A theory of generalized functions based on holomorphic semi-groups: part B : analyticity spaces, trajectory spaces and their pairing

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Graaf, de, J. (1983). A theory of generalized functions based on holomorphic semi-groups: part B : analyticity spaces, trajectory spaces and their pairing. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8307). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1983

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TECHNISCHE HOGESCHOOL EINDHOVEN NEDERLAND

ONDERAFDELING DER WISKUNDE

TECHNOLOGICAL UNIVERSITY EINDHOVEN THE NETHERLANDS

DEPARl'MENT OF MATHEMATICS

Memorandum 83-07

J. de Graaf

A THEORY OF GENERALIZED FUNCTIONS BASED ON HOLOMORPHIC SEMI-GROUPS

Part B: Analyticity spaces, trajectory spaces and their pairing

In a complex separable Hilbert space with inner product (-,.)

X

we consider an unbounded non-negative self-adjQint operator

A.

This operator will be fixed throughout chapters 1, 2, 3. Since

A

is self-adjoint it admits a spectral resolution

For details on such spectral resolutions, see [y], [LS]. By O~a<b~oo, a we mean 00

f

Xa,b(A)$(A)dEA -00 with on (-l,b] _{if a}

=

0 X b (A)

=

t

on (a,b] if a > 0 a, elsewhere. We define

-tA

f

-At E e

=

e d A t ~ <I: •

o

For t € IR the operator e

-tA

is self-adjoint, e

-tA

is bounded iff Re t

~

o.

-tA .

.

'ff 0

-tA..

'bl d

e 1S um tary 1 Re t

= .

Further e 1.S 1.nverti e an

-tA -1 tA

Yt€~ (e )

=

e

+ -tA

On ( , i.e. the set of t ~ Q: with Re t > 0, the operators e establish a one-parameter semi-group of bounded injective operators on

X.

We now introduce our test space

SXtA'

This space equals the set of analytic vectors for the operator

A.

Cf. [Ne].

Definition 1.1. 5 _X,A

₌

_U { _e -tA _x

I

_{x € X}}

₌

Re t>O U Re t>O e -tA(X) •

SX,A is a dense linear subspace of

X.

From the semi-group properties the following equalities immediately follow for any 0 > 0

U 1 n€IN,O<-<o

n

-tA

Each of the spaces e (X), t > 0, can be considered as a Hilbert space. The inner product is

tA tA

= (e u,e v) X ==

The completeness of

e~tA(X)

follows from the closedness of etA e-tA(X) consists of exactly those u € X for which

< QO •

B.2

If locally no confusion is likely to arise we suppress as many subscripts as possible. In chapters 1, 2 and 3 we shall write consistently:

5

for

5

X,A' Xt -tA

for e (X), ( • , .) for (.,.)

X.

Definition 1.2. The

strong topoLogy

on S is the finest locally convex topology on

5

for which the natural injections it : X

t +

5,

t > 0, are all continuous.

In other words:

5

is made into a locally convex topological vector space by imposing the inductive limit topology with respect to the family {Xt}t>O' Cf. [SCH] Ch. II.G. Since the natural injections X

T + Xt, T > t > 0, are all continuous the inductive limit topology is already brought about by the family {X₁/} IN' A subset

U

c

5

is open iff for every t > 0 the set

-1 n n€

i t (U) == U n X_t is open in X or, equivalently, iff for every n € IN the set

is not strict! A closed subset of X

T when considered as a set in

X

t,

o

< t < 'r, is not necessarily closed. We shall see that open sets in S are

always unbounded.

We introduce some notations:

- B denotes the set of everywhere finite real valued Borel functions on IR such that Vt>O the function w(x)e-tx is bounded on [o,~).

- B+ C B contains those W for which there exists € > 0 such that w(x) ~ € > 0,

voor.alle x ~

o.

Let W ~ B. Then weA) is defined by

The domain of w(A}, denoted by D(w(A}, consLsts of exactly those x ~

X

with ~ 2

of

w (A)d(EAx,x) <

=.

It follows that S is contained in the domain of each operator w.(A) •

-tA

Further: VW~B, Vt>O the operators weA)e are bounded and self-adjoint, The sets of operators corresponding to the sets B and B+ will be denoted by B(A) and B+(A), respectively. The operators in B+(A) are all strictly

positive •

Definition 1.3. For each W € B+ we introduce the (semi-) norm Pw by ~ ":!

pw{u)

=

II w(A) ull

= {

f

w(A)2 d(EAU,U)} u ~

S .

o

Further for W E B+ and € > 0 we define the set

U,J,

=

{u E S

I

IIw(A)ull <

d .

