A) is a Fr~chet space.

The space 0(X,A) is introduced as

tA

Here e (X) denotes the completion of X with respect to the norm

-tA

lie

.II

X. Hence o(X,A) can be seen as an inductive limit. In [EJ the set of functions F(R) has been defined as follows

¢ E F(R)

So the operators

¢(A)

are bounded (even smoothing). We have the f01-lowing relations T

(X,

A)

and o

(X,

A)

U

</>(A)

(X) ¢H(R)

n

X

</>EF (R) ¢

Here X</> denotes the completion of X with respect to the norm

II

¢

(A) ·11 •

These relations lead to a description of T(X,A) as an inductive limit and o(X,A) as a projective limit. We observe that each element of o(X,A) is in unique correspondence with a mapping

F

F(R) ~ X with the

property F(¢lji) == ¢(A)F(lji) , ¢,ljJ EF(R) . To see thi,s, write formally

tA

F == e y , with t > 0 and y C X and then define

-

tA

F(¢)

== e

¢(A)y,

¢

E F

(IR)

Both in [GJ and in [EJ a central role is played by a semigroup $ of functions which generate the test space in the Gelfand triple as an inductive limit. In particular in [GJ

- ~)

-{cjl \ cjl Borel,

_Vt>O

sup

xE:IR

The distribution space can be seen as the space which consists of all mappings F from ~ into X with the property

F(cjllj!) '" cjl(A)F(I(I) 1(1 (A)F(cjl) cjl,lj! E~.

Related to the function semigroup

~

is the set

~*

which consists of all everywhere finite Borel functions with the property that

sup \f(A)cjl(A) \ < 00 for all cjl

E

~.

Each function f (

~*

defines an

A(JR

unbounded (normal) operator f (A) in X. The seminorms sf : W 1-+

II

f (A)wll

induce a locally convex topology equivalent to the inductive limit topology for SX,A (T(X,A».

'J'hi~; new vi,t'W on tll<' yeneration of tll'~ C!'lLmd I.cipll'~; 5

X

,A

c X c IX,A and T(X,A)

c

X

c

o(X,A) inspired us to look for conditions on a set of bounded Borel functions ~ which lead to a generation of Gelfand triples with similar properties. The conditions we impose on~, reflect on the properties of the Gelfand triples it generates.

The triples occuring in this paper are denoted by

Here S~(A) '"

u

¢(A) (X) with inductive limit topology, and T~(A)· ¢E~

consists of all mappings F from ~ into X such that F(¢lj!)

=

¢(A)F(X)

=

I(I(A)F(¢), cjl,lj!

E

~.

The set

~*

consists of all everywhere finite Borel functions with the property that sup \f(A)¢(A) \ < 00 • The elements of

~*

induce seminorms

A(:IR

on S,]J (A) and hence a locally convex topology. This topology is equivalent to the inductive limit topology. Also, we impose conditions on ~ such

that each member of T~(A) can be represented as F(¢)

'"

¢ E ~, for some x E X and f

E:

~

•

We note that the inductive limit S~(A) is not strict, in general.

The sections

1-4

contain a detailed discussion of the topological

aspects of the spaces S~(A) and T~(A)' their duality and corresponding

algebras of continuous linear mappings.

In the sections 5-7, the space S¢(A) is used as a core for certain

related algebras of closable operators. Here we consider so called GB*-algebras. The notion of GB*-algebra has been introduced by Allan.

(ef. [Ai 1] and [Ai 2]). It is a useful generalization

of the usual notion of B*-algebra. A summary of Allan's theory is given in Section 5.

In the sections 6-7 we study the algebras <p(A)c and

~(A)cc,

which

Sx

_,

A

c X c

-tA I

{e t >

to the triple

arc subalgebras of L (S,p (A))' (See also [AET].) Here

~

(AJ

c consists

of all operators L E L(S<p(A)) with the property that L¢(A)w

=

¢(A)Lw

for all

¢ E

~

and all w

C

S¢(A)' Similarly, L'

~

¢(A)cC if and only if

L'Lw = LL'w for all L

E:

~(A)c

and all w

E:

S~(A)'

The algebras

~(A)c

and ~(A)cC have roused our interest in earlier research with respect

TX,A' We have proved that both

<jJG(A)c~

{e-tAlt > o)c cc

O} are GB*-algebras. Moreover the inductive

limit topology for SX,A is generated by the seminorms

\ (w) = IILwll

X ' vJ E

s

X,A

where L

E:

¢G(A)CC. In this paper both aspects of ¢(A)cC will be taken

7

-c cc

In Section 6 we prove that ~(A) and ~(A) are GB*-algebras. More-over

~(A)cC

is commutative. The algebras

~(A)c

and w(A)cC are extensions of the usual commutant ~(A) I and bi-commutant ~(A)" which are

C*-sub-algebras of

B(X).

We have

and

The Gelfand-Naimark theorem for ~(A)" extends to the whole algebra

cc

In Section 7 we develop a functional calculus for ~(A) . In a natural way the set of functions ~ generates a B*-algebra with the usual supre-mum of bounded functions as its Banach norm. Next we consider monotone sequences of functions in the real part of that B*-function algebra. We obtain a so-called Borel*-algebra W~(~) of bounded Borel functions on ~. We prove that

W<!>(A) := {8(A)

10

E W<l>(]R)}

=

1>(A)" •

A natural extension of

W~ (~)

is the function algebra

W~* (~)

which consists of all everywhere finite Borel functions with the property that for all <p E: 1> the function <p.f CW~ (~) . There exist conditions on ~ such that

elements

Q

of ¢(A)cC induce seminorms S on S A) which generate the

Q ¢(

inductive limit topology for S¢(A) .

The results of the last part of Section 7 are derived from non-trivial measure theory concerning Souslin spaces and Borel spaces.

9

-In the first sections of this paper we intend to describe a Gelfand

triple S

cjJ

(A)

c

X

c

T

cjJ

(A). Let us first explain the meaning of the

indicated symbols.

X

denotes a separable Hilbert space which is

taken fixed throughout the paper, and A denotes an (unbounded)

self-adjoint operator in

X.

The partially ordered set of functions cjJ is

defined as follows.

(1.1)

Ve6inLtion

The set

cjJ

of generating functions on

lR

is a p.o. set with respect to

the usual ordering of functions. Moreover it has the following

prop-erties:

A

I

Each

¢

E

<jl

is a nonnegative Borel measurable function bounded

by

1-A I I

V

¢E<Jl

=1-1jJ

E<Jl

:

4>1

,,;:; 1jJ

A III V V, ::J- :=I

.pE'l>

l)E:lR

XH c>O

A IV

V

¢E:<Jl 3ljJ EcjJ 3

c>O

,\ C

lR

.

Let

(EA)AClR

denote the spectral resolution of the identity corresponding

to A. Then for any

¢

E

cjJ

the operator <jJ(A)

=

J

<jJ('\)dE,\ is bounded and

JR

even "smoothing".

