A theory of generalized functions based on holomorphic semi-groupsCitation for published version (APA):Graaf, de, J. (1979). A theory of generalized functions based on holomorphic semi-groups. (EUT report. WSK,Dept. of Mathematics and Computing Science; Vol. 79-WSK-02). Technische Hogeschool Eindhoven.

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TECHNISCHE HOGESCHOOL EINDHOVEN

NEDERLAND

ONDERAFDELING DE~ WISKUNDE

TECHNOLOGICAL UNIVERSITY EINDHOVEN

THE NETHERLANDS

DEPARTMENT OF MATEEMATICS

A THEORY OF GENERALIZED FUNCTIONS BASED ON

HOLOMORPHIC SEMI-GROUPS

by

J. de Graaf

T.H.-Report 79-wSK-02

March 1979

Contents

Abstract

Chapter O. Introduction

Chapter 1. The space SX,A

Chapter 2. The space TX,A

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

Chapter 4. Characterization of continuous linear mappings between

the spaces SX,A ' SX,A' Sy,B and SY,B

Chapter 5. Topological tensor products of spaces of type SX,A ' SX,A

Chapter 6. Kernel theorems

Appendix A. Functions of self-adjoint operators

Appendix B. Holomorphic semi-groups

Appendix C. A Banach-Steinhaus theorem

Acknowledgement

References

1

2

12

26

33

39

48

59

65

67

69

70

71

Abstract

In a Hilbert space X consider the evolution equation

du dt = - At:!

wit,h A a non-negative unbounded self-adjoint operator. A is the infinite­

simal generator of a holomorphic semi-group. Solutions u(') : (0,00) + X of this equation are called trajectories. Such a trajectory mayor may

not correspond to an "initial condition at t = 0" in X. The set I

of trajectories is considered as a space of generalized functions. The

= U e -At (X) • test function space is defined to be SX,A t>O

For the spaces SX,A,SX,A I discuss a pairing,topologies, morphisms,

tensor products and Kernel theorems. Examples are given.

AMS Classifications 46F05 46F10

- 2 -

CHAPTER O. Introduction

In a very inspiring paper, De Bruijn [B] has proposed a theory of

generalized functions based on a specific one-parameter semig~oup

of smoothing operators. This semi group constitutes what is known as

a holomorphic semigroup of self-adjoint bounded operators on

L2 (lR) •

This observation has enabled me to generalize De Bruijn's theory

and to place it in a wider context of functional analysis.

By this approach, we can introduce De Bruijn type generalized functions

and corresponding spaces of test functions on manifolds and

also on duals of nuclear spaces. We can construct the spaces of

test functions even so as to be invariant under one or more given

operators. In [B] such a given operator is the Fourier transform.

Even so, there are many possibilities, allowing us to impose further

conditions. In the sequel I study extensively the spaces of test

functions and distributions that arise in this fashion. I characterize

their topologies and morphisms. Moreover, I give necessary and sufficient

conditions for the validity of Kernel Theorems.

In this introduction I illustrate the ideas by a few examples. The first

example is very simple and I discuss it in some detail. I use it to give

an outline of the general theory. The remaining examples merely

illustrate how to apply the general theory; they are not worked out

in detail. They could be the subject of further investigation.

- 3 -

Example 1

Let us consider the elementary diffusion equation

(1) X E JR, t > 0 •

A classical solution F(x,t) with the property Vt > 0 F(-,t) E L2

(JR)

will be called a trajectory. Corresponding to any "initial condition"

f E L2

(JR) there is a trajectory given by the well-known formula

F(x,t)

00 ... 2

J f(~)exp(- (x - ~) )d~

4t t > 0 .

From this expression it follows that the special trajectory FC-,t) Mtf,

f E L2 (JR) has the properties

(i) Vt > 0 VT > 0 MtF(.,T) = F(', T + t) = MTF(·,t) •

(ii) Vt > 0 F(x,t) is an entire analytic function of x.

With the aid of the maximum principle for parabolic equations, [PW] p.160,

it follows that (i) is true for ~ trajectory and hence also (ii) is

true. From the positivity and linearity of the integral operator Mt it

follows that for a given trajectory F(·,t) there exists at most one

g E L2 (JR) such that F(',t) = Mtg. In general no such g does exist at

all, consider e.g.

G(x,t) 2

-~ (x - a) 2(nt} exp - 4t

Note that G(x,t) "tends" to the a-function a (x - a) for t '" O. Any

derivative of G is again a trajectory.

Following De Bruijn we gather all trajectories of (1) in a complex

vector space S' . The elements of $' are to be interpreted as generalized

functions. As we just have seen, a generalized function sometimes

corresponds to an element of L2 (JR) and, if so, the correspondence is

unique. The test function space $ is defined to be the dense linear

subspace of L2 (JR) consisting of smooth elements of the form Mth,

h E L2 (JR), t > O. The densely defined inverse of Mt is denoted by

M . For each (jJ E S there exists T > 0 such that M Q) makes sense. -t -T

The 2airing between Sand $' is defined by

- 4 -

(2) cp € 5 , F E S'

Here (. I .) denotes the inner product in L2 (lR) •

The definition makes sense for e positive and sufficiently small.

Since Mt is a symmetric operator and MtMT

= Mt +T

, the definition (2)

does not depend on the specific choice of e. In the following chapters

we provide S with suitable topologies and we prove

(i) For each fixed F E 5' the map,» <"F> is a continuous linear

functional on S. (ii) For each continuous linear functional t on 5 there exists exactly

one trajectory F~ such that for all cp € 5 we have i(cp) = <cp,Ft>.

Suppose P is a densely defined linear operator in L2 (lR) with a densely

defined adjoint P* which leaves 5 invariant, i.e. P*(5) c 5. Then P can

be extended to a continuous mapping from 5' into itself by

* <~,PF> = <P cp,F> •

Examples of such operators Pare M , R , T , Z" V and compositions of . a a a 1\ ~ax these. Here (R f) (x) = e f (x), (Tbf) (x) = f (x + b), (Z). f) (x) = f eX. x) I

df(xT . (Of) (x):= dx w~th a,b,A € ]R.

Finally we want to show that certain strongly divergent Fourier integrals

can be interpreted as elements of 5' •

This interpretation is closely related to the Gauss-Weierstrass summation

method. Let g be a measurable function on ]R such that for each e > 0 the -ey2

function g(y)e

J:ro

g(y)eiyxdy can

is given by G(x,t)

is in Ll (lR) • The possibly divergent Fourier integral

be considered as an element of 5'. The trajectory G

f oo -ty2 iyx := _00 g(y)e e dy. This simpe illustration of

our theory certainly has some eleg.ant features, but in some respects

it is too simple. Since Mt is not a Hilbert-Schmidt operator there is

no Kernel Theorem in this case. That is to say, there exist continuous

linear mappings from S into S' which do not arise from a trajectory of

the diffusion equation in ]R2. Cf. Chapter 6 case b.

- 5 -

Partial sketch of the general theory

We consider an evolution equation in a Hilbert space H

(3) du dt :::: -Au

Here A is a densely defined unbounded operator such that the exponential -At

e can be given a meaning. A trajectory of (3) is defined to be a

mapping F : (0,00) ~ H with the property Vt > 0 V, > 0 e-A'F(t) = F{t + T).

Examples of trajectories are F(t) = e-Ath, h € H.

Further we introduce a test function snace S = U e-At(H). ... t>O

In order to make all this into a non-trivial interesting theory of

generalized functions which generalizes the situation of example 1, we

must meet some requirements. -At

(i) -A must be an infinitesimal generator. In other words e must

have a meaning and it is an everywhere defined bounded operator.

-At (ii) e must be injective for each t > o.

At -At -At (iii) The inverse e of e has to be unbounded in order that e is

"smoothing".

These conditions are fulfilled if -A is the infinitesimal generator of a -At

one-parameter holomorphic semi-group e t ~ O. Here holomorphic

means holomorphic in t (as a bounded operator-valued function). This is

not related to any kind of smoothness or analyticity of the "test functions"

in S. The next chapters deal with the case that A is a non-negative

unbounded self-adjoint operator in a separable Hilbert space H. In example

1 we had H = L2 OR), A = - d2

2 with domain D(A) = H2 (JR) , the 2

nd order

Sobolev space. dx

Example 2

ex > 0

In this case

with

Kt(aix,y)

- 6 -

00

-At J (e u) (x) = Kt (cqX,y)u(y)dy

-00

00

! J exp(-lkI2at)cos k(x -y)dk •

o

For a = 1 the kernel Kt has been calculated in example 2.

For a = ~ we have

1 t Kt(~ix,y) =; 2 2

t + (x - y)

which is just the Poissonkernel for solving the Dirichlet problem in a

half plane t ~ O. This is not surprising, since, at least formally,

The following properties of the corresponding test function spaces S are

easy to verify

a > ~ S consists of entire analytic functions.

S consists of functions which are analytic in a strip around the

real axis. 00

a < a < l:1: The elements in S are C but non-analytic in general.

Finally we wan.t to show that for each a > 0 certain strongly divergent

Fourier integrals can be interpreted as elements of S·. Cf. Example 1.

Let g be a measurable function on lR such that for each £ > 0 the function'

1 12a foo iyx g(y)exp - (;: Y is in Ll (JR) • The possibly divergent integral _00 g(y)e dy

is in S·, The trajectory G is given by

G(x,t)

-00

For a = l:1 this corresponds to Abel's summation method.

- 7 -

Example 3

H ::: L2 ([ 0 , 2 11 J ) I Cl > 0

In this case

with

For Cl 1

u(O) == U(21T),

21T

-At f (e u) (x) == Kt

(Cl; x,y) u (y) dy

1 21T

oo

n==-""

o

exp{- Inl2Cl

t + in(x - y)} =

u· (0)

=_1 {1+2 21T I 2Cl

exp{-n t}cos n(x - y) • n=l

1 -t Kt(lix,y) = 2; 83 (x - y,e )

Here 83

is one of Jacobi's theta functions [WW] p. 464.

1 sinh t 21T cosh t - cos (x - ~) •

U'(21T)}.

In the latter case the smoothing integral operator is the Poisson kernel

for the solution of the Dirichlet problem for the unit disc. For a

general discussion of this phenomenon see Example 5. 2 -At

For each Cl > 0 Kt(a j o,') E L2

([0,21T] ). In other words e is a Hilbert-

Schmidt operator. This implies that for generalized functions of this type

a Kernel Theorem holds true. See Chapter 6. Finally, we remark that

certain strongly divergent Fourier series do have a meaning within this

theory. oo

Let a > 0 be fixed. Consider a sequence of complex numbers {c } such n n=-""

that {Ic lexp - E\nI2Cl

} is a bounded sequence for each E > O. Then n. 00 l.nx

E c e can be viewed as a generalized function. The corresponding n-- oo n trajectory is given by G(x,t) ::: roo c exp{- Inl 2Clt + inx}.

n=-oo n

- 8 -

Example 4 1 1

H d 2 2(1 2 2S

A = (- -2) + (x) , (1 > 0, B > 0 , dx

1 ex

D (A) = H OR)

here F is the Fourier operator. -At For ex = e = ~ an explicit representation of e. by means of an integral

operator can be found in [B].

This special case has been extensively investigated by De Bruijn and

Janssen [BJ, [JJ. A part of our results, especially in chapters 3 and 4

€onsists of generalizations and adaptations of their work.

Whenever ex a the test function space is invariant under Fourier .. transformation. We conjecture that the Gelfand-Sllov test function classes B S , [GSJ Vol. II Ch. 4 are the same as the test function spaces 5 of 0.

this example.

The preceding examples can all be generalized to lR n in a direct obvious way.

These generalizations can also be obtained by application of the theory of

Chapter 5 where topological tensor products of the spaces 5 and 5' are

formed. Repeated formation of tensor products of the Sand 5' spaces of

example 3 leads to distribution theories on n-dimensional tori. Our type

of distributions can be introduced on any, not too bad, differentiable

manifold M by taking for A a positive elliptic differential operator which

is self-adjoint in an L2-space over M. On a compact Riemannian manifold

one may take the Laplace-Beltrami operator. This can be done on the

q-dimensional unit sphere in lR q+1 for example. In the latter case how-

ever a very nice semi-group of smoothing operators can be chosen which

is closely related to the Poisson-integral solution of the Dirichlet­

problem for the unit ball in lR q+1• This is the subject of the last

example.

