Some Results on Continuous Linear Maps between Frechet Spaces

Functional Analysis: Surveys and Recent Results I II K.D. Bierstedt and B. Fuchssteiner (eds.) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

349

SOME RESULTS ON CONTINUOUS LINEAR MAPS BETWEEN FRECHET SPACES Dietmar Vogt

Universitat-Gesamthochschule Wuppertal Fachbereich Mathematik

D-5600 Wuppertal 1, GauBstr.

20

1 The p r e s e n t a r t i c l e i s m a i n l y concerned w i t h t h e d e r i v e d f u n c t o r s E x t (E,.) t h e f u n c t o r s L(E,.),

of

E f i x e d , a c t i n g f r o m t h e c a t e g o r y o f F r e c h e t spaces t o

t h e c a t e g o r y o f l i n e a r spaces ( s e e Palamodov [17] ), w i t h c o n d i t i o n s f o r 1 1 E x t (E,F) = 0 (see [31]) and t h e i r a p p l i c a t i o n s . E x t (E,F) = 0 means t h a t a l l e x a c t sequences 0

--*

F

+

G

--*

E

+

0 s p l i t ( c f . Thm. 1 . 8 . ) .

I n a f i r s t s e c t i o n we g i v e a s h o r t i n t r o d u c t i o n t o t h e t h e o r y o f t h e f u n c t o r s 1 E x t (E,.) ( c f . , c31.) w i t h s p e c i a l emphasis on t h e c o n c r e t e r e p r e s e n t a t i o n s 1 f o r E x t (E,F) and t h e c o n n e c t i n g maps i n case E o r F i s n u c l e a r . The necessary 1 r e s p . s u f f i c i e n t c o n d i t i o n s f o r E x t (E,F) = 0 (Thm. 1 . 9 . ) a r e p r e s e n t e d w i t h o u t 1 p r o o f ( s . [31]). I n s t e a d we g i v e i n $2 a d i r e c t p r o o f f o r E x t ( s , s ) = 0 which l e a d s v i a t h e permanence p r o p e r t i e s d e r i v e d i n $ 1 d i r e c t l y t o t h e s p l i t t i n g theorem f o r e x a c t sequences 0

+

F

+

G

+

E

+

0 where F i s a q u o t i e n t and E a

subspace o f s ( c f . 1231, 1341, 11243 ) . The d e s c r i p t i o n o f t h e c l a s s e s 1 1 E x t (E,s) = 0 and E x t (s,F) = 0 g i v e n i n Thm. 2.5. shows t h e s t r o n g c o n n e c t i o n 1 between t h e t h e o r y o f E x t and t h e s t r u c t u r e t h e o r y o f n u c l e a r F r e c h e t ( - , a )

spaces as developed f o r t h e case o f s i n [23]

and [34].

I n $ 3 we p r o v e by use o f t h e p r o p e r t i e s o f Ext'(w,.)

and a lemma f r o m [293

(Lemma 2.1.

i n t h e present paper) t h a t a h y p o e l l i p t i c p a r t i a l d i f f e r e n t i a l

o p e r a t o r on

IR" w i t h c o n s t a n t c o e f f i c i e n t s has no r i g h t i n v e r s e i n Cm(Q),

Q c Din open ( s e e [34 w i t h a d i f f e r e n t p r o o f ) . T h i s extends a r e s u l t o f Grothendieck on e l l i p t i c o p e r a t o r s .

1 I n $ 4 we show how t h e knowledge o f c o n d i t i o n s f o r E x t (E,F) = 0 can be used f o r t h e i n v e s t i g a t i o n o f t o p o l o g i c a l p r o p e r t i e s ( b a r r e l l e d n e s s e t c . ) o f spaces

Ei

F '2 Lb(E,F),

E,F F r g c h e t spaces, one o f them n u c l e a r ( c f . Grothendieck [ 8 ] ,

11, $ 4 ) . I n p a r t i c u l a r we g i v e i n Thm. 4.9. a complete b a s i s f r e e d e s c r i p t i o n o f t h e c l a s s e s {E;Lb(E,F,) b a r r e l l e d } and { F : Lb(Eo,F) b a r r e l l e d } i f Eo o r Fo i s a power s e r i e s space s a t i s f y i n g a s t a b i l i t y c o n d i t i o n , so e x t e n d i n g t h e r e s u l t s o f Grothendieck l o c . c i t . . We a l s o g i v e a s o l u t i o n f o r t h e problem o f c l a s s i f i c a t i o n o f complex m a n i f o l d s V a c c o r d i n g t o t o p o l o g i c a l p r o p e r t i e s ( h e r e

350

D. Vogt

barrelledness of Lb(E,F))as proposed in Grothendieck [81, 11, p. 128, a t l e a s t i f V i s a Stein manifold. A more direct and systematic treatment i s contained in 1331. In 5 4 we make use of the r e s u l t s of [31], 5 4 and 5 5 on E x t 1( E , F ) = 0 which are * * based on (S1) and ( S 2 ) . The proofs are rather complicated so we do n o t present * * them here. Instead we use in the final sectionconditions (S1) and (S2) t o investi1 gate which spaces E or F can occur in a nontrivial way in the relation E x t ( E , F ) = O . These are essentially the countably normed spaces E (more precise: the spaces E satisfying a condition (DN ) ) and the quasinormable spaces F . A special role play cp the spaces F which are quojections or do n o t s a t i s f y the condition ( * ) of BellenotDubinsky [3]. Some r e s u l t s of !j 5 are strongly related t o the r e s u l t s of

"1.

0. We will use standard notation of the theory o f locally convex spaces a s in [12] , 1201. For nuclear spaces we refer t o [ 8 ] , [19], for sequence spaces t o [ 5 ] and f o r concepts of homological algebra t o 1151 and p 5 ] .

Throughout the paper E,F,G,H always denote locally convex Frechet spaces, over a fundamental system of seminorms. L ( E , F ) i s the IK = R o r E, 1) I l l c 11 112 5 linear space of continuous linear maps from E t o F. On the dual space E'=L(E,IK) of E we consider the dual (RtU{t-)-valued) norms IIyIlk = SUP \ y ( X ) ) : X E, Ilxl/k 5 1).

...

We p u t

*

El; := { y C El : lIyl/k 4

t

m)

.

A Frechet space E i s said t o have property ( D N ) (resp. ( Q ) ) i f there e x i s t s a 2 *2 fundamental system of seminorms such that I1 l l k - 5 II 1tk-l /I IIk+l (resp. II Ilk 5

II

II

for all k

(see [ 2 4

1341

POI ) .

A nuclear Frgchet space i s isomorphic t o a subspace of s i f f i t has property (DN)

( s . [23]), i t i s isomorphic t o a quotient space of s i f f i t has property (a), ( s . [ 3 4 ) . s denotes the nuclear Frschet space of a l l rapidly decreasing sequences: S = {X =

(x1,x2,

... )

:

[lx/lk =

4 J

lxjl j

k

<

+

m

for all k}

be an i n f i n i t e matrix such t h a t 0 More generally l e t A = ( a j . k ) j ,k > 0 for a l l j and k . Then we define sup a k

5

.

aj,k5aj,k+l,

j,k

E N these are Frechet spaces. They Equipped with t h e i r respective seminorms (11 are called Kothe sequence spaces. They are nuclear i f f f o r every k there i s p such t h a t (with 0 = 0 ) :

351

Continuous linear maps between Frechet spaces

2

j

a

j , k a.

(Grothendieck-Pietsch c r i t e r i o n ) .

< + m

J,k+P

I n t h i s case we have h ( A ) = h m ( A ) , and v i c e v e r s a . a.

...~r

...

(a.). : a 1 5 a 2 s f+-, J J t h e n h ( A ) ( r e s p . h m ( A ) ) i s c a l l e d power s e r i e s space and denoted by n r ( a ) ( r e s p .

o2

and

I f aj,k

= okJ w i t h sequences 0 c

!,;(a)).

I t depends o n l y on r and t h e sequence a. F o r f i x e d a a l l spaces Ar(a)

+

(resp. nF(a)) w i t h r < t o t h e cases r = 1, +

m.

-

p1

<

<

a =

a r e i s o m o r p h i c . T h e r e f o r e we can r e s t r i c t o u r a t t e n t i o n

n l ( a ) i s c a l l e d power s e r i e s space o f f i n i t e t y p e , !,_(a)

o f i n f i n i t e t y p e . Power s e r i e s spaces o f f i n i t e t y p e

and o f i n f i n i t e t y p e can

n e v e r be i s o m o r p h i c . By

w :=

by #(V)

lK’

we denote t h e p r o d u c t o f c o u n t a b l y many c o p i e s of t h e s c a l a r f i e l d ,

t h e n u c l e a r F r k c h e t space o f holomorphic f u n c t i o n s on a complex mani-

f o l d V. 1. I n t h i s f i r s t s e c t i o n we p r e s e n t some b a s i c f a c t s on t h e d e r i v e d f u n c t o r s k E x t (E,.),k = O , l , o f t h e f u n c t o r L(E,.). A l l these functors are considered

...,

t o a c t f r o m t h e c a t e g o r y o f F r k c h e t spaces t o t h e c a t e g o r y o f l i n e a r spaces over

lK

=

IR o r a. E denotes a f i x e d F r 6 c h e t space.

We do n o t g i v e a c o n s t r u c t i o n f o r these f u n c t o r s b u t t a k e t h e i r e x i s t e n c e as g r a n t e d by homological a l g e b r a ( s e e Palamodov [17]).

I n s t e a d we t a k e an

a x i o m a t i c approach, i . e . we s t a t e p r o p e r t i e s o f t h e s e f u n c t o r s and t h i s w i l l be t h e o n l y i n f o r m a t i o n w h i c h we w i l l use.

k The f u n c t o r s E x t (E,.)

have t h e f o l l o w i n g p r o p e r t i e s :

(0)

E x t o ( E , * ) = L(E;)

(I)

To e v e r y s h o r t e x a c t sequence 0 maps

k

tik : E x t (E,H)

such t h a t 0

+

L(E,F)

*

L L(E,G)-

+

q*

+

F

L

+

E x t k + l (E,F),

L(E,H)-

G

q --f

H

+

0 t h e r e a r e assigned

k = 0,1,

...

1 1 1 ql Ext ( E , F ) L Ext ( E , G ) - - + Ext

€io

i s e x a c t and depends f u n c t o r i a l l y on t h e s h o r t e x a c t sequence.

(11) F o r e v e r y i n j e c t i v e space

k I we have E x t ( E , I )

= 0 for k =

1,2,

... .

352

D. Vogt

We used (and w i l l use) t h e f o l l o w i n g n o t a t i o n s : c p O A = :Ay assigned t o A by t h e f u n c t o r s E x t

k

(E,.), k

(pkA

... . A

= 1,2,

i s t h e map

space I i s c a l l e d

i n j e c t i v e i f f o r each space E, each c l o s e d subspace Eo c E , and each cp c L ( E o , I )

t h e r e e x i s t s an e x t e n s i o n 9

c

L(E,I).