(1.2)

Ve6inLtion

Let ¢ E ¢. Then the subspace ¢(A) (X) of X becomes a Hilbert space if we introduce in ¢(A)X the following inner product

-1

<u,w>¢

=

<¢(A) u, u,w E ¢

(Al

(X)

-1

where

ep(A)

denotes the unbounded self-adjoint operator

f

¢(A) - l dE\

supp(¢)

(supp(<P) = {~EJR

I

¢(A) > O}.)

Now we define the space Sr[)

(Al

as follows

(1.3) Ve6~nition

S

(I) (A)

where the topology 0

ind on S¢(A) is the inductive limit topology

generated by the spaces ~(A) (X) .

We note that the inductive limit topology is the finest locally convex topology for which the inj.::,ct:Lons

it/> :

(4)CAl (X) C+ S<p(A) are cont.inous.

Hence a set V c S<p(A) is open iff V

n

¢(A) (X) is open in the Hilbert

space ¢(A) (X) for all ¢ E ¢. Because of A III and A IV in Definition

(1.1), the space S¢ (A) is Q dense A-·invar iant subspace of X.

Together with the set 1) we introduce the set of everywhere finite

- 11

-sup

I

f (A)¢(A)

I

< 00 •

AElR

We note that

~*

is a function algebra by

A II.

Let f E

~*.

Then f is associated to the normal (unbounded) operator f (A) ,

Because of the definition of

~*,

the operator f(A)¢(A) is bounded on X for all <p E Qi.

(1.5) Ve..M~on

Let f f

¢*.

The seminorm sf on SQi(A) is defined by

IIf(A)wll ,

where

II· II

denotes the norm of X.

We observe that sf is well-defined on S¢(A).

( 1.6)

The..ahem

1. The seminorms Sf' f E l'

*

, are (lind-continuous on Sl'(A) .

II. Let a convex set

n

C SQi (A) be such that for each <p f Qi the set

~

n

¢(A)

(X)

contains an open neighbourhood of 0 in ¢(A) (X). Then

n

contains a set

I. Prom the inequality

the continuity of sf follows.

wE cjJ(A) (X) , II. Put P n n

f

dE>..

+ n-l -n+l

J

dE,\

-n

n E IN . For each n E IN we define r := sup{p n

p

_n(X) A II Pull_n < p ] => u F P_n(Q)}

From A III i t follows that r is well-defined and non-zero. The n

function f is defined as follows

f (A) 2 2

max(~-

,1) n

I>..

I ( [n - 1, n) . ::F We shall prove that f E: <!> •

So let cjJ

E:

<!>. Then there exists X ( <!> and c > 0 such that for all n

E:

IN 2 n sup ¢(I.) i

>..

I

([n-l,n) Lnf

!

A

i

Eln-l,n) ( A) •

To see this we observe that following axiom A IV 2 n sup ¢(A) i

>..

I([n-l,n) < c 1n 2

for some well chosen

Xl E:

<!> and c

1 > O. Applying A IV again, the existence of the indicated X E <!> follows.

- 1 'l

-J

-2 2

{u E X(A) (X)

I

'

x

Pc)

d ( EAU,u) < E: } c Q

n

X(A) (X)

supp(x)

because

Q

n

x(A) (X) contains an open neighbourhood of 0 in x(A) (X).

Thus we get that r

> E:

inf

X

(A) for all n (:

l-l

with

n

IA/EEn-l,n)

supp(~)

n

(En - l,n)

U

(-n,

-n +

lJ)

~

0.

Hence

by

+ for all

1..,11..1

E

En - l,n),

2

n

f(A)¢(A)

< 2(

sup

¢(A))max{-- ,

1}

<

IAIEEn-l,n)

r n

n2 -1

<

2

(sup

~ (A) ) (- ( inf

X(Ie))

+

1) < 2

(~ +

1) •

!A!H n -l,n)

E:

!A!r::[n-l,n)

E:

We next show that

(*.)

Ilf(A)ull

< 1 ~

u

E:

Q •

Let

u

E:

'HA)

(X)

for some

,~

c: '\'.

Then

~

liP

ull

2 < ,"

and

n=l

n

¢

(**)

(~(A)-lp

,u,p u)

<

n n X

sup

(¢(A))

I

A

I

Hn-l

,n) 2

lIulle!>

Because of our assumption

(*)

for all n

E

R

we have

2n

2

P

u

E

Q.

n

Now we express u as follows

U =

N 1 00

I

(2n

2

p

u)

+ (

I

__

l__)u

where

1 -1

( I

~-)

j=N+l 2j2

By (**) and by A II of Definition (1.1) there exists a function ~ E ¢

such that

00

~

4N4

I

(¢(A)-l p u,p u)x

n=N+l n n

4 _ 2

~ 4N sup (¢(A)) (~ lIuil

IAI~

N

¢

2

Since lIu II, -+ 0 as N-+ 00, we obtain uN EQ

n

~(AJ(X) for sufficiently

N 1jJ

2

large N. Moreover for all n E ]['1, we have 2n

P

u Eli. Hence u is a

n

subconvex combination of elements in the convex set li. So u E Q. It is

clear that u

<:

Q

n

¢(A) (X).

(1.7)

The-altern

The locally convex topology generate.s by the seminorms sf' f ( ¢* is

equivalent to the topology Ii.

1 f · '

**

1 L ib natura to de lfle c>'

o

x

(.p**:

~

X is a Borel function such that V

*:

sup

I

f(A)X(A)

I

f f::<t

HlR

- 15

-In the remaining part

of this section we assume the following extra condition on the lattice ¢.

(1. 8) A V

Axiom A V gives rise to a description of bounded subsets and null

sequences in S¢(A) as for strict inductive limits, and also to the

proof of its completeness. With emphasis we note that in general the

inductive limit S¢(A) is not strict. But nevertheless the following

results can be proved.

(1. 9)

Lemma

Let

Be X. B

is a bounded subset of ¢(A)

(X)

for some ¢

E

¢ iff

Be

D(f(A))

and the set {f(A)xl x E B} is bounded in X for all f E:

<!J*.

*

~) Suppose

B

is a bounded subset of ¢(A)

(X).

Then for all f

E:

¢ and

all wEB

Hence each set {sf(w) Iw E B} is bounded.

=I:

<:) By assumption for all f

E:

¢ there exists K

f > 0 such that V

wEB IIf(A)wll ~ Kf

Now define r :0= sup

II P

wll , where again

P

n wEB n X n

define the function X on lR by

n

( f

n-l -n+l +

f )

dE>..

-n , and

x

(A) nr

n for

I

A

I (

[n-1,n) .

We prove that X E

¢* .

So let f E

¢ .

Then we observe that

f

defined

by

f(A) _{n sup}

_I

_{f ( A)}

_{I '}

IAI([n-1 ,n)

I A

I

C [n - 1,n) ,

is also a member of ¢ . Indeed, this follows from the estimation

sup I

f (

A)<p (A)

I

AfJR

sup n ( sup I f (A) I ) ( sup <p (A» ~

nON , AIC[n-1 ,n) IAIE[n-1 ,n)

<

K sup ( sup (

I

f (A)

!liJ(

A) )

neJN I A! EJn-1 ,n)

K sup If(A)\j!(A)

I

< 00 •

AOR

The existence ofK > 0 and l/J f ¢ is derived from A IV.