'I',€

The next theorem is very fundamental, It tells that the strong topology in SX,A is generated by the semi-norms Pw'

B.4

Theorem 1.4.

I. V1jJeB , Ve>o U", is a convex, balanced, absorbing open set in the strong

+ '1" e:

topology. In other words the semi-norms P1jJ are continuous.

II. Let a convex set " c S be such that for each t > 0

"n

X_t contains a neighbourhood of 0 in X

t • Then" contains a set UtjJ,e: with 1jJ e B+ and € > O. Proof.

I. A standard inner product argument shows that U,,, is convex, balanced and 'I',e:

absorbing. For the terminology see [Y] Ch. 1.1. We now show that for each t > 0 UtjJ,e: n X_t is an open set in Xto

By definition

U n X

t = {u

I

u eXt' IItjJ (A) ull < d . 1jJ, e:

Because of 111jJ (A) ull :s IItjJ (A) e -tAli lie tA u II and the boundedness of 1jJ (A) e -tA, the norm P1jJ is continuous on X

t "

II. Introduce the operator p... In dE" n € IN. Let r be the radius of the

n n-l A n

largest open ball in P (X) which fits in " n P (X). SO

n n

r = sup {p

I

[u e P (X) 1\ liP ull < p] .. u E P (m} "

n n n n

Next define X : IR +IR as follows

X (A) =

o

if A < 0

2

max

(~2

,1)

if

A

E (n-l,n] n X (~) if A ... 0 • n e lN

We prove X e B +" Let t > O. Then there is e > 0 such that

eo

{u

I

feAt d(EAu,u) < e2} c " n

X~t

o

because" n X~t contains an open neighbourhood of O.

Thus we find that for all n E lN, r >

€e-~nt.

Hence for A e (n-l/n] n

-At

sup e X(A) <: 00 •

A 2:0 We now show

Suppose u € \ for some t > O. Then E:=l

nPnult~

<: <:0 and for some 1', 0 < l' <: t

(*) IIPuU 2

s

e- 2 (n-l) (t-1') lIu II 2t • n l'

Further, because of our assumption (*)

II P ull

n

-2

s

~ min(n r ,1) n

Therefore 2n2 P u €

n

n X for every n E IN.

n 1: In

X

we represent u by 1: u

=

I

~

(2n2 P u) +

(~

_1_)

~

n=l 2n n n=N

2i

with ( 01)

)-1

01)

=

L

-L

L

P

u •

~

j=N 2j2 n=N n Wi th (*) we calculate

Since

n

n

X

contains an open neighbourhood of 0 for N sufficiently large

T

u E

n

n

X .

N T

Finally we gather that u is a sub-convex combination of elements in

n

n X

T

which is a convex set. A posteriori i t is clear that u €

n

n

X

t • Remark. Similar to the proof of part I one proves that the sets

$ E B+, E > 0 ,

are .closed.

B.6

Defini tion 1.5. A subset W c: S is called bounded if for each O-neighbourhood

U

in

S

there exists a complex number A such that

W

c:

AU.

Cf. [SCH] 1.5. The next theorem characterizes bounded sets in

S.

Theorem 1.6. A subset W c

S

is bounded iff

and

Proof •

3t>O W c

X

t

3M>O 'v'ueW lIullt :S M •

.. ) Let 1/1 E Band u E: W. Then

+

.) Taking 1/1

=

1 we observe that W is a bounded set in X. Denote its bound by P. Suppose the statement were not true, then

tA

'v't>O 'IM>O 3UEW lie ull > M •

By induction we define two sequences of real numbers {t }, {N }, t ~ 0,

n n n

N

+

~ as n ~ ~ and a sequence {u } c:

W

as follows:

n n

n = 1 Choose tl > 0 and M

=

2. Then take N1 > 0 and u₁ E:

W

such that

n

=

t + 1: Suppose

is true. Then

W

is a bounded set in

X

_t, t s ~tt' because

co

J

s

p2e2tNt + t + 1 .

N~

If our sequence terminates for a certain value of n then

W

is a bounded set. If not, define the function n(A) on (O,~) by

neAl

=

eAtn on the interval [N _n-1,N _n

J ,

n

=

1,2, ••••

Since neAle-At is a bounded function for t > 0 we have n € B+. Then because

of the assumption the sequence n (A) u should be bounded. However

n

~

IIn(A} u_nIl2

=

J

n2(A)d(E_Au_n,u_n) > n + 1 •

o

Contradiction!