Examples o f i n j e c t i v e spaces a r e t h e

spaces l m ( M ) , M a n i n d e x s e t , and t h e i r p r o d u c t s .

I f F i s a F r e c h e t space t h e n an e x a c t sequence

I k i n j e c t i v e f o r a l l k, i s c a l l e d an i n j e c t i v e r e s o l u t i o n . k We can use any i n j e c t i v e r e s o l u t i o n o f F t o c a l c u l a t e t h e space E x t (E,F),

as

the f o l l o w i n g p r o p o s i t i o n shows. The p r o o f i s s t a n d a r d . I n f a c t u s u a l l y one uses k i n j e c t i v e r e s o l u t i o n s t o prove t h e existence o f f u n c t o r s Ext ( E , . ) w i t h the

-

p r o p e r t i e s d e s c r i b e d above.

1.1. P r o p o s i t i o n : If 0

k = 0,1,

F

+

Lo

I0

11-

L1

I2

-t

... is

an injective r e s o l u t i o n ,

... .

for k = 1,2,

proos:

+

We p u t Zo = F , Zk = k e r

... a 0

L~

= im

L

~

...

f o- r ~k = 1,2,

and o b t a i n f o r each

s h o r t e x a c t sequence

+

Zk

-

Ik

Lk

zk+l

+

0

which l e a d s by ( I ) and ( 1 1 1 ) t o a l o n g e x a c t sequence

*

0

+

L(E,Zk)

L(E,Ik)-

--t

S i n c e L(E,Zk+l)

= ker

Lk

L

*

L(E,Zk+l)

1 Ext (E,Zk)

f o r k = 1,2,

L;+~

/ im

ExtJ+'(E,Zk)

k (E,F) 2 E x t 1 (E,Zk-l)

... .

0

+

+

1

+

E x t (EyZk+l)

2

E x t (E,Zk)

+

... , j

0

+

... .

= 1,2,

...

:

*

L~

.

Combining these e q u a t i o n s and u s i n g Zo = Ext

+

t+h i s~ g i v e s us f o r a l l k = 0,1,2,

~

1 E x t (E,Zk) 2 '' ker Extj(E,Zk+l)

+

F we o b t a i n

ker

L;

/ im

LL-~

I t can e a s i l y be seen t h a t e v e r y Banach space F has an i n j e c t i v e r e s o l u t i o n o f Banach spaces. We need o n l y t o know t h a t e v e r y Banach space X can be n a t u r a l l y imbedded i n t o l " ( B o ) where Bo i s t h e u n i t b a l l i n X ' . We a p p l y t h i s t o F and p u t

353

Continuous linear maps between Frechet spaces

t h e n t o I o / F and o b t a i n 1 1 , e t c . .

I. = l " ( B o ) ,

1 . 2 . C o r o l l a r y : If E is nuclear and F then E x t k (E,F) = 0 for k = 1,2, ,..

=

J

F . is a product of Banaeh spaces F . , J J

Proof: We a p p l y t h e above d e s c r i b e d procedure s e p a r a t e l y t o t h e F . and o b t a i n an J i n j e c t i v e r e s o l u t i o n o f t h e form

~

o

+

n j

F . --t J

n

n ~j

j

l ajl

j

n

0

j

where t h e IJk a r e Banach spaces. From t h e p r o p e r t i e s o f a n u c l e a r space ( s . Grothendieck [83 , I I , § 3, nO1) and Prop. 1.1. we conclude t h e a s s e r t i o n .

The p r e c e d i n g remark i s v e r y u s e f u l as one sees f r o m t h e f o l l o w i n g lemma which i s contained i n

[3u

[ln Thm. 5 . 2 ( o r Cor. 5 . 1 . ) and a p r o o f o f which can be found

in

(Lemma 1 . 1 . ) . We need t h e f o l l o w i n g n o t a t i o n :

A p r o j e c t i v e spectrum P

~ : , F1 ~+

Fk (1 2 k ) o f Banach spaces i s c a l l e d a

fundamental system o f Banach spaces f o r t h e F r P c h e t space F i f ( i ) F = l i m p r o j Fk ( i i ) f o r e v e r y k t h e r e i s an 1 2 k such t h a t pk

: F

+

pk

F i s dense i n

p1

,k F1, where

Fk denotes t h e c a n o n i c a l map.

1 . 3 . Lemma: If P , , ~ : F1

+

Fk is a fundumenta2 system of Banaeh spuces f o r F

then the sequence ("eanonieal r e s o l u t i o n " )

is exact, where

L

: x

--+

( P ~ x and ) ~ q : ( x ~ )* ~( P k + l , k ' k + l

-

'k)k

.

T h i s y i e l d s as an i m e d i a t e consequence (see [ 3 q , Thm. 1 . 2 . ; c f . [16],

[17]

, p-

7.1.,

51) :

1.4. Theorem: If E o r F is nucZear then E x t k (E,F) = 0 f o r k 2 2.

Proof: I f

F i s nuclear, then F

morphic t o 1".

has a fundamental system o f Banach spaces i s o -

Hence t h e Fk i n Lemma 1 . 3 . can be chosen i n j e c t i v e , which means

t h a t t h e s h o r t e x a c t sequence i n Lemma 1 . 3 . i s an i n j e c t i v e r e s o l u t i o n o f l e n g t h 1 . The a s s e r t i o n f o l l o w s f r o m Prop. 1.1. I f E i s n u c l e a r t h e n we choose any fundamental system o f Banach spaces and

354

D. Vogt

a p p l y ( I ) t o t h e c a n o n i c a l r e s o l u t i o n (see 1.3.).

... + 0 ... .

f o r k = 2,3,

-+

0

k E x t (E,F)

-+

+

0

+

0

We o b t a i n

... .

The z e r o s come f r o m Cor. 1.2.

From Thm. 1.4. we o b t a i n a permanence p r o p e r t y :

If E o r F i s nuclear, E x t 1(E,F)

1.5. C o r o l l a r y :

then a l s o Extl(E,Fo)

M:L e t

q : F

+

=

= 0 and Fo

a quotient of F,

0.

Fo be t h e q u o t i e n t map, G := k e r q. Then

s h o r t e x a c t sequence

O + G + F + F

0

(I) a p p l i e d t o t h e

-+O

gives

...

-+

1 E x t (E,F)

1 E x t (E,Fo)

+

2 E x t (E,G)

7-

+

... .

The f i r s t term i s z e r o by assumption, t h e t h i r d by 1.4.,

hence a l s o t h e m i d d l e

term.

1

Our n e x t t a s k i s t o g e t more i n f o r m a t i o n on E x t (E,F)

i n case one o f t h e spaces

i s nuclear. 1.6. Theorem: F,

Let $ =

be a fundamental system of Banach spaces f o r

{Fk, pl,k}

l e t E o r F be nuclear. Then

where

1 E x t (E,F) 2r1 L(E,Fk) k B(E,F) = {(Ak)k

c

/ B(E,3)

n

L ( E y F k ) : ZI(Bk)k E rI L(E,Fk) such t h a t k

k

Ak = Pk+l,k B k + l

proos: I f

E i s n u c l e a r we a p p l y ( I

obtain

... + n k

L(E,Fk)

q"

n k

- Bk

f o r a l l k).

t o t h e c a n o n i c a l r e s o l u t i o n (see 1.3.).

L E,Fk)

’ 6

1 Ext (E,F)

+

0

+

We

...

where t h e z e r o comes f r o m Cor. 1 . 2 I f F i s n u c l e a r and p

: F1 -+ Fk i s any fundamental system o f Banach spaces t h e n 1Y k : FY + F i o f i n j e c t i v e Banach spaces and we can f i n d a fundamental system po 1 ,k maps qk : FL -+ Fk such t h a t

355

Continuous linear maps between Frechet spaces

i s commutative. Hence we o b t a i n a c o m n u t a t i v e diagram

where

cp

i s i n d u c e d by t h e q k . We a p p l y ( I ) t o t h i s diagram and o b t a i n

. .. + n

... + n

Q

1."

L(E,Fk)

n

q

0

FQ

0

6

L(E,Fk)

+E x t

1(E,F)

+

...

+

0

/id

L ( E , F i ) L n L(E,Fi)k k

So

1

E x t (E,F)

The z e r o comes f r o m ( 1 1 ) . O b v i o u s l y t h e a s s e r t i o n f o l l o w s f r o m t h i s diagram. From Thm. 1.6. a g a i n we o b t a i n a permanence p r o p e r t y :

1.7. C o r o l l a r y : If E or F is nuclear, E x t 1(E,F) = 0 and Eo a closed subspace of E, then Extl(Eo,F)

proos: We

=

0.

choose a fundamental system o f Banach spaces f o r F which f o r n u c l e a r

F we assume t o c o n s i s t

o f i n j e c t i v e Banach spaces. Hence i n b o t h cases t h e

r e s t r i c t i o n maps Rk : L(E,Fk)

0 = E x t 1(E,F)

-+

L(Eo,Fk) a r e s u r j e c t i v e f o r a l l k

( f o r E nuclear

The Rk induce t h e map R i n t h e f o l l o w i n g diagram. R i s s u r j e c t i v e .

see [8],11,93).

k

L(E,Fk)

/ B(E,F)-n

R

k

L(Eo,Fk)

/ B(E,,,.3)

1 E x t (Eo,F)

T h i s shows t h e a s s e r t i o n . I n t h e f o l l o w i n g p a r t o f t h i s s e c t i o n we want t o e x p l a i n , what i n terms o f t h e c o n c r e t e r e p r e s e n t a t i o n o f 1.6. (and 1.4.) sequence ( I ) "means".

1

E x t (E,F)

resp. t h e long exact

L e t us remark t h a t t h e isomorphism which we used i n 1.6. i s

356

D. Vogt

i n a c a n o n i c a l way determined by t h e fundamental system o f Banach spaces. I t i s induced by t h e ’6 assigned t o t h e c a n o n i c a l r e s o l u t i o n (see 1 . 3 . ) .

I n the following

d i s c u s s i o n we w i l l always t a l k a b o u t t h i s isomorphism. Let 0

--f

F

L

G

q

H

-f

+

0 be an e x a c t sequence. We assume t h a t e i t h e r E o r F,G

3 ,b be fundamental

and H a r e n u c l e a r . L e t 5 ,

systems o f Banach spaces such t h a t

f o r e v e r y k we have an e x a c t sequence o f Banach spaces Lk

0 + Fk

Gk-

f

'k

Hk

0

+

and such t h a t f o r a l l 1 > k t h e diagram L

q

0-F-G-H

4

J

L1

0-F-G 1

0

1

4

1'O

-4

Lk qk F k - + Gk

--t

+O

.1

91 -ti

Hk

+

0

i s commutative. We can o b t a i n such fundamental systems e.g.

by t a k i n g for

Banach spaces generated by a fundamental system o f seminorms on G and f o r

fi?

the

5 and

@ t h e Banach spaces generated by t h e induced r e s p . coinduced fundamental systems

o f seminorms on F r e s o . H. We have t h e f o l l o w i n g s i t u a t i o n Sor

L(E,H)

/

\ 0

L

E x t 1(E ,F)

1 t

It?

n

I

L(E,Fk)/B(E,F)-

k

Ext'(E,G)

n k

q1 +

Il? L(E,Gk)/B(E,J)-

Q

1 E x t (E,H)

n k

It?