Now we proceed as follows. Let n E :N and A (JR with

I

AI ~ [n - 1 ,n) •

Then

I

X(A) f (A) I

~

nr sup

I

f (A)

I

nI A!E[n-1,n)

SCll)

ilf

v,CEl

**

Hence X E ¢ . By A V, there exists <p f ¢ and c > 0 such that X~ c·<p. It follows that for all wEB.

n c

2

I [__

1

r-

2 (

r

+ 2 n

J

n=1, n n-1 r

1-0

1:1. \ 1 I !_, p. 1 n -n+1

f

)d(EAw,w)]

-n

~o B is a boundc~

- 17

-(1. 10)

TheM_em

Let

~

fulfil

also axiom

A V.

A

subset

BC

S(jJ

(A)

is bounded iff there exists

~

a bounded subset of

~(A)(X).

(1.11)

Co~ottahY

f

(jJ such that B is

Hence S

-

n

(jJ (A) - f("(jJ*

+

Let u

EX. Let u

E

S(jJ(A) iff u

E

D(f(A»

for all f

E

(jJ:.

D(f(A»

.

In the previous result we have introduced the notation (jJ*

_+'

The subset

<\J*

C

i\l* consists of all positive functions in i\l* which are larger than 1.

+

It clearly makes sense to impose

generated by the spaces D(f(A»,

on S(jJ(A) the projective limit topology

f E <p*. It is not hard to see that

+

D(f(A»

is a Hilbert space with the inner product

(u,w)f

(f(A)u,f(A)w)

,

u,w (" D(f(A»

We denote this projective limit topology on S

A) by

0 ,.

From the

(jJ (

pro)

previous results and Theorem (1.6) it follows that the topologies

o. d

~n

and

G

,are equivalent.

pro)

( 1. 12)

Lemma

Let

Be

S(jJ(A) be a bounded set containing null. Then there exists

~

E (jJ

such that B is a bounded subset of

~(A)

(X). Moreover, a net (wa)aEI

c

B

converges to null in S(jJ(A) if

IIwall~ -+

O.

~)

Let (w )

~I

be a net such that

Uw

U~ ~

O.

Then for all f

E

¢*,

a a _ (j. 'j'

IIf{A)w

II

~

IIf(A)rjJ(A) IllIw

II~ -+ 0 .

(j. a 'j'

~) Since the set

B

is o. d-bounded, there exists ¢

E

Q such that

B

In

(*)

is bounded in the Hilbert space

<p(A) (Xl.

Let s >

O.

Then there is N E IN such that [sup (~( A) 1

I

A

13N

2

< r . It follows that

[sup (rjJ(A» + s2]

IAj;?N

where the positive constant KB depends on

B.

-~

*

since A ,-+

CPU)

"EXt ) (A) + X{

I ( )'

2,U) ] E

(jl -N,N 11 <p p ,?cJ Moreover, there exists a

O

( I

such that (**) <pU..) d(E;\wa,w a}

1

-1 2 < € [IA

<NJ

1\ [¢ 0.)

>s

2 ]

for all a > 0'.0. Thus i t follows that for all a > a O

By A II there exists ~ ( ~ such that

IIw

II

< [. (1 +

21',:,

- 1<)

-From the above lemma the following result

is

immediately clear.

( 1. 13)

TheM.em

Let <P fulfil also axiom A V.

Any bounded subset B of S¢(Al is homeomorphic to a bounded subset of X , where a suitably chosen

¢

E ~ establishes the homeomorphism.

(1.14l Co~ott~y

I. A subset K of S~(A) is compact iff there exists ¢ E ~ such that K is a compact subset of ¢(Al (X). Hence a subset K

c

S~(A) is compact iff i t is sequentially compact.

II. A sequence (unlnON in S~(Al is a Cauchy sequence iff there

exists ¢ ( ~ such that (unlnEIN c ¢(Al (Xl is a cauchy sequence in ¢(Al{Xl. Hence S~(A) is sequentially complete.

In the following theorem we use the standard terminology of locally convex topological vector spaces in order to make a link to the general literature on this subject.

(1.15) _The.o~em

Let <f> _{fulfil also axiom A V.}

I .

_S~(A)

is complete

II.

ScHAl is barreled I I I .

IV. _S~(A) is Montel iff each operator ¢(A), ¢ ( ~, is compact as an operator from X into itself.

V.

S~(A)

is nuclear iff there exists f E

~:

such that f(A)-l is a nuclear operator from X into itself.

I. Let (wa)afI be a Cauchy net in S~(A)' Then (f(A)wa)aEI is a

*

Cauchy net in X for each f E ~ . Since X is complete there

+

exists w(f) E X such that (f(A)w ) ( converges to w(f). Take

a a I

,rCA)

=

1, A E JR. Then ,T(A) is the identity on X, and w -+ w(~)

a

in X-sense. Since the operators f(A) are closed, one has

w(~) E D(f(A)) and w(f) = f(A)w(~r). The result follows by

applying Corollary (1.11).

II. Let

V

be a barrel, i.e. a radial convex circled and closed

subset of S~(A)' Then

V

n

¢(A) (X) is a barrel in the Hilbert

space ¢(A) (X) for any ¢ E~. SO there exists an open null

neighbourhood L¢ C V

n

¢(A) (X). Then following Theorem (1.6.11),

V contains an open set {wlllf(A)wll < s} for some f E

~*

and E: > O.

III. Every circled conV2X SUbset. ~! c: S, " d',at abso:ct>s ("very bounded

y (Ai

subset Wc S~(A) is to be a neighbourhood of

O.

Let

B¢

denote

the unit ball in ¢(A) (X), ¢ E ~. Then clearly one has E:B¢ c ~

n

X¢

for some E: > O. Again applicat lon of Theorem (1.6.11) lectds to the

- L1

-IV.

~) Consider the following scheme.

x

ljJ (A) ) X

" (Ai

2

j

j"

_(AI

1/I(A)2{X) C r 1/1 (A) (X)

From this scheme the following is clear. Let B be a closed and bounded set in S~{A). Because of Theorem (1.10) and axiom A II there exists 1/12 E

~

such that B is closed and bounded in ljJ{A)2{X). Since

~(A)-2B

is closed and bounded in X and since 1/1 (A) is a compact operator from X into itself, it thus follows that B is a compact subset of ljJ{A) (X).

<=)

V.

~)

Let (u ) (_{n n -:IN} be a bounded sequence in X. Then (1/1 (A) u ) elN is a_n n,,-bounded sequence in 1/1 (A) (X). SO the closure of the set C

C = {1/1 (A) u

I

n ( IN} n

is compact and hence sequentially compact in S~{A). It follows that the sequence (1/I{A)u_n)nE:IN contains an S~{A)- and hence X-convergent subsequence.

