Analyticity spaces of self-adjoint operators subjected to perturbations with applications to Hankel invariant distribution spaces

Analyticity spaces of self-adjoint operators subjected to

perturbations with applications to Hankel invariant distribution

spaces

Citation for published version (APA):

Eijndhoven, van, S. J. L., & Graaf, de, J. (1983). Analyticity spaces of self-adjoint operators subjected to perturbations with applications to Hankel invariant distribution spaces. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8315). Technische Hogeschool Eindhoven.

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

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Department of Mathematics and Computing Science

Memorandum 1983-15 November 1983

ANALYTICITY SPACES OF SELF-ADJOINT OPERATORS SUBJECTED TO PERTURBATIONS WITH APPLICATIONS TO HANKEL

INVARIANT DISTRIBUTION SPACES

by

S.J.L. van Eijndhoven and J. de Graaf

Eindhoven University of Technology

Department of Mathematics and Computing Science PO Box 513, 5600 MB Eindhoven

INVARIANT DISTRIBUTION SPACES

by

S.J.L. van Eijndhoven and J. de Graaf

Abstract

A new theory of generalized functions has been developed by one of the authors (De Graaf). In this theory the analyticity domain of each positive self-adjoint unbounded operator A in a Hilbert space X is regarded as a test space denoted by SX,A. In the first part of this paper, we consider perturbations P on A for which there exists a Hilbert space Y such that

A + P is a positive self-adjoint operator in Y. In particular, we investigate for which perturbations P and for which v > 0, SX,Av c SY,(A+P)v. The second part is devoted to applications. We construct Hankel invariant distribution spaces. The corresponding test spaces are described in terms of the

S8-n spaces introduced by Gelfand and Shilov. It turns out that the modified Laguerre polynomials establish an uncountable number of bases for the space of even entire functions in S~ (~ ~ ~ ~ 1). For an even entire function

~

we give necessary and sufficient conditions on the coefficients in the Fourier expansion with respect to each basis such that ~ E S~.

~

Introduction

Let X be a separable infinitely dimensional Hilbert space and let

L

be a linear operator 1n X. Then DW(L), the analyticity domain of L, consists of

00

all vectors v E

n

D(Ln) satisfying n=1

3 3 V

a>O b>O nElN

For a positive self-adjoint operator A in X, Nelson ([13J) proved that DW(A)

can also be described as

U e - t A (X) t>O

-tA

{e

wi w

E X, t > O} •

Instead of DW(A) we use the notation

Sx

_,

A introduced by De Graaf. The spaces of type

Sx

_,

A are called analyticity spaces. They are non strict inductive limits of Hilbert spaces. Together with their strong duals

TX

_,

A they establish the functional analytic description of the distribution theory in [GJ.

For each positive constant v the operator AV is well-defined, positive and self-adjoint in X. So it makes sense to write

Sx

_,

AV. The question arises for which perturbations P on A there can be found a Hilbert space Y such that

A

+ P is a positive self-adjoint operator in Y and Sx,AV c SY,(A+P)V. In the

paper ([IJ) the case v = I has been considered. Also some results concerning analytic dominancy can be found there.

In the second part of this paper we study a class of Hankel invariant

test-and distribution spaces, and, also their relations to the S8-spaces of

Gel-a

fand and Shilov ([9J). With our papers [2J and [4J we have started this study.

1

There we have shown that the space of even functions in

S!

remains invariant

•

under the modified Hankel transfonns E , a > -I, defined by

a

00

(R f) (x)

J

(xy) -a J (xy) £(y) Y 2a+1 dy

a a

o

Horeover, for each a > -I the space of even functions in

st

equals the ana-2a+1

lyticity space

Sx

A where X =

L

₂

«O,00),x dx) and

2 a' a a

d 2 d

A = - - - + x - (2a + I) x - . The operator A has an orthononnal basis of

ex dx2 dx a

eigenvectors (l(ex»oo with eigenvalues 4n + 2a + 2. So for each even fESt

n n=O ~

t > 0 such that there exists an l2-sequence (w )00 0 and

n n= 00

f =

L

exp(-(4n + 2a + 2) t) w £ (a) • Here

n=O n n we prove similar results for the

spaces

Sx

(A)V with v ~

!

and ex > -I.

ex' a

It will follow that for all a,S> -I and all v ~

!

For v E [~,IJ the analyticity space S (A)V contains just the even

func-X_!'

