Math. Nachr. 120 (1985) 203-216

A New Class of Perfect FRECHET Spaces

By FERNASDO COBOS of Madrid

(Received July 6, 1983)

Abstract. This paper deals with a new class of perfect FRECHET spaces which can be obtained by interpolation of echelon spaces: Zp,q[am,n]. We determine the reflexive, XONTXL, SCHWARTZ, totally reflexive, totally YONTEL and nuclear spaces Zp.q[am,n]. We also derive results on closed sub- spaces of the spaces (Zp,q)(v.

Many authors have worked on the echelon spaces of order p h 1, introduced by G. KOTHE ( p = l), J. DIEUDONNJ~: and A. P. GOMES (psi) (see [7], Q 30 and [14], Chap. 11). Generalizations of these spaces have also been extensively studied. For example, E.DUBINSKY generalized in [2] the construction of the echelon spaces and showed that one obtains all perfect FRECHET spaces in this way. He also studied those perfect PRECHET spaces that are MONTEL spaces. G. CROFTS completed in [l] the work of E. DUBINSHY and characterized the perfect FR~CHET spaces that are SCHWARTZ spaces. Another generalization of the echelon spaces was studied by C. FENS&, E. SCHOCK [3] and J. PRADA-BLANCO [ll] : the (not necessarily locally convex) spaces A:.

This paper deals with the spaces Zp,q[am,n] (l-=p-c.o, 1 sqs-), constructed by replacing the space Zp with the LORENTZ sequence space lp,q (see [S], 4.e, [lo], 13.9 and [13], 1.18.3) as defined by echelon space of orderp. The spaces Zp,q[am.n] can be obtained by interpolation of echelon spaces and form it new class of perfect FRECHET spaces. Purthermore the echelon space of order p is equal to Zp,p[um,n]. We determine the reflexive, MONTEL, SCHWARTZ, totally reflexive, totally MONTEL and nuclear spaces Zp,q[um,n]. We also derive results on closed subspaces of the spaces (Zg,q)(N) in the line of those obtained by A. GROTHENDIECX and 31. V A L D I V ~ ~ for the spaces ( Z p ) ( w (see [14], Chap. 11, S 5 , 7).

Part of the content of sections 1,3 and 6 has been taken from the author’s doctoral thesis, written, under the direction of Professor MANUEL A. FUGAROLAS at the University of Santiago de Compostela, but the proofs given here of those results are almost always distinct.

The problem treated in section 4 has been suggested to us by Professor MANUEL VALDIVIA.

204 Math. Nachr. 120 (1955)

0. Terminology

Subsequently, almost all notations and some basic definitions are adopted from [ 7 ] and [14]. With regard to interpolation theory we refer to [13] for BANACH spaces and [5] for locally convex spaces.

We designate by en the sequence which is zero at all co-ordinates but the nth co-ordinate where i t is one. If 6 = (tn) is a sequence of complex numbers ( {tn : n E

EN} cC) then we define the rnth section, trn, of 5 by E m = Enen. For [, q se-

quences of complex numbers, t q (resp. E/q) denotes the sequence (tnqn) (resp. (fn/qn) provided the terms of q are not zero) and when the following series conver-

ges we write 4 5 , q=- = 2 &,rb. Noreover, if n is any one-to-one map from N into N, we write J,(E) = ([n-1(14)) where Ee = 0 and &([) =

complex numbers 5 = ( E n ) converging to zero and having st finite norm

m

n=i

- n = l

We denote by Z p , q the LORENTZ sequence space, formed by all sequences of

where (5:) designates the rearrangement of the elements of (tn) by magnitude of the absolute values: IETIzi(zIz... (see [8], 4.e, [lo], 13.9 and [13], 1.18.3). For p = q , Z p , q is equal to the space of p-summable sequences (Zp, I [ \ I p ) , both spaces having equivalent norms. The norm / I I \ ( p , q ) is equivalent to the quasinorm

lltIlp,g=S~P (m l lP IEnI) * if l<p-=m q = - .