*

-1

Suppose there exists f E ~ such that f(A) is nuclear. Then the +

operator A has a discrete spectrum because f- 1 {O(A» = O{f- 1 {A» by the spectral theorem. Let {ej)jClN denote the corresponding

orthonormal basis of eigenvectors _{and 0,.). its eigenvalues.} J ]ON

::j: ~

Let y ( ~+ and let Sg denote the completion of S~(A) with respect

to the seminorm Sg. Put h

=

f.g. Then the injection Ih,g : Sh ~Sg

is nuclear, because 1

w

~ h,g

)~

j=1 -1 - 1 - 1 EO.) (h(A)w,h(A)(h(A.) ('))<j(,\.) \'. J J J ] ]

-1

We note that (h (;..) e ) 1.S an orthonormal basis in Sh and

j j JON

-1

-(g(;".) e.) is an orthonormal basis in S .

] ] g

~) Let S~(A) be a nuclear space. Take g =~. Then there exists f f ~::j: such that the mapping

I

:

Sf y..X

-1

23

-Since the p.o. set

¢

consists of Borel functions bounded by 1, we

may as well suppose that ¢ is closed under pointwise multiplication.

Then it makes sense to introduce the following space.

The space T¢(Al consists of mappings

F

:¢

+

X

satisfying

The projective limit topology

T

proj

on T¢(Al is the locally convex

topology induced by the seminorms

The Hilbert space

X

can be embedded into T¢(Al. To this end, we

define ernb ;

X

+

T

¢

(Al by

ernb(xl

(~) = ~(A)x

,

x E X,

~

E

<P.

( 2 • 2 1

T he.M,em

emb

(X) is a

dense subspace of T

A .

(', ( 1\)

Put

Since

x

_(-n,n) ~ c .<p •

J J

for suitably chosen c. >

a

and 4>. (: <P, we derive that

J J

-1

c.<p.(A)) IT ]

] ] n c.F(¢.)_{] ]}

EX.

So i t is clear that G , defined by

n

G

n <p t+ IT F(¢)n

is an element of emb(X), for each n

E

IN. Moreover, for all

¢ (

¢

IIG (

¢ ) - F (<p)

II

=:

II

or -

I) F ( <p)

II

-+

0 ,

n

n

*

Let f E <l> , and let x EX. Then fox defined by

n -+ 00

o

(fox) (¢) _{¢ E ¢}

is an element of

T

¢ (A). As a conseq'le:nce of the first part of the next

theorem i t follows that any element of T¢(A) arises in this way.

We introduce the following notation. Let V C XI then foV =: {fox/xC V}

(2.3)

The.oJtem

I.

A

set B C T¢(A) is bounded iff the set {F(<p) IF ( B} is bounded in

X

for all <p

E

¢.

is bounded iff there exists f

II. A set B C T¢(A)

set So c X such that B =: foSO.

*

25

-Observe first that by axiom

II

~)

As usual we take P

n

The proof of I and II

~)

is trivial and it is omitted.

n

-n+1

(f

+f

)dE

A•

n-l

-n

-1

A IV there exists

~

E

¢ such that

~(A)

P

n -1

is bounded on

X.

Put PnF =

~(A) PnF(~)

for each F

E

T¢(A). Then

P F

E X

and r

;= sup liP FII

~ II~(A)-lp

II sup

IIF(~)II

< 00

n

n

FEB

n

n FEB

Define the Borel function f by

f(A) =

nr

n

IA[

E [n - l,n)

,

nElN.

We prove that f E ¢*. To this end notice that for all

A, IAI

E [n - l,n)

and for each

~

E ¢

nr

~ (A) ~

nr

sup

(~ (A» ~

c sup II P F

(X)

II

.

n

n

I

A:

EI

n

-1,

n)

FfB

n

Here

X

E ¢ and c

>

0 have to be chosen as indicated in axiom A IV. The

'*

choice does not depend on n E IN. It follows that

f

E ¢ .

Now let FEB and put

00

P F

I

n

X

F

nr

n=1,

n

r

r!O

n

Then we have

00

liP FII

2

00

I

n

~

I

1

2 2

2

n=l,

n r

n=l n

r

r!0

n

n

Hence x

F

(

X

and the set B

O

{XF!F E B} is bounded in

X.

It is clear

We can replace ¢* in the above theorem by ¢*. +

(2.4)

Lemma

Let B be a bounded subset of T¢(A) which contains zero. Then there

exists 9 E ¢* with the following properties

+

I. There exists a bounded subset

8

0

c

X

such that B = gOB

O

.

II. Let (F ) E- be a net in B. Then (F ) c converges to zero iff

(J. a _I a (J.cI

F gox , a E:' I and

II

x

II

+

o.

a

a

a

In virtue of Theorem 2.3 there exists f E

~*

and a bounded subset

+

B

₁

c

X

such that B = foB

1

•

Put g(A)

2 2 -1

(1 + A ) f (A) and put B0

=

(1 +

A)

B1 .

Then clearly B _gOB

O·

For each a E I there exist xa

E

Band y0 a

E

B

₁

such

that F

a

foya goxa .

2 -1

Now let f: > 0, and take N E IN so large that (1 + A) < € for all A, with

(* )

IAI ~ N. Then we obtain

f

d(EAxa,x_a)

IAI~N

2 (; E

_f

_d(EAya,y a) (;

IAI~N

Next observe that followi'l:) dxi2El p, III dTlU .A PI l:her~; ,"'dst" constants

c. > 0 and functions ¢. E ~ such that

J J -1 g(A) X(-N,N) (\) (; We thus derive ":J=1. ~;_-6 .

_,

i (** )

r

~

d(E,x,x) <c, /\ a a IA <N L _j

II. A

- 27

-2 Hence there exists a

O

E I

such that _A<N

5

d(EAx_a IX_a) < E as soon as

a > aO. Together with

(*)

we get

for all a > aO.

Now we obtain by the previous lemma.

(2 • 5)

TheM.em

Each bounded subset B of T~(A) is homeomorphic to a bounded subset

of X. The homeomorphism is established by a well chosen 9 E

~:.

(2.6) COJtOUM!1

I. A subset K of

T~(A)

is compact iff there exists f (

~:

and a

compact subset V of X such that K

=

f-V. Hence K c T~(A) is

compact iff i t is sequentially compact.

sequence (Fn)nElN c T~(A) is fundamental iff there exists

*

f E ~ such that (F ) c is a Cauchy sequence in foX.

+ n n,-lN

Hence T~(A) is sequentially complete.

(2.7)

TheoJteJn

The space

T$(A)

with topology lproj is complete.

net in X for all ¢ E <P. Hence there exists x¢ ~ X, ¢ E <P , with Fa (,~)+ x¢.

Now put F : ¢ 0-+ x<p, <P (<P • Then

lim F (¢oljJ)

a a ¢ (A) lim Fa a (ljJ) ¢ (A)F(ljJ) .

Hence F E T¢(A) and the space T¢(A) is complete. o

In the remaining part of this section we assume that the lattice ¢

satisfies axiom A V, i.e.

::J

c" 0

x

~ c¢

The function space ¢* consists of all functions in ¢* which are +

larger than 1. It can be seen easily that (¢*)* = ¢**.

+

*

Let f

E

<P+.

