On the nuclearity of the spaces $S_{\Phi(A)}$ en $T_{\Phi(A)}$

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On the nuclearity of the spaces $S_{\Phi(A)}$ en $T_{\Phi(A)}$

Citation for published version (APA):

Martens, F. J. L. (1985). On the nuclearity of the spaces $S_{\Phi(A)}$ en $T_{\Phi(A)}$. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8602). Technische Hogeschool Eindhoven.

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

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 86-02

ON THE NUCLEARITY OF THE SPACES

S<p(A) AND T<p(A)

by

F.J.L. Martens

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

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ON THE NUCLEARITY OF THE SPACES SQ>(A] AND T Q>(A) by

F • J. L. Martens

Eindhoven University of Technology,

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

ABSTRACT

This paper contains manageable necessary and sufficient conditions for the nuclearity of a class of locally convex topological vector spaces which are both projective and inductive limits of Hilbert spaces,

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- 1

-1. INTRODUCTION

In the first section we consider dual systems of topological vector spaces, S<t>(A) and T <p(A)' as described in [EGK]. These dual systems are characterized by a directed set <t> of Borel functions on mn and a setA of n strongly commuting self-adjoint operators.

In the second section we give necessary and sufficient conditions for the nuclearity of the spaces presented in section 1.

We start with the introduction of special sets of real valued Borel func-tions.

DEFINITION 1.1. Let cP denote a set of Borel functions on mn, bounded on bounded Borel sets of

m

n , which is directed in the sense of the usua 1, partial ordering of functions. The set 9 has the following properties: A.I. A. II. A.III. Each tP E <t> is is bounded on Here

!2

denotes function of ~. -1

nonnegative and the Borel function A ~tP(A) XtP(A) bounded Borel sets.

the set

m

n " tP + ({ O}) and X the characteristic ~

n

The supports ~ of tP E cP (01)er the whole

m ,

i. e.

(l+lml) sup tP(A)

AE~

;;; c inf l/J(A)

AEQ

m

In the above definition we use multi-index notation. So

m = (m₁, ••• _{,mn ) and Iml}

=

Imll + Im21 + ••• + Imnl. Further Q m

denotcs the eube {A E lRIl

I

Ak E [~-1 ,ll'k)' 1 <'. k :;; Ill.

Starting from the set cp we define a second set cp+ of Borel functions on mn.

DEFINITION 1.2. l'he set cP+ consists of functions f on mn with the proper-ties:

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2

-(i) f is a nonnegative BOf'el function such that A I-- f(A) -1 Xf(A) is

bounded on bounded BOf'el sets.

(H)

LEMMA 1.3. The set ill+ is a dif'ected set of Borel functions which are bounded on bounded Borel sets of JRr: The set ill + satisfies A. I-III.

PROOF. See [EGK], lemma 1.5.

Let

X

denote a separable Hilbert space and let

A

₁, •••

,A

_n denote n strongly commuting self-adjoint operators in

X

all taken fixed throughout this paper.

o

Let

(E~k»aEJR

denote the spectral resolution of the identity corresponding

to the self-adjoint operator ~, 1 ~ k ~ n. Then to any Borel set ~ in JRn. of the form ~ = ~1 x ••• x ~n we link the spectral projection

n

E(~) =

n

k=l

Since the sets ~ generate the whole a-algebra B(JRn) of Borel sets in JRn, the above defined mapping

E

extends to a projection valued measure on the who Ie of B (JRn). We denote this extens ion by E.

E is called the joint resolution of the identity belonging to the n-tuple

A = (A

1, ... ,An)'

Corresponding to each Borel function ~, bounded on bounded Borel sets we

introduce the nonnegative self-adjoint operator

~(A)

=

~(Al,···.An)

=

J

~(A)d[A'

IRn

-1 -1

By ~ we mean A '-~(A) X (A). (We agree that 00.0 = 0.)