-

L(E,Hk)/B(E,*)

where D,I,Q denote t h e maps induced by means o f t h e isomorphisms f r o m F o , We want t o d e s c r i b e these maps. We a p p l y t h e r e f o r e ( I ) columnwise t o t h e f o l l o w i n g diagram:

0 J

F

.1

0

0 .

-

L

G

-

J

H

J

0

0

9

J

5.

0

0

+

0

L',

q

1

.

357

Continuous linear mam between Frechet spaces

where t h e columns a r e c a n o n i c a l r e s o l u t i o n s . We o b t a i n by use o f ( I ) and ( t h e p r o o f o f ) 1.6. t h e f o l l o w i n g commutative diagram w i t h e x a c t columns. The rows a r e e a s i l y

seen t o be e x a c t .

L

nL k

$

1

Ib ",;

------in

ll L(E,Fk) k

k

-n

L(E,Gk)

k 1

16O

E x t 1(E,G)

Extl(E,F)

L(E,Hk) J16O

q

Extl(E,H)

1

1

.1

0

0

0

We see i m m e d i a t e l y t h a t t h e maps I and Q a r e t h e n a t u r a l maps between t h e r e s p e c t i v e spaces, i . e . t h e maps induced by

n k

n

ck and

k

qk on t h e q u o t i e n t s .

Moreover we can by a s t a n d a r d procedure o f homological a l g e b r a ( s . [15]) 1 a map d : L(E,H) + E x t (E,F) as f o l l o w s : F o r 9 E L(E,H) we have, because o f t h e n u c l e a r i t y assumptions, acp E i m

*

b

define

* (n q k ) . b

We

choose j, E n L(E,Gk) such t h a t (Il q k ) j, = acp. Since (TI q k ) b 9 = c ( n q k ) j, = c acp = 0 we can f i n d x E n L(E,Fk) such t h a t (n L,;) x = b 9. We p u t d cp := 6 ’ x .

I t i s n o t d i f f i c u l t t o p r o v e t h a t d i s w e l l d e f i n e d , l i n e a r and makes t h e

f o l l o w i n g sequence e x a c t

... Hence ’6

+

L(E,G)

*

J--+ L(E,H)

d

+

1 E x t (E,F)

A o d where A i s an automorphism o f k e r 1 f r o m L(E,H) + E x t (E,F). =

w

We now p u t Dcp :=

[x]

where

that also

L~

1

E x t (E,H)

+

..

and ’ 6 i s t h e one a c t i n g

[ ] denotes t h e e q u i v a l e n c e c l a s s i n r~ L ( E , F k ) / B ( E , Y ) .

1 k L(E,Fk) + E x t (E.F) induces an isomorphism) k i s w e l l d e f i n e d and l i n e a r . S t r a i g h t f o r w a r d c a l c u l a t i o n shows t h a t

I t f o l l o w s e a s i l y (so a c t i n g f r o m

O

1

N

n

N

= B o 0 where B i s some automorphism o f k e r I , o r e q u i v a l e n t l y : t h a t i m 0 = i m 0, N

k e r D = k e r D. We have t o e x p l a i n

E.

I t d e s c r i b e s t h e f o l l o w i n g ( M i t t a g - L e f f l e r - ) approach t o t h e

s o l u t i o n o f t h e l i f t i n g problem

3 58

D. Vogt

F i r s t we s o l v e a17 t h e " l o c a l " l i f t i n g problems 0

Fk

+

Gk qk+ Hk

+

0

--t

E

which we can do b y o u r n u c l e a r i t y assumptions. The qk a r e t h e maps i n d u c e d by qY i . e . t h e components o f aq. The q k g i v e t h e components o f acp. Then we a p p l y b, i . e . o I$f we ~ + t h i n~ k o f L as an imbedding, t h e s e can be f o r m t h e maps ~ ~ + ~ , -~ qk. c o n s i d e r e d as maps i n L(E,Fk),

X

E

t h e y d e f i n e t h e components xk o f an element

i.e.

f L(E,Fk)*

Our o r i g i n a l l i f t i n g problem i s e a s i l y seen t o be s o l v a b l e i f and o n l y i f we can f i n d (Bk)k E

nk

-

L(E,Fk)

($k+l - Bk+l)

pk+l,k

=

such t h a t P k + l , k o $ k + l - $k = Pk+l,k 0 B k + l BkA hence - Bk f o r a l l k . T h i s means n o t h i n g e l s e t h a n EQ=[x] = 0

qk

and t h a t i s e q u i v a l e n t t o EQ = 0.

Hence t h e a x i o m a t i c approach d e s c r i b e s (under o u r n u c l e a r i t y assumptions) v i a t h e c o n c r e t e r e p r e s e n t a t i o n n o t h i n g e l s e t h a n t h e p o s s i b i 1 it y o f t h e " n a t u r a l 'I way o f 1 s o l u t i o n o f t h e l i f t i n g problem. k e r L d e s c r i b e s t h e o b s t r u c t i o n a g a i n s t t h i s i n 1 t h e c o n c r e t e s i t u a t i o n . E x t (E,F) d e s c r i b e s t h e " s t r u c t u r a l " o r "maximal" obs t r u c t i o n i n any s i t u a t i o n . I f i t i s z e r o t h e procledure always works. We a d m i t w i t h o u t f u r t h e r p r o o f t h e f o l l o w i n g theorem ( s .

pl],

Thm 1.8.):

1.8. Theorem: The following are equivaZent: 111

Ext'(E,F)

(2)

Every ezact sequence 0

= 0

0 splits.

F

+

G

+

E

131

For every exact sequence 0

+

F

+

G

+

H

+

0 and

cp E

L(E,H)

(4)

For every exact sequence

+

H

4G

+

E

+

o

cp E

L(H,F) there e x i s t s

+

E

L(E,G) m t h

cp =

$ E L(G,F) w i t h q =

-f

q o $.

Ji o

o

--f

9

and

there e x i s t s

L.

The r e s t o f t h i s paper w i l l be m a i n l y devoted t o a d i s c u s s i o n o f c o n d i t i o n s f o r 1 E x t (E,F) = 0 and t h e i r a p p l i c a t i o n s . We use t h e f o l l o w i n g c o n d i t i o n s on two

3 59

Continuous linear maps between Frkchet spaces

*

F r g c h e t spaces E and F ( s . [31]

and f o r (S1) A p i o l a [ 2

] ) . 11 (I1 5 11 Il2 5 ... llyll," = s u p 1 / y ( x ) I :

denotes a fundamental system of seminorms on E o r F r e s p . , Jlxllk 5 11 t h e dual norm o f

*

(S1)

/Ixllm llYllk

*

(S2)

Vp

I n [31],

Thm. 3.7.

then

s

*

(S1)

*

( /Ixlln llYllK +

3no,k VK,m

1.9. Theorem: I f

I n [13]

5

for y c

E’ o r F ' .

3n,S Vx E E, y c FL

3no Vw 3k VK,m

*

/I / I k

3,s

Vx E E,

*

llYllw

IIXII

1

y CFA

i n c o n n e c t i o n w i t h 3.9. i t i s shown:

E is countabZy normable o r F r e f l e x i v e and one of them nuclear, =

1 E x t (E,F)

i t i s shown t h a t

( S *2 )

=

0

*

(S2)

3

.

i s necessary and s u f f i c i e n t f o r E x t 1(E,F) = 0 i n t h e

case o f Kothe spaces. A necessary and s u f f i c i e n t c o n d i t i o n f o r E x t 1(E,F) = 0 i n t h e g e n e r a l case i s c o n t a i n e d i n [33]. F u r t h e r r e s u l t s on s p l i t t i n g r e l a t i o n s , g e n e r a l i z a t i o n s and i n v e s t i g a t i o n s o f 1 c o n d i t i o n s f o r an e x a c t sequence t o s p l i t i f E x t (E,F) # 0 a r e c o n t a i n e d i n [1

, ~ 1 8 1[XI, ~ [XI.

1,

2. I n t h i s s e c t i o n a l l spaces a r e assumed t o be n u c l e a r . I n s t e a d o f p r e s e n t i n g d e t a i l s o f t h e r a t h e r c o m p l i c a t e d p r o o f o f Thm. 1.9. we d i s c u s s w i t h f u l l p r o o f s

1

an example which i m m e d i a t e l y l e a d s t o t h e most i m p o r t a n t cases o f E x t (E,F) = 0 w i t h r e s p e c t t o a p p l i c a t i o n s . I t shows i n a v e r y n i c e way t h e i n t e r p l a y between t h e s t r u c t u r e t h e o r y o f ( s ) as developed i n [231,

[34]

( c f . a l s o [5],

[24],

[35])

and t h e p r e s e n t t h e o r y . The f o l l o w i n g lemma 2 . 1 . i s i n an e q u i v a l e n t f o r m (s. Thm. 1.8.) c o n t a i n e d i n [23],

1.5..

We g i v e a d i r e c t p r o o f u s i n g t h e m a t r i c e s o f t h e

r e s p e c t i v e maps. 2.1. Lemma:

E x t 1( s , s ) = 0

proof: We a p p l y 1.3. o r 1.5.

t o t h e fundamental system

of i n j e c t i v e Banach spaces. L e t Ak E L(s,Fk), i s r e p r e s e n t e d by a m a t r i x (ai!j)v,j

~

with

k = 1,2,

...

be given. Each Ak

360

D. Vogt

f o r some n = n ( k ) , d = d ( k ) . We have t o determine m a t r i c e s (b(k!) ;J - b(k! f o r a l l k,v,j. V,J

s a t i s f y i n g analogous e s t i m a t e s such t h a t

ack! = b ( k + f ) v,J V,J

We p u t D ( l ) = d ( l ) , D ( k + l ) = max ( d ( k + l ) , 2 D ( k ) + k

+

3 ) and N ( l ) = n ( l ) ,

N ( k + l ) = max ( n ( k + l ) , 2 N ( k ) ) f o r a l l k. We w i l l determine i n d u c t i v e l y m a t r i c e s ( u ( ~ ! ) f o r k = 2 , 3 , k = 1,2, (1)

...

lUiyI v;pj

V

with

,J

...

and (v!;)

z ( k - 2 ) j 5 2-k 2kj

p(k)+N(k)v

1 k,v,j. We s t a r t w i t h v ( ' ) = 0 f o r a l l v , j . V ,J

1, =

t(v,j)

I1 = t(v,j)

: D(k)

+

L e t vi:j

N(k)v

E I.

we o b t a i n

k

: D(k) + N(k)v + k

Hence ( 3 ) i s f u l f i l l e d by d e f i n i t i o n . For (v,j)

+

be determined. We d e f i n e

+ 2

+

< j}

2 2 j }

for

361

Continuous linear maps between Frhchet spaces

~

-

21+D( k)+N( k ) v + j 22D(k)+2N(k)v+k+3

5 2D ( k + l ) + N ( k + l ) v which p r o v e s ( 1 ) and (2) Now we d e f i n e

The second e q u a l i t y comes f r o m ( 3 ) . The s e r i e s converges because o f ( 1 ) . We o b t a i n t h e f o l l o w i n g e s t i m a t e :

f o r a p p r o p r i a t e C = C ( k ) and a l l k , v , j .