We recall that foX denotes the subspace of all F f T¢(A)

for which there exists x ( X such that F(¢) = f(A)¢(A)x , ¢

E

<P.

The space fo X becomes a Hilbert space with the inner product

(2.8) (x,y) F,'G (.-= £

X

-1 -1

f(A) F and y - f(A) G iff fox F and goy G.

From Theorem 2.3 we obt2in

- 29

-It IIIdkl':; :;('11:;(' LUlnlruducl' III<' l.ndIICLi.vl.' .limit I Ul)()J(Hlyl , d ull

~n

TW(A)

generated by the Hilbert spaces foX. So a set VC T~(A) is

:j:

open iff

vn

foX is open in foX for all f f W . + The following theorem is similar to Theorem (1.6).

(2.9)

The-ahem

1. The seminorms t¢1

¢

~

W,

are continuous with respect to the inductive limit topology Lind on T~(A)'

*

II. Let a convex set ~ c

TW(A)

be such that for each f E ~+ the set ~

n

foX contains an open neighbourhood in foX. Then ~ con-tains a set

v

<PIE:·= {F E T

~

(A)

IIIF(<p) II <

d

for some E: > .0 and ¢

E:

~.

I. Let <p E ~. Then for all F C foX

From this observation the wanted continuity of t¢ follows. II. The proof of this part is similar to the proof of part II of

Theorem (1.6).

For each n C ~, put

p

n

and define

f

dE'

r "= sup{p > OI[F C emb(P (X)) /\ liP FII < pJ ~ P FEst}.

n n n n

We recall that P F can be interpreted as an element of

X

in the

n

following way: P F

=

~(A)-l

P

F(~)

where

~

€

~

is taken such that

n

n

(X[ + X J) ~ eli; for some c > O. Define the function G as

n-l,n) (-n,n+l

follows

G (:\) i f !AIE [n - 1,n) .

(ef. the

To this end, let f

E

¢*.

Then f

+

- *

element of ¢ .

+

We shall prove that

G E

¢**

sup (f(A)) is an

I

A

I

EJn-l,n)

proof of Theorem (1.6)). So there exists 0 > 0 such that

- 2

defined by f(A)

=

n

{F E

f·

XIII FII

f

<

o}

C $(,

n

f·X

because

n n

~·X contains an open neighbourhood of

O.

Thus we find

2 > cS n that r n So for each sup (f(A)) IAIEIn-l,n)

n E :IN and all

I

A

I

E

for all n C IN.

[n - 1,n) 2n2

o

(A)flIt) ~ r n 2 sup ( f ((p )) ~ ().

1,\

JEIn-l ,n) ::f::4:.

It follows that 8 E ¢ . By axiom A V there exists c > 0 and

¢

E ¢

such that G ~ c¢.

We next show that i t is enough to put s 1

c i.e. we prove'

1

II F (¢)II < - ~ F E ,L c

\1 ~

Let F E

foX

for some f E

~:.

Then

w

I

/lPnF/I~

< 00 n=l and

II

A

2)-3PnFllf 2 -3

11 II

(*) (I + ~ (1 + (n - 1» F f 1

Because of our assumption /IF(<j>)/I < - we get c

(**) 2n 211P FII

~

r cIIP F (¢)II

~

r

n n n n

for all n

E

IN. Now we express F as follows

where F N

L

1 (2n2p F) + (

L

~l~)F

n=l 2n2 n j=N+1 2n2 N 00 00 Put g(A)

( I

~)-1(

I

P

F) j=N+1 2j n=N+1 n

(1 +

A

2

)3 f

(A).

Then g

E

~*,

because of A IV. By (*) we get

4 2 -3

~F /I ~ 4N (1 + N) /lF~f N g

and so i t follows that F

N

E

~

n

goX

for sufficiently large N E ~ .

Moreover, by (**) for all n E :IN , 2n P F E2 [2.

n

Hence F is a subconvex combination of elements in the convex set ~.

So F ~ ~. It is clear that F E Q

n

foX.

An immediate consequence of the previous theorem is the following interesting result.

III.

(fog) 0 X

bOUllded in T

cp(A) - 'l'hcn 8 is closed and bounded ¢*. From the above scheme i t is clear that B is

+

is Montel iff there exists f

E

¢* such that f(A)-l

+ is barreled

is bornological

is a compact operator from X into X.

and hence Hilber-Schmidt.

T¢(A) is nuclear iff for all ¢

E

¢ the operator ¢(A) is nuclear

f(A)-l X ~~---~~---~+ X q(A)

1

1

f(A)g(A) II. I. (2.11)

The.OfLe.m

I I I . (2. 10)

The.OfLe.m

The topology 'proj on T¢(A) is equivalent to the topology Lind.

in yoX for some g ( Let B be closed and IV.

The proof of I and II runs completely the same as the proof of

Theorem (1.15. II and III). Therefore i t is omitted.

- II

-a comp-act subset of (f·g)·X. Cf. Coroll-ary (2.6).

~) Let (x )_{n nE:IN} be a bounded sequence in X. Then the sequence (xn )nElN is also bounded in T<p (A)' and hence ~I n E :IN} is a

compact subset of T<p(A)' SO for each ~ E <p the sequence (~(A)xn)nEm contains a converging subsequence. It follows that each operator ~(A), ¢ E (~ , is compact. Hence the projection nn

=

X (-n,n] (A) is a compact operator, cf. the proof of Theorem (2.2). Thus we obtain that A has

discrete spectrum without accumulation points. Put f(A) = 1 +

;.2,

A

E:

lR.

*

-1

Then f E <P and f(A) is a compact operator on X. +

IV

=» Assume that all operators ¢(A), ¢

E

<P, are Hilbert-Schmidt. It follows that A has a discrete spectrum

and eigenvectors (e.). ON' Let ~ E <P. Then ]

J-with eigenvalues (A.) ''IN

J J-'

J :

t

¢

~

t

;p2,

where

T

~

2

-and T~ denote the completion of T<p(A) with respect to t¢2 and t~, respectively. Since] can be written as

JG

co

I

~(A) (G,~(A

)-l e

)~ ~(A

)-2 e

n=l n n n ~ n n

-1

where (G,¢(A) e ) = (¢(A)G,e ),

J

is a nuclear map.

n n <p n

=:l

For each X

E:

<P the embedding

J

Then there exists ~ E <P such that

x

e+t

is bounded. Let <p E <P X

is nuclear. Hence the injection X e+T¢ is nuclear. It follows that

In order to make S¢(A) and T¢(A) into a dual pair, we introduce the sesquilincar form <,> on S,]>(A) x T<j'(A) by

~lcre ~ E ¢ is such that u f ~(A) (X). This definition does not depend

on the choice of ~. To see this, we note that for any u C ~(A) (X)

n

1jJ(A) (X)

The sesquilinear form is non-degenerate. I f <u,F> 0 for all F _{C T<P(A) ,}

then u = 0 because

_X

_{c T<p(A)' Conversely} i f <u,F> _{0 for all u E:}

S<p(A)'

tlwn (f(A)u,x) - _{0 for all u E}

S¢(A) and F = fox. Since D(f(A)) is

dense in

X,

we obtain x 0 and hence F =

O.

