Convolution Operators and Surjective Limits

Advunces i n Holornorphy, J . A . Barroso (ed. 1 @ North-Holland Publishing Company, 1979

CONVOLUTION OPERATORS AND SURJECTIVE LIMITS

PAUL BERNER

Convolution operators on a space of entire functions have been studied by a variety of authors including Boland[5] who showed that if

E

is a quasi-complete nuclear and dual

nuclear space, then a non-zero convolution operator on continuous for the compaot open topology

'Go

,

H(E),

satisfies

Malgrange's charactorization of the kernel [8] , and if

E

is

also a dual Frechet nuclear space, then such an operator is also surjective.

In this paper we study the case in which

E

is an open and compact surjective limit of appropriate spaces. Our results enable us to draw Bo1and"s conclusions for nucleaspaces

E

nuclear.

which are not dual metric and not necessarily dual Our result includes the important case of

E = Q',

Schwartz'ts space of distributions, and for this case and others we show that the kernel characterization and surjectivity result holds for the larger class of convolution operators.

d6

continuous

Thus we give two affirmative answers

to a question of Boland [6, problem 2c].

1. DEFINITIONS

We recall (see r 7 ] ) that

93

E = surj limuEA ( E u , n a )

is

P.

94

BERNER

called an open surjective limit of the complex 1.c. spaces if for each a in the directed set ( A , > ) , na:E + E

EEala€A n

R

and

n

that

EB +

=

a

TT

aB

Ea’ B en

a, are open and surjective maps such

2

E

and

B

has the projective limit topology

generated by TT a’ a E A . If also for each a E

2

there exists a compact set

E

we say

an: f E H(Ea) I--

a E A

E H(E).

fon

‘n of

H(E)

f E H(E)

H(E) =

limit.

DEFINITION

u

aEA

H(Ea),

F

Let

A:

H(F)

f(y)

3

(K)

= K , then

denote the transpose map

n

-X

‘n.

It is known (see 171 o r [2])

when

E

is an open surjective

be any complex 1.c.s. %,

H(F) + H(F)

f o r all

A(f)

H(F)

If

has a

then any continuous linear

is called a

%-convolution operator

x E F

and

That is, if

f E H(F)

where

y E F.

f(y+x) all

We say that

is surjective,

Ct

will be regarded as a subspace

if it is translation invariant.

A(T-xf) = 7 -x

Ct

K c Ea

is contained in one of these subspaces.

locally convex topology

on

such that

Since each

H(Ea)

via the inclusion

that every

‘n

let

a is injective, thus

operator

c E

,”

is an open and compact surjective limit.

For each

That is

and each compact set

F

has property M1 for

%-convolution operator on

H(F),

A,

if f o r every

we have that

Ker(A)

the closed linear span of all the exponential-polynomials which belong to

Ker(A).

We say that zero

F

has property M 2 for % if every non-

.G -convolution operator is surjective. It is a result due to Malgrange that every finite

dimensional space

F

has properties M 1 and M2.

is

95

CONVOLUTION OPERATORS AND SURJECTIVE LIMITS

'i$o-CONVOLUTION OPERATORS

2.

LEMMA

E = surj l i m

If

l i m i t and then

H(E)

A:

Irn(AIH(Eu))

U

i t s image

+

H(E)

T

-x

U

)

-

A(fOnu)) = 0

[,I,

A(foTT

f o n

where

U

Dx(A(fon

U x

(A(fon

-X

E nil(0).

[21) it with

= gonu

Now f o r a l l Hence by t h e

E

E

So now by l e m m a 1 . 2 of

a+ 0 n-'(O).

H(Eu).

(

T

- ( ~A ( ~f 0 n

A(fona)

i v e l y : M2 for

%o) then

M2 for

16~).

PROOF

Let

A

a E A

E

E

Un ( H ( E a ) )

i s a n open and

(Eu,nu)

(E,}

uEA

a l s o has M1 f o r

be a non-zero

By t h e l e m m a , f o r e a c h

For

E

That i s ,

t h i s form.