In the next theorem we give a characterization for the convergence of sequences {u } in the strong topology of

S.

n

Theorem 1.7. u + 0 in the strong topology of

S

iff

n

Proof •

.. ) For any 1/! E B we have 111/!(A)u II S It1/! (A>e-tAnnetAunl[ + 0 as n +

~

because

-tA + n

1/!(A)e is a bounded operator on

X

•

• ) Suppose u + O. Then V1/!EB II1/! (A) u " + O.

n + n

o

We conclude that the sequence {u } is a bounded set. So 36>0 VnElN !tu

II

S 6.

n n l'

Further, taking tjI = 1, it is clear that l[unll

O + 0 as n + ~. From this we derive

+0 as n + <Xl •

Now we show

II

un" t + 0 for any t < 1'.

L

J

e 2At _d(EAun,un) +

B.S

The second integral can be estimated uniformly

co co

f

e 2At d(C"u,u )

₌

J

-2A(T-t) 2AT deE ) ~

n n e e "un,un L L co -2L (T-t)

J

e2

"T

d(E"u ,u ) ~ -2L(T-t) 2 ~ _e e

e .

n n L

By taking L sufficiently large the second term in (*) can be made smaller than ~€ uni formly in n.

The first term in

(*>

tends to 0 as n -+ "" because of (*).

o

Theorem 1.8.

I. Suppose {u } is a cauchy sequence in the strong topology of S. Then n

3t>O {un} c X

t and {un} is a cauchy sequence in Xt •

II. S is sequentially complete, i.e. every Cauchy sequence converges to a limi t point.

Proof.

I. An argument similar to the proof of the preceding section. II. Follows from I and the completeness of

X

t • Next we characterize compact sets in S. Theorem 1.9. A subset XeS is compact iff

3t>Q X c X

t and

K

is compact in Xt • Proof •

• ) Let {Qa} be an open covering of

K

in

S.

Then {Qa n

X

t} is an open covering of K in X

t • Since X is supposed to be compact in Xt there exists a

N

finite subcovering {Qa.}, 1 ~ i ~ N, with U

i=l (Qa~ n

X

t) ~

K.

But then

N ~ •

certainly U_i=l Q_a ~ K. i

o

.. ) Since Xis compact i t is bounded and therefore i t is a bounded set with bound

e

in X_T for some T> O. We show that K is compact in X_t whenever t < T. Consider a sequence {u } c

K.

_{There exists a converging subsequence {unj }}

J J Put

Unj -

u

=

V

j •

{Vj } is a bounded sequence in Xt and II Vj II ~ 0 as j ~

=.

Now we find ourselves in exactly the same position as in the proof of the only-if-part of Theorem 1.7. We conclude Ilv jll t ~ 0 whenever 0 S t < T. Our

subsequence {un_j } converges to u in Xt-sense. This shows the compactness of

K in X_t for 0 s t < T.

0

The next theorem gives an alternative description of the space SX,A' The simple proof is omitted.

Theorem 1.10. Let u € X be such that u € D(~(A}) for all ~ € B+. Then u € S.

In other words

o

In order to make a link with the literature on topological vector spaces we now describe the properties of our space S by using the standard terminology of topological vector spaces. [sea].

The terminology is explained in the proof. Theorem 1. 11 •

I. SX,A is complete. II. SX,A is bornological. III. SX,A is barreled.

-tA

IV. SXtA is Montel iff for every t > 0 the operator e is compact as an operator on X.

-tA

v.

SX,A is nuclear iff for every t > 0 the operator e is a

as

(=

Hilbert-Schmidt) operator on X. Proof.

I. Let {X_{a } be a Cauchy net. The a's belong to a directed set} D. For each neighbourhood Q 3 0 there is y €

D

such that whenever a >y and S

7

y one

has xa -

Xs

€ Q. We now prove that there exists an x € S such that xa ~ x

in the strong topology. Let x be the limit of xa in X-sense. For each ~ € B+

closed-B.I0

ness of ~(A) one has x~ € D(~(A}) and x~

=

~(A}x. The result follows by

applying Theorem 1.10.