-1 ~ -1

The meaning of ~ (A) is clear if we require that ~ is bounded on bounded Borel sets of JRn•

DEFINITION teA) = {~(A)

I

~ E ill} •

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3

-DEFINITION 1.5.

The

dense subspace G of X is defined by

G =

u

E(lI)X lIEBb (lRn)

n .

f

n

Here Bb (lR ) denotes the set of aU bounded Borel sets a lR.

The space

G+

consists of all mappings F from Bb(lRn) into

X

with the property

Such mappings F are called spectral trajectories. The Hilbert space

X

is embedded in

G+

by the mapping emb defined by

emb(x): fI,., E{fI)x , l:I E Bb (lR ) . n

Let ~ be a Borel function, which is bounded on bounded Borel sets of lRn, For each

l:I

E Bb(lRn) the operator ~(A)E(l:I) is bounded in

X.

The space

~(A)X is defined to be the subspace of

G+

which consists of all mappings of the form

x E X

(A)X •

1£

So each element w E ~(A)X corresponds precisely to one element x E X

(A)X.

w ~ .

We turn ~(A)X into a Hilbert space by defining the nondegenerate inner product (w 1,w2) ~ = (x w ,x )X' 1 w2 w. E ~(A)X • 1

Now we are in a position to define two special subspaces of

G+.

DEFINITION 1.6.

S~(A)

=

u

~(A)X

•

~Eip

llhe topology 0ind for S<l'>(A) 1:S the inductive limit topology generated by

the Hilbert spaces ~(A)X.

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4

-DEFINITION 1.7. Let f E <1>+. The seminaI'm sf on S<jl(A) is defined by

LEMMA 1.8. + f E <P •

The topoZogy crind for S~(A) is generated by the seminorms sf'

PROOF. See [EGK]t theorem 1.7. o

DEFINITION 1.9. The spaae Tq;(A) denotes the subspaae of G+ whiah aonsists

of aU speatral. trajeatorics F 1J.iith the property

V~E<jl: ~(A)F

E

emb(X) •

The topol.ogy T

proj is the l.oaaUy aonvex topoz.ogy for T HA)generated by

the seminorms t~, ~

E

<Pt defined by

F E

T.p(A) •

We recall that a locally convex vector space V is nuclear iff for each

continuous seminorm p there exists a continuous seminorm v such that p ~ v

and such that the canonical embedding

V

~

V

is a nuclear mapping. Here

v p

V

denotes the Banach closure of

VI

{v E

V I

p(v)

=

O} etc. See [T],

p

definition 50.1.

2. THE NUCLEARITY OF S<Il{A) AND T <Il(A)

This section contains the main result of this paper. The following concept

is important:

DEFINITION 2.1. A nonnegative funation m on JR is a iP-multipl.ier iff

THEOREM 2.2. The following three statements are equival.ent:

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(ii)

(iii)

PROOF.

5

-V~E~ VfE~+: ~(A)f(A) is a nuclear mapping from

X

into

X.

There exists a

~+-multiplier m such that

m(A)

~

1 and m- 1(A) is a

nuclear mapping from

X

into

X.

+ +

(i)

*

(ii). Take ~ E ~ and f

E

~

.

There exists g

E

~ such that ... ...

g ~ f and the canoni~ embeddin~1(S~(A»sg + (S~(A» is nuclea~A

representation of (S~(A»s is g (A)X and a representation of (S~(A»Sf

is 1(A)X. g

So g-1(A)f(A) is a nuclear operator from

X

into

X.

Since

~(A)g(A)

1S a

bounded operator, the operator ~(A)f(A) is a nuclear mapping from

X

into

X.

(ii)

*

(iii). It follows from A.II and A.III that for each m

E

~n there exist d > 0 and ~ E ~ such that XQ

m ~ d~. Hence XQm(A) 1S nuclear for all m E ~n. So the joint spectrum of A is discrete and has no accumulation points

I

0 E JRn•

Let (Ak)k be an enumeration of the eigenvalues of

A.