Hence

a 1)

But s i n c e

gonu.

e a c h o f which

h a s p r o p e r t y M 1 f o r t h e compact open t o p o l o g y

a

-

g-1

E = s u r j l i muE.,

If

A

=

A(T-x(fona))

)) = l i m

i s g l o b a l l y of g

a)) =

It f o l l o w s t h a t the d i r e c t -

compact s u r j e c t i v e l i m i t o f s p a c e s

so that

with

i s complete.

and t h e p r o o f THEOREM 1

x

A,

i s l o c a l l y o f t h e form

i s convex ( s e e A(fomu)

E H(Ea).

f

a

0

for a l l

a)

H(Eu)

(fan ) ( y ) = f o n ( y + x ) = f o n , ( y ) .

for a l l

ional derivative

) i s a n open s u r j e c t i v e

a E A.

all

t r a n s l a t i o n i n v a r i a n c e of

= A(fon

a

i t follows t h a t a n element of t h i s s u b

n(H(Ea)),

E Uil(0),

,TI

S i n c e w e have i d e n t i f i e d

s p a c e i s of t h e f o r m x

a

i s a t r a n s l a t i o n i n v a r i a n t operator,

c H(Eu)

a E A.

Let

PROOF

(E

uEA

fixed, l e t

%o

(respectivel~

Z 0 - c o n v o l u t i o n o p e r a t o r on H ( E ) .

a E A,

c (an)-loAou~:

(respect-

%o

H(Ea) + H(Ea)

x , y E Ea

1=

Im(A

and l e t

U

n(H(Eu))

i s w e l l defined.

x,y

E E

be s u c h

96

P.

that

So

= x

nu(;)

A

(H(E),50)

i s j u s t the

A

we

cor,l,l]).

E

Now s u p p o s e e a c h

g =

g E Ker(A)

forr

I

p

Now

)

C N

I$ E E d ,

Eu,

and

i s a p o l y n o m i a l on

'c

and s i n c e

g = an ( f ) E % ( c l

f E H(Eu).

N

= c l sp(N),

E

(p-e'

I

p

Let

and A(p*eCP)=O}.

i s j u s t the

Therefore

sp(N) C K e r ( A ) ,

thus

now

induces w e have t h a t

sp(Nu)) C c l sp(N).

Clearly

Clearly

Zo

U

do,

we

O}.

,

ep E E'

H(E,)

on

=

A(p-e')

E,

(H(E),'Co)

C c l sp(N).

and

i s Hausdorff,

do

which p r o v e s t h a t

Ker(A)

E

has property

,-Go.

M1 for

g = fon

H(Eu),

Property M 1 f o r

a

r e l a t i v e topology t h a t

Ker(A)

and

u

a€A

b e l o n g s t o t h e c l o s e d s p a n of

f

N = (p.ecp

a

Go- convo lu t i o n

H(E) =

u E A

E K e r ( A ).

f

i s a p o l y n o m i a l on

0:

a s c a n e a s i l y be

has p r o p e r t y M 1 for

Since

for some

U

implies

implies t h a t

%(N

v i a the inclusion

T h i s f a c t immediately implies

a E A,

a'

g E Ker(A) c H(E).

must have

H(Ea)

the r e l a t i v e

H(E~).

o p e r a t o r on

and t h a t

H(Ea)

; -continuous hence i t i s a

is

0

i n d u c e s on

t o p o l o g y on

!lo

v e r i f i e d (or s e e [ 3 ,

all

f E H(Eu)

Then f o r e a c h

i s a compact s u r j e c t i v e l i m i t ,

E

topology t h a t

that

nu(?) = y .

and

is translation invariant.

a

Since

n'

BERNER

A

f 0

so there exists

2

ao.