If q<m then (en) is a SCHAUDER basis of lp,g and if l-=q-=m the space lp,q is reflexive. Moreover the LORENTZ sequence spaces are ordered lexicographically. This means that

lp,g is continuously embedded in lr,s if 1 -=p< r e m

and Z p , g is continuously embedded in lp,s if 1 s q c s Sm.

The subspace of Zp,- (1 <p-= m) consisting of all sequences E with lim I][ - En1[@,+ =

= 0 , is denoted by l"p,.,. If 1 s r s p it is easily checked that $,- is the closure of I , in Zp,-, whence (Z&, 1 1 is a BANACH space whose dual space is lpl,i where l /p+l/p '=l (see [13], Thm. 1.11.2 and Thm. 1.18.3/2).

We designate by (lp,n)(N) (resp. ( Z p ) " ) ) the locally convex direct sum of a countable infinity of spaces equal to Zp,@ (resp. Zp) .

n

Cobos, A New Class of Perfect FrBchet Spaces 205

If (a,) is a sequence of real numbers greater than zero, we denote by Z,,,(a,) (resp. Z,(a,)) the BANACH space formed by all sequences of complex numbers

B subspace D of a sequence space A is said to be a sectional subspace of A if i: = (ln) having a finite norm II,!lll,, *(an) =lI(~nlnn)lJ(p,*) (resp. l1611zp(an) =ll(~nl*)ll,).

there is an increasing one-to-one map n : N -N such that

Q= (,!€A : J , (& , (O)=E} - If E and F are linear spaces which form a dual system (E, F ) we denote by

E[%,(F)], E[Zk(F)] and E[F,(F)] the space E with the weak, MACKEY and strong topology from F respectively.

If O<r9< 1, 1 S ~ S - and (Ao, A, ) is a compatible couple of locally convex separated spaces, we denote by (Ao, Al)a,g the interpolation space obtained by means of the function norm (see [ 5 ] , 3.5)

1. General Properties of the Spaces

1.1. Definition. Let (a,,,,) be an infinite matrix formed by real numbers such that

and let l-=p<-, 1 sqa-. The space Zp,4[am,,] consists of all sequences t of complex numbers such that t~Z~,~(a,,,) for every m EN, and is endowed with the topology defined by the sequence of norms

O-=a,,n<am+l,n rn, n = l , 2, ...

ym(t) =Il~!llp,q~sm,,~ m = 1, 2 , ... Clearly, ZP,,[u,,,] is a FR$CHET space and for I, = q it is equal to the echelon

space of order p , denoted by &[am,,] (in the notation of [7], p. 420, this space is that obtained by the steps (a&J m= 1, 2 , ...) where the system of norms (v,) is equivalent to the system of norms

B,(E) =IItllzp,am,,, m= 1, 2, *I.

We have the following interpolation formula 1.2. Theorem. As6urne that 1 sqs-, 1 spa, pl-==rn, po=kpI, 0-=6-=1 and thcrt

1 / I , = (1 -9)/po +$/pi. The% (Z,,[a,,,], Zp,[am,n])a,p = ~,,4[cb,,nI (with equivalent systems of norms).

Proof. It is easily checked that (Z,,[am,,], 2p,[am,n]) is the strict projective limit (see [ 5 ] , p. 41) of the family (Zp,(am,n), Zpl(ak,n))(m,k)cNxN therefore [5], Prop. 3.3. and Remark 3.1

(ip,tn,,iI, Z,[am,nI)a,* = proj lim (&, (am,n), &,(am,,) )a,g. m

206 Xath. Nachr. IS0 (1985)

So then, in the case po-=pi, the result follows from [a], Satz 5 , and if po>pl , from the fact that