- 9 -

Example 5

(4 )

Here ilLB denotes the Laplace-Beltrami operator, 1 denotes the identity

operator.

After introducing orthogonal spherical coordinates xi = rF. (91

, ••• ,6 ), . q+1 ~ q

1 :::; i :.;; q + 1 ~n lR we obtain for the Laplacian

(5) q+1

f:, = I i=l

a Cl 1 .... _+-f:, r ar 2 LB r

From this it follows, see [M] p. 4, that an m-th order spherical hal:oonic

is an eigenvector of ilLB with eigenvalue equal to -m(m + q - 1). Then a

simple calculation shows that each m-th order spherical harmonic is an

eigenvector with eigenvalue m of our operator A. -t

Introduction of r = e in (5) transforms the Laplacian into

2t 32 a

6. '" e {- - (q - 1)- + 6.LB

} • at2 at

The expression between { } can be factored into two evolution equations.

Thus

2 ~ ~(q - 1)1 + {~(q - 1) 1 - lI

LB} ]

[Jl - ~(q - 1)1 + {~(q - 1)2 - II }~J at LB

3 The second factor can be written as at + A I with the A from (4).

From these considerations it follows that substitution of r = e-t

in

the Poisson integral formula, cf. [M] p.41, leads to an integral ex--At

pression for e •

- 10 -

(6)

Here t;. and T1 denote elements of sq, 1. e. unit vectors in lR q+ 1. The

"surface measure" of sq is denoted by dw , while F;.'n denotes the inner q

product of t;. and n. For fixed n the kernel in (6) denotes the trajectory of the o-function

centered at n. For a representation of the kernel involving zonal harmonics. See -At

[G Se]. Here the operator e is obviously Hilbert-Schmidt. Then from

Chapter 6 it f 11 th t f th di o ows a or e stribution theory arising from the operator

A in (4) Kernel Theorems are available. Similar to example 3 certain

strongly divergent sequences of spherical harmonics can be "summed".

I conclude this introductory chapter by summarizing briefly the contents

of the next chapters.

In Chapter I the test function space SX,A is discussed. Here X is a

separable Hilbert space, A is a non-negative unbounded self-adjoint

operator in X. SX,A is provided with a non-strict inductive limit

topology. A base of the neighbourhood system of 0, is given. Bounded and

compact sets are characterized. A necessary and sufficient condition for

the convergence of a sequence is given. In Theorem 1.11 the properties

of SX,A are summarized using the standard terminology of the theory on

locally convex topological vector spaces.

In Chapter 2 the program of Chapter 1 is repeated for the space of

trajectories T X,A' A characterization of the trajectories is given.

In Chapter 3 the pairing between elements of SX,A and TX,A is discussed.

Weak topologies are introduced. Banach-Steinhaus theorems are proved.

Strong and weak convergence of sequences are compared. It is proved

that SX,A and TX,A are both strong and weak duals of each

This motivates the choice of the notation TX,A = SX,A' In Chapter 4 continuous linear mappings between the spaces

SY,B and SY,B are extensively discussed and characterized.

of extending linear operators is considered,

other.

SX,A' SX,A' The problem

Chapter 5 deals with completions of topological tensor products of the

spaces of Chapter 4.

- 11 -

Chapter 6 is devoted to Kernel Theorems. We give necessary and sufficient

conditions on A and B in order that all linear mappings of Chapter 4 are

themselves element of spaces SX®Y,A EE1B ' etc.

- 12 -

CHAPTER 1. The space SX,A

In a complex separable Hilbert space X with inner product (.,o}X we

consider an unbounded non-negative self-adjoint operator A. This operator

will be fixed throughout chapters 1,2,3. Since A is self-adjoint it

admits a spectral resolution

For details and conventions on such spectral resolutions we refer to

Appendix A •. We define

-At e

o

00

tEa: •

For t E lR the operator Mt is self-adjoint, Mt is bounded iff Re t ~ O. -1

Mt is unitary iff Re t O. Further Mt is invertible and 'It E a: Mt M_t + On a: , i.e. the set of tEa: with Re t > 0, the operators Mt establish a

one-parameter semi-group of bounded injective operators on X. For more

details on such holomorphic semi-groups see Appendix B.

We now introduce our space of test functions SX,Ao

Definition 1.1.

SXA= U {MtXlxEX}= U Mt(X) , Re t>O Re t>O

SX,A is a dense linear subspace of X. From the semi-group properties the

following equalities immediately follow for any 0 > 0

Each of the spaces M (X), t > 0, can be considered as a Hilbert space. t

The inner product is

(u,V)M (X) t

- 13 -

The completeness of Mt(X) follows from the closedness of M_to

Mt(X) consists of exactly those u E X for which 00

< 00 •

a

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

SX,A' Xt for Mt(X), (0,0)0 or (',.) for (",)X'

Defini tion 1. 2.

The strong topology on S is the finest locally convex topology on S for

which the natural injections it Xt

+ 5, t > 0, are all continuous.

In other words S is made into a locally convex topological vector space

by imposing the inductive limit topology with respect to the family

{Xt}t>O • Cf. [SCHJ Ch. 11.6. Since the natural injections XL + Xt T > t > a, are all continuous the inductive limit topology is already

brought about by the family {X1

/ n } nEJN' A subset U c S is open iff for

-1 every t > 0 the set it (U) = U n X

t is open in X or, equivalently, iff

for every n E IN the set U n X1

/ n is open in Xl/n0 The inductive limit

here is more complicated than in the case of the Schwarz test function

spaces because our inductive limit is not strict! A closed subset of X

when considered as a set in Xt

, a < t < ., is not necessarily closed.

We shall see that open sets in 5 are always unbounded.

We introduce some notations:

- B denotes the set of everywhere finite real valued Borel functions -0

on lR such that Yt > 0 the function ~(x)e is bounded on [0,(0)'

- B c B contains those ~ with ~(x) > +

a Yx ~ 0

T c B consists of those ~ which are zero for x < a and piece-wise +

constant on the intervals [O,lJ, (n,n + 1J, n E IN. The value of ~

on such an interval is a natural number.

- B (A) , B (A), T (A) denote the corresponding sets of self-adjoint +

operators in X defined by

1"

- 14 -

00

A foo 2 The domain of ~( ) consists of exactly those x € X with 0 ~ (A)d(EAx,x)

It follows that S is contained in the domain of each operator ~(A). -At

Further: V~ € B Vt > 0 the operators ~(A)e = ~(A)Mt are bounded

and self-adjoint. The operators in B+(A) are all strictly positive.

Definition 1. 3.

For each ~ € B we introduce the following semi-norm (which is actually +

a norm)

= (~(A)u,u)~ = {J o

Further we define the sets

U € S •

u = {u € S I (~(A) u, u) <: E:2 , ~ € B, E: > O} •

~,E: +

The next theorem is very fundamental.

Theorem 1.4.

I. V~ E B V£ > a U is a convex, balanced, absorbing open set in + ~,E:

the strong topology. In other words the semi-norms p are continuous.

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

contains a

neighbourhood of 0 in X t' Then n contains a set U with ~ € Band £ > O.

~,£ +

III. That set of U,', ' ~ E T, £ > 0, establishes a base of the neighbourhood ,/"E:

system of 0 in S. In other words the strong topology in S is generated

by the sets U ~,E:

IV. For any open set V c S , 0 E V and for each t > 0 the intersection

Xt

n V is an unbounded set in Xt

•

< 00 •

- 15 -

Proof

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

and absorbing. For the terminology see [yJ Ch. I.1. We now show that

for each t > 0 U~,£ n Xt is an open set in Xt

•

By definition

2 U n Xt = {ulu EXt' (lJ!(A)u,u) < E } ~/£

2 Because of (ljJ(A)u,u) '" (1jJ(A)M2t

M_t

u,M_t

u) ::; 1IlJ!(A)M2t

IlIlM_t

ull ""

= 11lJ! (A) M2tllil u II~ and the boundedness of ljJ (A) M2t the sesquilinear form

(lJ!(A)u,u) is continuous on Xt • Therefore the set of u's in Xt

for which 2

(lJ!(A)u/u) < E is open in Xt*

II. We proceed in six steps

a) Introduce the projection operator P = n

the radius of the largest open ball in IN. Let r be

n,t

Thus

r n,t

[u E ~ n P (X) n X J} n t

b) For t > T ~ 0 we have e(n-l) (t-')r n,.

The proof of this runs as follows

n

n

fits in ~ n P (X ). n t

I 2At 2 e d(EAu,u) < p ] ~

n-1

.",. n (t-.) '" r ::; e r n,t n,c

II P u 112 n • .f

2AT -2 (n-l) (t-"r) e d(EAu/u)::; e

n-l n-l

Therefore if we take u such that

- 16 -

II P u 1\ n t (n-l) (t--r)

< e r (n-1) tt--r) n,T

it follows that II Pull < r • Consequently ,n l' n,T

r n,t

~ e r. The n,t

proof of the other inequality runs similarly,

00 p -1 c) For any fixed p > 0 and t > 0 the sequence E -1 n (r t) is convergen=. n- n,

Indeed, there exists an open ball in X , l' < t, with sufficiently small l'

radius E and centered at 0 which lies entirely within Q n Xt

• Then

Vn E ID r ~ E and with the first inequality in b) it follows that -1 n,T -1 - (n-1) (t--r) -1 - (n-1) (t-t)

(r t) ::; (r ) e ::; E e • n, n, T

d) We define a function X on ]R by

x(x) n E: ID

x(O) = X(~)

x(x) = 0 for x < 0

-EX and claim that X E B • For the proof we have to verify that X(x)e is

+ bounded on [0,00) for every E > O. On (n-l,n] and for every 1', 0 S T S 1

applying b) with t = 1 we find

-EX 4 2n-£(n-l) -2 X (x) e s 4n e (r 1) S n,

< 4 4 2+(n-l) (21'-E) ( )-2. - n e r n,T

Taking l' < ~E and invoking c) the result follows.

e) We now prove

(x (A) u, u) < 1 .. U E Q •

00 2 Suppose u E X

t for some t > O. Then E II Pull < 00 and for some 1',

n=l n t o <: , < t it follows with b)

(ft )

Further because of our assumption (*) and b)

II Pull n , n, -2 n,-n -2

sell Pnu 110 < ~n r 1 e S ~n r n, n, T

Therefore 2n2 p u E Q n X n T

for every n E ID.

- 17 -

In X we represent u by l'

N I;X)

I 1 2 I 1 u = 2n2

(2n P u) + ( -)u n=l n n=N 2n2 N

with

00

'I _1_)-1 'I P L L u. j=N 2j2 n=N n

With (*) we calculate

Since n n X contains an open neighbourhood of a for N sufficiently large T

u E Q n X . N .-

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

n n X which is a convex set • .-(Aposteriori it is clear that u E n n X

t).

f) We finish the proof of II by taking $ E T such that on (n-1,nJ it is 4 2n -2 equal to the smallest natural number greater than 4n e (r 1) •

n,

III) Any convex open neighbourhood of a satisfies the conditions of the set Q in II.

Therefore it contains a set V,I, with 1ji € T and £ > a. '/"IE:

IV) V contains a set V,I, • The results lIb and lIe show that V,I. n X contains ,/",£ ,+,,£ t

arbi trary large elements. [J

Remark. Similar to the proof of part lone proves that the sets

are closed.

Definition 1.5.

2 s €: €: > a}

A subset W c S is called bounded if for each a-neighbourhood U in S there

exi~ts a complex nunilier A such that W C AU • Cf. [SeB] 1.5.

- 18 -

In the next theorem we characterize bounded sets in S.

Theorem 1.6.

A subset W c S is bounded iff

3t > 0 3M > 0 Vu E: W II u II :s; M . t

Proof.