II. Every circled convex subset fl c S that absorbs every bounded subset W c S

has to be a neighbourhood of O. Let

B

t be the open unit ball in

X

t, t > O. B_t is bounded in S, therefore for some E: > 0 one has E:B_t c fl n X_t •

We conclude that for every t > 0 the set fl n

X

_t contains an open neighbour-hood of O. But then according to Theorem 1.4.11

n

contains a set~" •

""E: III. A barrel V is a subset which is radial, convex, circled and closed. We have

to prove that every barrel contains an open neighbourhood of the origin. Because of the definition of the inductive topology V n

X

_t has to be a barrel in X_t in the X_t -topology, for every t > O. Since X

t is a Hilbert space there exists an open neighbourhood of the origin L with LeV n

X

_t • Again the conditions of Theorem 1.4.I1 are satisfied so that V contains a set U,I. •

""E:

IV. We have to prove that every closed and bounded subset of S is compact iff

-tA

for every t > 0 the operator e is compact.

-tA

.) Suppose e is compact for every t > O. Let TN be a closed and bounded subset in

S.

Then

W

c

X

t for some t > 0 and

W

is closed and bounded in

X

t• See Theorem 1.6. Take.,

a

< • < t. We claim that

W

is a compact set in

X

.•

T

To see this, consider the following commutative diagram where ~ denotes the natural injection:

X

T

r

e -TA

The vertical arrows are isomorphisms. So

G

is a compact map and TN is compact in X • Consider an open covering {C } of TN in S, then {C n X } is an open

T a a T

covering of

W

in

X

.•

Because of the compactness of TN in

X

there is a finite

T •

subcovering:

N

W

c U

(Ca.

n

X) •

i=1 l. T

., Suppose

S

is Montel, Le. each closed and bounded set is compact. Let {un} be a bounded sequence in

X.

Pick any fixed t > O. Consider the

sequence {e-tAu }. Consider the closure in

S

of this sequence. This closure is a bounded

se~

and, according to our assumption, compact. So {e-tAu }

n

.

S

-tA

contains a converging subsequence in : e Un. -+ v. This sequence certainly

_~ J

converges in X. SO e must be a compact operator. -tA

V • • > Since e is HS and self-adjoint for all t > 0 there exists an ortho-normal basis {e } of eigenvectors of A in _{n .}

X.

We order the eigenvalues with repetition according to multipliCity 0 S 1.1 S A2 ::;; 1.3 S •••• We have An -+ QO

as n -+ QO. Thus we get the representation

-tA e u

=

~

n=1 -t). e n(u,e)e n n -t).

where {e n} is an 1₂-sequence for each t > 0 and hence an t

1-sequence.

-tA

So e is a nuclear operator for each t > O.

S

is a nuclear space iff for every semi-norm p~, ~ E B+, there is a

semi-norm p , X E B+, X ~ ~, such that the canonical injection J : Sp -+ Sp is

X X 1Ji

nuclear. Cf [y]. Here the Banach space Sp is the competion of S with respect to the norm p.

The canonical injection can be written

Note that _{_}

{x.~l(A)ej}

and _{_}

{~-l(A)e.}

_J are orthogonal bases in the Hilbert spaces Sp respectively _{X .}

c:

_-p~

•

Now suppose there exists a E B+ such that a-1(A) is a nuclear operator. If

we take X

=

alJi then the canonical injection can be written

Ju =

-1

Since {a (A_j

>}

is an 1

1-sequence J is a nuclear operator.

It only remains to show that a E B+ with the desired properties can be con-structed. To this end define integers N(n), n

=

0,1,2, ••• , such that

B.12

N(O)

= 0,

N(n + 1) > N(n),

AN(n)+l > AN(n) and

Q)

-A./n

I

e J

_<"2'

1

j=N(n) n

Next define the sequence {"k} by

N (n - 1) < k

:s:

N (n) , n = 0,1,2, ••••

00

It is clear that L

k=l vk is convergent. _A

_In

Define a step function cr € B+ by setting cr(A)

=

e k on the interval

(\-1

2 + \ ' \

+ \+1]

2 ' k

=

2,3, ••• , and cr(A)

=

e

A

1 on 0,

[1..1

+ 2

A2]

•

a-leA} is nuclear because it is self-adjoint, it has discrete spectrum, and the eigenvalues v_k establish an t

1-sequence •

• ) Suppose S is nuclear. There exists 1/J € B such that the injection

Sp.

-'l- X 1s

-1 + 1/J

a nuclear map. Therefore 1/J (A) must be a nuclear operator and consequently also

as.

But then e-tA

=

1/J-l(A)[1/J(A)e-tA] must be

as

since the operator

We now introduce our space of trajectories T X ,A' The elements in this space are candidates for becoming "generalized functions fl (or fldistributions").