If e a c h

a

property M2 f o r Aa(h) = f .

a

f o r some

Hence

Go,

E, 2

uo

and

f

E

t h e r e e x i s t s an

Go.

such t h a t

h a s M2 f o r

A(honu) = g

has property M2 f o r

a, E I\

and

-Go

H(Eu). h

A U

g

E

H(E)

Since

E H(Ea)

which p r o v e s t h a t

f 0

Ea

for thgl

has

such t h a t E

also

97

CONVOLUTION OPERATORS AND SURJECTIVE LIMITS

Let

COROLLARY

be a compact s u r j e c t i v e l i m i t o f d u a l s o f

E

Frgchet nuclear spaces M 1 and M 2 f o r PROOF

‘6

.

E

(DFN spaces) then

limit

Every s u r j e c t i v e

has p r o p e r t i e s

o f DFN s p a c e s i s n e c e s s a r i l y

a n open s u r j e c t i v e l i m i t and Boland h a s shown DFN s p a c e h a s p r o p e r t i e s M 1 and M2 f o r

I::

m

E =

1.

1 1

i=O

C

i s a n open and compact s u r j e c t i v e l i m i t of

t h e f i n i t e d i m e n s i o n a l DFN s p a c e s m

E =

2.

1-1

i=O

m

c

C x

i=O

l i m i t of DFN s p a c e s Let

3.

t W i 3i c N

I]

Cn,

n

E

N.

i s a n open and compact s u r j e c t i v e

C

II c

i=O

be open i n

m

x

c

C ,

i=O

Eln,

n

E

N.

f o r some

n,

and l e t

be a s e q u e n c e o f o p e n , r e l a t i v e l y compact s u b s e t s

ii c

all

Wi+l,

i,

compact s u r j e c t i v e l i m i t of

4.

z0.

The f o l l o w i n g s p a c e s have p r o p e r t i e s M 1 and M2 f o r

EXAMPLES

with

[ 5 ] t h a t every

Let

F

then

E = Q’(I])

t h e DFN s p a c e s

i s a n open and i

€!’(Wi),

E

IN,

be a s t r i c t i n d u c t i v e l i m i t o f a s e q u e n c e of

Fr6chet n u c l e a r spaces

(FnlnEN,

then (see

[7])

E = F‘

is

a c o u n t a b l e ( s e e below) open and compact s u r j e c t i v e l i m i t o f the DFN spaces REMARKS

Examples

.

1-3 a r e s p e c i a l c a s e s o f example 4 .

T h a t t h e above examples have p r o p e r t y M 1 f o r a l r e a d y known from a theorem o f Boland. does n o t r e s t r i c t t h e i n d e x i n g s e t

I\,

However,

Z

is

Theorem 1

and a n a r b i t r a r y open

and compact s u r j e c t i v e l i m i t o f DFN s p a c e s i s not n e c e s s a r i l y co-nuclear cases.

s o t h a t Boland’s r e s u l t does n o t a p p l y i n such

BERNER

P.

98

That t h e s e examples have p r o p e r t y M 2 f o r be p r e v i o u s l y unknown, spaces r e q u i r e s

Indeed B o l a n d t s r e s u l t f o r n u c l e a r

H i s d u a l m e t r i c e c o n d i t i o n i s needed t o i n s u r e

t h a t the

‘%o t o p o l o g y i s a Ilgood” t o p o l o g y .

(when

i s a DFN s p a c e ) ,

example 1,

i s FrBchet.

‘do

I n h i s case However i n

i s n o t F r e c h e t , and i n example

%o

i s rrbadlri n t h a t

nor b o r n o l o g i c a l

3

seems t o

t o be d u a l m e t r i c which i s e v i d e n t l y n o t

E

the case here.

E

‘Go

5o

2

do

i s n o t even b a r r e l l e d n o r semi-complete

( s e e [ 31 )

and a s shown i n [ 41

Zob

#

‘dub.

‘%* -CONVOLUTION OPERATORS

Whereas

‘Go may sometimes be a I1badl’ t o p o l o g y , we

ZUb = G b

know t h a t

i s a s t r i c t (LF)-nuclear space topology

i n examples 2 and 3 , t h e r e f o r e i t might be i n t e r e s t i n g t o

look a t

‘t;

b

-convolution o p e ra t o rs i n t h e s e cas es .