_!

tions 1'n SI/2v

1 /2v'

(I) General theory

Let A be a positive self-adjoint operator in a Hilbert space X and let v > O. It makes sense to write

A

V and the operator

A

V is positive and self-adjoint 1n X. So the space

Sx

_,

AV is well-defined. Its elements are characterized by

(I. I) Lennna

For each f E D(A

oo

) c X the following statements are equivalent

(i) 3 3 V a>O b >0 kE liI

Proof

(i) ~ (ii). Let N E liI and let T >

o.

Consider the following estimation

N k IIAvk fll N k II A-I +v k - [ v k J II II A [ v k J + I f II (*)

I

T

I

T k! ~ k! ~ k=O k=O N k «[vkJ + I)!)vk ak ~ b l

I

T

kT

k=O

where b I b sup ( II A -I +vk-[ vkJ II ). The following inequalities are valid

ktJNu{ O}

([vkJ + I)!

~

([vkJ+t)([vkJ+I)[vkJ

~

e([vkJ+I)(vk) vk •

So ([vkJ+ I)!

~

(e([vkJ+I»I/V(ve)kk!, and for T < (vea)-I the series

(*) converges. I t imp lies that f E exp (-T AV) (X) .

(ii) ~ (i) Suppose g E

Sx

_,

AV. Then there exists s > 0 and w E X such that

g = exp(-s AV)w. Let k E lL Then we estimate as follows

k k v

II A f II ~ II A exp (-s A ) II II w II II w II (~) k/v e -k/v ~ vs

~

II w II (1/ v s) k / v • (k!) 1 / v •

With a (vs) -I/v and b IIwll the implication (ii) ~ (i) has been proved.

•

•

2 3 Let

L

be an unbounded linear operator in X. Then the operators

L ,L , ...

are well-defined. As a corollary of the previous theorem we get

(1.2) Corollary

Let n E ~ and let f E DW(L). The following statements are equivalent.

(i) 3 3 V

a>O b>O kEJN

As mentioned in the introduction we investigate perturbations P on A such

W v

that D «A+P) ) ::> Sx A

_,

V ' For v the following result has been proved in

[I]. Here we consider general v > O.

(1.3) Theorem

Let

P

be a linear operator in X with D(P) ::> SX,A_{v ,} Suppose the following

conditions are satisfied

(i) There exists a Hilbert space Y such that exp(-tAv) maps X into Y for

all t > 0;

(ii) In addition, A + P defined on Sx A

_,

_V is positive and essentially self-adjoint in Y.

(iii) There exists an everywhere defined, monotone non-increasing function ~

on (0,1) such that

V

r:O<r<1

v -I v

Proof

We note first that Sx AV = U exp(-t AV) (X). So let 0 < t < 1, and let , O<t<)

o

< T < t. Put s

=

t - T. We want to estimate the norm of the operator

v k v

exp(TA) (A+P) exp(-tA) for each k E :N. Therefore we factor as follows

v k v

exp (T A ) (A + P) exp (- t A )

k-) {

1

v 1

1

v s v}

=.n

exp«l' + k s)A ) (1 + PA- ) exp(-(T + k s)A )A exp(- k A ) • J=O

This factoring yields the estimate

A

v

A

k v

lI exp(l' )( +P) exp(-tA)1I ~ II A exp (-

k

s AV

)

II k

k-\

j v -) j v

n

Ilexp«l'+ks)A )(1+PA )exp(-(T+kS)A )11 ~

j=O k-) ~ (k!)l/v ( __ I )k/v

n

vs j=O k-I

o ,

1 , ••• , k - I , we ge t

n

(I +CP(T +ts»

~

j=O k (I +CP(T» .

Thus we have proved that

v k v

V OV 0 3 OV

k .. " {a} : Ilexp(TA )(A+P) exp(-tA)1I

t> T, <T<t a> E.L'U ~

(k!) I

I

v

} .

Let t > 0 and let w E X. Set f exp(-tAv)w. Then for 0 < T < t fixed there

exists a > 0 such that

II ( Av) II II w II X ak (k '.) 1 Iv .

~ exp -, X-+Y

From Lemma (1.1) it follows that f E SY,(A+P)v.

o

-k

Remark: Suppose there exists k E IN such that the operator A maps X

conti-nuously into Y. Then Condition (1.3.i) is fullfilled because

v

II exp(-t A ) II X-+Y ~ _{II A} -k _{II X-+Y} II A exp ( - t A ) II X k v

(1.4) Corollary

Let P be an operator in X and let n E IN with D(P) ~ Sx An. Suppose there

_,

exists an everywhere defined monotone non-increasing function cp on (0, I) such that Proof V O<r<1 n -I n II exp(r A ) PA exp(-r A ) II ~ cp(r) •

As in the proof of the previous theorem: V V 3 V :

t>O "O<,<t a>O kE1'l

A

n

A

k n

II exp ( , ) ( + P) exp (- t A ) II ~ (k!) 1

I

n a k .