Definition 2.1. TX,A denotes the complex vector space which consists of all mappings F : (0,=) + X such that

(i) F can be extended to an analytic function on the open right half-plane ¢+ •

+

-tA

(11) Vt/'tE:¢ e F('r) = F(t+T) •

Such a mapping F will be called a trajectory. If for some ~ > 0 F1 (~)

=

F2(~)

then F1

=

F

2• This follows immediately from the semi-group and analyticity properties. We shall use the notation e -t'\ for F (t). It is immediate that

-tA

Vt>O e F € S. In chapters 2 and 3 T

XtA is abbreviated by

T.

-tA Defini tion 2.2. The embedding emb

for all x € X and t > O.

X +

T

is defined by (emb x) (t) = e x

Sometimes we shall omit the symbol emb and loosely consider X as a subset of T. Thus SeX c

T.

The question arises whether there exist trajectories F € T such that F(t)

does not converge or I even II F (t) I[ t = if t

+

O. The answer is affirmative: simply take x E X \ D(A) and F(t)

=

Ae-tAx. More general examples are

obtained by taking

~

€ B, u € X and defining G(t)

=

~(A)e-tAu,

t > O. The next theorem shows that all elements of

T

arise in this way.

Theorem 2.3. For every F €

T

there exists w € X and ~ € B+ such that

F(t)

=

~(A)e-tAw

for every t > O. We write F

=

~(A)w.

n Proof. Let F €

T.

Let P denote the projection f dE,.

n ~1 A

Put a = ileA P F(1) II.

n n Let ~ be defined by { max (a ,1) ~(A)

=

n 1 for A € (n-l/n] I n E IN elsewhere. Then eA P F(1) € X. n

B.14

We shall show that ~ E B+. Let t > O. Then for A E (n-1,nJ with an ~ 1 one estimates n -2(n-1)t e

J

d(EAF(t),F(t» Se2tUF(t>l12 n-1

Defini tion 2.4. The strong topology in T is the topology induced by the semi-norms p , n E IN,

n

1

p (F) = I!F(-) IfX •

n

n

In other words a base of open O-neighbourhoods is given by

Remark. The strong topology is equivalent to the topology of uniform

con-+

vergence on compacta in f; •

Theorem 2.5.

o

I. T endowed with the strong topology is a Frechet space, Le. i t is metrizable and complete.

II. A base of open sets {OU,t} is given by 0U,t

= tF

I

F(t) E

U}, U

open in

X,

t > 0 and fixed.

(In words: the set of trajections which pass at t through

U.>

Each open set in

T

is a denumerable union of sets of this type. Proof.

I. The topology. is generated by a countable number of semi-norms. Hence

T is

metrizable. See [yJ.

Further, since

T

is metrizable i t is complete iff each Cauchy sequence converges. Let {F } be a Cauchy sequence in

T.

m n I ( (l/n) - (l/v»

A

1

an element hl/ t

X.

For v > n one has F (-) "" e F (-). Since

n «l/)_(l/»Amv mn

both sides converge and e n v is a closed operator we have

L" D( «l/n)-(l/v»A d h "" «(1/n)-(1/v»A h N define

hl/n <:. e ·1 an l/v e l/n' . ow we

F(t) "" e-(t-(l/v»A hi/V' _{Re t e}

_{[-V '}

1 _{v-l' ,} I Clearly F t

T

and is the limit of {F }.

m

veIN.

II. It is enough to prove that the semi-norms F ..

II F (t)" are continuous. This

follows simply from the estimate

for 0 <

l

< t, n

Theorem 2.6. emb(S) is everywhere dense in T.

Proof. For any F €

T

the sequence emb(F(l/n» converges to F in the strong

topology.

o

o

Theorem 2.7. A set

BeT

is bounded iff for every t > 0 the set {F(t)

I

F €

B}

is bounded in

X.

Proof •

.... ) Each continuous semi-norm has to be bounded on B. Therefore p (F) "" IIF (l/n) II

n is a bounded function on B.

--rA

Because of the boundedness of e for each T > 0 i t then follows that {F(t)

I

F € B} is a bounded set for each fixed t > O.

.) Trivial.

Theorem 2.8. A set

BeT

is bounded iff there exists a bounded set

V

c

X

and ljI € B+ such that B

= ljI(A)

(V) •

Proof.

-tA

.) For each t > 0 the set ljI(A)e (V) is bounded in

X.