(3.1) Theo~0m (Riesz-Fischer)

1. _{A linear functional} 9, on

S

is continuous in the topology o;nd

,]>(A) L

iff there C'X~f' \1(' Lhe,,!:

9,(u) = <u,G> ,

II. A linear functional m on

_T

, i s

¢ (A) c:on-;:.~_~c.nuous in the topology T_pro].

of T¢(A) iff there exists w

E S

stech that

- 35

-1. =» Let GET¢ (Al. Then for all w E~ (Al (Xl

Hence £ defined by £(w) <w,G> is continuous on each ~(A) (X) and hence on _{(S<j>(Al' oind) .}

1. <=) Let Q, be a continuous linear functional on (S1l (A) , o. d)_l.n ~ Then £ 0 _{~(A) is a continuous linear functional on X for all ~ f ¢. So}

there exists x~

E

X such that

(2 0 ~ (A) ) (x)

=

(x,x¢)

Put G ~ ~ x~, ~

E

¢. Then i t is easy to see that G(~~) Hence G (T¢(A). Moreover

1/I(A)G(~).

<u,G> .

II. =» Let w 1/1(Alx with x E: X and 1jJ E ¢. Then we have

So the linear functional F ->- <w,F> is continuous on (TI' (A)' T .)

( pro]

II.<=) Let

m

be a continuous linear functional on (T¢(A) ,T

proj)' Since 1l is a p.o. set there exists c >

a

and a function ¢ E ¢ with

(*) !m(F)

I

~cIIF(¢)IIX·

f

-1

f

-1

Put

P

dE

A

and ¢(A)

¢(A)

dE

A

.

It is clear that ¢ sUPP(<jl) supp(<jl)

¢(A) (X) is a dense subspace of

P

¢ (X) (a Hilbert subspace of Xl. Moreover, the linear functional m 0

¢(A)-l

is bounded on

¢(A)

(Xl

by (*l. Hc'ncC' there exists y (

r

(Xl such t.hat the continuous 0xt"c'nsion

'I'

-1 -1

m

0 ¢(A) of

m

0 ¢(A) to

P¢

(X) satisfies

m

0 _{(x,y) ,}

x

EX.

-1 .

Hence m(F) =

m

0 ¢ (A) (F(¢)) = (y,F(¢))

Now put w

=

¢(A)y. Then m(F)

=

<w,F>.

From the previous theorem i t follo\,s that the spaces S<jl(A) and T<jl(A) a dual pair. Thus it makes sense to introduce the weak topologies

are []

o (Sq, (A) ,Tq, (A)) and oCTq, (A) ,Sq, (A)) on Sq; (A) and T<jl(A)' respectively. I t is

a natural question whether weak boundedness of a subset of Sq,(A) (Tq,(A)) implies its boundedness.

(3.2)

TheOJteJn

(Banach-Steinhaus)

I . _{Each weakly bounded set B in}

Sq,(A) is bounded. II. _{Each weakly bounded set}

_V

_in

Tq, (A) is bounded.

I. Let B be a weakly bounded set in S i~\' So for each G ET<p(A) there exists M

G > 0 such that

'*

Now let f E <p • Then fot d.ll. x E X IJe have

- 37

-From the Banach-Steinhaus theorem for Hilbert spaces i t follows that there exists M

f

>

0 such that

wEB.

*

Since f ( ¢ is taken arbitrarily, i t follows that

B

is bounded.

II. Let V be a weakly bounded set in T¢(A)' Then for each u E S¢(A)

there exists M > 0 such that

u

V

G(B

I

<u,G>

I

~Mu

Let

¢

C

W. Then for all x E X we have

V GCB

So there exists K¢ > 0 such that

G (

V .

o

In the remaining part of this section we assume that the p.o. function

set ¢ satisfies axiom A V, see (1.8). Then we have the following theorem

which is similar to Corollary (1.14).

(3.3)

The.altern

Let ~ fulfil also axiom A V.

1. A sequence (un)nCN C S~(A) is weakly convergent to zero iff there

exists

¢

E ~ such that the sequence (u ) ( converges to zero

n n ]N

weakly in ¢(A) (X), i .e. for all x E: X.

( ¢ (

A)

-1 u , x ) -+ 0 . n

II. A sequence (G )

E:

eTA

is weakly convergent to zero iff nnlN ¢ ( )

there exists f E ¢* such that (G ) c

foX,

G

fox,

and (x ) ON

+ n n n

nn-tends to zero weakly in

X.

1. Since (un) nON is weakly bounded in S¢(A) i t is also strongly

bounded in S¢(A). SO by Lemrrla (1.9) there exists

¢ (

¢ such that

(u ) EJN is a bounded seql1ence in the Hilbert space ¢(A)2 (X). It

n n

follows that there is a positive constant M such that for all

n CIN

r

-4

J

¢(A)

d(EAun,u n) supp(¢) Now take A

O

so large that for

IAI

> A

O

'

¢(A)

< s, where S > 0 is

taken arbitrarily. Put

A

= {A E

JRI(I"A!

> A

O) v (¢(A) <

s)L

AO

,

E:

Then we have

2 2 < E: r.~

Moreover, let x E X be taken arbitrarily. Then there exists N E IN

~II

Hence

I

«I)(A)-lu ,x)

I "

c(l + Mllxll) for all n " N.

n

The converse of statement I is trivial, i.e. any weakly convergent

sequence (un)nE~ c ¢(A) (X) is a weakly convergent sequence in S~(A)'

II. Let (Gn)nEIN C T~(A) be weakly convergent to null. Then the sequence

=l=

(G)_{n nE:}-IN is T_pro].-bounded. So there exists g E ~₊ such that (G ) is_n

bounded in g-X. Put G ~ goy. Then there exists D > 0 such that

n n k

Ilynll

~

D for all n E::IN. Put Ilk

=

f

dE

A, k E::IN. Then

-k

2 -1

Now let € > 0, and take k E IN so large that (1 + k ) D < €.

By A IV of Definition

(1.1),

there exist constants c

1

, ...

,cjk and functions ¢ 1 " " ' ¢ ' ( ~ such that

]k

(*) _{(1 + A)}2 -1_g(A)-1_{X(-k,k) (A)}

_~

_{c.¢.(A) ,}

] ] A E JR •

Now let x E X. By (*) we obtain

So there exists NEill such that for all n > N

< €

Put x

+

With f

:( c(l +

Ilxll)

2

A~ (1 + A )g(A), thG proof is complete. LJ

As a consequence of the previous result we obtain.

(3.4) COfLOUMIj

1. Each bounded sequence in S~(A) contains a weakly convergent sub-sequence.

ones,

II. Each bounded sequence in T¢(A) contains a weakly convergent sub-sequence.

Comparing the results of this section with the results in the previous i t is not hard to see that

a.

d ~ (J on S a n d T. . ~ T. d

~n proj ~(A) proJ ~n

on T¢(A) are equivalent to the Mackey topology on each of these spaces. We note that this observation seems to be no longer true if axiom A V is dropped.