So for f exp(-tAn)w, t > 0, W E X, we get

k An k An

II (A+P) fllx ~ IIexp(, )(A+P) exp(-t ) II IIwil ~

Remark: If P satisfies the conditions in Corollary (1.4), then An analyti-cally dominates (A + P) n. (For the tenninology, see [6]).

In order to prove the converse statement of Theorem 3, i.e.

we have to interchange the roles of A and A +

P.

Put differently, if we write

B

=

A +

P

and hence A

=

B - P,

then we have to check whether the pair

B,P

satisfies the conditions required in Theorem (1.3).

2. Hankel invariant distribution spaces

In our papers [2J, [4J on Hankel invariant distribution spaces the following

results have been proved.

Let A

y

d2 2

denote the differential operator - ---2 + x 2y + 1 ~ and let X

x dx

denote the Hilbert space L2 «0,00) ,x Y 2 +1 dx) where we take dx y > -I. Then for every a,S> -I we have shown that

Moreover, f £

Sx

A if and only if f lS extendable to an even function in

I y' y

S~ . Also, it has been proved that the space

Sx

A

:1 y' y

the modified Hankel transform E defined by

y

(JH f) (x) y

o

00

J

£(y) (xy) -y J (xy)y 2y+1 dy

y

remains invariant under

•

Here J denotes the Bessel function of the first kind and of order y. The y

Hankel transfonn Ii extends to a unitary operator on X and Ii A

=

All.

y y y y y Y

It follows that for all a,S> -I, Ii maps the space Sx A onto itself.

a s '

e

By duality, each Ii leaves invariant each space of generalized functions

a

Tx A

corresponding to Sx A . The functions ley) defined by

S' S S, S n

( 2r(n + I)

)~

e-!x 2

L (y) (x 2)

r(n+y + I) n n € ~ u {a} x > 0

establish an orthononnal basis in X and they are the eigenfunctions of the

y

self-adjoint operator A with respective eigenvalues 4n + 2y + 2. Here L (y)

y n

denotes the n-th generalized Laguerre polynomial of order y. We note that Ii l (y)

y n We recall that for each a,S> -I the functions 00

f w £, (S) where w = a(e -nt) for some

n n n

€

Sx

A can be written as f =

I

a' a n=O

t > O.

With the aid of the theory presented in the first part of this paper we ex-tend the mentioned results and prove that

for all v ~ ! and all a,S> -I. In addition, we show that for each v € [!,I]

and all a > -I the space

Sx

(A)v contains just the even functions of the

a' a 1/2v

space SI/2v' So each even

~

(a)£,(a) 'th (a) L _Pn n w~ _Pn n=O . 1/2v funct~on f € SI/2v v a(exp(-n t)). admits Fourier Gelfand-Shilov expansions f

=

Let a,S > -I . Then A can be written as a

I d

where we put R =

x

dx' Obvious ly, Aa can be obtained from AS by means of the 'perturbation' 2(a-S)R, and AS from Aa by means of 2(S-a)R. In order to show that R and hence c R, c E C, is a perturbation in the sense of

Theorem (1.3) we bas i s (£ (y) ) 00

n n=O

R£ (y)

n

compute the matrix of R with respect to the orthonormal To this end, we mention that

-£(Y) - 2 £ (y)

n n-I

d L (y)

-L(y+l)

where the relation n used.

dx n-I ~s

L (y+ I) k _L~Y)

Now

I

and hence

k

j=O J

_ (U(n+I))! [(r(n+Y+I»)!£CY) + 2 nI l (rcm+y+I»)!!CY)] r (n + Y + I) r (n + I) n m=O 2 r( m + I) rn •

Thus we obtain the matrix of R with respect to the basis (£(y»oo n n=O

•

-I if f k , k E :IN

o

if f > k k E :IN u {a} I

2(

r

(k +

I)

r (k + Y + I)

r

(f

ra

+ Y + 1») 2 • f 0 < 0 < k k c -"'1 • + I) ~ - .{.. , "- .. "