B.16

-) See [ETh], Ch.II cor. 2.5. It is not difficult to give an ad hoc proof in

the spirit of Theorem 2.3.

o

Theorem 2.9. A set K c T is compact iff for each t > 0 the set {F(t)

I

F ~ K} is compact in X.

Proof.

-) If

K

is compact then each sequence {F } c

K

has a convergent subsequence.

n

This means that in the set K

t

=

{F (t)

I

F € K}, t fixed, each sequence has a

convergent subsequence, which says that K

t is compact in X.

-) Let {F } be a sequence in K. We must prove the existence of a converging

n

subsequence. Consider the sequence {F

n(l)} c

Kl

c X.

Kl

is compact, therefore a convergent subsequence in

Kl

exists. Denote it by {F~(l)}. The sequence

{F~(~)}

hasa convergent subsequence in

K~.

Denote it by

{F~(~)}.

Proceeding in this way we arrive at sequences {Fm} c K such that {Fm} c {FR.}

ml n . n n

for m > R. and {Fn(m~} converges to an element in K

1/m• The "diagonal

sequence" {F~} has the property that {F~(t)} converges to F(t) E K

t• But then

also Fn + F in the strong topology.

0

n

Theorem 2.10. A set

K

c T is compact iff there exists a compact set

W

c.X and ~ ~ B+ such that

K

=

~(A)

(W).

Proof •

• ) For each t > 0 the operator

~(A}e-tA

is bounded. Therefore

~(A}e-tA(W)

is a compact set in X.

-) See [ETh], Ch. II cor. 2.13.

In the last theorem of this chapter we describe the properties of our topological vector space T in the standard terminology of topological vector spaces [SCH].

Theorem 2.11.

I. TX,A is bornological. II. TX,A is barreled.

operator on X.

IV. TX,A is nuclear iff for every t > 0 the operator e-tA is a

as

operator on X. Proof.

I,ll. T is bornological and barreled because it is metrizable. For a simple proof see [SCB] II.S.

III • • ) Suppose T is Montel. Take a bounded sequence {x } c X. The sequence {F }

n n

defined by F (t)

=

e-tAx is a bounded set in T. Because of our assumption

n n

the closure of this set is compact. But then, by theorem 2.9, for each t > 0 the sequence F (t) contains a converging subsequence. This shows the

compact--tA

n

ness of e

-tA

.>

Suppose e is compact for every t > O. Let

B

be a closed and bounded set. Then the set

B

t

=

{F(t)

I

F €

B,

t > 0 and fixed} is bounded. Since Bt+T

=

e-tA(B~)

it follows that Bt+T is precompact in X. Now take any sequence

{G

_n} c

B.

Because of the pre compactness of each

B

_t the "diagonal procedure" of the proof of theorem 2.9 yields a converging subsequence of {G }. This subsequence converges to an element in _n

B

because

B

is closed. We conclude that B is compact.

IV • • ) Suppose T is a nuclear space. Take n € IN. There exists a semi-norm

Pm ~ _{Pn such that the natural injection fpm} ~ Tpn is nuclear.

This natural injection is realized by the map e-(l/n-l/m)A which must therefore be nuclear. It follows that e-

tA

is

as

for each t > O.

-tA

.) Suppose for each t > 0 e is HS. Because of the seud-group property e-

tA

is also nuclear. Let m > n. The natural injection Tp

~

Tp

is realized

-(l/n-l/m)A . . m n

by e and 1S therefore nuclear. This is enough for T to be nuclear.

CHAPTER 3. The pairing of SX,A and TX,A

We now consider a pairing of S-and T. It turns out that S and T can be considered as each others strong duals.

Defini tion 3.1. On Sx ,A x T X ,A we introduce a sesquilinear form by tA

<u,F>X

=

(e u,F(t»X'

Note that this definition makes sense for t > 0 sufficiently small. Note

B.18

-tA also that because of the semi-group property and self-adjointness of e

the definition does not depend on the choice of t. We remark that <UO,F>

= 0

for all F € T implies U

_o

=

0 and <u,F

O>

=

0 for all u € S implies FO

=

O. These two facts easily follow by taking F

=

emb(u

O)' respectively the denseness of emb(S) in T.

Theorem 3.2.

I. For each F € T the linear functional <u,F>, which maps S onto ~, is

con-tinuous in the strong topology of S.

II. For each strongly continuous linear functional 1 on

S

there exists G € T

such that t{u)

=

<u,G> for all u € S.