- 41

-Let 1>1 and 1>2 be two generating function lattices as introduced in

Definition (1.1), and let

Al

and

A

2 be self-adjoint operators in the

Hilbert spaces Xl and

X

2

,

respectively. In this section we characterize

the continuous linear mappings from S~

(A )

into S~

(A )

and from

1 1 2 2

T~

(A )

into T~

(A ).

Also we introduce the notion of extendibility

1 1 2 2

for linear mappings from

S

into S~

(A ).

~

1

(A

1) 2 2

( 4 • 1)

T

he.O!te.m

-~----Let R be a locally convex topological vector space, and let Lbe a

linear mapping from S~

(A )

into R. The mapping

1 1

L~1 (AI) : Xl ~ R is continuous for all ~1 E ~1·

L

is continuous iff

By the definition of inductive limit topology, ~1(AI) (Xl) is

continuous-ly injected into 51>

(A ).

So if

L :

51>

(A )

~ R is continuous i t follows

1 I 1 1

that L~l (AI) : Xl ~ R is continuous.

~) Let L~ denote the restriction of

L

to ~1

(A )

(Xl). Then L~ is

1 1 '1'1

continuous on ~l(AI) (Xl). So if ~ is an open null neighbourhood in R,

then

L-1(~)

n

~1

(AI) (Xl)

=

L;l(~)

is an open neighourhood in

1

~1

(AI) (Xl)· I t follows that L-1

(~)

is open in 5q, (A ). 0

In the following theorem we characterize the space

L(S¢

(A

),S¢

(A ))

1 1 2 2

of continuous linear mappings fnyu) Scp1(Ai) into S¢2(A 2

) •

(4.2)

The.OfLe.m

I. Let

P

be a linear mapping from

S¢ (A )

into

S¢ (A ).

Then

P

is

1 1 2 2

continuous iff for all ¢1 E CPl and for all f

2 E

¢i,

the linear II. Suppose ¢2 satisfies axiom A V.

Let

P

be a densely defined operator in Xl wi-th

P :

D(P) -+ X

2 and D(P) :::J

S<jJ (A )' Then P is a continuous linear mapping from ScjJ (A)

1 1 1 1

into S iff one of the following conditions is fulfilled. cjJ2(A

2)

(i) _{For all ¢1} E _{¢1 and for all f 2}

E:

_{<1>2 the linear operator}

r/'

(1\)

1',1'

1

(1\)

It· ( )I~I

_Xl

iIII ( )

_X;>

i ,; I )()lllld,·<I .

(ii) For each ¢1 E '1>1 there exists ¢2 € cjJ2 such that the linear operator -1

¢2

(A

2) P¢ 1(Ai) is a bounded linear operator from X1 into X2·

I. 'l'he proof of thispar' olLy"c; >;''lTk;:',:i" f J'( from the previous

theorem and the fact that the inductive limit topology on 5 '1>2

(A

2)

is equivalent to the locally convex topology generated by the seminorms Sf

\'1~"

il C ..

(A,.,)

II,

f E <1'*

' 2 ';::

2

-) Let wI

= ¢

(A)x

E S A ) ' Then for all

f

2

C

~~2_

it

folloWA

1

1

1

'1'1 (

1 "

that

P¢l

_{(A 1)x 1}

E

_{D(fZ(AZ))' So by Corollary (1.11)}

P¢1

(A

1

)x

1

E S¢

(A )'

2 2

Hence

P

maps

S¢

(A ) into S¢ (A ). Applying Part I of this

1 1 2 2

theorem, i t follows that

P

is continuous.

II.il . •) The set

{P¢l(Al)xl~xl E

X

1

,Ox

l

O= 1}

is bourided In S¢l(A

1

)'

So there exists ¢z

E ¢2

and c

> 0

following Lemma (1.9) with

operator from Xl into

X 2•

¢2

E

¢

such that the operator

2 .

'*

Then for all f

2 f 4'2

the operat.or

-J

If for each ¢1

E

¢l

there exists

-1

A

¢2(AZ)

~Pl ( 1) is bounded.

o

(4. 3)

COfWUaJtlJ.

Let

Q :

Xl

+

X

z

be a closable operator. If

D(Q)

~

S

and if

<t> 1 (A

1

)

Q(S<t> (A ))

C

S~

then

Q

is a continuous linear mapping from S

1 1

""z

(A

Z) <Ill

(AI)

into S¢ (A ).

z z

*

Let ¢l E <PI and let

f

_{Z E <P Z}

,+

Then the operator f

2

(A

Z

)Q¢l (AI) is an

everywhere defined operator from Xl into X

operator (f

2

(A

2)Q<P 1

(A

1

»*

is densely defined (its domain contains

-1 f

2

(A

2)

(D(Q*)}),

It follows that f2

(A

2

)Q<P

1

(A

l) is closable and :I'

hence a bounded operator from

Xl

into

X

2

·

Since

<P

1

E

~1 and f

2

E

~2,+

are taken arbitrarily, the result follows from the previous theorem, - 0

Next we characterize linear mappings from

(4,4)

TheoJtem

of continuous

Let

R

Xl

7

X

2

,

Then

R

can be extended to a continuous linear mapping

from such

T~

(A )

into T~

(A )

iff for all

<P2 E

~2 there exists

<P l E

~l

1

1

2

2

-1

that the operator

<P

₂

(A

2

) R<P

l

(A

l) is bounded in

X

10n

<P

l

(A1)X

l .

If the function lattice ~l satisfies axiom

A V,

then the previous

condition can be replaced by: for all

~) Let

<P

2

E

~2' Then there exists <P

1

E

~1 and c >

0

such that

into

X2'

=-) Wu definc~

I<

on T,p1(A,

L 1

45

-This definition does not depend on the choice of

¢1

E

<I>1.

It is clear that RF E

T<I>2(A

2

).

The continuity of

R

follows from the inequality

II~) We define R in

T<I> (A )

as follows 1 1

<1>2

E

<I>2

is the continuous extension to . Xl of the operator

¢ 2

(A

₂₎

Rf

~

(Ai)· Hence R maps

T<I> (A )

into

T

1>

(A ).

1 1 .2 2

The continuity of R follows from the definition of inductive limit

*") Trivial.

o

(4.5)

CoJWliaJl.lj

Let

<I>2

satisfy axiom A its dual mapping

P' :

v. Then P :

S<I> (A )

~

S<I> (A )

is

1 1 2 2

T1>

(A )

~ Tq,

(A )

is continuous.

2

2

1

1

continuous iff

Finally we introduce the notion of extendibility.

(4.6)

VeOi.-iUUon.

mappings by

E(S<I>

(A ) ,

5<I>

(A

».

1 1 2 2

T"

(A )

such that

ff

=

E

I 2 . :>. 5<I' (A) •

1 1

We denote the space of extendible

A continuous linear mapping

E

from 5

<I>

1(Ai) into 5<I>

2 (A

2

)

is called

ex-tendible if there exists a continuous linear mapping

E

from

T<I>

(A )

into

1 1

(4.7) Theoltern

Let

E

be a continuous linear mapping from S~l

(A

l )

~l satisfy axiom A V. Then

E

E E(S~

(A )

,S~

(A ))

*

1 1 2 2

and

E

(S~

(A ))

c

S~

(A ).