The inequality (cf. [IIJ) yields I-s n ~

r

(n + I) ~

r

(n + s) (n+I)I-s O~s~l,nElN if Y ~ 0 , 0 ~ I < k , k E IN u {a} if -I < y < 0 , 0 ~ I < k , k E :N u {O}. v -I v

For each \) ~

!,

the operator exp(r(A ) ) R(A) exp(-r(A)) has to satisfy

y Y Y

Condition (iii) of Theorem (1.3). We define the weighted shift operators

W (n) (r) n E IN u {O}, Y,\) , wi th norms (w(n) (r) £ (y) £ (y)) y,v k ' I Y II Wen) (r) II X Y,v Y + n exp(-r( (l + n) Y - t Y)) I 4

a

+ n) + 2y + 2 (0) I

SO IIW (r) II ~ - 2 2 . Now let n E IN. The inequality

y,v Y +

v v

l !

(t + n) - I ~ (t + n) 2 - t

elsewhere

is valid for all

t

E IN

u

{O} and all

v

~

!.

In addition, the matrixelements I(R£(Y) £(Y)I are smaller than 2(t+n)-y/2 for -I < Y < 0 and smaller than

t+n'

t

2 for Y ~ O. If -I < Y ~ 0 we therefore get

IIW(n) (r) II ~ y,v 2 (l + n) -Y / 2

!

!

sup 4 (t + n) + 2y + 2 exp (-r «t + n) - t )) ~ tElNU{O}

~

sup (!(t+n)-h- I exp(-!rn(l+n)-!))

~

tElNU{O}

Since

co

w(n)(r)

y,v

v -I v

exp(r(A ) ) R(A) exp(-r(A»

y y y

I

n=O

we can use the following straightforward estimate for all r > 0

v -I v

lIexp(r(A) )R(A) exp(-r(A»1I

y y y where d y (2.1) Lennna co :0:; :0:; :0:; co II Wen) (r) II

I

--n=O y,v co 1 _{d (2-) 2+y}

I

- - + 2y + 2 1 r n=1 d (2-) 2+y _{+ - - -}1 Y r 2y + 2

Let y > -] ,Then there exist constants d

y > 0 and Py > 0 such that

Proof V r>O \) -] v II exp(dA ) ) R A exp(-r(A» II y y y 1 + -2y + 2 (2-)2+y n

For -1 _ y S 0 the assertion has already been proved. For y > 0 it follows

from the matrix expressions for R that

v -1 v

II exp r (A) R A exp (-r (A ) ) II

y y y

In addition, we show that given r > 0, y,o > -1, the operator exp(-r(A )v)

y

maps Xy into

Xo.

In [2J, p. 17, the following result has been proved

V 3

sEJN lElN II Q2s (A ) -{ II y y < co •

:0:;

•

Here

Q

denotes the multiplication operator in X given by

y

(Qf)(x) = xf(x)

o-v

Now let 0 > -I and let f E X

y• Put s := [max{O ' 2 } J + 1. Then there exis ts fO E :IN such that II Q2s

A~fll

y < 00 for all f

~

f

O• So we derive 00 00 (*)

_f

\«A )-ff)(x)\2 x 20+l dx =

f

x 2 (0-y) I«A )-ff)(x)\2 x 2y +ldx ~ y y 1 00 ~

f

x 4s 1 «A ) -f f) (x) 12 x 2y + 1 dx

~

Y

Following [12J, p. 248, there exists fl E :IN and d > 0 such that

max

I£~Y)

(x) \ xdO,1 J f ~d(k+l) 1 For f > fl it yields (**) 1

f

o

1 «A) -f f) (x) 12 20+1 x dx ~ ( max y xE[O,IJ I 1 1«\)-ff)(x)I)2

f

x20+ldx~

o

< _ "_I _

(I

(f.c (y) ) ( 1

)f

max

- 20 + 2 k=O ' k y 4k + 2y + 2 xdO, 1 J

\.c~y)

(x)

I)

2

~

U (k + I) 1 ) II f II 2

it

(4k + 2y + 2) y ( 00 < 1 2 - 20 + 2 d

k~O

v

V 3 3 V

y>-I 0>-1 £.EJN C>O fEX

Y 00

f

o

cIIfll 2 y

i.e. (A )-£.

~

s

a continuous linear operator from X into

X~.

y Y u

(2.2) Lemma

Let y > -I. Then for every r > 0, v > 0 and 0 > -I the operator exp(-r(A )v) y is a continuous linear operator from Xy into Xo'

Proof

Let r > 0, v > 0 and let 0 > -I. Then there exists£. E J.I such that (A )-£.

y

is a continuous linear mapping from Xy into Xo' Hence exp(-r(Ay)V)

=

-£. £. v

=

(A) {(A ) exp(-r(A) )} is also a continuous linear mapping from X

y Y Y Y

into Xo'

Lemmas (2.1) and (2.2) yield the following important result.