E N d e f i n e s a map Bk E L(s,Fk).

Hence (bi!j)v,j

We f u r t h e r have because o f ( 3 )

which means t h a t

pk+l,k

0

Bk+l - Bk = Ak

.

From 1 . 5 . and 1.7. we g e t i m m e d i a t e l y

2.2. Theorem:

1 E x t (E,F) = 0.

I f E i s a subspace o f

s and F a q u o t i e n t space of s t h e n

F o r t h e f o l l o w i n g r e s u l t we use t h e method o f [23], t h e remark a t t h e end o f p r o o f o f S a t z 1.7.

2.3. Lemma:

rf

Extl(E,s)

= 0 then E

u: There e x i s t s a n e x a c t O - s - s ~ s - ~0

Satz 1.7.,

i t i s indicated i n

.

i s isomorphic t o a subspace of

sequence ([23]

, 1.6.)

S.

362

D. Vogt

set E

According t o T. and Y . Komura's theorem ( [ l l ] ) =

-1 E and o b t a i n an e x a c t sequence q

we can assume E imbedded i n sN. We

N

O+s+E+E+O which s p l i t s by assumption (and Thm. 1.8.). The E x t ' ( s , - )

Hence we have an imbedding E

case i s a l i t t l e b i t more d i f f i c u l t .

N

+

E

+

s.

The p r o o f uses t h e method f i r s t

used i n [3q i n t h e s l i g h t l y changed f o r m o f [24]. 2.4.

Lemma:

?

F is isomorphic to a quotient space of s.

= 0 then

We c o n s i d e r F as imbedded i n s',

Proof: as

If E x t 1(s,F)

N

c a l l Q t h e q u o t i e n t space and o b t a i n Q

i n t h e p r e v i o u s p r o o f . kle g e t t h e f o l l o w i n g diagram

0

0

Since

5

1

c s we g e t f r o m E x t (s,E)

H

second row s p l i t s which g i v e s

= 0 and =

E

.:+a

1.7. t h a t E x t 1(Q,E)

= 0. T h e r e f o r e t h e

The f i r s t column o f t h e diagram above g i v e s t h e f i r s t row o f t h e diagram below. Our s t a n d a r d e x a c t sequence g i v e s t h e r i g h t column. The r e s t i s c o n s t r u c t e d a l o n g t h e same l i n e as above

.

0

0

t

9

O + s + H + s N + O

t

t

t

T

O + S + G + S S

S

t

t

0

0

+

O

363

Continuous linear maps between Frkhet spaces

E

Because o f 2.1. t h e second row s p l i t s . Hence H

s

G

61s

@

Q i s a quotient o f

2’ s and t h e r e f o r e a l s o E.

Now we have a l l i n g r e d i e n t s f o r t h e f o l l o w i n g theorem which d e s c r i b e s t h e e x a c t

1

s o l u t i o n c l a s s e s o f E x t (E,s)

1

= 0, E x t (s,E)

= 0. Remember t h a t a l l spaces i n t h i s

s e c t i o n a r e assumed t o be n u c l e a r .

2.5. Theorem: Ial E x t 1(E,s) = 0 if and o n l y if E is isomorphic to a subspace of s. Ibi Extl(s,E) = 0 if and onZy if E is isomorphic to a quotient space of 5. The c l a s s e s of subspaces and q u o t i e n t spaces o f s have been d e s c r i b e d i n [23J,

[34]

by t o p o l o g i c a l l i n e a r i n v a r i a n t s ( D N ) and (9). I t i s i n t e r e s t i n g w i t h r e s p e c t t o t h i s t o compare Thm. 2.5. w i t h [31], 5 4 where we d e s c r i b e e.g. t h e c l a s s e s 1 1 { E : E x t (E,s) = 0 1 and { F : E x t (s,F) = 01 by t o p o l o g i c a l i n v a r i a n t s .

3. I n t h i s s e c t i o n we use t h e t h e o r y o f

5

1 t o show t h a t h y p o e l l i p t i c p a r t i a l

d i f f e r e n t i a l o p e r a t o r s w i t h c o n s t a n t c o e f f i c i e n t s have no r i g h t i n v e r s e s i n C"(Q), where o i s an open s e t i n R n . T h i s extends a r e s u l t o f Grothendieck on e l l i p t i c o p e r a t o r s ( s . [24, [ 2 2 ] ) . I t i s shown by a d i f f e r e n t p r o o f and i n g r e a t e r g e n e r a l i t y i n [32]. L e t P ( D ) be a h y p o e l l i p t i c l i n e a r p a r t i a l d i f f e r e n t i a l o p e r a t o r w i t h c o n s t a n t c o e f f i c i e n t s , Q c lRn open and P(D)-convex. We p u t

f(Q)

= { f E C"(s)

: P(D)f = O j

From t h e e x a c t sequence

we o b t a i n by ( I ) f o r any n u c l e a r F r 6 c h e t space E an e x a c t sequence 0

--f

L(E.

&a))

8

L(E,Cm(B))-

P(W* L(E,C"(a))

0

6

E x t 1 (E,fl(a))

L1

E x t 1(E,C"(B))+..

From [ Z q , 2 . 1 . we t a k e t h e f o l l o w i n g lemma. I t i s shown t h e r e under t h e assumption o f e l l i p t i c i t y b u t t h e p r o o f needs o n l y h y p o e l l i p t i c i t y . F o r t h e sake o f completeness we g i v e a p r o o f . 3 . 1 . Lemma.:

Proof:

L'

=

0.

We choose a sequence Q1 cc

a2

and denote by Hk t h e c l o s u r e o f J ( n )

c c . . . c c Q o f open s e t s such t h a t B =

1 Ek

i n C(3,)

c o n s i d e r e d as a space o f

u slk k

f u n c t i o n s on Q ~ W . e o b t a i n i n a n a t u r a l way a commutative diagram w i t h e x a c t l i n e s :

.

364

D. Vogt

0

+

Crn(Q)

n

--t

k

t L

0 L

--f

d(a)

q

cm(")

tj

C"(Qk)

k

9

Hk

+

n

-+

nHk

--f

+

fj

k

0

0 .

+

- fk)k.

and j a r e t h e i d e n t i c a l imbeddings, q i s d e f i n e d by q ( ( f k ) k ) = (fk+llek

The lower l i n e i s a c a n o n i c a l r e s o l u t i o n (see 1 . 3 . ) .

To show exactness i n t h e

upper l i n e we have t o p r o v e s u r j e c t i v i t y o f q. We even g i v e a r i g h t i n v e r s e . T h e r e f o r e we choose q k c D ( Q k ) , ((fk)k) = (

qlk

k v=l

-

qVfv

= 1 on 52 k - l fk)k

( Q :~= pl

) and p u t

*

n

T h i s d e f i n e s a c o n t i n u o u s , l i n e a r map R :

k

C"(Q,)

-+nCrn(Qk). k

Since

on Q~ we have q o R = i d . A p p l i c a t i o n o f ( I ) g i v e s t h e f o l l o w i n g commutative diagram w i t h e x a c t l i n e s

... + n

*

L n

L(E,Crn(Gk))

k

... + n k

L(E,Cm(Qk))

6O E x t 1(E,Crn(Q))

--+

-

tj*

’ 6

L(E,Hk)

...

-+

?L1

Extl(E,

. d(~)) 0 +

q * i n t h e upper l i n e i s s u r j e c t i v e s i n c e q has a r i g h t i n v e r s e t h e r e . Hence ’ 6 = 0 i n t h e upper l i n e and t h e r e f o r e

L~

o ’6

= 0 where 6'

(lower l i n e ) i s s u r j e c t i v e .

T h i s proves t h e r e s u l t .

I f P f D ) has a r i g h t i n v e r s e , t h e n c l e a r l y P(D)* i s s u r j e c t i v e f o r e v e r y F. Hence 1 E x t ( F , ~ ( Q ) ) = 0 f o r e v e r y F, i n p a r t i c u l a r f o r F = w : = (I;' . 3.2. Lemma:

If H is a Fr6chet space such that E x t 1 (w,H)

Banach space f o r every continuous seminorm

=:

=

0 then H/ker

1)

11 i s

a

I1 II on H.

1 Since t h e assumption i m p l i e s E x t (w,H/L) = 0 f o r any c l o s e d subspace L c H i t s u f f i c e s t o show: If H i s a F r i k h e t space w i t h c o n t i n u o u s norm such t h a t 1

E x t (w,H) = 0 t h e n H i s a Banach space. We may assume t h a t

11

5

11 112

5

...

i s a fundamental system o f c o n t i n u o u s norms

on H. We c a l l Hk t h e c o m p l e t i o n o f t h e normed space (H,ll (for k

3

1),

and P ~ :, Hk ~

-+

H,

1 ) t h e c a n o n i c a l e x t e n s i o n o f t h e i d e n t i t y . Then we have t h e c a n o n i c a l

resolution

365

Continuous linear maps between Frkhet spaces

where q ( ( x k ) k )

=

(Pk+l,k

-

'k+l

.

'k)k

If H i s n o t a Banach space t h e n f o r each k pk,l

ak f H. Otherwise we would have

>

1 we can f i n d ak

C

Hk such t h a t

Hk = H f o r some k which, by t h e c l o s e d

pk,l

graph theorem, would i m p l y t h a t H i s i s o m o r p h i c t o t h e Banach space Hk/ker okk,l

(c1, c2,

.

Then by E~ az, ... ) c n H k k' assumption f o r A E L(w, n Hk) t h e r e e x i s t s B E L ( w , n Hk) such t h a t A = q o B. We k p u t B e = (bl,j, bz,j, ... ) where e j = ( 6k , j )k and ko b t a i n : j For 5 =

...) E

w

we p u t A(5) =

bk + l , j

'k+l,k

al,

f o r k > j and k

bk,j

=

(cl

=.

j

=

0 for j

- b. .