W is bounded iff V~ E: B+ 3M ~ 0 Vu E: W (~(A}u,u) ~ M. Therefore by taking

~ = 1 we observe that W is a bounded set in X. SO we have

3p > 0 Vu E W (u,u) :s; p

and

M

f 2At 2Mt

(*) Vu E: W VM > 0 Vt > 0 e d(tAu,U):S; e p

o

~) Vu E: W (~(A)u,u) = (~(A)M~ M u,M u) "s -s -s

which is a constant only depending on ~ •

• ) Suppose the statement were not true then

Vt > 0 \1M > 0 3u E W II Mull > M • -t

By induction we define two sequences of real numbers {t }, {N }, t + 0, N + 00 as n n n n

n ~ 00 and a sequence {u } c W as follows n

n = 1

n '" 9. + 1

= 2. Then take Nl > 0 and u 1 E W such that

> 1 •

Suppose Vt ~ ~t9. Vu E: W VK > 0 2tA

e d(tAU/U) ~ 9. + 1 is true.

Then W is a bounded set in Xt

, t $ ~t , because with the aid of (*) 9.

00 00

f 2At

e d(EAu,u) J 2+N JI'p

$ e + £ + 1 •

o o

- 19 -

If our sequences terminate for a certain value of n then W is a bounded set.

If not, define the function X(A) on (a,"") by

2At n neAl = e on the interval (N 1,NJ n- n

-At Since n(A)e is a bounded function for t > 0, nEB

n = 1 ,2, • •• •

+ and the sequence

{(n(A)u ,U )} should be bounded. However (n(A)u iU ) = n n n n f~ n(A)d(E,u ,u ) > n + 1.

I\. n n 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.

Proof.

u + 0 in the strong topology of S iff n

3t > C {u } c X and II u II -+ 0 • n tnt

~) For any ~ E B+ we have (~(A)u ,u ) = (~(A)M2tM t U 1M u) n n - n -t n ::; II1/! (A) M2tllil Ai u II 2

-t n + 0 as n + 00 because ~(A)M2t is a bounded operator on X.

-*) Suppose u . .,. O. Then VI/! E B+ (1/;(A)u ,u) .. O • n n n We conclude that the sequence {u } is a bounded set. So 38 > 0 'In E :IN lIu II n n T

::;

Further, taking ~ 1 , it is clear that lIunl~ + 0 as n + "". From this we derive

+ 0 as n -+ "".

Now we show II un I~ -r 0 for any t < 1'.

A.

- 20 -

The second integral can be estimated unifo::mly

= f -2A(T-t) 2ATd (E ) _< e e ,u ,U 1\ n n

L L

00

e -2L (-r-t) J L

2AT . _< -2L(T-t) 2 e d (E, u ,U) e e.

1\ n n

By taking L sufficiently large the second term in (,1) can be made smaller

than ~~ uniformly in n.

The first term in (*) tends to 0 as n + 00 because of (*).

Theorem 1.8.

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

3t > a {u } c Xt

and {u } is a Cauchy sequence in X • n n t

II. S is sequentially complete, i.e. every cauchy sequence converges to a limit

point.

Proof

I. An argument similar to the proof of the preceding theorem.

II. Follows from I and the completeness of Xt •

Next we characterize compact sets in S.

Theorem 1.9.

A subset K c S is compact iff

3t > 0 K c Xt

and K is compact in Xt

•

Proof.

<=) Let {rl } be an open covering of K in S. Then {n n Xt

} is an open u u

covering of K in Xt

• Since K is supposed to beN compact in Xt there exists

a finite subcovering N

{rl }, 1 sis N, with i;;1 (n n X ) :> K. But ai

ai

t

then certainly i~1 rl a ,:> K. ~

~) Since K is c~mpact it is bounded and therefore it is a bounded set with

[

- 21 -

bound e in XT

for some T > O. We show that K is compact in Xt

whenever

t < T. Consider a sequence {u } c K. There exists a converging sub-n

sequence {u } with u ~ u ~ K as j + ~, convergence in S. Then also nj nj

u ~ u in X-sense. Put u nj nj

- u = vj

• {vj

} is a bounded sequence in

Xi and IIVj II ~ 0 as j ~ 00. Now we find ourselves in exactly the same

posi tion as in the proof of the only-if-part of Theorem 1.7. We conclude

II Vj lit + 0 whenever 0 s; t < T. OUr subsequence {un} converges to u in

Xt-sense. This shows the compactness of K in Xt

jfor 0 s; t < T. 0

Lemma 1.10.

Let u E X be such that u ~ D(w(A» for all WEB. Then u E S. +

Proof d _

If not, then for every t > 0 the integral f~ e2At

d(EAu,u) is divergent.

Pick sequences of real numbers {til, {Ni

} with ti ~O and Ni t ~ such that

for all i E ]\j

N i 2A

( t.

. e ~d(EAu,u) > 1 • . N. 1 3.-

At i Define W on (0,00) by W(A) = e

2 Joo 2 111fi (A) u II = 0 1jJ (A) d (E AU, u) = 00

Con tr adi cti on!

on

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. [SCH).

The terminology is explained in the proof.

Theorem 1.11

I SX,A is complete

II SX,A is bornological

o

- 22 -

III. SXtA is barreled

IV.

V.

Proof

SX,A is Montel iff for every t > 0 the operator Mt

as an operator on X.

SX,A is Nuclear iff for every t > a the operator Mt

HS (=Hilbert-Schmidt) operator on X.

.. At e is compact

-At = e is a

I. Let {x } be a Cauchy net. The a's belong to a directed set D. For each ~

neighbourhood rt :3 0 there is Y E D such that whenever a >-- y and S >- y

one has xa - Xs ~ rt. We now prove that there exists an XES such that

x ~ x in the strong topology. Let x be the limit of x in X-sense, a a For each tjJ E B the net {tjJ(A)x} converges in X-sense to a limit x,/,,

+ a 'i'

Because of the closedness of tjJ(A) one has xtjJ E D(tjJ(A» and xtjJ = tjJ(A)x.

The result follows by applying Lemma 1.10.

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

III.

has to be a neighbourhood of O. Let Bt

be the open unit ball in Xt

, t > O.

Bt

is bounded in S therefore for some E: > o one has E:Bt

c rt f1 Xt

•

We conclude that for every t > 0 the set n f1 Xt

contains an open neighbourhood

of O.

But then according to Theorem 1.4.II rt contains a set U,/, • 'i',E:

A barrel V is a subset which is radical, 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 '1 Xt

has

to be a barrel in Xt in the Xt-topology, for every t > O. Since Xt is a

Hilbert space there exists an open neighbourhood of the origin [ with

[ c V n Xt ' Again the conditions of Theorem 1.4.1I are satisfied so that

V con tains a se t iJ", • 'i"€

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

for every t > a the operator Mt

is compact.

~) Suppose Mt is compact for every t > O. Let W be a closed and bounded

subset in S. Then W c Xt

for some t > 0 and W is closed and bounded

in Xt , See'I'heorelll 1.6. Take 1",0 < 1" < t. We claim that W is a compact

set in X , To see this, consider the following commutative diagram where C; T

v.

- 23 -

denotes the natural injection

Xt c: X

T

M' i i M t T

Xo ,.. Xo M t-T

The vertical arrows are isomorphisms. So c:;. is a compact map and W

compact in X . Consider an open covering {C } of W in 5, then {C T a a

is an open covering of W in X • Because of the compactness of W in T

there is a finite subcovering:

W c:

N u

i=l (C

a. ~

n X ) • T

N But then certainly W c: u C , which shows the compactness of W.

i=l a i

is

n X } T

X T

~) Suppose S is Montel, i.e. each closed and bounded set is compact. Let

{u } be a bounded sequence in X. Pick any fixed t > O. Consider the n

.. )

sequence {M u }. Consider the closure in S of this sequence. This closure t n

is a bounded set and, according to our assumption, compact. Thus

{Mtun

} contains a converging subsequence in S: Mtu ~ v. This sequence nj

certainly converges in X. SO Mt

must be a compact operator •

We

a)

proceed in four steps: a,b,c and d.

Suppose that for every t > 0 Mt

is H.S., i.e. for any orthonormal

basis {e.} in X ~

00

Vt > 0 I II Mte.112 < 00.

i=l ~

It follows that M t

-At e is compact. Hence A has a purely discrete

spectrum with no finite accumulation point and all eigenspaces are

finite dimensional. We order the eigenvalues in a non-decreasing

sequence in which each eigenvalue occurs as many times as the

b)

- 24 -

dimension of its eiger.space

A -+ 00 n as n -+ 00 •

We choose an orthonormal base of eigenvectors {e }. Together with 00 - 2Ak t n 00

(*) we obtain for each t > 0 Lk=l e < 00 and aposteriori L

k=l

Let B1 and B2 be Banach spaces. A bounded operator T lZalled nuclear if there exists a sequence {f'} C 8' ,

n - 1 {a } c 82 and a sequence of complex numbers {c } such ~n

and

Our

sup (II f' II, II g II) < 00

n n

I Ie i < ro

n=l n

Vx E

operator

-At e :u

B1

e

m Tx '" lim I c <X,f I>g

m-+<» n=l n n n

-At can be represented by

00 -:\ t I e k (u,e )e

n=l n n

and is obviously nuclear.

n

in 82

•

: Bl -+ 82 is

a sequence

that

-A t k

e

c) Our aim is now to construct an invertible nuclear operator veAl of

the form

00

v (A) u

-1 in such a way that v(A) E B+(A).

Define integers N(n), n 0,1,2, ••• , such that N(O);:: 0, N(n + 1) > N(n),

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

00 - A . lin L e J

j =N (n)

1 < '2 .

n

< 00

- 25 -

Now define the sequence {Vk

} by

Ak n

N (n - 1) < k S; N (n) , n = 0,1,2, ••••

00

It is clear that E vk

is convergent. k=l

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

Ak_ 1 + Ak Ak + Ak+1

on A1

and cr(A) = e (2 2 3, k = 2,3, ••• -1

It is easily seen that cr(A) = veAl.

the interval A + A

on [0, 1 2 J 2

d) S is a nuclear space iff for every continuous semi-norm. p on S there

is another semi-norm q z p such that the canonical injection S ~ S q r'

is a nuclear map. Here the Banach space S is defined as the completion p

of the quotient space S;{p-l(O)}. Since the composition of a nuclear

operator with a bounded operator is again nuclear, $ is a nuclear space

iff for every semi-norm P1J!' 1jJ € B +' there

such that the canonical injection J : S

This canonical can be Px

injection written

(I':) Ju 00

I X-~(A.)1J!~(Aj) (x(A)u, j=l J

{x-~(A>e.} ~ J

We remind that

S respectively Px

Hilbert spaces

becomes

00

is a semi-norm p , X E.B+,X X

~ S is nuclear. P1jJ.

are orthonormal bases in the 2

we take X = a 1J! then (*)

\' -1 -~-~ Ju I... cr (A.)(x(A)u,x (A)e.)o1J! p .. )e.

1 ] J. J J

-1 And this operator is nuclear because cr (A

j) = V

j•

?

~) Suppose S is nuclear. See the definition in d} above. Take for p the X-norm. -Then for some semi-norm q the injection S ~ X is nuclear map. Further we q

can take 1jJ f. B such that p,/, ? q. (Apply Theorem 1.4.II). The injection + 'I' -1

S ~ X must be mlclear. Therefore tjJ (A) must be a nuclear operator and Pw -At -1 -At

consequently also H.S. But then: e = 1J!(A) [1J!(A)e ] must be H.S.

since the operator in square brackets is bounded for every t > O.

1jJ

Il

- 26 -

CHAPTER 2. The space TX,A

We now introduce our space of trajectories TX,A" The elements in this

space are candidates for becoming 'generalized functions" (or "distributions").

DeUni tion 2. I •

TX A denotes the complex vector space which consists of all mappings , +

F : ~ + X such that

(1) F is holomorphic.

(ii) + Vt,T E: a: M F(T) t

F (t + T) •

Such a mapping F will be called a trajectory_ If for some ~ E a:+ FI~) = F2(~) + then FI = F2 throughout a: • This follows immediately from the semi-group

and analyticity properties. We shall use the notation MtF for F(t}" It is

immediate that Vt > 0 MtF E S. In chapters 2 and 3 TX,A is abbreviated

by T.