(2.3) Theorem

Let a,S> - I. Then for every v ~

Sx

(A)V

a' a

o

Proof

Let v ~ ~. We have shown that

v

exp(-t(A

a) ), t > 0, maps Xa continuously into XS;

D(R)

- There exist constants d,p > 0 such that for all r > 0

a a

I

v -I v

IIexp(r(A) )R(A) exp(-r(A))1I

p

~ d (J.) a + -2a + 2

a a a a a r

So by Theorem (1.3),

Sx

(A)V c

Sx

(A )v. Interchanging a and S we get the

a' a

S,

S

wanted result.

Let a > -I. Since :JH A

a a v v

=

A l l , also E (A)

=

(A ) E . a a a a a a

o

So the Hankel

transform Ea is a continuous bijection on the space S~,(Aa)V' v ~

!,

and

hence on the spaces SXS,(AS)V' v ~

!,

s

> -I. By duality each transform Ea

leaves invariant the spaces of generalized functions TX

(A

)v· For a =

-!

i

2 .S' S(_D

I

--2 + x . The funct~ons £k are the

dx

Hermite functions. With the aid of the papers [8J and [IOJ the following

characterization of the spaces Sx ,(A )v, v E [!,IJ, can be obtained,

-!

-!

: . . f is extendable to an even function in the

SI/2v

space I /2v'

even

The spaces

sq,

p + q ~ I, p,q ~ 0, are introduced by Gelfand and Shi10v in

p

[9J. In this connection we note that in our paper [5J we have proved that the

spaces

S~~~::

are analyticity spaces; explicitly

Sk/k+1

l/k+1 with

B

_k = ( -

~

_dx2

2k)(k+ I) /2k

Relevant for the present paper are the spaces S~

!

~ ~ ~ I. We have

~'

if and only if ~ is an entire function satisfying 3

A,B,C>O : Icp(x+iy)1

~

Cexp(-Alxl l

/lJ +

Bly!I/I-~)

and

I

~ E SI if and only if ~ is analytic on a strip about the real axis say of width r > 0 and satisfying

3_A,C>O : sup !~(x+iy)! ~ Cexp(-A!xl)

Iyl <r

Now Theorem (2.3) leads to the following important results.

(2.4) Corollary

Let a > -I and let v E [!,IJ. Then f E Sx (A)V if and only if f is

extend-a' a

Sl/2v

able to an even function in the space 1/2v.

(2.5) Corollary 1/2v

Let f E SI/2v be even, with v E [!,lJ. Then for each y > -I, there exists an 00

.t

2-sequence

(w~

y)

):=o

and t> 0 such that f

I

exp(-n

V

t)w~Y).c~Y)

where

n=O the series converges pointwise.

Appendix

The set of so-called entire vectors for a positive self-adjoint operator A in a Hilbert space X is equal to

n

t>O

- t A

e (X).

_ 00 A

In [3J, Van Eijndhoven has used the Frechet space D (e ) as the test space ~n a theory of generalized functions which is a kind of reverse of the theory

00 A

in [7J. The space D (e ) is denoted by T(X,A) and it may be called the entire-ness space. To our opinion the well-known theory of tempered distributions

( d2 2 )

is considerably generalized in

[3J.

(Put A

=

log \- --2 + x + I . Then dx

T (L

2 (:IR) ,A) is the space S (lR) of functions of rapid decrease.)

Similar to Theorem (1.3) we prove.

(a. I) Theorem

Let P be a linear operator in X with D(P) :> exp(-a AV) (X) for some a > 0

sufficiently large. Suppose the following conditions are satisfied.

(i) There exists a Hilbert space Y such that exp(-tAv) maps X into Y for all t > O.

(ii) Also, A + P defined on exp(-a AV) (X) is a positive essentially self-adjoint operator in Y.

(iii) There exist positive constants rO ~ I, d > 0 and 0 ~ q < I/v such that for all r > rO

v -I v q

II exp (r A ) PA exp (-r A ) II X < d r .