(2)

a j= o j + l , j

(3)

F o r each k we have j k such t h a t b

bj+l,j

j.j

k ,j

2 jk

( 1 ) i m p l i e s t h a t t h e r e i s b . F H such t h a t b = b . f o r a l l k 2 j . I f we choose J k,j J 3 j,, we have because o f ( Z ) , t h e second p a r t o f (1) and ( 3 )

j

P

~

a ,j =~ b.

J

-

'j,l

b

. = b . - b = b . E H J 1,j J

j,J

which c o n t r a d i c t s t h e c h o i c e of

a

j'

We r e t u r n t o o u r h y p o e l l i p t i c o p e r a t o r P ( 0 ) . I f i t has a r i g h t i n v e r s e i n C"(Q) 1 we know t h a t E x t ( w . ~ ( s ? ) ) = 0 . We choose a compact K c M w i t h non empty i n t e r i o r

Ilfll = su I f ( t ) 1 . Then ~ ( Q ) / k e rI1 112 d ( Q ) IK . i s a n u c l e a r Banach space, hence f i n i t e i i m e n s i o n a l . T h i s i s p o s s i b l e o n l y f o r

and a p p l y Lemma 2 t o t h e seminorm

c-R

n = 1. T h e r e f o r e we p r o v e d t h e main r e s u l t o f t h i s s e c t i o n :

3.3.

Theorem:

If n

in c - ( Q ) .

2

2 and P(D) is hypoeZZiptic then P(D) has no r i g h t inverse

F o r t h e e l l i p t i c case t h i s r e s u l t was f i r s t p r o v e d by Grothendieck ( s . [21J, Appendix C o r [22],

p . 40 f ) . Theorem 3 i m p l i e s t h a t a l s o p a r a b o l i c o p e r a t o r s

( t h e h e a t e q u a t i o n e.g.)

have n o r i g h t i n v e r s e s . F o r h y p e r b o l i c o p e r a t o r s i t i s

known t h a t t h e y have r i g h t i n v e r s e s a t l e a s t i n C m ( W n ) ) .

4 . I n Grothendieck [83, I1 § 4 and a l s o i n [27]

r e l a t i o n s between Kothe m a t r i c e s

r e s p . F r c c h e t spaces a r e i n v e s t i g a t e d which l e a d i n c o n c r e t e examples t o r e s u l t s very s i m i l a r t o t h e r e l a t i o n

1

E x t (E,F) = 0.

366

D. Vogt

We explain,up t o some extent,the connection. In p a r t i c u l a r we show how the res u l t s on E x t 1(E,F) = 0 can be used t o solve a problem o f Grothendieck i n [ 8 ] , 11,

5

4, p . 118 ( s . Thm. 4.6. below) and t o complete h i s i n v e s t i g a t i o n of the r e l a t i o n E b Gfl F i n the case t h a t one of t h e spaces i s a power s e r i e s space s a t i s f y i n g a c e r t a i n s t a b i l i t y condition. I n f a c t we get a r a t h e r complete answer. A more systematic approach can be found in 1331. The crucial lemma 4 . 4 . i s a s p e c i a l ized version of a r e s u l t proved t h e r e . For spaces with basis see [13].

We assume E

= h"(A),

F

= h(B), E

a Schwartz space and p u t P = E b

Gn

F. P can be

considered as a space of double indexed sequences o r of matrices. The dual P ' can be i d e n t i f i e d with the space of a l l matrices belonging t o bounded l i n e a r maps from F t o E. By P*we denote t h e Kothe dual o f P , i . e . t h e space o f a l l matrices v = (v. .). such t h a t 1,J 1 , j E N v. . j

i;j

1

,J

<

+

"

f o r a l l u = ( uI. , J. ) i , j c l N E P . E, means t h e s e t of nonnegative real sequences i n E = k(A). 4.1. Theorem (Grothendieck): The following are equivalent: ( a ) P is bornological

ibi icj

(di

P is barrelled P ' = P* Vno 3no Vm,

h c E+ 3 R

0 Vi,j

:

I t i s easy t o see t h a t we can w r i t e down (Si) in t h e following way:

*

(S2)

Vno 3mo Vm 3 n , R >

0 Vi,j :

Both conditions on the matrices a r e obviously r e l a t e d . We have even (see [13)):

*

4.2. Proposition: Conditions 4.1. ( d i and (S2) are equivalent.

A t l e a s t f o r one p a r t of 4.2. we shall now give a proof not involving any special assumptions on E o r F, in p a r t i c u l a r they a r e not assumed t o be Kothe spaces.

W e recall t h a t f o r any complete l o c a l l y convex space X t h e following implications

367

Continuous linear maps between Frkchet spaces

hold:

X bornological

3

X b a r r e l l e d = X b s e q u e n t i a l l y complete.

We s h a l l show t h a t s e q u e n t i a l completeness o f

*

hV

P i , P = Lb(E,F) o r P = E '

implies (S2). F o r t h e p r o o f which has some s i m i l a r i t y w i t h t h e p r o o f o f Theorem 7 i n

[ 41

F

we

use t h e f o l l o w i n g n o t a t i o n : A p r o j e c t i o n P i n F i s c a l l e d 12 - a d m i s s i b l e i f t h e range o f P i s f i n i t e dimensional and

z x J.

Px =

j

y.(x) J

with y. t F'

.

4 . 3 . Lemma:

( S 2 ) i s equivalent t o the following condition: For euery

J

P

*

there

e x i s t no.k such t h a t f o r a22 K,m there e x i s t n,S and a F-adniissible p r o j e c t i o n P with

f o r a l l x E E and y E ( k e r P ) '

F.

Proof:

We p u t

f o r j = cl,k,K

Q

* 1 lw

e t c . denote the dual norms i n ( k e r P ) ' .

K. With C such t h a t 11Qx11. * J C I y IJ. f o r t h e s e j .

= i d - P and assume IJ. < k <

we have

*

I y l j 5 lly

*

o Qllj

5

5

Cllxllj

We o b t a i n

for a l l x

c

E, y E ( k e r P ) '

IJ.'

hence f o r a l l y E F '

IJ.'

S i n c e P has f i n i t e dimensional

range an analogous i n e q u a l i t y w i t h Q r e p l a c e d by P i s t r i v i a l . We add t h e s e i n e q u a l i t i e s and observe t h a t Ily o it

llyllj f o r

j = p,k,K.

*

QIIj +

*

1Iy o P l l j i s e q u i v a l e n t t o

T h i s proves t h e a s s e r t i o n .

4.4. P r o p o s i t i o n : If P b i s sequentially complete f o r P=Lb(E,F)

or P = E i

&,,

F,

then E and F s a t i s f y condition ( S 2 ) .

Proof:

*

Under t h e assumption t h a t ( S 2 ) does n o t h o l d we c o n s t r u c t d o u b l e indexed

sequences x

k ,n

i n E and y

k ,n

i n F ' such t h a t t h e s e r i e s

k$n < Y k , n ' ' P X k , n ) converges u n i f o r m l y on e v e r y e q u i c o n t i n u o u s s e t o f maps t h e map A E L(F,E)

d e f i n e d by

cp

c.L(E,F)

and such t h a t

368

D. Vogt

i s n o t bounded. S i n c e t h e e q u i c o n t i n u o u s subsets o f L(F,E) a r e t h e bounded s e t s i n Lb(E,F),

since

f u r t h e r m o r e t h e c a n o n i c a l map E ' F + Lb(E,F) i s c o n t i n u o u s and under t h e I -b n, c a n o n i c a l i d e n t i f i c a t i o n s (Eb @n F ) corresponds t o t h e bounded maps i n L(F,E) ( i f t h e i r image i s c o n t a i n e d i n E c E " ) t h i s proves t h e a s s e r t i o n .

W e a p p l y lemma 4 . 3 .

Then by assumption we have p (we can assume IJ. = 1) such t h a t

f o r e v e r y k ( w i t h no = k-1) t h e r e e x i s t K (we can assume K = k + l ) and m (we can assume m = k ) such t h a t f o r a l l n and C and f o r a l l 1 - a d m i s s i b l e p r o j e c t i o n s P we have x E E, y E ( k e r P ) ; w i t h

We choose a b i j e c t i o n

on

V.

For

v

=

v

+

( k ( v ) , n f v ) ) f r o m D\1 o n t o h’

a c c o r d i n g t o ( * ) . We choose a

k ,n

E F w i t h llak,nlJk

L e t x ~ , ~Y ,~ , ~ ak,n , be chosen f o r k = t h a t (*) i s s a t i s f i e d and

W e p u t ~ ( x =) put k =

x

IN

and s e t up an i n d u c t i o n

1 and k = k ( l ) , n = n ( 1 ) we p u t C = 8, P = 0 and choose x ~ , ~yk,n ,

,dl

k,l

1, Yk,n(ak,n)

n = n ( p ) and p

k(p),

= 6

5

6

n,m

=

1,

=

1.

...

,V

such

'

x ( x ) . P i s a 1 - a d m i s s i b l e p r o j e c t i o n . We k ( P ) ,niIJ.) Y k b ) ,n(lL) k ( v + l ) , n = n ( v + l ) , Q = i d - P and choose Co such t h a t IlQxll. 5 Co llxllj,

f o r j = l,k,k+l.

Now we s e t C = Co 2k+n+1 and f i n d x

a c c o r d i n g t o (*), i . e . such t h a t w i t h y

k ,n =

yk,n o

k ,n

Q

E E,

yk,n c

J

(ker P)

l

,.W e choose ak,n E k e r P = i m Q such t h a t /lak,nllk = 1 and yk,n (ak,n) = Yk,n ( a k y n ) = 1.

369

Continuous linear maps between Frechet spaces

Every e q u i c o n t i n u o u s subset o f L(E,F) i s c o n t a i n e d i n one o f t h e f o r m

{v

6 L(E,F)

and C ( k )

B

=

:Ilqxilk 5 C ( k ) ~ ~ x l ~ n f( ok r) a l l k } w i t h i n c r e a s i n g sequences n ( k ) i n

z 0.

Given B we p u t 1 = n ( l ) , rn

=

n ( l + l ) and o b t a i n f o r

ip

N

E B

S i n c e t h e e s t i m a t e i s termwise, t h e s e r i e s on t h e l e f t hand converges u n i f o r m l y on B. F o r A E L(F,E) as d e f i n e d above we have A ak,n

=

x

~ hence , ~ IIA ak,nllk

= n while

Jlak,nllk = 1 f o r a l l k and n. T h e r e f o r e f o r e v e r y k t h e s e t { A x : llxllk 5 11 i s

n o t bounded, i . e . A i s n o t bounded.

1 S i n c e a l l t h e necessary c o n d i t i o n s f o r E x t (E,F)

it

=

0 i n 1311 a r e based on ( S p )

t h e y a l l a r e necessary c o n d i t i o n s f o r t h e b o r n o l o g i c i t y o r b a r r e l l e d n e s s o f Lb(E,F).