Definition 2.2.

The embedding emb + and t t- ~

X + T is defined by (emb x) (t) = Mtx for all x 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 E T such that F(t)

does not converge or, even II F (t) II t 00 if t -Ii O. The answer is affirmative:

simply take x E X\DCA) and F(t} = Mtx. More general examples are obtained + by taking ~ B, u E X and defining G(t) = ~(A)Mtu, t E ~ • The next

theorem shows that all elements of T arise in this way.

Theorem 2.3.

For every F E T there exists w E X and $ E B+ such that F(t) + for every t E ~ •

Proof

-At = ljJ(A) e w

If the integral f~ 2A A e d(E

AF(1) ,F(1» converges then F(1) E D(e ).

- 27 -

A Take w = e F(l) and ~ = 1. Now suppose this integral is divergent. Take

00 2A - 2Tn a sequence {Tn}' Tn ~ O. All the integrals foe e d(EAF(l),F(l»,

n = 1,2, ••• , converge. Define a sequence of real numbers {N(n)},

n c 0,1,2, ••• , N(n) t 00 such that N(O) = 0, N(n) > N(n - 1) and

00

f 2" -2T 1A n+

e e d(EAF(1),F(1»

1 < 2'"

n N(n)

TIA Define IjJ on [0,"") by e

+ n = 1,2,... • Then IjJ E B

1-'(1) Eo: D(1/I-l (A)e~. Take

Tn+1A on [O,N(l)J and e on (N(n),N(n + 1)J,

and f~ e2A

IjJ-2(A)d(E"F(1),F(1» < 00 , which means

-1 A w = IjJ (A)e"F(1) •

In T we introduce the topology of uniform convergence on compacta in ~+. Because of the semi-group property F + F in Tiff vm E ~ F (l) + F(l) n n m m in the X-norm. More precisely

Definition 2.4.

n

The strong topology in T is the topology induced by the semi-norms Pn' n Eo: m I

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

{FIIIF(';')11 < c., c > 0, 1 ~ j ~ N , ] ] j

Theorem 2.5.

I. T endowed with the strong topology is a Frechet space

II. A base of open sets {OV,t} is given by 0V,t = {FIF(t) E V, V open in X, t > 0 and fixed}.

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

Each open set in T is a denumerable union of sets of this type.

- 28 -

Proof

I. Frechet means metrizable and complete. We define the distance of F from

the origin in the standard way by

00

d (F) = L 2 -~I F (l) 1I{1 + II F (l) II } -1 • n=l n n

The set of F with d(F) <

W • On the other c

1c

2",c

N

2-N- 1min(l,c1

, ••• ,CN

) certainly lies within

hand each W lies within a ball centered c 1c 2 ···cN

at the origin with radius

N 1 c 1 I n +-2

n 1 + cn 2N n=l

Therefore the sets n = {F\d(F) < 1 n € IN} establish a base of O-neighbour-n n

hoods. Further, since T is metrizable it is complete iff each Cauchy sequence

converges. Let {F } be m

Then for each n IN

a Cauchy sequence in T, 1 {F (-)} is a Cauchy sequence in X which converges to

m n an element h

1/ n E X. For V > none

sides converge and e«1/n)-(1/V»A

has F (l) = e«l/n)-(l/V»AF (1)· Since both m V m n

is a closed operator we have

h - D( «1/n)-{1/v»A) d h lin E e an 1/v = e«1/n)-(1/V»~1/n' Now we define

F(t) = e-(t-(1/V»~1/V'

Clearly F E T and is the limit of {F }. m

II. We proceed in three steps.

Re t € [1, 1 1) , V E IN • v v-

a} First we prove that, given 0V,t' then for each T, 0 < T < t, there

exists an open set V c X such that ° = a • For V we take the V,tA(t V,. A(t )

inverse image of V under the mapping e- -TJ, then e- -T (V) c V.

It is clear that each trajectory which passes at t through V, passes

at T through V and conversely.

b) We prove that aU,l/n' U c X open, n € IN, is an open set: V can be

covered by open balls lying inside V, therefore aV,l/n is a union of

translates of sets of t'l'pe W • cl,··eN

- 29 -

c) From a) and b) we,conclude that the sets aU t are open. Further , a n a with 1: < t can U,t V,T be written as ~U n °v = a~U V .

,1' f: n ,1:

Here U is the inverse image of U under e-(t-1')A. Now it is easy

to see that each translate of a set of type W is a set of c ••• c

N type aU,t' We are ready if we show that an arbItrary union of

sets aU,t can be reduced to a denumerable union of sets of this

type. This can be achieved by the construction

U (j

1 1 U,t n + 1 !S t<n

(j~1 ..L 'n+l

with

v U U 1 1

< t<-n + 1 n

1 1.) ..... where for each t £ [-;;r , U denotes the reduction of U at n

1 the point t to the point - in the way we described in a) • n+l "

Theorem 2.6.

emb(S) is everywhere dense in T.

Proof

For any F E T the sequence emb(F(l/n» converges to F in the strong

topology.

Theorem 2.7.

A set BeT is buunded iff for every t > 0 the set {F(t) IF E B} is

bounded in X.

Proof

~) Each continuous semi-norm has to be bounded on B.

Therefore Pn (F) ;: II F (lin) II is a bounded function on B.

r

Ll

- 30 -

Because of the boundedness of M for each T > a it then follows 't

that {F(t) IF E B} is a bounded set for each fixed t > D.

-) Suppose for each fixed t > 0 the set {II F (t)1I IF E B} is bounded.

We have to prove that each open a-neighbourhood can be blowed up

that B is contained 1n it. Let at = sup II F(t)1i 1 t fixed. FEB

00 -n within an open ball with radius E n=l 2 Cl 1/ n (1 + Cl 1/ n )

Example

Let B be a bounded set in X. Let ~ E B. Then the set

{FIF(t) = ~(A)Mtx , x € a} is bounded in T.

Theorem 2.8.

-1

Then B

A set K c T is compact iff for each t > 0 the set {F(t) IF € K} is

compact in X.

Pl:oof. -~) If K is compact then each sequence {F } c K has a convergent sub-

n sequence. This means that in the set K

t = {F(t) IF E K, t fixed}

each sequence has a convergent subsequence, which says that Kt is

compact in X •

such

lies

0

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

subsequence. Consider the sequence {Fn (1)} c Kl c X. Kl is compact

therefore a convergent subsequence in Kl exists. Denote it by {F~(l)}. The sequence {F~(~)} has a convergent subsequence in K~o Denote it

bY{F2(~)}. n

Proceeding in this way we m t {F } c {F } for m > t and n n

arrive at sequences m 1 } {F (-) converges to n m

The "diagonal sequen ce II {1<,n} has the property n

{Fm} c K such that n

an element in Kl/mo

that {Fn(t)} converges n

to F(t) E Kt • But then also F~ + F in the strong topology.

I.

II.

III.

- 31 -

Example

Let K be a compact set in X. Let I/J E B. Then the set {FIF(t) == I/J(A)Mtx, x Kj

is compact in T.

In the last theorem of this chapter we describe the properties of our

topological vectorspace T in the standard termlnologyoftopological

vectorspaces. [SCH].

Theorem 2.9.

TX,A is bornological

TX,A is barreled

TX,A is Montel iff for t > 0 the operator Mt

-At is every = e

compact as an operator on X.

IV. TX,A is Nuclear iff for every t > 0 the operator Mt a H.S.-operator on X.

-At ;: e is

Proof

I,ll. T is bornological and barreled because it is metrizable. For

a simple proof see [SCH] 11.8.

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

sequence {Fn} defined by Fn(t) == Mtxn

is a bounded set in T. Because

of our assumption the closure of this set is compact. But then, by

theorem 2.8, for each t > 0 the sequence F (t) contains a converging n

subsequence. This shows the compactness of Mt

•

-) Suppose Mt is compact for every t > O.

Let B be a closed and bounded set. Then the set

Bt

= {F(t) IF E B, t > 0 and fixed} is bounded. Since BtH - M .. /Bt >

it follows that B is precompect in X. Now take any sequence {G } c B. t+T n

Because of the precompactness of each Bt

the "diagonal procedure"

of the proof of theorem 2.8 yields a converging subsequence of {G }. n

This subsequence converges to an element in B because B is closed.

We conclude that B is compact.

- 32 -

IV =» Suppose T is a nuclear space. Take n E :IN. There exists a

semi-norm P ~ P such that the natural injection T ~ T . = Xisn p p

nuclear. However there exists m > n and a > 0 such thattRe semi-

norm aPm~ p. Since the composition of a nuclear and a bounded

map is nuclear the natural injection t ~ f is nuclear.

This natural" injection is realized by Pm the Pn

map M(l/n)-(l/m) which

must therefore be nuclear. It follows that Mt

is H.S. for each t > O.

is

of the semi-group property

injection t ~ TPm Pn

is therefore nuclear. Now let p be anrealized by M(l/n)-(l/m) and

arbitrary semi-norm on T.There e?Cist a > 0 and n E :IN such that ap > P on T. Then for In > nn_ _we have ap > p and the natural injection T ~ T is nuclear. 0

m aPinP

• ) Suppose for each t > 0 Mt

is H.S. Because

Mt

is also nuclear. Let m > n. The natural

I

- 33 -

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

We now consider a pairing of Sand T. It turns out that S and T can

be considered as the strong topological dual spaces of each other.

Defini tion 3.1.

On SX,A x TX,A we introduce a sesquilinear form by

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

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

Mt the definition does not depend on the choice of t. We remark that

<UO,F> = 0 for all F E T implies U o = 0 and <U,FO

> = 0 for all u E S

implies FO = O. These two facts easily follow by taking F = emb(uO)'

respectively the denseness of emb(S) in T.

Theorem 3.2.

I. For each F E T the linear functional <u,F>, which maps S onto 0;, is

continuous in the strong topology of S.

II. For each strongly continuous linear functional t on S there exists

GET such that t(u) = <u,G> for all u E S.

III. For each v E S the linear functional <V,G> , which maps Tonto 0;, is

continuous in the strong topology of T.

IV. For each strongly continuous linear functional m on T there exists

w E S such that m(F) = <v,F> for all F E T.

Proof

I. <u,F> is continuous on S iff for all t > 0 it is continuous in the

Xt-norm when restricted to this subspace. Indeed

- 34 -

II. For fixed t > 0 the mapping Mt

: X ~ S is continuous. Therefore the

mapping t(Mtx) : X ~ ~ is continuous. From Riess' theorem follows

the existence of bt E X such that l(Mtx) = (x,b t) for all x E X.

If we replace x by MtYr y E: X, it follows from the self-adjointness

of M that b == M b We take G such that G (t) = bt

, Then 1: t+T l t'

R..(u) = <u,G> for all u E S.

III, Since T is metrizable it is sufficient to prove continuity for

sequences in T, LetG ~ ___ n

small t we have <v,G > n

o in the strong topology. Then for sufficiently

(M tV,G (t» ~ 0 because G (t) ~ 0 in X-sense. - n n

+ . IV. Take WEB. Take a sequence {x } c X, x ~ 0 in X. Define w(A)x E T

n n n by (w(A)x ) (t) == w(A)M x • Since w(A)M

t is a bounded operator w(A)x ~ 0

n t n n strongly in T. By taking W = 1 we conclude first that the restriction

of m to X has the Riess representation m(x) = (x,S) 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 weAl it follows that a E D(w(A)l + for each W E: B • But then, by Lemma 1.10, e E S. Finally, as X is

denae in T the representation m(F) = (F,a) = <e,F> is valid for all

F t. T.

Definition 3.3.

The weak topology on S is the topology induced by the semi-norms

PF{u) = I<U,F>I, F E T. The weak topology on T is the topology induced by the semi-norms

p {G} = !<v,G>!, v E S. v

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.

T S I. In the sequel we shall denote by

o

In the next theorem we show that in both spaces Sand SI weakly bounded

sets are strongly bounded.