( lp , (am,n) , lpl(am,n))a,g== (lpl(am,n)! 4p(am,n) ) i -6,q

and again [4], Satz 5 . * show that Z,,, is a step (in the sense of [2] , p. 188).

the a-dual (I,,,)” of the space lPsq is l,.,,, and

For the purpose of proving that Z p , q [a,,,] is a perfect space, we shall first

1.3. Lemma. Let l / p + l / p ’ = l / q + l / q ’ = l with l i p < - and l s q s - . T h e n

P,,q* /I ll(P,d = ~ P , Q r w ? P , q ) >I * Proof . Suppose first q<w. It is verified by interpolation [13], Thm. 1.18.312

and Thm. 1.11.2, that T(/) = ( / ( e n ) ) is an isomorphism from the dual space (lp,,)’ of Z p , q onto lp,,9” hence (Z,,,,,) c ( Z p , g ) x . The converse inclusion follows from the BANACH-STEINHAUS theorem. Therefore

(lP,,ll Il(P,,)) = ~ P , q [ % ( V P , A X >I *

4f,, c ( (%,A ) = (l,,=.=) ’

Suppose now q=m. By the result just proved we get

Conversely, if (En) E (Z,,.,)’ c ( Z p ) x = Z p r , the sequence (E , ) converges to zero, then there is a one-to-one map ?t : N -+N so that (5,) =J,( ( E : ) ) . Let (7,) =Jn( (n(l’p’)-l)) E Elnsm, we have

hence (tn) € Z p , , i . Now, since Zlclp,qcZ-, it follows from the first part of the proof that is a step, whence lpr,- is also a step [2] , p. 188, and thus lp,.,[Sb(Zp>,l)] is a BAXACH space. But every sequence of lp,,i defines a continuous linear functional on Z,,-, therefore [6], Thm: 1.9.1

(4,d II Il@,-J = lp,- [S&,AI * * A s an immediate consequence of Lemma 1.3 and [ 2 ] , Cor. 1 we obtain

1.4. Corollary. Assume that 1-=p<w, 1 sqsm and 1 /p+l /p‘=l /q+l /q’=l . - T h e n the spaces lPsq [am,n] and u lp)3q, (a;.;) are perfect and u-dual to each other.

It may occur that Z,,,[a,,,] is normable. For example, if am,,=m + n i t can be checked that T ( ( E n ) ) = ( (n + 1) 6,) is an isomorphism from Z,,,[a,,J onto Z p , q .

The following result, direct consequence of [7], 5 31, 4. (l), characterizes those spaces Zp,,[am,n] which have this property.

1.5. Proposition. Let l-=pd:... and 1 sqs;-. T h e space Zp,q[am,n] is not nor- mable if and only if it has a quotient isomorphic to the topologicab product of a count- able infinity of spaces equal to the field C.

m = l

Cobos, A New Class of Perfect Fr6chet Spaces 207

2. MONTEL Spaces I , , , [ U ~ , ~ ]

2.1. Definition. (see [7], p. 421 and [2], p. 188). We shall say that the matrix (a,,J is strongly increasing provided that there is no sequence of positive integers

n1-cn2< ... -cnk-= ... and'r iN such that for all m sr

inf {ar,nk/am,,tk : k EN) =- O . It is not hard to verify that the topology defined by the sequence of norms

(vn) coincides with the strong topology on Zp,q[am,n] relative to the dual system formed by Zp,q[am,n] and its a-dual. Then, using Thm. 4 and Thm. 5 of [%I, we get

2.2. Corollary. Let l-=p-=- and 1 ~ q s r n . The space Zp,q[am,n] is a MOIUTEL space if and only if the matrix (am,n) i s strongly increasing.

The proof of the next result is parallel to that of [14], Chap. 11, Q 3, 2. (5) .

3.3. Proposition. Let l-cp-=- and 1 sq S m . The space Zp,q[am,n] is not n MONTEL space if and only if it has a sectional subspace isomorphic to lp,q.