2 2 1 1

~) Observe that

E*

is contir.'lOl by Corollary (4.3). Put

clearly

E

is continuous by

-

*

1 E = (E ) : Corollary T~

(A )

~ T~

(A )'

Then 1 1 2 2 (4.5) and E~S (A) 00 E • I <P1 1

~) For x

E

D(E) and w

E

S~

(A )

we have

2 2 (w,Ex)X 2 <w,Ex>X 2

-*

- I

*

It follows that E ~ E and

E

(S~

(A ))

c

S

2 2

~l(Al)

o

(4.8) COltOliCULlj

E E E(S", (A)' S", (A ,) iff O(E) :) SI; '.~. \ '

'" 1 1 '" 2 'L ' ,co .1· 1·

E(S~

(A)) cSq> ( A ) ' F tS_cP (A)) c SCP1(A

l) .

47

-5.

A

~~vey

06

Atta~'~ theo~y O~ GB*-atgeb~

The and

c space S¢(A) establishes a natural core for the *-algebras ¢(A)

cc c

¢(A) of closable operators. Here ¢(A) denotes the commutant in L(S¢(Al) of the set ¢(A), and similarly ¢(A)cC its bicommutant. In Section 6, precise definitions of ¢(A)c and ¢(A)cC will be given and i t will be proved that both *-algebras are GB*-algebras. The notion of GB*-algebra has been introduced by Allan in his papers [Al 1J and [Al 2].

We give a short survey of his theory here.

Let A be a locally convex topological *-algebra over the field of complex numbers. So the mappings p~ pq, p~ qp and p~ p* are

continuous in A. An element pEA is said to be bounded if there exists s E

~

, {O} such that the set {(sp)nln E

~}

is bounded in A. The

set of bounded elements is denoted by A O.

Let

B

denote the family of all bounded absolutely convex closed subsets of A with the property that for all

BEB

(S.l)

(5.2)

B

=

B*.

Now a GB*-algebra is introduced as follows.

(5.3) Ve6~nition

A locally convex topological *-algebra A with unit ~ is called a GB*-algebra if the folloWing conditions are fulfilled.

(i) A is sequentially complete.

(ii) A is symmetric, i.e. for every pEA the element (~ + p*p)

is invertible in A and

(~

+ p*p)-l is bounded.

(iii) The family

B

defined by (5.1) and (5.2) has a maximal element

B

O with respect to set inclusion.

Remcvdz

Allan uses the notion of pSE:\l<:lncompleteness in his definition of a GB*-algebra. However for our aims i t is enough to consider sequential completeness which implies pseudo-completeness.

Let A be a GB*-algebra. Then the *-algebra

is a B*-algebra with respect to the Minkowski norm induced by B O. (Cf. [Ai 2], Theorem 2.6.)

(5.4) P~opo~~on ([Ai 2], Proposition 2.9)

is also a GB*-algebra. The maximal element of the family

{B c C , B bounded, absolutely convex closed, B2c B, B*

is just the set

B,

~ = 8l.__0

n c.

(ii) If A is commutative then A

For a commutative GB*-algebra Allan has proved a "non-bounded"

extension of the Gelfand-Naimark theorem. In order to show this,

let A denote the set of all multiplicative linear functionals on

AO

A(B O)' i;e. the spectrum of the commutative B*-algebra A

_O

.

So A is a compact set in the topology o(A,A

O

). The Gelfand

transform on A

O

is an isometric *-isomorphism from A

_O

onto the

B*-algebra of continuous complex valued functions C(A), with norm

II

fll '" sup

I

f (X)

I .

Each element X E: A has an extension to the whole

XfA

algebra

A.

(5.5) P~aro~~t{on

([Ai 21, Proposition 3.1)

Let A be a commutative GB*-algebra with identity and let A be the

spectrum of AO. Then corresponding to each X E A there is an

ex-tending complex-valued function X

e

: A

~ ~*

'"

~

U

{oo}

such that

(i) (ii)

e

X

is an extension of X

for pEA,

~

E

~

with the convention 0

0 00

o.

e e

provided that X

(Pi)

and X

(P

(iv) for Pi ,P2 E A, provided that (Xe (Pl) ,X8(P 2))

~

(00,0) or (0,00). (v) for pEA, e X (p*) Xe (p)

with the convention 00 = 00

(vi) the set N = {X E Alxe(p) ~ oo} is a nowhere dense closed subset of p

A.

The explicit formula for Xe can be obtained as follows. Let pEA be hC'rmitGi.m. Then Allan proved that there exists tl E a: such that (p,-r - p)

A -1 e X ( (tl,-r -1 -1

is invertible in and (tl,-r

-

p) (" _A

O'

We put X (p) = tl

-

p) )

-1 e

if X(tl,-r - p) ) ~ 0, and X (p)

=

otherwise. So we can extend X to all hermitean elements of p.. ,md next t,y 1 Lnearity to the whole of A.

(5.6) Ve6~~a~

A collection F of a:*-valued functions on a LOpOlOglcdL space f is

called a *-algebra of functions if each f E F takes the value 00 on an at most nowhere dense subset of

f.

Moreover for any f,g (

F

and

a,S

r-

a: the functions o.f+ 13g, 1'-g and f* = ar", pointwise well defined on

a dense subset of f on which f and g are finite.

Each of all those functions has a unique extension to a a:*-valued continuous function jp •

- 51

-(5.7)

The-altern

([Al 2], Theorem 3.9)

Let A'be a commutative GB*-algebra with identity and let

A

be the

spectrum of A

O

. Define the mapping

on A,

p H·

p,

p

fA, by

P

(X)

co

Xe (p)

~

where X E A. Then

~

is a *-isomorphism of A onto a *-subalgebra A of

continuou~;<r:*-Vil1ued funct.ions on II. 'l'hp Illappinq ~ ('xt('n(l~; the \1Slli11 Gelfand transform for A

let <Jl(A)

G.

The. aXgeb!Lcu

1'(A)c

and

1'(A)cC

Let l' be the function lattice as introduced in Definition (1.1), and

{ ¢(A)

I

¢ E 1)}. Further, ll't L(5 A)) denote the set of

(I) (

continuou~; Unc:ilr mappinqs on 5(1)(A). We note' tlwt 'I)(AJ c L(S,!)(AJ)·

In the following definition we introduce two subalgebras of L(Sw(A))

which are connected to w(A) in a natural way. As a main result we

prove in this section, that these snbalgebras are GB* in the sense

of Allan. In the next section we shall construct a functional calculus

for one of these GB*-algebras.

(6.1) Ve6i~~on

is called its bicommutant.

(6.2)

Ve6inition

The uniform topology

T

u for L(S<j)(AJ) is the topology induced by the

seminorms r f ,¢.

II

f

(A)

L

¢

(A) II ,

where f E

,~*

and ¢ E

w.