III. For each v €

S

the linear functional <v,G>, which maps Tonto C, is con-tinuous in the strong topology of T.

IV. For each stronglt continuous linear functional m on T there exists w €

S

such that m(F)

=

<v,F> for all F € T.

Proof.

I. The function u ~ <u,F> is continuous on

S

iff for a l l t > 0 it is continuous in the Xt-norm when restricted to this subspace. Indeed

II.

I

<u,F>

I

=

I

{e tAU,F (t)}

I

s

[e tAu II IIF (t) II

=

IIF (t) II lIull t • -tA

For fixed t > 0 the mapping e : X +

S

is continuous. Therefore the mapping

tee-tAx) : X +

~

is continuous. From Riess' theorem follows the existence of

X

-tA

-tA

bt € such that tee x)

=

(x,b

take G such that G(t) = b_t• Then t(u) = <u,G> for all u E S.

III. Since T is metrizable it is sufficient to prove conti nul.. ty for sequences in

T. Let G + 0 in the strong topology. Then for sufficiently small t we have

n

fA

<v,G >

=

(e v,G (t» + 0 because G (t) + 0 in X-sense. IV.

n n n

+

Take ~ E B • Take a sequence {x } c

X,

x + 0 in X. Define ~(A}x E

T

b~

-tA n An n

(~(A)x ) (t)

= ~(A)e x. Since ~(A)e-t is a bounded operator ~(A)x

+ 0

n n n

strongly in

T.

By taking ~

=

1 we conclude first that the restriction of m to

X

has the Riess representation m(x)

=

(x,a) for some a E

X.

Secondly we conclude that m(w(A)x) is a continuous function on

X

for each

W

E B+. Using

the self-adjointness of w(A) it follows that a e D(~(A» for each

W

E B+. But then, by Theorem 1.10, a e S. Finally, as

X

is dense in

T

the

representa-tion m(F)

=

(F,e)

=

<e,F> is valid for all F E

T.

0

Definition 3.3. The weak topology on S is the topology induced by the semi-norms PF(u)

=

l<u,F>!, F e

T.

The weak topoLogy on T is the topology indiced by the semi-norms

pv(G}

=

l<v,G>!, v e S.

A simple standard argument, [CH] II § 22, shows that the weakly continuous

functionals on S are all obtained by pairing with elements of

T

and vice versa. Together with Theorem 3.2 it follows then that Sand T are reflexive both in the strong and the weak topology.

In the next theorem we show that in both spaces Sand

T

weakly bounded sets are strongly bounded.

Theorem 3.4 (Banach-Steinhaus).

I. Let ::: c

T

be such that for each g E

S

there exists M > 0 such that for g

every Fe::: one has l<g,F>1 S M , then for each t > 0 there exists C t > 0 g

-such that for every FE::: one has II F (t)1I S C

t , So weakly bounded sets in T are strongly bounded,

II. Let 9 c S be such that for every F e

T

there exists ~ > 0 such that for every f E 9 one has

I

<f IF>

I

S ~, ~ there exists, > 0 and c > 0 such that 8 c XT- and IIf II, < C for all f e 9. So weakly bounded sets in S are

B.20

Proof.

I. From the assumption it .follows that for each h € X and each t > 0 there

exists a constant

~,t

> 0 such that for every F € E,

l<e-t~,F>1

s

~,t

or

I

(h,F(t)}

I

s

~,t' From the Banach-Steinhaus theorem in Hilbert space it then follows that the set {F(t)

I

F €

E},

t fixed, is a bounded set in

X.

II. Let tjJ E: B+. Let w E D(tjJ(A». Then tjJ(A) 2 w, defined by (l/J(A)2w) (t)

=

2

-tA

== tjJ (A) e w belongs to T. From our assumption it follows that for every

f E:

a

From the Banach-Steinhaus theorem in Hilbert space it then follows that

a

is a bounded set in the Hilbert space XtjJ' i.e. the completion of $ with respect to the norm IItjJ (A) • II. But this means that

a

is bounded in $, since each semi-norm PtjJ' tjJ € B+, is bounded on it.

In the next two theorems we give characterizations of weak convergence of sequences both in

S

and

T.

Theorem 3.5. u + Q in the weak topology of $ iff n

3t>Q {u } c

X

n t

tA

Proof. Weak convergence of {un} in X_t means weak convergence of e un in X.

-) Let F E:

T.

<u ,F>

=

(etAu ,F(t}). Since etAu + 0 weakly in

X

and

n n n

F(t) €

X,

it follows that <u ,F> + O.

n

o

->

First we remark that weak convergence in $ implies weak cORvergence in

X.