First let

us n o t e t h a t t h e s e two c l a s s e s o f c o n v o l u t i o n o p e r a t o r s d o not coincide,

zob f zuJb= z6

I n example 2 s i n c e

it follows

from Mackey’s theorem t h a t t h e r e i s a z b - c o n t i n u o u s l i n e a r f u n c t i o n a l T on A

by

H(E)

which i s n o t z - c o n t i n u o u s . 0

Af(X) = T(T_,~) a l l

x

E E,

f

E

H(E),

then

Define A

i s a’

t 6 - c o n v o l u t i o n o p e r a t o r b u t n o t a ’G0-convolution o p e r a t o r . We r e c a l l t h a t a s u r j e c t i v e l i m i t i s c a l l e d c o u n t a b l e

if t h e i n d e x s e t

A

is just

N

and i f t h e r e e x i s t c o n t i n u m .

99

CONVOLUTION OPERATORS AND SURJECTIVE LIMITS

maps

TT

nm : Em

+

En

nn = nnm~TTm

such t h a t

m , ncN

whenever

m > n.

and

W e s h a l l s e e n e x t t h a t i n t h e a b o v e s i t u a t i o n t h e class

of G6-convolution o p e r a t o r s i s s t r i c t l y larger t h a n t h e c l a s s ' of

convolution

THEOREM 2

operators.

E = s u r j l i m nEN

Let

(En,nn)

compact s u r j e c t i v e l i m i t of DFN s p a c e s - c o n v o l u t i o n o p e r a t o r on operator on PROOF

E

H(E),

and

is also a

G6 - c o n v o l u t i o n *Gb.

the subspaces

and o n l y i f

An:

continuous f o r each

is

H(En)

W E n ) , Go>

Hence e v e r y

i s 'E - c o n t i n u o u s

A

"rr c

H(En)

6

(H(En)

J b ) is

But s i n c e t h e i n d u c t i v e l i m i t i s

n.

s t r i c t , the r e l a t i v e topology t h a t is just

i s a s t r i c t inductiw

so that 4

of A

An

In [ 3 ]

30 - c o n t i n u o u s .

(H(E),$)

(H(En),'Lo),

then

zo '

t h e o r e m 1, t h e r e s t r i c t i o n

of

l i m i t of t h e s p a c e s

Z6)

Then e v e r y

h a s p r o p e r t i e s M 1 and M2 f o r

theorem 4 . 1 i t i s shown t h a t

H(E),

.

i s a convolution operator f o r

A

a s shown i n t h e p r o o f

if

'En' nEN

i s n e c e s s a r i l y a n open s u r j e c t i v e l i m i t .

Suppose

t o e a c h of

E

H(E)

be a c o u n t a b l e

'do.

Thus

A

H(E,)

i n h e r i t s from

is also

' & o - c o n v o l u t i o n o p e r a t o r on

t -continuous,

H(E)

6

is also a

0 -convolution operator. d Next s u p p o s e

i s a non-zero

A

( n o t n e c e s s a r i l y 'Go-continuous). operators

(H(E,L

An,

zo)

to

n

- c o n v o l u t i o n operator

The t r a n s l a t i o n i n v a r i a n t

a r e c o n t i n u o u s as o p e r a t o r s f r o m

E IN, H(En)

"6

=

(H(E),Z6)

because

(H(E,),

a r e the d e f i n i n g subspaces of t h e i n d u c t i v e l i m i t

to) nE N Z6.

And

100

P.

BERNER

because t h i s i n d u c t i v e l i m i t i s s t r i c t , each a s a n o p e r a t o r from

E

that

(H(En),Zo)

to

A

i s continuous

n

N o w t o show

(H(En),Zo).

h a s p r o p e r t i e s M 1 and M 2 f o r

w e may p r o c e e d a s

Zb

i n t h e p r o o f o f t h e o r e m 1 n o t i n g t h a t e a c h DFN s p a c e p r o p e r t i e s M 1 and M 2 f o r REMARKS

If

F

i s a DFN s p a c e t h e n

p e r t i e s M 1 and M 2 f o r the space

Z

on

=G8

0

H(F)

(see

E

n E N.

to

ncN

s a t i s f i e s t h e h y p o t h e s i s o f t h e o r e m 2 and

i s a t o p o l o g y on 9

f l l i f t f lfrom t h e s p a c e s

Z8

E.