Proof . If Zp,q[am,n] is not a MONTEL space, the matrix (a,,,J is not strongly increasing, then we have an increasing one-to-one map n : N -3, r EN and a se- quence ( J f n ) of positive numbers such that

I

am,n(n) S N ~ U ~ , ~ ~ , , ) n = 1, 2, ... m = 1, 2, ... For every m EN, let don) be the sequence (am,%), let i2 be the sectional subspace of Zp,q[am,n] formed by all sequences t with Jz(Qn(t)) = E and let T(6) ~ J - ( t ) / d ( ~ ) . The operator T is linear, one-to-one and continuous from lp,q into lp,q[nm,n] because, for every mEN

vrn(T(t)) =IId(m)J,(~)/S(r)II(p,q) sJfmIIJn(t)II(p,g) = ivmIItII(p,q) .

lbIl@J,q) = llQ*(~(r)~)ll(p,q) = llfT4l(p,q) = Y A P )

Xoreover if pEi2, then z=&,(d(')p) € l p , q being

and T(t) =p. So T is then an isomorphism from lp,q onto 9. The converse impli- cation is obvious. .t.

If I-=q-=m, the space Z9,q[nrn,n] is the projective limit of the sequence of reflexive spaces (lp,p(am,,)), then it follows from [7], Q 23, 3. (7) that Z p , q [ ~ m , n ] is reflexive. On the other hand, if 1 ~ q - = m it is easily checked that (en) is a SCHAUDER basis of lp,rl[n'm,n]. For the remaining parameters we have :

2.4. Theorem. Assume that 1 - = p i 00 and q = 1, m. If Zp,g(am,n] is reflexive, then it i s a MONTEL space.

Proof. If lp,,[nm,~] is not a MONTEL space, i t follows from Proposition 2.3 that it has a closed subspace which is not reflexive, therefore the space ~,,,[u,,,] is not reflexive, by [7], $ 23, 5. (10). +

208 Math. Nachr. 190 (1985)

2.5. Theorem. Let 1 <p< 00. Then the following conditions are equivalent. (i) The sequence (en) is a SCHAUDER basis of lp,Jam,n]. (ii) The space lp,,[am,,] is a MONTEL space.

Proof. Suppose first that lPJa,,,] is not a MONTEL space. Applying Propo- sition 2.3 we obtain a one-to-one map ;z : N +N and r €3 such that T(6) = Jn(c) /d(r) is a continuous operator from lP,- into ~ , , J U ~ , ~ ] . Let 6 = (n-i'p) and p = T(E) . For every n E N and every m z r we have

vm (p -$(9 = 11 J, !&,(s'm'/d'7') (E-t?) =ll&n(c~(m'/d'r') ( E --E+)Il(,,..) zllt -5+ll(,,-) 1 .

Therefore ( en ) is not a basis of lp,- [a,,,].

coincides with Sk((lp,Ja,,n])x), hence (en) is a basis of ZPJa,,+], by [7], 5. (10.). *

Reciprocally, if Zp,- [a,,,] 1s a MONTEL space by [2], Lemma 1, its topology 30,

3. SCHWARTZ Spaces Zp,q [a,J

In order to determine the SCHWARTZ spaces lp,lI [am,+], we shall first prove

3.1. Lemma. Let a= (a+) be a bounded sequence of positive numbers and let D be the continuous linear operator o n lp,q (l-=p-=-, 1 sqsm) defined by D(E) =a[. Then D is compact if and only if a converges to zero.

P r o o f . If a converges to zero, given any E=-O there is an NEN such that a, S E whenever n ZLN. Let P,&(t) =tn, then we have

iim - ~ + ( E ) l l ( p , q ) 4 I E l l ( p , p ) for all n Z=-N - Consequently D is the limit of a sequence of operators of finite rank and is there- fore compact.