- 35 -

Theorem 3.4. (Banach-Steinhaus)

I. Let :.: c $' be such that for each g EO S there exists M > o such that g for every F ( ::: one has I <g ,F> I :£ M , then for each t > 0 there exists g C

t > 0 such that for every FE::: one has II F (t) II :£ Ct

II. Let 8 c S be such that for every F EO SI there exists MF > 0 such that

for every f EO 8 one has ! <f ,F>!:£ MF

, then there exists T > 0 and c > 0 -such that 8 c X and II f II < c for all f E 8.

T T

Proof

I. From the assumption it follows that for each hEX and each t > 0 there

exists a constant Mh > 0 such that for every FE::: !<M h,F>! ~ ~ t ,t t n, or I (h,F(t»! :::; ~ • From the Banach-Steinhaus theorem in Hilbert h,t space it then follows that the set {F(t) IF E :::}, t fixed, is a bounded

set in X.

+ 2 2 2 II. Let t/i E: B • Let w EO D(t/i(A». Then t/i(A) w, defined by (t/i(A) w) (t) =l)!(A) I\W,

belongs to 5'. From our assumption it follows that for every

fEe l<f,l)!(A)2w>1 = I (l)!(A)f,l)!(A)w) I < M". • 'I' ,w

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

e is a bounded set in the Hilbert space \p' i. e. the completion of S with

respect to the norm 1Il)!(A)· II. But this means that 8 is bounded in S,

since each semi-norm Plj!' + .

l)! E B ,is bounded on it.

In the next two theorems we give characterizations of weak convergence

of sequences both in S and S'.

Theorem 3.5.

Proof

U ~ 0 in the weak topology of S iff n

3t > 0 {U } c X and Vw EO Xt n t

Weak convergence of {Un} in Xt means weak convergence of M_tun in X.

- 36 -

~) Let F E S I. <u ,F> = (M u ,F(t». Since M tU + 0 weakly in X n -t n - n

and F(t) £ X, it follows that <u ,F> + O. n

.) First we remark that weak convergence in S implies

in X. Therefore for any w E X any L > 0, t > 0 fOL

as n .~ 00 {u} is a bounded set in S. Therefore n

weak convergence At

e d(E>..un,w) + 0

3T > 0 38 > 0 Vn E :IN II u II ~ e. We shall prove that n T

u + 0 weakly n

in XT

• Denote the projection operator J~ dE>.. by TIL' TIL commutes with

M • Take w £ X then -T

L

(M u ,w) 0 -T n I e>..td(E>..un,w) + (TILM_run,TILw)o·

o

We have II fiLM u II:;; 8 and II ITLw II + 0 as L + 0). Therefore if we take -r n L large enough the second term in (*) is smaller than ~E uniformly

for all n. As we have just seen the first term in C*) can be made

smaller than ~E by taking n large enough. This finishes the proof. 0

Corollary 3.6.

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

II. Any bounded sequence in S has a weakly converging subsequence.

Theorem 3.7.

Proof

F + 0 in the weak topology of S' iff n

Vt > 0 Vv E X

.) For any v E X <M v,F >= (v,F (t»O + 0 as n + 00. t n n

.. ) For any <P E Sand t sufficient.ly small

because M -t qJ X.

- 37 -

Corollary 3.8.

I. Strong convergence of a sequence in $' implies its weak convergence.

II. Any bounded sequence in S' has a weakly convergent subsequence. (Use

a diagonal argument).

It looks reasonable to conjecture that the weak topology on S is the

inductive limit topology with respect to the Hilbert spaces Xt , now

endowed with the weak topology. It is easily seen that the weak

topology on 5' is induced by the semi-norms Pt

' t > 0, V € X ,v and Pt,v(F) = I {v,F{t})ol. We will not pursue these things further

here. The next theorem deals with the question: When does weak convergence

of a sequence imply its strong convergence?

Theorem 3.9.

The following three statements are equivalent.

I. For each t > 0 -At

e is a compact operator on X.

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

III. Each weakly convergent sequence in $' converges strongly in S'.

Proof

I - II. A weakly convergent sequence in S converges, for some t > 0,

weakly in Xt • See Theorem 3.5. Because of the assumption the natural

injection Xt

c Xa

, 0 < a < t, is compact. Cf. the proof of Theorem 1.11.

But then our sequence converges strongly in X • a

II ~ I. Take any sequence {f } C n

t > 0 and each F E S' we

X, f -+ 0 weakly in X. For each n

have <M f ,F> -+ O. Thus Mtf -+ 0 weakly in 5. t n n

Because of the assumption Mtfn -+ 0 strongly in S. And so IIMtf

n '6 -+ O.

This shows that Mt must be compact.

I ~ III. Let {F } C S'. Suppose Vg € S <g,F > -+ O. n n

Then Vh E X Va > 0 <M h,F > (h,F (a» -+ O. This means that Va > 0 ann

Fn(a) + 0 weakly in X. Using the compactness of MS' S > 0, we find that

F (a + B) = MoF (a) -+ 0 strongly in X. Therefore Vt > G IIF (t) \I -+ O. n ~n n

- 38 -

III ~ I. Let {v } c X be such that v n n 7 0 weakly in X. Then v 7 0

n weakly in 5' because Yg E S <g,v> = (g,V )0 7 O. Now the n n assumption says vn 7 0 strongly in S'. This means Vt > 0 Mtv

n 7 0

strongly in X. The compactness of Mt follows for each t > O.

One might wonder whether in the case that Mt

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 O-neighbourhoods

always contains a closed subspace of finite codimension • Generally

speaking such "large" sets do not fit in a strongly open O-neighbour­

hood.

o

It is relatively simple to show that both in Sand $1 the weakly compact

sets are just the (weakly) closed and bounded sets. Then it is not

difficult to prove that Sand S' with their strong topologies are

Mackey spaces. That is the strong topology is the finest locally

convex topology for which the dual of S (resp. 51) is 51 (resp. S). (All spaces which are either barreled or bornological are Mackey.

See [SCH] IV.4).

- 39 -

CHAPTER 4. Characterization of continuous linear maRpings between the soaces

Let B be a non-negative self-adjoint operator in a separable Hilbert

space Y. As before A is a non-negative self-adjoint operator in a

separable Hilbert space X. In this chapter we shall derive conditions

implying the continuity of linear mappings SX,A + Sy,B' SX,A + Sy,B ' S~,A + SY,B' S~,A + Sy,B' Further we investigate which linear

operators, defined on a subset of X, can be continuously extended to

operators on S~,A' The next theorem is an immediate consequence of the

fact that SX,A is bornological. Cf. Theorem 1.11, For completeness we

give an ad hoc proof.

Theorem 4.1.

A linear map Q : SX,A + V, where V is an arbitrary locally convex

topological vector space, is continuous

1. iff for each t > 0 the map Q!\ : X + V is continuous.

II. iff for each null sequence {un} c SX,A ' un + 0 in SX,A ' the sequence

{Qu } is a null sequence in V. n

Proof

1 .. ) Mt is an isomorphism from X to Xt

' By the definition of the

inductive limit, Xt

is continuously injected in SX,A' SO if Q is

continuous it follows that QMt is continuous •

• ) Let ~ denote the restriction of Q to Xt

• From the continuity of

QfAt on X follows the continuity of Qt on Xt ' Now let rt 3 0 be open in -1 -1 V. For each t > 0 Q Crt) n X

t = ~ (n) is an open a-neighbourhood in

-1 Xt ' Thus Q (rt) is open in SX,A'

II) Follows from I because null sequences in S are always null

sequences in some Xt

' t > 0 and vice versa.

In the next theorem we characterize continuous linear mappings from -Bt SX,A to SY,B' We denote Nt = e ,

r' l'

- 40 -

'l'heorem 4. 2 ,

Suppose P : SX,A ~ Sy,B is a linear mapping. The following six conditions

are equivalent.

I. P 1s continuous with respect to the strong topologies of SX,A and SY,B'

II. un ~ 0, strongly in SX,A' implies Pun ~ 0, strongly in SY,B'

III, For each a > 0 the operator PM is a bounded linear operator from a

X into Y.

IV. For each t > 0 there exists B > 0 such that PMtCX) c Ne{Y) and N_ePMt is a bounded linear operator from X into Y.

V. There exists a dense linear subspace E c V such that for each fixed

y € - the linear functional Lp,y(f) = (Pf,y)V is continuous on SX,A"

* VI. For each t > 0 (PMt ) is a bounded linear operator from Y to X. * (Remark: One has (PMt ) (V) c SX,A).

Proof

We proceed according to the scheme

.............,. ~ -;::::~~ ~ I II III IV V VI ~~

I ~ II. See Theorem 4.1.

II .. III. If {x } is a null sequence in X then for any a > 0 {M x } is nan a null sequence in Sx A' By II {PM x } is a null sequence in , a n SY,B and hence in Y.

III .. IV. Let a > O. We define a class A of continuous linear a

functionals on X in the following way

Obviously a 1 < a 2 .. Aa1 c Aa2 since for g € Na1 CV) we have

N q N (N g) where N g € N (V) and 1\ N g lIy :$ 1. We -a1 -a2 u2-a1 a 2-a1 a 2 a 2-a1

claim

Vx € X 3a > 0 3M > 0 V£€ A a I R, (x) I :$ M •

III - V.

- 41 -

Indeed, since PMtx E Sy,B' for et sufficiently small PMtx € Vet and

N PM x € Y so that -0. t

1R.(x) I = I (N PM x,g)vi $ liN PM xliV -0. t -0. t

Applying a modified version of the Banach-Steinhaus theorem, see [J],

see our Appendix C, from (*) it follows

30. > 0 3M > 0 Vx € X V~ € A 0.

But then for all x E X and all g € Y 0.

which means PMtx to D(N ) and N -clMt -0. We can 1!ake - :: Y. Take y € Y fixed.

t > 0 we have

I Lp (M x) I = I (PMtx ,y) V I s ,y t

I R. (x) I ~ MIl x \I

bounded.

For each x € X and each

Ct,yll xliX

Together with Theorem 4.1 the result follows.

V - VI. According to Riess' theorem for each y € 3 and each t > 0 there

exists f t € X such that for each h € X

Replacing h by M x, X € X we observe that f = M f t so that f t E Sx A' T * t+T T . ,

From (*) we obtain ft

= (PM) y. SO D«PM )*) ~ E which is dense in V. t t *

Since PMt

is defined on the whole X the operator (PMt

) is defined

on the whole V and bounded. Repeating the argument with arbitrary

* y € X shows (PMt ) y E SX,A' VI - III. PMt is bounded because (PMt )* is bounded.

IV - II. Let {un} be a nUll-sequence in SX,A" Then for some 0. > 0

{M u} is a nUll-sequence in X. But then, using the boundedness of -0. n N_SPMo.' {N_SPun } = {N_SPMo.M_o.un} is a null-sequence in V. Hence {PUn}

S [1 is a nUll-sequence in Y,So-

I.

II.

III.

- 42 -

The next corollary is important for applications.

Corollary 4.3.

Suppose Q is a densely defined closable operator: X ~ Y. If

D(Q) J SX,A and Q(SX,A) c SY,B' then Q maps SX,A continuously into

SY,B'

Proof

Q is closable iff D(Q*) is dense in V. Since QMt : X ~ Y is defined

* on the whole X, it.s adjoint (!2.Mt) is bounded. T*he adjoint however

* * is densely defined since on D(Q*) one has (aMt ) = MtQ • Hence (~\{t)

is defined on the whole Y and bounded. From this the boundedness of

!2.Mt

follows. Application of Theorem 4.2 III yields the desired result. n

Theorem 4.4.

Let K : SX,A + Sy,B be a linear mapping. The following four conditions are

equivalent.

K is continuous with respect to the strong topologies of SX,A and SY,B'

For each t > 0, a > a NtKMa is a bounded linear operator from X into Y.