Put

a _k

=

and

Define

I =

2

(1+b

hQ

mE~n m m

We show that I is a ~+-multipller.

b = m

_A

2

_EQ a

_k

k m trace(X Q _m(A» + ~+

Take ~ E ~, fEw • There exist ~ E ~, g E ~ and d > 0 such that for all

m E ~n Put Then sup ~(A) AEQ _m ::;; d lnf lj!(A) AEQ m M

:=

sup I(A)f(A)~(A) AEJRn and

M sup sup (1 +b )f (A)~(A)

m AEQ

m

sup (l+b ) _m sup f(A) sup

AEQ AEQ m m

'"

m sup f(A);? AEQ m ~O.) inf g(A) • AEQ_m

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6

-~ d sup (l+b ) inf g(A) inf ~(A) ~

m

m _AEQm AEQro

~

d sup (1 +

La)

inf g(A) inf

~(A) ~

ro

AkE~

_{k AEQm} _AEQm

~

d sup (sup

g(A)~(A)

+

I

a_k

g(Ak)~(;\k»)

ro _AEQm AkE~

~

d (sup g(A)ljJ(A) + trace(g(A)1)!(A») < 00 •

AElRn

Hence £ is a ~+-multiplier.

Choose k so large that

L

(l+lml)-k < 00, The function

m

is also a ~+-multiplier because ~+ satisfies A.III. Further it is clear that meA) ~ 1 and

trace(m-1

(A»

=

trace

(I

(l+lml)-k(l+bm)-l XQ (A»)

m m

L

(l+lml)-k < 00 •

m

-1

So

m

(A)

is a nuclear operator from

X

into

X.

(iii) ~ (i). Take f E ~+. Then mf E ~+, mf ~ f and the canonical embedding

(S~»Smf

-r

(S~»Sf

is nuclear because (mf)-l(A)f(A)

=

m-1(A) is a nuclear

operator from

X

i into

X.

[J

DEFINITION 2 The set ~ satisfies the sy~~try condition iff

A.IV.

REMARK. The set ~+ satisfies the symmetry condition.

THEOREM 2.4. Let ip aatisj'y A.IV. J'hen T~(A) equaZG S<I>+(A) as a topological vector space

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7

-THEOREM 2.5. Let ~ satisfy A.IV. The foZlowing three statements are equivalent:

(i) _The _spac;e _{T q,(A)} _{is nwZear.}

(H) _V~Eq:,_VfEq,+: _~(A)f(A)_{is a nucZear operator from}

X

_into

X.

(iii) There exist$ a

~muZtipZier

m

such that

meA)

~

1 and m-'(A) is a

nuclear operator from

X

into

X.

PROOF. Since Tq,(A) equals S~+(A) topologically, we apply theorem 2.2 with q,+ and q,++ in the role of q:, and ~+. Since q:, satisfies A.IV it follows that @++-multipliers are q,-multipliers and vice versa. So (i) and (iii) are equivalent.

++

Finally, since q, c q, and since q, satisfies A.IV, it also follows that

(i) and (ii) are equivalent. [J

REMARK. A convenient sufficient condition of nuclearity of Tq,(A) is

k

trace(x~(A»

=

O«l+lml) ) for some k.

In this case the plier.

ACKNOWLEDGEMENT

function

I

(1 + trace(XQ (A»)XQ is a

@-multi-m m m

The author likes to thank J. de Graaf and S.J.L. van Eijndhoven for valuable discussions.

REFERENCES

[EeK], Eijndhoven, S.J.L. V<.ln, J. de Graa[ and P. Kruszynski, Dual systems of inductive-projective 1 imits of Hilbert spaces originating from self-adjoint operators, Proc. Koninklijke Nederlandse Akademie van Weten-schappen, A88 (3), 1985.

[T], Treves, F., Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, London, 1967.