I n s t e a d o f a converse o f Prop. 4 . 4 .

wecomplement i t by a s u f f i c i e n t

condition.

4.5. P r o p o s i t i o n :

(B), then Lb(E,F)

If E and F are nuczear, E has property ( D N ) and F has property I an F is bornoZogica2.

= Eb

d .,

prooS: From G r o t h e n d i e c k ' s r e s u l t [81,11,§4,n03, we know t h a t Lb(s,s) i s b o r n o l o g i c a l . From [24], sequences

and 4 . 5 . ,

Prop. 3.3.

Cor. 2

( c f . 4.1. and 4.2.)

by a d d i n g complements we o b t a i n e x a c t

L

O + E + s + G - 0 q

O+H-s-F+O where G and H a r e complemented subspaces o f s . Since E x t 1(E,H) = 0 and E x t 1( G , s )

= 0 we o b t a i n t h a t t h e map

370

D. Vogt

i s s u r j e c t i v e . By [ 8 ] , I , Hence E ' b

6n

5

1, no 2, Prop. 3

L*

8 q i s a t o p o l o g i c a l homomorphism.

F = Lb(E,F) i s b o r n o l o g i c a l .

L e t V be a complex a n a l y t i c m a n i f o l d . I n [ 8 ] , 11, that Lb(X(V), %(V))

i s b o r n o l o g i c a l if V =

0

4, p . 128 Grothendieck s t a t e s

Ic, whereas f o r V

=

D (unit disc) i t

i s n o t . He asks f o r a c l a s s i f i c a t i o n o f complex m a n i f o l d s w i t h r e s p e c t t o l i n e a r t o p o l o g i c a l p r o p e r t i e s o f X ( V ) . Now f o r t h e p r e s e n t case we can g i v e a complete answer.

4.6. Theorem: The following are equivaZent:

i s bornoZogicaZ i s barrelzed

(1)

L ~%(V), (

(2)

Lb(X(V), & ( V ) )

(3)

g ( V ) has property (DN)

(4)

every pZurisubhamonic f u n c t i o n on V,

&(V))

which i s bounded from above i s constant

(strong LiouvilZe p r o p e r t y ) .

Proof:

We have (1) = 1 2 )

=

Pt; s e q u e n t i a l l y complete f o r P = L b ( & ( V ) , & ( V ) ) .

t h i s implies (S2) f o r E = F

4.4.

=a(V). Since

on (S;) we know f r o m t h a t theorem t h a t ( 3 ) = ( 1 ) f o l l o w s f r o m 4.5.

x(V)has

[3l],

Thm. 7.2.

By

i s c o m p l e t e l y based

(DN).

s i n c e x ( V ) i s n u c l e a r and has p r o p e r t y

( a ) ( c f . [31]

§ 7,B). The e q u i v a l e n c e o f ( 3 ) and ( 4 ) i s shown i n L371. B e f o r e we t r e a t t h e case o f power s e r i e s spaces we need some i n f o r m a t i o n about t h e c o n n e c t i o n between t h e r e s u l t s o f [27]

and t h e p r e s e n t paper. By LB(F,E) we

denote t h e space o f a l l bounded l i n e a r maps f r o m F t o

E, i . e . t h o s e maps which

send some neighbourhood o f z e r o i n t o a bounded s e t . We o b t a i n f r o m [27],

1.3.

and 1.4.

4.7. Theorem: If E = x"(A) o r F = X(B) then the foZZowing are e q u i v a l e n t : (1)

L(F,E) = LB(F,E)

(21

f o r every sequence K(N) t h e r e e x i s t s K such t h a t f o r each n we have

and C w i t h

f o r alZ x f E, y

F'.

No

371

Continuous linear maps between Frtichet spaces

One p a r t o f

[Zq,

7.3. says t h a t a t l e a s t f o r r e f l e x i v e E t h e e q u a l i t y L(F,E)

i m p l i e s t h a t E b @n

LB(F,E)

=

F, hence a l s o E L &n F i s b a r r e l l e d .

We r e c a l l t h a t a F r g c h e t space F has p r o p e r t y

( 5 )i f

t h e f o l l o w i n g h o l d s ( s . [36]

,

c2q 1: (5)

*2 * * VY E F ' : I l ~ l l K5 C IIYIIL IIyllk.

Vk 3K VL 3C > 0

One c o n n e c t i o n between

[21

and t h e p r e s e n t paper i s g i v e n by t h e f o l l o w i n g lemma

w h i c h e x p l a i n s i n a v e r y s a t i s f a c t o r y way t h e c o i n c i d e n c e o f t h e c o n d i t i o n s i n [31]

, 4.2. and [ 2 n , 4.3. as w e l l as i n [31],

4.3.

( r = l ) and [27],

2.1. a t l e a s t

f o r nuclear iil(u).

(L)then

4.8. Lemma: If E = X m ( A ) or F = X(B) and F has property L(F,E)

=

Proof:

*(ST) y i e l d s n o .

f r o m (S1)

LB(F,E).

(S;)

impZies

F o r a g i v e n sequence K(N) we p u t p = K(no) and o b t a i n

a number k such t h a t f o r a l l K and m we have n and S w i t h

f o r a l l x E E, y E F ' . F o r k we choose K (S;),

a C

3

k according t o

0 such t h a t

(L)and

o b t a i n f o r L = K ( n ) , n = n(K,m)

as i n

for all y E F'. We o b t a i n f o r x E E, y E F ' :

*

I I X I I ~IIYIIK

5

(C

*

+

S ) max (lIxlln I I Y I I K ( ~ ,lIx/I )

*

/ I Y I I K ( ~ ~1 )

which p r o v e s t h e a s s e r t i o n .

*

*

A s i m i l a r p r o o f shows t h a t (S1) and ( S 2 ) a r e e q u i v a l e n t i f E has p r o p e r t y (DN).

Remark:

I n t h e f o l l o w i n g theorem we assume f o r ( 3 ) , a.and

p. t h a t

SLPI

an+,/

an < +

m

372

D. Vogt

and f o r ( 3 ) , y. t h a t l i m / an = 1 ( s t a b i l i t y assumptions). n A F r 6 c h e t space F has p r o p e r t y (E) i f t h e f o l l o w i n g h o l d s ( s . [27] ; [31] ) :

(E)

Vk,

E

> O

3K VL

Obviously property

3C > O

(n) i m p l i e s

*ltE

Vy E F ' : IIYIIK

C-

*

+€

C IIYIIL IIYIIk

.

('5').

The theorem extends ( i n t h e n u c l e a r case) Prop. 15 i n 181, 11, 54, n03 which g i v e s t h e case a. f o r E a Kothe space. 4.9.

Theorem: If E and F are nuclear and one o f them a power s e r i e s space

satis-

f y i n g our s t a b i l i t y asswlrptions, t h e n t h e following a r e e q u i v a l e n t : (1)

Lb(E,F) i s bornological

(2)

Lb(E,F) i s b a r r e l l e d

( 31

a . i f F = h r ( a ) : E has property (DN) p . i f E = AJu)

: F has property ( Q )

y . i f E = A l ( a ) : F has property

As i n 4.6. we have (1)

proos: 4.1.,

4.2.,

*

3

(2) = (S2).

(1) f o l l o w s f r o m 4.5.,

3

S i n c e t h e n e c e s s i t y p a r t s o f thms.

a r e based on ( S z ) , t h e s e theorems y i e l d ( 2 )

4.3. i n [31]

t h e cases and p. ( 3 )

-

(5).

5

(3). I n

i n t h e case y . c o n d i t i o n (S;)

is

s a t i s f i e d as we know from [3], 4.2. t h e s u f f i c i e n c y p a r t o f w h i c h i s based on (S1).

4.8. y i e l d s L(F,E) = LB(F,E), which i m p l i e s b a r r e l l e d n e s s o f E L 6~~F Lb(E,F) ( s . [2g, 7.3.). F i n a l l y Theorem 14.3. b i n [ 8 ] , 11, 5 4 no 2 t e l l s t h a t i n t h i s case ( 2 )

3

(1).

5 . We want t o d e t e r m i n e t h o s e F r 6 c h e t spaces E and F which can o c c u r i n a non-

1

t r i v i a l way i n t h e r e l a t i o n E x t (E,F)

= 0. We s h a l l use t h e t h e o r y o f 131).

t h e main t o o l i s an i n v e s t i g a t i o n o f t h e r e l a t i o n s (S1)

*

Hence

and ( S 2 ) .

F o r a g i v e n fundamental system o f seminorms i n F we denote by Fk t h e c a n o n i c a l Banach spaces. The dual o f Fk can be i d e n t i f i e d w i t h t h e space F;C, i t s norm with

[3]

/I . ;1

We use t h e f o l l o w i n g c o n d i t i o n i n t r o d u c e d by B e l l e n o t and Dubinsky

t o c h a r a c t e r i z e t h e Fr'echet spaces which have n u c l e a r Kothe q u o t i e n t s w i t h

c o n t i n u o u s norm

(*I

3w Vk 3K :

SUP

*

{llyIJk : y E F;L

*

, I J y / J5K 1)

=

+

-.

T y p i c a l examples f o r spaces n o t s a t i s f y i n g (*) a r e p r o d u c t s o f Banach spaces o r , more genera1,quojections s a t i s f y (*) i s

W.

(s. [3]).

The o n l y F r 6 c h e t Monte1 space which does n o t

373

Continuous linear maps between Frichet spaces

5.1. P r o p o s i t i o n :

If F does n o t s a t i s f y ( * ) then (S1) i s s a t i s f i e d f o r every

FrQchet space E. The

proof

i s obvious.

W i t h r e s p e c t t o t h e f o l l o w i n g a n a l y s i s (Lemma 5.3. and Thm. 5 . 5 . )

the p r o p o s i t i o n

*

means t h a t f o r F n o t s a t i s f y i n g ( * ) we cannot g e t any i n f o r m a t i o n on E f r o m (S1)

*

.I).

o r f r o m ( S 2 ) . However i t i s o n l y known under a d d i t i o n a l assumptions t h a t (S1)

1

p l i e s E x t (E,F)

= 0 ( s e e e.g.

Thm. 1.9. o r

139,

5

im-

3).

I f we r e s t r i c t o u r a t t e n t i o n m a i n l y t o t h e cases where e i t h e r E o r F i s c o n t a i n e d i n t h e c l a s s o f n u c l e a r spaces t h e n t h e case where F i s n u c l e a r causes no d i f f i -

*

1

c u l t y s i n c e i n t h i s case (S1)

i m p l i e s E x t (E,F)

= 0.

I n t h e o t h e r case we can g i v e a p r e c i s e c h a r a c t e r i z a t i o n o f t h e spaces F f o r which 1 E x t (E,F) = 0 f o r e v e r y E. F o l l o w i n g B e l l e n o t and Oubinsky we c a l l q u o j e c t i o n s t h e F r g c h e t spaces F where F / k e r norm

11 11.