For each t > a NtK is a continuous map from SX,A into Sy B' , IV. There exists a mapping Z of the interval (o,~) into the continuous linear

maps from SX,A into SY,B such that

(i) For each t > 0, T > a Z(t + t) = NtZ(T) •

(ii) For each u E SX,A Ku = lim emb(Z(t)u) t-l-O

strongly in Sy B' ,

Proof

I - II. Let {x } be a null-sequence in X. Then {M x } is a null sequence nan in SX,A' Since K is continuous {KMaxn} is a null sequence in Sy,B' which means that for each t > 0 {(N KM )x } is a null sequence in Y.

tan Hence N KM is bounded.

t a

- 43 -

II ~ I. Let {Un} be a null sequence in SX,Ao Then for some a > 0

{M u } 1s defined and is a nUll-sequence in Xo But then -et n {Kun } = {KMetM_aun } is a null-sequence in Sy,B since for each t > 0

(KM M u) (t) = (NtKM)M u ~ 0 in Y et -a n a -a n

II ~ III. Apply theorem 4.2 with P = NtKo III ~ IV. Define Z(t) = NtK. It obviously has the property (i). we

must show that for each, > 0 lim N Z(t)u = (Ku) (,). But this is also HO -r

obvious because N~Z(t)u = Nt Ku and the semi-group N is strongly ,+, u continuous.

IV ~ III. Trivial because NtK Z (t) •

Theorem 4.5.

Let r : SX,A ~ Sy,B be a linear mapping.

Let r r

X ~ Y denote the restriction of r to X. The following five

conditions are equivalent.

I. r is continuous with respect to the strong topologies of SX,A and

SY,B"

II. r;(Y) c SX,Ao

III. There exists t > 0 such that r;(Y) c Mt(X) and M_tr; is a bounded

operator from Y into X.

IV. There exists t > 0 such that r M r -t

with domain Xt

c X is bounded as

an operator from X into Y.

V. 'l'here exists t > 0 and a continuous linear map Q SX,A ~ Sy B such

that r = w\.

Proof

I - II. From the continuity of r it * continuous. Its adjoint f , r

immediately follows that r is r

,

* defined by (x,rrY)X = (rrx,y)y for all

x E X, Y E Y, is a bounded operator as well. Now consider the dual

operator r' : SY,B ~ SX,A defined by

<F,r'G>X <fF,G>y for all F E SX,A'

o

- 44 -

For fixed G E Sly,B the righthand-side defines a continuous linear

functional on SX,A' r; is a restriction of r 1 and therefore maps

Y into the set SX,A C X. II ~ III. For e > a consider the following class of continuous linear functionals

on y

As in the proof of Theorem 4.2 one shows that

We claim

Vy E Y 313 > a I R, (y) I ~ M •

* Indeed, since rrY E SX,A one has for 13 sufficiently small

Applying the modified Banach-Steinhaus theorem, see Appendix C, it

follows

3S > a 3M > 0 Vy E Y V2 E Be I R, (y) I ~ M Uyll y •

But then for all f E MS(X) and all y E Y

which implies r;y E D(M_ S) = Xs and M_er; is bounded.

* * * * * III - IV. M_arr has a bounded adjoint (M Qr) but (M Qr) ~ r M Q ~ -~ r -~ r r -~

with domain Xe C x. * * IV - V. Take Q = (M or ) . For each a > 0 ~M is the extension of

-~ r ~ a r M Q which is bounded, So Q satisfies condition III of Theorem 4.2.

r -Io'+a We have r = Q.M~.

V - I. If r = Q.Mt the mapping r is obviously continuous. n

- 45 -

Theorem 4.6.

Let 4l: S X,A -+ SY,B be a linear mapping.

Let ~r X -+ Sy,B denote the restriction of 4l to X.

The following six conditions are equivalent.

1. 4l is continuous with respect to the strong topologies of

SX,A' SX,A and

II. For each g E Sy,B the expression <g/~F>yt F € SX,A is a continuous

linear functional on SX,A

III. For each t > 0 the operator Nt 4l is a continuous map from SX,A into

SY,B' *

C SXtA • IV. For each t > 0 (Nt

4lr

) (y)

* c Me (X) V. For each t > 0 there exists e > a such that (Nt 4lr ) (y)

M-/3(Nt~r) * operator from Y into X. and is a bounded

VI. For each t > 0 there exists 13 > 0 such that N 4l M 0 = N ~M Q on t r -p t-p

the domain X/3 C X is bounded as an operator from X into Y.

Proof.

We proceed according to the scheme

~~-....~ I II III IV V VI ~

I ~ II. Trivial.

I ~ III. Trivial because Nt : Sy,B -+ Sy,B is continuous.

III ~ IV. Theorem 5.4. condition II.

IV ~ V. Apply Theorem 4.5 to N ~ . t r

* V - VI. The adjoint of the bounded operator M-13(Nt~r) is an extension of (Nt

4lr

}M_S

'

Therefore the latter is bounded.

VI ~ I. Let {Fn} be a null sequence in SX,A' Then for each t > 0

there exists S > 0 such that we can write (~F ) (t) = Nt~ M of (13). n r -p n This converges to zero as n -+ ~ because of the boundedness of Nt 4lr M_S'

II ~ V.

- 46 -

<q,~F>y has the representation <f,F>X' see Theorem 3.2.IV,

here f = ~Ig E SX,A' Taking F = U E SX,A we observe that

(u,~'g)X = <u,~'g>X = <~u,g>y is, as a function of g, a continuous

linear fUnctional on Sy,B' Then with Theorem 4.2.V it follows that

~' maps SV,B continuously into SX,A" Then, by Theorem 4.2.IV, for

each t > 0 there exists S > 0 such that M_B~'Nt is a bounded

* operator. But M_e~INt = M_B(Nt~r) because for all x E X, Y € V

* (~INtY'x)X = <NtY,$rx>V = (YINt~rx)y = «Nt~r) y,x)X

The interesting question arises which densely defined (possibly

unbounded) operators from X into Y can be extended to a continuous

mapping from SX,A into SY,B'

Theorem 4.7.

Let E be a linear map X ~ D(E)

to a continuous linear map t : + Y with D1tf = X. E can be extended

SX,A + Sy,B iff E has a densely

D{E*, + X with E*(Sy,B) C SX,Ao E* .' Y S defined adjoint ~ V,B ~

Proof

-) For each ~ E D(E) and g € Sy Bone has (E~,g)y = (~,E*g)X.· , *

It follows from Theorem 4.2.V that E maps Sy,B continuously into

SX,A' We define EF, F € $'

X,A by

<v,IF>y * <E V,F>X

This functional is continuous as a function of v, so IF € Sy B' , It is obviously continuous as a function of F. Then by Theorem 4.6.11

the continuity of I follows.

-) If E exists as a continuous map, its dual operator E' maps Sy,B

into SX,A'

For each x € D(E) and g E Sy,B one has <g,Ex>y = (g,Ex)V * -It follows that E ~ E' and E*(SY,B) c SX,A'

n

- 47 -

Corollary 4,8,

A continuous linear map Q : SX,A + SY,B can be extended to a

continuous linear map Q: Sx A + Sy B iff Q has a Hilbert space

* * ' *' adjoint Q with D(Q ) ~ SY,B and Q (SY,B) c SX,A'

- 48 -

CHAPTER 5. Topological tensor products of spaces of type 5' 5' . X"A'. X(A

For two separable Hilbert spaces X and V we consider the complex

vector space consisting of all Hilbert-Smidt operators Z from X into

V • vJe shall denote this vector space by X ® V. For any Z e: X ® V

and any orthonormal basis {ei

} C X we have

"" Olzlll2 = .I IIzeil~ < "".

~=1

The double norm III • III does not depend on the choice of the orthonormal

basis { e. }. We introduce an inner product in X e V by ~

00

(Z,K) XQV

Endowed with this inner product X ® V is a Hilbert space. See [RS]

cr.. VIII.1D. Examples of elements in X ® V are!; ® n, ~ e: X, n e: V

defined by (~en)f: (f'~)Xnf for all f E X, and finite linear N

combinations of these: l: j=l (F,; j ® ~) with f; j E X, nj E Y. The linear subspace

of X e Y which consists of all H.S. operators of the type just

mentioned will be denoted by X ®a V, i.e. the (sesquilinear)

algebraic tensor product of X and Y. X ® V may be regarded as the

completion of X ® V with respect to the double norm 111·111 • Therefore a

X®Y is called the completed (sesquiliriear.) topological tensor product

of X and Y. For later reference we mention the following properties

taken from [RS] Ch. VIII.

Properties 5.1.

(a)

(b)

Vx,.; £ X Vy,n E Y

VA E ~ VF,; E X Vn £ V

Thus the canonical mapping X x Y ~ X e V, defined by [x;y] ~ x e y

is anti-linear in x and linear in y.

- 49 -

( c) VZ E X ® Y Vx E X Vy E Y (Z,x ® y)X®Y = (ZX,Y)y

Let H resp. ] denote bounded linear operators on X, resp. Y into

themselves. H ® ] denotes the linear mapping of X ® Y into itself

defined by (H ® J) (x ® y) = (Hx) ® (Jy) and linear extension,

followed by continuous extension.

(d) The uniform operator norms of H, J and H ® J are related by

II H ® J II = II H1111J II •

(e) (H ® J)Z = JzH*

(f) Hand J injective -H ® J is injective.

The theory of closable tensor products of unbounded closable operators

and the description of their properties in terms of corresponding

properties of their factors presents greater difficulties. Only

rather recently significant general results have been attained [TJ.

Definition 5.2.

Let A with domain DCA) be a densely defined closed linear operator

X. Let B with domain D(B) be the same in Y. On D(A) ® D(B) c X ® Y a

we introduce the operator A ® r + r ® B by (A ® . r + 1 ® B) (x ® y) a a a a

(Ax) ® y + X ® (By) and linear extension. This extension is well

defined and closable, [RS] p. 298.

Lemma 5.3.

Let A resp. B be self-adjoint operators in X resp. Y.

I. A ® 1 + 1 ® B is essentially self-adjoint in X ® Y • We denote the a a

unique self-adjoint extension by A ® 1 + 1.:0 B or, briefly, A EEl B.

II. A ~ 0 and B ~ 0 implies A EEl B ~ 0

Proof

I. We follow Nelson's approach [SJ. The operator C = A ® r + I ® B a a

is obviously symmetric. C 1s essentially self-adjoint iff C*z = ± iZ

implies Z = O.

- 50 -

-iAt iBt On X ® Y we introduce the bounded linear operator e ® e

t E lR. This operator is isometric and

For Z E DCA) e D(B) we have a

* * Now let K E D(C ) and C K = ±iK

Then for each Z E D(A e r + I® B) a a

d -iAt iBt 'dt(Z, (e ® e )K) X®Y

-iAt iBt = =F (Z, (e ® e ) K) XeY

Thus

+t e (Z,K)XeY

But

1 -iAt iBt I

(Z I (e ® e ) K) X®y. $ II zll X®yll 1<11 X®Y

This contradicts (*) except when (Z,K)X®Y = 0 for all Z E D(A) ®a D(B),

which 1s a dense set. Therefore K must be zero.

II. We must show that sums of type

are non-negative. This follows by taking orthonormal bases in the

span of x1"",xN and Yl""'YN and straightforward calculation.

Since the graph of A e 1 + 1 ® B is dense in the graph of A EEl B a a

it follows that also the latter must be non-negative.

Theorem 5.4.

On X ® Y we have for t ~ 0

- (AEEl B)t -At -Bt e = e ® e = Mt ® Nt •

[1

- 51 -

Proof

-At -Bt e 0 e is a strongly continuous one-parameter semi-group of

bounded non-negative hermitean operators. Obviously

(e-At ® e-Bt ) (D(A) ® D(S» c D(A) ® D(S). The infinitesimal a a

generator of Mt ® Nt restricted to D(A) ®a D(B) is -C. The

infinitesimal generator must therefore contain the closure of -C

which is the self-adjoint operator ~A ~ B). The infinitesimal

generator however has to be self-adjoint. Hence it must be equal

to -(A ~ B). Mt ® Nt is, of course, holomorphic. 0

Applying the results of th~ preceding chapters we can introduce

the spaces SX®Y ,A~ H' SX®Y ,A IEB and, by taking A = 0 or B = 0,

the spaces SX®Y,A®I' SX®Y,I®B' etc.