Av v

Proof

Since T(X,Av) n exp(-tAV) (X), we consider t> rO only. Let 0 < T < I

t>rO

wi th s = t - T > I. The factoring used in Theorem (1.3) yields the following

estimate Put b T Set a Av k AV II exp (T ) (A + P) exp (-t ) II 1 + dTq• Then k-I

n

j=O (J + d) 2q (.!.) 1 /v. Then v

A

v

A

k v II exp ( T ) ( + P) exp ( - t A ) II X v

For £ E exp(-tA ) (X) it yields

k-I

~k!(_I)k/'I)

n

(l+d(T+js/k)q). 'l)S • 0 J= b (1 + d) k 2qk s qk T ~ (k!)I/v (J..)(-q+l/V)k _ak _b s T k AV v k AV v

II (A+P) flly ~ Ilexp(-T )II

X+y lIexp(TA )(A+P) exp(-t )IIX Ilexp(tA )fll ~

Thus r (t)

. v 1 -q + 1 /v

we hnd that f E exp (-r(A + P) ) (Y) for all r < - - s . Now put

v a e

1 -q+l/v . I

- - - s w~ths

=

t + - - 1 for instance. Then we get

v a e +l t v T(x,A ) (exp(-tAV)(x» c

n

t>rO v (exp(-r(t) (A + P) ) (Y)

n

(exp(-r(A+P)'I) (y»

r>O

T (Y , (A + P) v)

o

It is not hard to see that the spaces T (X , (A ) }, V

a a a > -1, are Hanke 1

in-V

cr (X , (A ) }.

a a

variant, and hence their strong duals The previous theorem and the Lemmas (2.1) and (2.2) lead to the following classification.

(a.2) Theorem

Let a,S > -I and let v ~

!.

Then

By [2J and [8J we obtain the following characterizations

and

f E T(X_~,A_!} iff f 1S extendable to an even entire function for which

Vo _<a< 1 3 _{C>O X+1YE(;} V • If(x+iy}1

1

f E T(X_₁,(A__1}2} iff f 1S extendable to an even entire function

2 2 for which V _r> 0 : sup Iyl<r,-oo<x<co \' er1x1If(x+iy)1 < co .

Finally, Theorem (a.2) gives the characterization lin classical analytic of the elements in each T(X ,A }, respectively T(X ,(A )!}, a > -I.

a a a a

References

[IJ Eijndhoven, S.J.L. van, Invariance of the analyticity domain of

self-adjoint operators subjected to perturbations. Preprint 1982.

[2J Eijndhoven, S.J.L. van, On Hankel invariant distribution spaces. EUT-Report 82-WSK-OI, Eindhoven University of Technology,

Eindhoven 1982.

[3J Eijndhoven, S.J.L. van, A theory of generalized functions based on

one-parameter groups of unbounded self-adjoint operators.

TH-Report 81-WSK-03, Eindhoven University of Technology,

Eindhoven 1981.

[4J Eijndhoven, S.J.L. van and J. de Graaf, Some results on Hankel

inva-riant distribution spaces.

Proc. Koninklijke Nederlands Akademie van Wetenschappen, A(86)I,

1983.

[5J Eijndhoven, S.J.L. van, J. de Graaf and R.S1• Pathak, A characterization

k/k+1

of the spaces SI/k+1 by means of holomorphic semigroups.

SIAM J. of Math. An., 14(6), 1983. [6J Faris, W.G., Self-adjoint operators.

Lect. notes in mathematics, Springer, Berlin, 1974, no 433. [7J Graa£, J. de, A theory of generalized functions based on holomorphic

o;emigroups.

TH-Report 79-WSK-02, Eindhoven University of Technology, Ei ndhoven, 1979.

Also to appear as a series of papers 1n Proc. Koninklijke

[8] Goodman, R., Analytic and entire vectors for representation of Lie groups.

Trans. Am. Math. Soc. 143 (1969) 55.

[9] · Gelfand, I.M. and G.E. Shilov, Generalized functions, Vol. II. Academic Press, New York, 1968.

[10] Gong-Zhing,Zhang, Theory of distributions of S-type and pansions. Chinese Math. 4(2), 1963, 211 - 221.

[ I I ] Mitrinovic, D.S., Analytic Inequalities, first edition. Springer, Berlin, 1970.

[12] Magnus, W., F. Oberhettinger and R.P. Soni, Formulas and theorems

for the special functions of mathematical physics, third edition. Springer, Berlin, 1966.

[13] Nelson, E., Analytic vectors.

Ann. Ma th. 70 (19 59), 5 72 - 6 1 5 .

Eindhoven University of Technology Department of Mathematics and Computing Science

PO Box 513, Eindhoven The Netherlands