131

11 11

i s a Banach space f o r any c o n t i n u o u s semi-

These a r e t h e spaces o c c u r i n g i n Lemma 3 . 2 . .

From Lemma 3 . 2 . and an

argument i n [ 3 g , we o b t a i n : 5.2. Theorem: 111

The following are e q u i v a l e n t f o r a FrQchet space F:

(31

F i s a quojection 1 E x t (w,F) = 0 E x t 1(E,F) = 0 f o r every nucZear FrQchet space E.

From

131

(2)

we know t h e f o l l o w i n g . I f F i s a q u o j e c t i o n t h e n i t does n o t s a t i s f y

( * ) . The converse i s t r u e f o r r e f l e x i v e F. I t i s unknown i f i t i s t r u e always, i.e.

if

F i s a quojection

i f f i t does n o t s a t i s f y ( * ) . Hence we do n o t know i f

( f o r E r e s t r i c t e d t o t h e c l a s s o f n u c l e a r spaces) 5.1. B u t 5.1.

says t h a t 5.5.

and 5.5. c o v e r e v e r y t h i n g .

(1) i s o p t i m a l as l o n g as we use c o n d i t i o n s (S1)

and (S;)

f o r our analysis. W h i l e i n t h e p r e c e d i n g we d i s c u s s e d cases where we cannot g e t any i n f o r m a t i o n on Y .I). 1 E from (S1) r e s p . ( S 2 ) o r f r o m E x t (E,F) = 0, o u r main o b j e c t i v e i s now t o i n -

v e s t i g a t e t h e case o f spaces w i t h (*). We need t h e f o l l o w i n g g e n e r a l i z a t i o n o f (ON) ( c f . [Z],

(O,+-)

--f

[34]). L e t

(O,+-),

cp

always denote a s t r i c t l y i n c r e a s i n g c o n t i n u o u s f u n c t i o n

l i m cp(r) = r=-

(DNcp ) 3no ’dm

+

m.

3, C

b'x

c

E, r > 0 :

374

5.3.

D. Vogt

( ONcp) f o r some

cp

.

We choose

Proof:

*

( S E ) then E has property

I f F s a t i s f i e s (*) and E and F s a t i s f y

Lemma:

)I

*

a c c o r d i n g t o ( * ) and o b t a i n f r o m (S1) no and k . ( * ) a g a i n

y i e l d s K and p u t t i n g t h i s i n t o (S;)

we f i n a l l y o b t a i n : F o r e v e r y m t h e r e e x i s t s

n and S such t h a t f o r a l l x E E and y f 0 E F ' we have:

*

*

E F ' such t h a t

We choose a sequence y,

and a f u n c t i o n

for all v

cp

c IN.

I-L

P

such t h a t f o r a l l S

3.

0 we have C

7

0 with

For

we o b t a i n lIxllm c

C'P ( r ) Ilxll

+

1

.

llxlln

0 which proves

By i n c r e a s i n g C i f necessary we have t h i s i n e q u a l i t y f o r a l l r (ONcp)

5.4.

*

Lemma:

I f E has property (DN ) f o r some cp

then there e x i s t s a nuclear

cp

it

Kiithe space k ( B ) with continuous nomi such t h a t (S1) i s s a t i s f i e d .

Proof: $(l)

=

-

We choose a s t r i c t l y i n c r e a s i n g f u n c t i o n $ : (0,tm)

1, l i m $(r) =

for all v

r-te-

c N.

+

+ (O,t-),

$(r)

5 r,

such t h a t

Moreover we choose a sequence p .

>

J -

1 such t h a t 2

J

-1

p. J

< t

-.

Cr

r

r

375

Continuous linear maps between Frkhet spaces

We p u t b. J9

1

= 1

for all j

+

bj,*

= $

-1

o

-1 ..o+ ( p j ) f o r a l l j

( j -1)ti mes

and d e f i n e b

j,k

f o r k > 2 i n d u c t i v e l y by

We o b t a i n f o r j 2 k b. J'k+l b j,k

$ o

=

...

(bj,2)

Q $

=

$

-1

... 0 ~r-1

(pj)

( j - k ) t imes

( k - 1) t i mes

Hence x(B) i s n u c l e a r . I t has c o n t i n u o u s norm.

*

To e s t a b l i s h (S1) we choose no such t h a t f o r a l l m we have n,C w i t h

(1)

llxllm

5

Cq (r)

llxll

+ "0

1

Ilxll,

f o r a l l r > 0 and x C E. F o r g i v e n p we p u t k = p + 1 and o b t a i n f o r a l l K > k and l a r g e j

I n (1) we p u t r =

hence

bj'Kand o b t a i n f o r l a r g e j

5

376

D. Vogt

f o r a l l j with appropriate S Lemma 5.3. and 5.4.

>

0, which proves t h e a s s e r t i o n ( c f . 4 . 2 . ) .

t o g e t h e r g i v e t h e f o l l o w i n g theorems. P r o p o s i t i o n 5.1.

t e l l s us t h a t t h e r e s t r i c t i o n on

F i s natural.

5 . 5 . Theorem: The f o l l o w i n g are e q u i v a l e n t : (1)

t h e r e e x i s t s F s a t i s f y i n g ( * ) such t h a t

1 E x t (E,F)

(2)

t h e r e exists anucZear Fnot isomorphic t o

~il such

131

E has property (DN ) for some ‘p

Remark:

= 0

1

t h a t E x t (E,F)

= 0

9.

A t r i v i a l consequence i s t h a t (1) o r ( 2 ) i m p l i e s t h a t E admits a

c o n t i n u o u s norm.

To g i v e a more c a r e f u l a n a l y s i s o f t h e meaning o f ( 3 ) i n 5.5. we use t h e

following condition (s.

1281):

(**) There e x i s t s po such t h a t f o r e v e r y p 2 p,

we have a q 3 p w i t h t h e f o l l o

E which i s Cauchy w . r . t . even converges t o 0 w . r . t . II II

wing p r o p e r t y : e v e r y sequence i n

11 1)

0 w.r.t.

Po

11 I1

9

-

and converges t o

P ’

T h i s c o n d i t i o n i s e q u i v a l e n t t o E b e i n g c o u n t a b l y normable ( s .

[ 7]),

i.e. to

t h e e x i s t e n c e o f a fundamental system o f norms on E whichmakes i t i n t o a c o u n t a b l y normed space i n t h e sense o f G e l f a n d 5.6. Lemma: (1) (2)

E

-

”

Silov.

The folzowing are equivaZent :

has property (DN ) f o r some cp

cp

t h e analogue t o ( * * ) w i t h “Cauchy” repZaced by ’%bounded” hoZds.

proos:

I t i s easy t o p r o v e t h a t (1) i m p l i e s ( 2 ) . We assume ( 2 ) and show t h a t t h e

property there implies:

for a l l x

6

E,

r > 0 and an a p p r o p r i a t e f u n c t i o n cp(r) = cp

P.Po39

F o r t h i s i t s u f f i c e s t o show t h a t f o r e v e r y r > 0 t h e r e e x i s t s all x E F llxllp

5

M llxll

PO

+

r1

IlXllq

E with

M such t h a t f o r

*

We assume t h a t t h i s were n o t t r u e . Then t h e r e e x i s t s a r M we have xM F

(r).

7

0 such t h a t f o r e v e r y

377

Continuous linear maps between F r k h e t spaces

+

11 > 0. Hence we can assume 11x 11 M P M P r f o r a l l M which c o n t r a d i c t s ( 2 ) .

= 1. But t h e n Jlx

I n p a r t i c u l a r IIx /Ix (1 mq

5

11

M Po

-+

0 and

We choose cp such t h a t

v PYP0.q ( r ) rf o r a l l p,po,q

= 0

Cp(t-1 which o c c u r . I t f o l l o w s t h a t E has p r o p e r t y (DN ) . cp

I f E i s a Schwartz space t h e n f o r e v e r y p we have which i s bounded w . r . t .

'i; such

t h a t e v e r y sequence

has a subsequence which i s Cauchy w . r . t .

p. T h i s proves

t h e second p a r t o f t h e f o l l o w i n g lemma, t h e f i r s t one i s obvious f r o m 5.6. ( 1 ) I f E has property (ON ) f o r some

5.7. Lemma:

cp

then it i s countably normatile.

(2) I f E i s a countabZynomable Schwnrtz space then i t hasproperty some

cp

.

Hence theorem 5.5. 5.8. Theorem:

(DN ) cp

For a Schwartz space

E t h e fofolowing a r e e q u i v a l e n t :

t h e r e e x i s t s a Schwarta spuce (nuclear space) F not isomorphic t o 1 t h a t E x t (E,F) = 0

121

E is countably normable.

[51,

for

g i v e s f o r t h e case o f Schwartz spaces:

(1)

I n Dubinsky

.

1 6 1 and i n

[7 1 ,

w

such

[28] t h e r e a r e g i v e n examples o f n u c l e a r

(F)-spaces w i t h c o n t i n u o u s norm b u t n o t c o u n t a b l y normable which t h e r e f o r e do n o t have t h e bounded a p p r o x i m a t i o n p r o p e r t y . From t h e p r e c e d i n g we conclude t h a t t h e r e a r e a l s o examples o f n u c l e a r spaces E w i t h c o n t i n u o u s norm f o r which t h e r e 1 i s no F s a t i s f y i n g ( * ) such t h a t E x t (E,F) = 0. F o r F s a t i s f y i n g ( * ) and n u c l e a r 1 E 5.1. i m p l i e s E x t (E,F) = 0 .

1 To determine c o n d i t i o n s on F f o r t h e e x i s t e n c e o f E such t h a t E x t (E,F) = 0 we r e c a l l t h e f o l l o w i n g concept which was i n t r o d u c e d by Grothendieck [ 9 ]

: A locally

convex space F i s c a l l e d quasi-normable i f f o r e v e r y e q u i c o n t i n u o u s s e t A c F ' t h e r e i s a neighbourhood o f z e r o V i n F such t h a t on A t h e t o p o l o g y o f FA c o i n c i d e s w i t h t h e t o p o l o g y o f u n i f o r m convergence on V . Equivalently (s.

[9],

p. 107): F i s quasi-normable i f and o n l y i f f o r e v e r y

neighbourhood of z e r o U t h e r e e x i s t s V such t h a t f o r each

a >

0 there exists a

378

D. Vogt

bounded s e t B c F such t h a t V C B In

[14]

t a u .

i t i s shown t h a t a F r g c h e t space F i s quasinormable i f and o n l y i f

t h e r e e x i s t s 9 such t h a t F has t h e p r o p e r t y

(QY) Vp 3k VK 3 C V r

>

0, y

c F'

:

* " 1 * l l ~ l 5l ~C cp ( r ) l l ~ l +l ~ IMlV -

5.9.

I f E and F s a t i s f y

Lemma:

*

( S p ) and E i s a proper inon normablel Frdchet

space, then F has property (9 ) f o r some 9 9

.