Definition 5.5.

The canonical sesquilinear map ® : SX,A x Sy,B -+ SX®Y,A ~B is

defined by [UiV] -+ u ® v. Here the symbol ® is the same as in

Properties 5.1. This definition is consistent because for u E SX,A'

v E SY,B there exist x E X, Y E Y and t > 0 such that u = Mtx ,

v = Nty • Further u 0 v = (Mt

® Nt) (x ® y) = (Mtx) 0 (Nty). So

that u ® v E SX®Y,A ~B'

Theorem 5.6.

SX®Y,A 83B is a complete topological tensor product of SX,A and Sy,S'

By this we mean:

I. SX®Y,A EElB is complete.

II. The canonical sesquilinear map ®

continuous.

III. The span of the image of 0, i.e. the algebraic tensor product

SX,A ®a Sy,B ,is dense in SX®y,A~S'

- 52 -

Proof

I. The completeness follows from Theorem 1.11.

II. It is enough to check the continuity of S at [0;0]. Let W be a

convex open neighbourhood of 0 in SX®YIA~B.Then for each t > 0

W n (X ® Y)t is an open O-neighbourhood in (X ® Y)t and it contains

an open ball centered at 0 and radius r t , 0 < rt

< 1. In Xt

resp.

Yt we introduce open balls At resp. Bt' centered at 0 and with

radius both r t • Let A = U At C Sx A and B = U Bt C SY,B' t>O I t>O

Then ® maps A x B in W since II x ® y II i (t ) ~ "x lit" y II ~ r i (t ) _ _ m n ,'r 'r m n ,T

whenever x € A, y € B. Let A resp. B denote the convex hulls of A A "-

resp. B. Then ® maps A x B in W. The set A is convex and A n Xt

contains an open a-neighbourhood in Xt " From Theorem 1.4.11 it

follows that A contains an open set V". • - ~,£

Similarly B contains an open set V r' We conclude that ® maps X,u v x V into W.

1f;, €'. X, a III. For each t > 0 X

t ®a Y

t is dense in (X ® Y)t' From this the desired

result follows.

Remark

Our strong topology on SX®Y,A ~8 is, generally speaking,not the

projective tensor product topology. Cf. [SCH] p. 93. Therefore the

universal factorization property for continuous sesquilinear maps

on this space does not hold in general.

Definition 5.7.

The canonical sesquilinear map ® : SX,A x SY,8 -+ SX®Y,A83B '

[FIG] -+ F ® G, is defined by (F ® G) (t) = F(t) ® G(t) •

LJ

Here ® is the same as in properties 5.1. The definition is consistent

because (F ® G) (t + T)

=: (M ® N ) (F ® G) (t) • T T

= M F(t) ® N G(t) = (M ® N) (F(t) ® G(t»= 'r T T T

- 53 -

Theorem 5.8.

SX®Y,A£:I3B is a complete topological tensor product of SX,A and Sy,B'

By this we mean:

I. SX®Y,A!EB is complete.

II. The canonical sesquilinear map ®: SX,A x SY,B -+ SX®Y,A£:I3B is continuous.

III. The span of the image of ®, i.e. the algebraic tensor product

SX,A ® SY,B' is dense in SX~y,A £:I3B •

Proof

I. The completeness follows from Theorem 2.5.

II. For each t > 0 we have IIF(t) ~ G(t) I~&y == IF(t)llX"G{t)lIy' From this the

continuity at [O;oJ follows.

III. X @a Y is dense in X @ Y which is dense in SX@Y,A £:I3B'

We now want to introduce mixed sesquilinear topological tensor products

of type Sx A @ Sy B' etc. The reader who is already content with the , , Kernel Theorems case a and case b of the next chapter may skip the

remainder of this chapter.

Defini tion 5.9.

We introduce the following linear subspace of SX®Y,I&B

If no confusion is likely to arise we use the shorter notation ~B

o

for this space. For ~ E ES we have for each t,. > a ~(t + T) = (1 ® NT)~(t).

On 1B we introduce semi-norms {Pt,~}, t > 0, ~ E B+, by

and the corresponding locally convex topology.

- 54 -

Definition 5.10.

The canonical sesquilinear map ® : SX,A x Sy,B + ~B is defined by

($ ® G) (t) = $ ® G(t). Here the s~~ol ® is the same as in Properties

5.1. The span of the image of ® is denoted by SX,A ®a SY,B'

Theorem 5.11.

~B is a complete topological tensor product of SX,A and SYtB' By this

we maan

I. ~B 1s complete.

II. The canonical sesquilinear map ® SX,A x Sy,B + ~B is continuous.

III. SX,A ®a Sy,B is dense in ~B .

Proof

I. Let {~ } be a Cauchy net. The a's belong to a directed set D. First a

take 1/1 = 1. ~a tends to a limit point ~ E SX®Y,I®B because the latter

is complete. It remains to show that for each t > a ~ (t) E SX®Y,A EBB"

For each 1/1 E B and each t > 0 1/1 (A EEl B)~ converges in X ® Y. + a

From the closedness of 1/1 (A EEl B) it follows that ~(t) € D(1/I(A EEl B».

This is true for each 1/1 E B+ and therefore by Lemma 1.10, for each

t > 0, <p(t) E SX®Y,AEElB'

II. It is enough to check the continuity at [O;OJ. Let Wt ,I, be the set ''f'11::

of elements in ES for which the semi-norm Pt,1/I is smaller than 1::; it is

an open O-neighbourhood. Let W denote the convex open O-neighbourhood 1/I,e:

in SXGilY,A EBB consisting of elements with semi-norm P1/l smaller than 1::.

Then W consists of those trajectories in ~B' which pass at t t'!ete:

through W . According to the proof of part II of Theorem 5.6 there ljJ,e:

exist open O-neighbourhoods U c SX,A ' V c SY,B such that

U ~ V c W . V n YLt contains an open ball, centered at 0, of radius 0, ljJ I e: ""2

say. Now let n {GIG E SY,B' II G(~t)1I Y < o}. n is open in SY,B and

the set {G(t) \G E n} c V. It follows that <P ® G E Wt ,I, whenever $ E U I'f',e:

and GEn. This proves the continuity of ®.

- 55 -

III. SX,A ®a SY,B can be considered as a subspace of SX,A ®a Sy,B'

SX,A ®a Sy,B is a dense subspace of SX®Y,AEBB' See Theorem 5.6, So

we are ready if we show that SX~Y ,A EBB is densely embedded in ES' Let <P € ES then for t > 0 <l> (t) E SX~y,A EBB' We claim that 4> (T) -+ 4>

in LS-sense as , f O. Indeed

({I ® N - I 0 I}4>(t), ,

The first factor in this inner product tends to 0 as T + 0 because

1 ® Nt is a strongly continuous semi-group. The second factor remains

bounded as T + O. This is so because it can be written.

with t: sufficiently small. The operators between { } are (uniformly)

bounded. This finishes the proof,

Remark.

In the proofs of parts III of Theorems 5.6 and 5.11 we circumvented the

problem whether the strong topology on SX®Y,A EBB is induced by semi­

norms of the form ~(A) ® X(B) with ~,X E B+,

Definition 5.12.

We introduce the following subspace of SX®Y,A®I

E' ={plp€S· Vt>O X 0 Y ,A01 ,A EBB X®Y ,A®I ,

o

If no confusion is likely to arise we use the shorter notation LA for

this space. For P E EA we have for each t,T >0 pet + T)"=: (M,0 I)P(t).

On EA we introduce the semi-norms Pt,~ of Definition 5.9 and the

corresponding locally convex topology.

- 56 -

Definition 5.13.

The canonical sesquilinear map ® : SX,A x Sy,B ~ LA is defined by

(F ® W) (t) = F(t) ® W. The span of the image of ® is denoted by

SX,A ®a SY,B'

The proof of the following theorem runs the same as the proof of Theorem 5. 11 •

Theorem 5.14 • . EA is a complete topological tensor product of SX,A and SY,B' By this

we mean

I. LA is complete.

II. The canonical sesquilinear map ® SX,A x Sy,B + LA is continuous.

III. SX,A ®a SY,B is dense in EA'

we conclude this chapter by introducing another candidate for a sesquilinear

tensor product of mixed type. For a > 0 the map M ® 1 which carries IX

SX®Y ,A EE1B into

SX®Y ,A EEl B into is realized by

itself can be extended to a continuous linear map from

itself. This follows from Corollary 4.8. The extension

«M ® I)K) (t) = (M ® I)K(t). The map M ® 1 is a IX a injective, see Properties 5.1.f, we denote its inverse by M ® 1.

-a

Definitions 5.15.

We introduce the locally convex topological vector space

The topology on (Ma ® 1) (SX®Y,A EEl B) is given by the metric d et • Here

det(T) = d«M_a ® 1)T) and d(') denotes the metric on SX®Y,AEElB' See

Theorem 2.5. The topology on EX®Y,A®I,AEBB or, briefly, EA is the

inductive limit topology with respect to the metries d , et > O. We will et

not look for an explicit system of semi-norms for LA' nor investigate

its completeness here.

- 57 -

On ~A x rA we introduce the sesquilinear form

for E > 0 sufficiently small. The definition does not depend on the

choice of E. Further we define the embedding of LA into LS by

(*) (emb T) (t) = (1 x Nt) T •

Theorem 5.16.

I. SX,A ®a Sy,S is dense in LA'

II. For each fixed P € LA the linear functional <P,T>A' as a function of T, is continuous on LA'

III. The embedding (*) of Definitions 5.15 is continuous.

Proof

I. X ® Y is dense in SX®Y A lEE' Therefore for a > 0 X ® Y is dense in a , a a

(Ma ® I) (SX®Y,A lEI B) " We conclude that U X ® Y which.1s a subset a>O a a

of SX,A ®a Sy B is dense in EA" t

II. It 1s sufficient to prove the continuity for restrictions to each

III.

(Ma ® I) (SX0 Y,A IES) , a > O.

We have \<P,T>A\ ~ cd(M_a 9 I)T) :::: cd (T) " a Here c is a constant which depends only on P and a •

P t ,II (emb T) = II {lidA lEI S) (M x 1) (1 0 Nt)} (M ® IYI'II for a sufficiently ''1' a -a X®y

small. The operator between { } is bounded. Therefore p (emb T) ~ C d (T) t,$. a

where c only depends on t, $ and a.

Definitions 5.17.

Similar to Definitions 5.15 we introduce the space

briefly dented by LS' with the appropriate inductive limit topology.

For [~;FJ € LS x LS we introduce the pairing ~,F>S = <~(£)/(1 x N_~)F> .. X®y

Further we define the embedding of IS into LA by (emb F) (t) = (Mt

@ I)F.

- 58 -

Analogous to Theorem 5.16 we have

Theorem 5.18. ~

I. SX,A e a SY,B is dense in LB'

II. For each fixed ~ E SX,A the linear functional <~,F>B' as a function

of F, is continuous on LB'

III. The embedding described in Def. 5.17 is continuous.

- 59 -

CHAPTER 6. Kerneltheorems

In this final chapter we show that the elements of the completed sesqui­

linear topological tensor products of the preceding chapter can, in a

very natural way, be interpreted as linear maps of the types we discussed

in Chapter 4. We give necessary and sufficient conditions on the semi--tA -tB groups Mt = e and Nt = e which ensure that the topological tensor

products comprise all continuous linear maps. In this case we say that a

Kernel Theorem holds.

CASE a: Continuous linear maps SX,A + SY,B' We consider an element

a E SX®Y,A fEB as a linear operator SX,A + Sy,B in the following way,

Let F E SX,A' We define aF by

(a) aF = N (N aM ) F (e::) , e:: -E: -€:

For €: > 0 and sufficiently small this definition makes sense and does not

depend on e::,

Theorem 6.1 • . I. For each a E SX®Y,A fEB the linear operator a

(a), is continuous.

II. For each 8 E SXIli!IY,A fflB ' F E S I X,A ' G E SY,B

SX,A+ SY,B' as defined by

III. If for each t > 0 at least one of the operators Mt , Nt is H.S, then

SXIli!IY ,A fEB comprises all continuous linear operators from SX,A into

SY,B'

IV. SX0X,AfEA comprises all continuous linear operators from SX,A into SX,A

iff for each t > 0 the operator Mt is H.S.