Whenever

(H(E)

Zo.

t h u s w e may r e g a r d t h e o r e m 2 as s h o w i n g t h a t t h e p r o -

[l]),

G

has

En

zo) IH(P;,)

H(E)

= (H(E)

9%)

such t h a t

then

6 C 'G

Zo 5

8'

IH(En) = ( H ( E ) , " b ) IH(E,)

From t h i s i t f o l l o w s t h a t one may r e p l a c e

f o r each -Gb

by

i n t h e s t a t e m e n t of theorem 2 .

ADDED I N PROOF

I t would s e e m from r e f e r e n c e convolution o p e r a t o r is a

-Gb

in

[4]

t h e d e f i n i t i o n of property

E

i s s a i d t o have p -r o p e r t y ( Q )

e x i s t s a continuous l i n e a r s u r j e c t i o n G,

zo

must be c o r r e c t e d t o r e a d :

An 1 . c . s .

space

t h a t not every

c o n v o l u t i o n o p e r a t o r i n exam-

p l e 3 a s w e l l a s e x a m p l e 2. However, (Q)

[!+I

where

G

i s isomorphic t o

n

from

E

i f there

onto a

m

C

i=O

C.

T h i s c h a n g e e l i m i n a t e s example 3 o f b o t h [ k ] and t h i s p a p e r from t h e c o n c l u s i o n t h a t show a d i f f e r e n c e i n

'Go

'ob

' U b

w h i c h is u s e d t o

and Z - co n v o lu tio n

b

operators.

The c o r r e c t i o n has t h e f o l l o w i n g e f f e c t on t h e o r e m

4.1

CONVOLUTION OPERATORS AND SURJECTIVE LIMITS

of [3]:

I f some

En

has property (Q)

then

#

Gob

1 01

ZWb

and

c o n c l u s i o n s ( 4 ) and

(5) are valid.

Conclusion ( 3 ) should be

corrected t o read:

( 3 ) zu f ZUb,

and

'5

UI

i s semi-montel,

etc. I a m e n d e b t e d t o Segn D i n e e n and P h i l i p B o l a n d f o r

p o i n t i n g out t h e s e c o r r e c t i o n s .

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and N A C H B I N ,

of holomorphic mappings, Proc. Holomorphy,

L.

On bounded s e t s

on I n f i n i t e D i m e n s i o n a l

L e c t u r e N o t e s i n Math.,

Vol.

364, S p r i n g e r

V e r l a g ( 1 9 7 4 ) , 123-133. 2. BERNER, P .

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L e c t u r e N o t e s i n Math.,

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3. BERNER, P.

T o p o l o g i e s on s p a c e s o f h o l o m o r p h i c f u n c t i o n s

of certain surjective l i m i t s ,

I n f i n i t e Dimensional

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4. BERNER, P.

( 1 9 7 7 ) , 75-92.

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Acad.

Sc.

Paris,

t . 280

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( 1 9 7 4 ) , 135-144.

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P.

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6. BOLAND, P.

BERNER

Holomorphic Functions on Nuclear Spaces,

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B, No. 16, Univ. de Santiago de Compostela (1977).

7 . DINEEN, S.

Surjective limits of locally convex spaces and

their application to infinite dimensional holomorphy, Bull. SOC. Math. France, t. 103 (1975).

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Existence et approximation des solutions des

Qquations aux deriv6es partielles et des Qquations des convolutions, Annales de 1’Institut Fourier, IV, Grenoble

(1955-56), 271-355.

DEPARTMENT OF MATHEMATICS

LE MOYNE COLLEGE SYRACUSE, NEW YORK 13214 USA