Therefore for any w €

X

any L > 0, t > 0,

+ 0 as n + co •

{u } is

n a bounded set in S. Therefore 3'£>0 39>0 Vn€lN lIu n '( II s

e.

We shall CII)

prove that u -,)0 0 weakly in X • Denote

n A T ~, llL commutes with e'( • Taken w E:

X,

the projection operator Lf dE_A by then

TA

We have IIL e un S

e

and II IIL wll -+ 0 as L -+ QO. Therefore if we take L large

enough the second term in (*) is smaller then ~€ uniformly for all n. As we have just seen the first term in (*) can be made smaller than ~€ by taking n large enough. This finishes the proof.

COroll¥X 3.6.

I. Strong convergence of a sequence in

5

implies its weak convergence. II. Any bounded sequence in

5

has a weakly converging subsequence.

Thearem 3.7. F _n -+ 0 in the weak topology of Tiff

Proof.

-tA

.) For any v to:

X,

<e v,F>

=

(v,F (t)}O -+ 0 as n -+ 00.

n n

.) For any , t

5

and t sufficiently small

tA

because e . ~ E

X.

COrollary 3.8.

tA

=

(e ~,F (t» -+ 0 n

I. Strong convergence of a sequence in T implies its weak convergence.

II. Any bounded sequence in T has a weakly convergent subsequence. Cf. Theorem 2.9.

It looks reasonable to conjecture that the weak topology on

S

is the

induc-t i ve limit topology with respect to the Hilbert spaces

X

t , now endowed with the weak topology. It is easily seen that the weak topology on T is induced by the semi-norms P_t , t > 0, V €

X,

and P

t (F)

=

I

(v,F(t})

I.

We will not

,v tV

pursue these things further here. The next theorem deals with the question: When does weak convergence of a sequence imply its strong convergence?

o

I.

II.

III.

B.22

Theorem 3.9. The following three statements are equivalent. For each t > 0, e -tA is a compact operator on X.

Each weakly convergent sequence in S converges strongly in S. Each weakly convergent sequence in

T

converges strongly in

T.

Proof.

I - II. A weakly convergent sequence in S converges, for some t > 0, weakly in X_t • See Theorem 3.5. Because of the assumption the natural injection X

t c Xa, 0 < a < t, is compact. Cf. the proof of Theorem 1.11. But then our sequence converges strongly in Xa'

II - I. Take any sequence {f }

-tA

n

each F €

T

we have <e f ,F> + O.

c X, f + 0 weakly in X. For each t > 0 and

n -tA

-tA

n

Thus e f + 0 weakly in S. Because of

n -tA .

the assumption e f + a strongly in S.

-tA

n And so lie f

II + a. This shows n

that e must be compact.

I - III. Let {F } c

T.

Suppose VgES <g,F > + O. Then

n n

Vh€X Va>O

<e-a~,F

>

=

(h,F (a» + 0 •

n n

This means that Va>O F (a) n B > 0, we find that F (a + n Vt>O IIF (t) II + a. n -SA

+ a weakly in X. Using the compactness of e ,

B)

=

e-B~

(al + 0 strongly in X. Therefore

n

III ... I . Let {v } c X

n be such that v n + 0 weakly in X. Then v n + 0 weakly in

T

because Vg€S <g,v >

n :: (g,v )

_-tA

_n + O. Now the assumption says v _n + 0 strongly in T. This means Vt>O e v + 0 strongly in

X.

The compactness of

n

follows for each t > O.

-tA

e

o

-tA

One might wonder whether in the case that e is compact for each t > 0 the strong and weak topologies coincide. This however cannot be true since a set which belongs to a weak base of a-neighbourhoods always contains a closed subspace of finite codimension. Generally speaking such "large" sets do not fit in a strongly open O-neighbourhood.

B.23

It is relatively simple to show that both in $ and

T

the weakly compact sets are just the (weakly) closed and bounded sets. Then i t is not difficul t to prove that $ and

T

with their strong topologies are

Mackey spaces.

That is the strong topology is the finest locally convex topology for which the dual of $ (respectvely

T)

is

T

(respectively $). (All spaces which are either barreled or bornological are Mackey. See [SCS] IV.4.)

REFERENCES. See Part A.

ACKNOWLEDGEMENT The author thanks Dr. S. J • L. van Ei j ndhoven for his contribution to the revision of the manuscript.