Proof

I. We shall prove that a satisfies condition IV of Theorem 4.5, Since

- 60 -

8 X 0 Y we have 8 r 8. Since 8 € SX0Y,A IE B we have for t sufficiently

small N_tGM_ t € X ® Y. Therefore OM_ t = Nt(N_tOM_t ) is bounded.

II. For E sufficiently small N 8M € X «I Y, therefore by Properties -e -e

S.l.c.

<8F,G>y = < N (N 8M )F(£},G>y = E -e -e

III. Let r : Sx,A 7 Sy,B be continuous. By Theorem 4.S.V there exists T > 0

and continuous Q : Sx,A 7 Sy,B such that r = QMT' By Theorem 4.2.IV

there exists S > 0 such that N-eQM~T is a bounded operator. Put

~ = ~ min(S,~T), then

r = N {No (N SQML )ML }M ~ ~-~ - ~T ~T-~ ~

The operator between ( ) is bounded. Further the operator between { }

is B.S. since No or ML is H.S. ~-~ ~T-~

It follows that r E SX0Y,A IEB'

IV. The if-part is a special case of III. For the only if-part concider the

special map r = M~ Sx,A 7 SX,A for some ~ > O. In order that

r ( SX®X,A lEA it has to be H,S.

CASE b: Continuous linear maps SX,A 7 SY,B' Let K E SX®Y ,A IEB' For f E SX,A we define Kf € SY,B by

(b) (Kf) (t) = Nt_EK (e) M_Ef

For any f E SX,A and t > 0 this definition makes sense for E > 0

sufficiently small. Moreover (Kf) (t) does not depend on E.

Theorem 6.2.

I. For each K E SX0Y ,A Il3B the linear operator K

by (b) is continuous.

SX,A + SY,B defined

o

- 61 -

III. If for each t > 0 at least one of the operators Mt

, Nt is H.S.

then SXSY,A~B comprises all continuous linear operators from SX,A into Sy,B'

IV, SX0X,A~A comprises all continuous linear operators from SX,A into

SX,A iff for each t > 0 the operator Mt is H.S.

Proof

I. We use condition II of Theorem 4.4.

For each t > 0, a > 0 NtKMa

is a bounded operator from X into Y because

for £ sufficiently small

All operators in the last expression are bounded.

II. For each T > 0 K(T) is a H.S.-map. For £ sufficiently small, with

Properties S.l.c.

<g,Kf>y

III. Let L : SX,A ~ Sy,B be continuous. According to Theorem 4.4.11 the

operators NtLMt are bounded for each t > O. However if Nt or Mt is H.S.

for each t > 0 then NtLMt

is H.S. for each t > 0 and it defines an element in

SX®Y,A 83B' A simple verification shows that this element reproduces L by recipe (b).

IV. The if-part is a special case of III. For the only-if-part consider

the special map L =emb = r. Here 1 is the identity map. Mt lMt == M2t

can be considered as an element of SXsY,A ~A iff Mt is H.S. for all

t > O. o

CASE c: continuous linear maps: SX,A ~ Sy,B' Let P E LA . For f E SX,A we define Pf E Sy,B by

•

- 62 -

Pf E Sy, since peE) E SX0y,A~B' The definition makes sense for £

suffici iy small and does not depend on the choice of E •

~ Theorem 6.3.

I. For each P E rA the linear operator P

is continuous.

SX,A + SY,B defined by (c)

II. For each P E rA ' each f E SX,A' each G E Sy,B

<Pf,G>X = <P,f 0 G>A

III. If for each t > 0 at least one of the operators Mt , Nt is H.S. then

rA comprises all continuous linear operators from SX,A into SY,B'

IV. Consider the special case Y = X and B = A. E' A comprises all continuous linear operators from SX,A into itself iff

for each t > 0 the operator Mt 1s H.S.

Proof

I. We use condition II of Theorem 4.2.

Let f ~ 0 strongly in Sx A' For some £ > 0 M f + 0 in X. n , -E n P(£) E SX0Y,A~B' therefore there exists c > 0 such that N_cP(£) is

a bounded operator. But then N ~Pf = (N ~P(£»M f + 0 in Y, which -u n -u -€ n

shows that Pfn + 0 strongly in SY,B'

II. For Band 6 sufficiently small and positive we have

III. Let Q : SX,A + SY,B be continuous. According to Theorem 4.2.IV for each

t > 0 there exist set) > 0 such that N_S(t)QMt is a bounded map from

- 63 -

X into Y. Now because of the assumption on M , N we find that a a

WAt :: NS(t) (N_S(t)QMt) is an element of LA . It reproduces the

operator Q if the recipe (c) is applied.

IV. The if-part is a special case of III. For the only-if-part consider

the identity map 1 : SX,A ~ SX,A' In order that IMt as a function of t

is an element of LA the operator Mt has to H.S. for all t > D. 0

CASE d: Continuous linear maps: SX,A ~ Sy,S' Let ~ € LB' For F € SX,A we define ~F E Sy,B by

(d) (~F) (t) = Ili (t) M -IS: (t) F (e: (t) )

This def1ni tion makes sense for t > 0 and e: (t) > 0 sufficiently sm'lll.

(~F) (t) E: Sy,B since <P (t) E: SxeY,A lEIS' Moreover

N,(<PF) (t) = N,Ili(t)M_e:(t)F(e:(t» :: CP(t + ,)M_e:(t/(e:(t» ::: (IliF) (t + c).

Theorem 6.4.

I. For each Ili € ES the linear operator Ili

continuous.

II. For each Ili € ES' F E S~,A' g € Sy,S

<g,IliF>y

SX,A ~ Sy,B defined by (d) is

III. If for each t > 0 at least one of the operators Mt

, Nt is H.S. then

LS comprises all continuous linear operators from SX,A into Sy,Be

IV. Consider the special case Y = X and B = A. LS comprises all continuous linear operators from SX,A into itself iff

for each t > 0 the operator Mt is H.S.

Proof

1. We use Theorem 4.6.III. For each t > 0 Ntlli E SXeY,AIEB' Then according

to case a, Ntlli is a continuous linear map from SX,A into SY,B'

- 64 -

II. For a and <5 sufficiently small and positive

== <~(a) ,(I ® N ) (F e g»X Y = -a e

<9,4>F>y

III. Let ~ SX,A + Sy,B be continuous, According to Theorem 4.6.VI for each

t > 0 there exists S(t) > 0 such that Nt¥rM_S(t) is a densely defined

and bounded operator from X into Y. If one of the operators N , M is a a

H.S. for arbitrary small positive a it follows that N ¥ is H.S. fer t r

t > 0, because Nt'¥r == Nt¥r(S(t)MS(trHerelfrM_S(t) denotes the extension of Nt'¥r to the whole of X.

Since Nt'¥r == Nl.:!t(Nl.:!tljl)i-S(~t»MS(t) it follows Nt'¥r € SXeY,AIBBo Hence Nt'¥rl as a fUnction of t, belongs to ES' By recipe (d) the operator '¥ is reproduced.

IV. The if-part is a special case of III. For the only-if-part consider the

identity map 1. In order that MtI, as a function of t, can be considered

as an element in LS the operator Mt

should be H.S. for all t > O. 0

- 65 -

APPENDIX A. Functions of self-adjoint operators.

A self-adjoint operator A in a Hilbert space H admits a spectral

decomposition

00

_00

Here tA is the so-called resolution of the identity, i.e. EA is a

monotonically increasing function with values in the projection operators,

EA is strongly continuous from the right, E(_oo) = 0, E(oo) = I,

E,E a E. (' ). Let ~ be a complex valued and everywhere finite A ~ m1n A,~

Borel function on lR. On the domain

00

the operator ~(A) is defined by

00

Vx € H

where the integration is carried out with respect to the Borel measure

determined by m«A1

,A2

])

We write formally

00

It can be proved, see [YJ Ch. XI. 12, that ~(A) is a normal operator.

~(A) is self-adjoint if ~ is real valued. Further (~)X) (A) = ~(A}x(A) where the usual precautions concerning the domains must be regarded.

- 66 -

In our case A ~ 0 and the functions $ are always piece-wise continuous

and piece-wise of bounded variation. We frequently employ the notations

o < a < b ::; 00 •

a

By this we mean

-00

with

[0

1 X b(It) = a,

on (a,b]

elsewhere •

By the notation

o < b ::; 00

we mean

The elementary and elegant presentation of spectral theory in [LS] is

quite sufficient for our work. We referred to [yJ because the theory

in [LS] is not completely dressed for our purposes.

- 67 -

APPENDIX B Holomorphic semi-groups

On a Banach space X we consider a one-parameter strongly continuous

semi-group of bounded operators Pt

, t ~ 0, with the properties

P ... I, P "" P P '" P P I o t+T t T T t t ~ 0, T ~ O. Such a semi-group

describes the solutions of the convolution problem in X:

duet) = _ Au(t), dt

Here -A is the so-called infinitesimal generator of the semi-group pt

"

A is a densely defined closed linear operator which enjoys the

following property (cf. [Y] Ch. IX):

There exist constants M and a such that for every A > a the resolvent -1 n -n R).. Ie (A + AI) exists and II R).. II S M(A - a) • Then

and

Our interest is centered upon holomorphic semi-groups then Vt > a Vx E H co

Ptx E D(A ), Ptx is an infinite differentiable function of t and has

the local representation

co co ( , _ t) n

..:..:.l.1\ --:~_ An P x = n! A L Vx E X I

n=O n=O

In this case Pt has a holomorphic extension into a sector larg tl < ~

in the complex t-plane.

A necessary and sufficient condition on Pt in order to be holomorphic

is

Vx E H Vt > a Ptx E D (A)

and

3a > o Vt,O < t S 1 II tP~11 S a.

Then <p '" arctan -1

(ae) • See [yJ Ch. IX. A necessary and sufficient

- 68 -

condition on the infinitesimal generator -A to generate a holomorphic

semi-group is;

3S > a 3£ > a VA, Re A ~ 1 + £

See [y] Ch. IX.

An interesting subclass of holomorphic semi-groups on a Hilbertspace H

is generated by operators of the form A = Q + B where Q is a positive

self-adjoint operator and B is subjugated to Q in a specified sense.

See [G].

In case B = a the operators Pt

, t ~ a are all selfadjoint and Pt

has

a holomorphic extension into the whole open right half plane and a

strongly continuous extension into the closed right half plane. Whenever

t > a the operator Pt

has a self-adjoint inverse which is unbounded iff

A is unbounded. In this paper we consider only the case that A is un­

bounded, self adjoint and non-negative.

- 69 -

APPENDIX C. A Banach-Steinhaus Theorem.

The following is taken from [JJ.

Theorem. Let (B,II II) be a Banach space. suppose that for every a > 0

there is given a collection A of bounded linear functionals of B a

such that 0.1

< a - A c A 2 0.

1 0.

2 statements are equivalent:

(0.1

> 0, 0.2

> 0). Then the following

Proof

It is easily seen that (ii) _ (i): if N > 0 and B > 0 are such that

I Lf I ::; N f for every fEB, LEA 6' then we can take a = 6 and M = Nil f II

in (i).

We now suppose (i), and we shall show that

yields a contradiction. To every n E ~ we can then find an fEB and an n

L E All such that I L f I > nil f II. Note that L is a bounded linear n n n n n n functional of B for every n E ~, and that (Lnf) nE~ is a bounded

sequence for every fEB. It follows therefore from the Banach-Steinhaus

theorem that there is an M > 0 such that I L f I ::; Mil f II for eve"ry n E ~ n

and every fEB. Contradiction. n

- 70 -

Acknowledgements

During the research I have profited from very stimulating discussions

with A.J.E.M. Janssen.

I wish to thank J.W. Nienhuys and J.J. Seidel for making valuable

suggestions on the presentation of the introduction.

Finally I wish to thank J. Boersma for showing me the way in Whittaker

and Watson.

- 71 -

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