Conversely, if D is compact then (D(e,) : nEN} is precompact. And so, given any E ~ O , there exists an NEN such that

1V

i = l W,) E U (D(ei) + {t : 11tll(p811~ 5 ~ ) ) n = 1 , 2, ..-

From where we get a, s min llD(e,) --D(ei)l\(p,T) z c for all n=-N . +

1SlCSLV

Now we can establish

3.2. Theorem. Let l-=p-= - and 1 sq s00. Then the following three assertions

(i) The space lp,g(am,+] i s a SCHITARTZ space. (ii) For each m EN, there exists a k C N such that

c u e equivalent.

1 p hA,,/am +k,n) = 0

(iii) Every weakly convergent sequence in (lp,g[am,,])' i s locally convergent.

Cobos, A New Class of Perfect Frhchet Spaces 209

Proof. The equivalence of (i) and (ii) is a consequence of [l], Thm. 3.2, Thm.

If (i) holds then Ip,Ja,,,] is a FRECHET-SCHWARTZ space, whence (iii) follows

Assume now that (iii) is satisfied. Given any m EN, let us consider the sequence

3.1, and former Lemma 3.1.

from [12], 4 4, (1).

of linear functionals

f r ( ( E n ) ) = a m , r t r r = i , 2, * * -

Since

l ~ , ( ( ~ n ) ) l = l a ~ , r ~ ~ l ~ ~ , ( ( ~ n ) ) ~ = 1 > 2, *..

we have that (fn)c(ZP,,[urn,+])’, and ( fn) also converges weakly to zero for if ( & , ) € Z , , , [ U , , ~ ] then (urn,&,) converges to zero. So, by (iii), there exist k c N and /I =- 0 such that if

v = { ~ € 2 ~ , q [ a m , n l : Il(am+E,nL)llp,q %PI and YO is its polar with respect to

(5 ,p [am,nIr (ip,q[urn,nI)’) 9

then - (fn) c (zp,q[am,n~)>o = U nvo

n=l

and (f,) converges to zero in the topology defined by the norm gvo (MINKOWSKI, functional of 7 0 ) . But

inf (e > 0 : f n E e vO) = & n , n / U m +k,n

therefore lim (~l,,,/a,+~,~) =O, that is to say, (ii) is satisfied. =+ n

In what follows we shall writme A=Zp,q[am,n] and we shall suppose A x =

= U ZP,,,,(a;,~) ( 1 / p + 1/p‘= l / q + l /q ’= 1) endowed with the inductive limit

topology E of the sequence of BANACH spaces (Z,.,,.(a,f)>. In this way A”[E] is an (LB)-space. We shall now compare 2 and Zb(A).

- m a 1

- Since for each [ € A the functional uS(p) = 2 $,,,LA, is continuous on Ax[%], it

n=l

follows from [7], $ 30, 5 . ( 5 ) that every bounded subset of Ax[%] is bounded in %&I). But Ax[%] is bornological, therefore E is finer than %,(A).

If i-=q-=o;, the space A is reflexive, whence Ax[Eb(A)] is bornological and as [2], Thm. 2

is a fundamental family of bounded sets inAx[Sb(A)], we obtain that %=%&A). For the remaining parameters we have: 14 Math. Nachr.. Bd. 120

210 Math. Nachr. 120 (1985)

3.3. Proposition. Assume thut l-=p-cm, q= 1, m and that the matrix (am,+) is strongly inc;easing. Then S= %&4).

Proof . Taking into account that A[Zk(Ax)] is a MONTEL space ([2], Lemma l) , we again obtain that Ax[Z,(A)j is bornological, from where the result follows pro- ceeding as before. *

We shall now characterize, in terms of the topology of the a-dual space. tbose spaces A= lp,cl[am,n] with (amJ strongly increasing, which are not SCHWARTZ spaces. For the case of the spaces ll[am,J see [7], p. 423.