If F does n o t s a t i s f y c o n d i t i o n (*) t h e n i t o b v i o u s l y has p r o p e r t y (9 )

proos:

for every

cp

cp

and hence i t i s quasi-normable.

I f i t s a t i s f i e s c o n d i t i o n (*) t h e n we know f r o m t h e remark a f t e r 5.5. admits a c o n t i n u o u s norm. We can assume t h a t a l l

that E

11 Ilk on E a r e norms. I n an

analogous way as i n t h e p r o o f o f 5.3. we show t h a t F has p r o p e r t y (Q ) f o r some

Given

9

.

cp

*

IJ-

we f i n d no and k a c c o r d i n g t o ( S p ) . We choose m

>

no such t h a t t h e r e

e x i s t s a sequence xv E E such t h a t

T h i s i s p o s s i b l e s i n c e F i s n o t normable. F o r g i v e n K and m we o b t a i n n and S such t h a t f o r a l l y E F ' and

\I

:

We c o n t i n u e as i n t h e p r o o f o f 5.3, The e x i s t e n c e o f a n u c l e a r h(A), even w i t h some a d d i t i o n a l p r o p e r t i e s , f o r g i v e n

1

F w i t h p r o p e r t y ( a ),such t h a t E x t ( X ( A ) , F ) = 0 i s i m p l i c i t l y shown i n [14]. 9 B u t s i n c e we do n o t need here t h e a d d i t i o n a l p r o p e r t i e s we g i v e a d i r e c t p r o o f . 5.10. Lemma:

I f F has property (Q ) f o r some cp then there e x i s t s a nuclear *'p

Kothe space E = h ( A ) such that (S1)

Proof:

We can assume cp(r)

2

is s a t i s f i e d .

r, 911) = 1. We choose a > 1 such t h a t Z a j j j

< t

m

379

Continuous linear maps between Frkhet spaces

and p u t a

j ,1

= 1 for all j

F o r k > 2 we d e f i n e a

j ,k

i n d u c t i v e l y by

aj,k+l = aj,k

cP(aj,k).

An i n d u c t i o n argument shows t h a t @ ( a . ) 2 Jlk

U.

J

f o r k 3 2 hence

which i m p l i e s t h a t h(A) i s n u c l e a r . The p r o o f c o n t i n u e s as t h e p r o o f o f 5 . 4 . The e q u i v a l e n c e o f ( 2 ) and ( 3 ) i n t h e f o l l o w i n g theorem i s c o n t a i n e d i n theorem 6. I t s h o u l d

c o n s t r u c t i o n o f x(A) g i v e n i n (141. The e q u i v a l e n c e o f i n Lemmas 5.9. and 5.10.

5.11. Theorem:

[lq,

be n o t i c e d t h a t h e r e we do n o t need t h e more s o p h i s t i c a t e d

(I) and

(2) i s c o n t a i n e d

For a FrBchet space F t h e f o l l m i n g are equivazent: 1 E x t (E,F) = 0

there e x i s t s a proper Frdchet space E such t h a t

(1)

(2) F has property (D ) f o r some ‘9

F i s quasi-normable

131

.

cp

References

[ I ] Ahonen, H.: On n u c l e a r Kothe spaces d e f i n e d by D r a g i l e v f u n c t i o n s , Ann.Acad. S c i . Fenn. Ser. A, I , Math. D i s s . 38 (1981) Apiola, H . :

[2

C h a r a c t e r i z a t i o n o f subspaces and q u o t i e n t s o f n u c l e a r Lf(a,m)-

spaces, t o appear i n Compositio Math. 13 ] Bellenot,S.,

Dubinsky, E.:

Fr’echet spaces w i t h n u c l e a r Kothe q u o t i e n t s ,

Trans. Amer. Math. SOC. 273, 579-594 (1982)

[4]

Dineen, S . , Meise, R.,

Vogt, D.:

C h a r a c t e r i z a t i o n o f n u c l e a r Fr’echet spaces

i n w h i c h e v e r y bounded s e t i s p o l a r , t o appear

[5

]

Dubinsky, E.:

The s t r u c t u r e o f n u c l e a r Fr’echet spaces, L e c t u r e Notes i n

Mathematics 720, S p r i n g e r 1979 I6

]

Dubinsky, E.:

N u c l e a r F r g c h e t spaces w i t h o u t t h e bounded a p p r o x i m a t i o n

p r o p e r t y , S t u d i a Math. 71, 85-105 (1981)

380 c7

I

D. Vogt

Dubinsky, E.,

Vogt,

D.: F r e c h e t spaces w i t h q u o t i e n t s f a i l i n g t h e bounded

a p p r o x i m a t i o n p r o p e r t y , p r e p r i n t 1982 Grothendieck, A.:

P r o d u i t s t e n s o r i e l s t o p o l o g i q u e s e t espaces n u c l c a i r e s ,

Mem. Amer. Math. SOC. 16 ( 1 9 5 3 )

19

1

Grothendieck, A.:

Sur l e s espaces (3) e t

(33,Summa

B r a s i l i e n s i s Math. 3,

57-122 ( 1 9 5 4 ) Ketonen, T.,

Nyberg, K . : T w i s t e d sums o f n u c l e a r F r g c h e t spaces, Ann. Acad.

S c i . Fenn. Ser. A,

I , Math. 7 , 323-335 ( 1 9 8 2 )

Komura, T . , Kornura, Y.:

Uber d i e E i n b e t t u n g d e r n u k l e a r e n Raume i n ( s )

Math. Ann.

162, 284-288 ( 1 9 6 6 )

Kothe, G . :

Topologische l i n e a r e Raume, S p r i n g e r 1960

Krone, J . , Vogt, Meise, R.,

Vogt,

preprin t N o r t h c o t t , D.G.:

A

,

D.: The s p l i t t i n g r e l a t i o n f o r Kothe spaces, p r e p r i n t

D.: A c h a r a c t e r i z a t i o n o f t h e quasi-normable F r g c h e t spaces, An i n t r o d u c t i o n t o homological a l g e b r a , Cambridge U n i v .

P r . 1972 Palamodov, V.P.:

The p r o j e c t i v e l i m i t f u n c t o r i n t h e c a t e g o r y o f l i n e a r

t o p o l o g i c a l spaces (Russian), Mat. S b o r n i k 75, 567-603 ( 1 9 6 8 ) , E n g l i s h t r a n s l . : Math. USSR-Sb. 4, 529-558 (1968)

Cld

Palamodov, V.P.:

Homological methods i n t h e t h e o r y o f l o c a l l y convex

spaces (Russian), Usp. Math. Nauk 26,1, 3-66 ( 1 9 7 1 ) , E n g l i s h t r a n s l . : Russian Math. Surveys 26:1, 1-64 (1971) Petzsche, H.J.: Some r e s u l t s o f M i t t a g - L e f f l e r t y p e f o r v e c t o r v a l u e d f u n c t i o n s and spaces o f c l a s s

A,

p. 183-204 i n " F u n c t i o n a l A n a l y s i s :

Surveys and Recent R e s u l t s I I " , K.D.

B i e r s t e d t , B . F u c h s s t e i n e r (eds.),

North-Holland Math. S t u d i e s 38 (1980) P i e t s c h , A.:

N u c l e a r l o c a l l y convex spaces, Ergebnisse d e r Math. 6 6 ,

S p r i n g e r 1972 Schaefer, H.H.:

T o p o l o g i c a l v e c t o r spaces, S p r i n g e r 1966

TrGves, F . : L o c a l l y convex spaces and l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , B e r l in-Hei d e l berg-New York : S p r i nger 1967 TrGves, F.: L i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , New York-London-Paris: Gordon and Breach 1970

[23

Vogt,

D.:

C h a r a k t e r i s i e r u n g d e r Unterraume von s, Math. Z . 1 5 5 , 109-117 (1977)

Continuous linear maps between Frgchet spaces

[24] Vogt, D.: Subspaces and q u o t i e n t spaces o f ( s ) , p . 167-187 i n " F u n c t i o n a l A n a l y s i s : Surveys and Recent R e s u l t s " , K.D. B i e r s t e d t , B. F u c h s s t e i n e r ( e d s . ) , N o r t h - H o l l a n d Math. S t u d i e s 27 (1977) [25] Vogt, 0. : C h a r a k t e r i s i e r u n g d e r Unterraume e i n e s n u k l e a r e n s t a b i l e n Potenzreihenraumes von endlichem Typ, S t u d i a Math. 7 1 , 251-270 (1982)

[26] Vogt,

D.: E i ne C h a r a k t e r i s i e r u n g d e r P o t e n z r e i henraume von end1 ichem Typ

und i h r e Folgerungen, m a n u s c r i p t a math. 37, 269-301 (1982)

[27] Vogt, D . : Frechetraume, zwischen denen j e d e s t e t i g e l i n e a r e A b b i l d u n g bes c h r a n k t i s t , J . r e i n e angew. Math. 345, 182-200 (1983)

[28] Vogt, D.: An example o f a n u c l e a r F r g c h e t space w i t h o u t t h e bounded a p p r o x i m a t i o n p r o p e r t y , Math. Z. 182, 265-267 (1983)

1291 Vogt, 0.: On t h e s o l v a b i l i t y o f P(D) f = g f o r v e c t o r v a l u e d f u n c t i o n s , t o appear

[30] Vogt, D.: Ober zwei Klassen von (F)-Raumen, p r e p r i n t 1 [31] Vogt, D . : On t h e f u n c t o r s E x t (E,F) f o r F r g c h e t spaces, p r e p r i n t [32] Vogt, D.: The n o n e x i s t e n c e o f r i g h t i n v e r s e f o r h y p o e l l i p t i c o p e r a t o r s , preprint

1331 Vogt, D.:

P r o j e c t i v e s p e c t r a o f (DF)-spaces,

preprint

[34J Vogt, D . , Wagner, M.J.: C h a r a k t e r i s i e r u n g d e r Quotientenraume von s und e i n e Vermutung von Martineau, S t u d i a Math. 67, 225-240 (1980) [35] Vogt, D.,

Wagner, M.J.:

C h a r a k t e r i s i e r u n g d e r Unterraume und Q u o t i e n t e n -

raume d e r n u k l e a r e n s t a b i l e n Potenzreihenraurne von unendlichem Typ, S t u d i a Math. 70, 63-80 (1981)

[36] Wagner, M.J. : Quotientenraume von s t a b i l e n P o t e n z r e i henraumen e n d l i c h e n Typs, m a n u s c r i p t a math. 31, 97-109 (1980)

[37] Z a h a r j u t a , V.D.;

Isomorphism of spaces o f a n a l y t i c f u n c t i o n s (Russian),

Dokl. Akad. Nauk. SSSR

2 2 , 631-634 (1980).

255 (1980), E n g l i s h t r a n s l . :

S o v i e t Math. Dokl.