3.4. Theorem. Suppose that l<p-=-, l s q s - and that the matrix (am,n) is strongly increasing. The space d i s not a SCHWARTZ space if and only if A"[%] contains compuct sets which are not compact in any l,.,,.(a,f).

Proof . If A is not a SCEWARTZ space, then Theorem 3.2 shows that there exists TEN such that (ar,n/ar+k.n) does not converge to zero for any k c N . Let K be the adherence in Ax[%] of the unit ball of l,.,,.(a;;). By Proposition 3.3 and the previous remark, we have that the set K is compact in A"[%], but i t is not compact in any z~,,~, (a,:).

Conversely assume that A is a SCHWARTZ space. Given any compact set K in 11x[2], there exists rEN such that K is bou,lded in Zp,,g,(a;i), by [2], Thm. 2. Then it follows from Theorem 3.2 and Lemma 3.1 that there exists k € N such that K is compact in Zp,,qz(aF2k,n). :+=

4. Totally Reflexive and Totally MONTEL Spaces Zp,n[a,,n]

Let i / p+l /p '= i /q+I /q '= l with l-=p-=m, l-=q-=m and let A=Zp,p[am,nJ. If A is a bounded subset of A x [ Z 8 ( r l ) ] given arbitrarily, we can find T E N and e=-O such that

by [ a ] . Thm. 2 . But B is compact in A"[Z,(A)] because ZP~,,/(a;~) is a reflexive space. Then i t follows from [14], Chap. 11, § 2, 3. (1) that ZPJam,J is totally reflexive.

We shall now determine, in terms of quotients, those spaces Zp,g[am,n] which are not SCHWARTZ spaces. This will allow us to characterize the totally reflexive spaces Zp,q[am,n] with q= l,= and the totally MONTEL spaces lp,q[am,n].

4.1. Theorem. Let 1+ p-cm and 1 sq-=- (resp. q=-). If ZP,,[am,+] is a M ~ N T E L space which is not a SCHWARTZ space, then it has a quotient isomorphic to lPvg (resp.

Proof . As the conditions on the matrix (am,n), characterizing the MONTEL or SCHWARTZ spaces Zp,q[am,n], are the same as the known conditions for the MONTEL or SCHWARTZ echelon spaces, we can use the construction of [14], Chap. 11, 5 2 , 3 t o find a positive integer r , an increasing one-to-one map n : N-N and a sequence

&J.

Cobos, A New Class of Perfect WBchet Spaces 21 1

(n,) of increasing one-to-one maps on N such that - N= U nz(N), n i ( N ) ~ q ( N ) = O when i+ j ,

i=L

and if b,,,fi =a, +m-l,x(n) then

inf {bl,,j(,,)/bl+i,,i(n) : nEN}=-O for all iEN

being (a) the space 0 = lp,q [b,,J ischorphic to a quotient of Zp,q[am,fi]. Let b(m) = (bm,,J and let d be the set of all sequences of complex numbers E

such that

Q,,(Nb'") €lp, ,qf 1 / ~ + l / p r = l / q + 1/q'= 1

and

Q x i ( t ) = ( l / i ? ) Q,i(b'2') QnI(E/b(')) i = 2 , 3, ... .A similar reasoning to [14], Chap. 11, 5 3, 3. (7) allows us to prove that

the set d is a linear subspace of 0'. Furthermore, taking into account that J2 is separable and that d is sequentially

closed in Qx[iz,(S)] (see [14], Chap. 11, § 3, 3.(8)), it follows from the theorem of KREIN-SMULLSN ([ 61, Thm. 3.10.2) that

the subspace d is closed in Sx[%,(0)]. Now we are able to establish

(b) T h e space A endowed with the toplogy induced by Z8(Q), is isomorphic to

For each [ C l p , , g J , let f ( E ) =q be the sequence defined by Qn,(q) =Q,,(b(") 6 ZP,,*' [z,(zp,q)] if q<-, and to lpJ,i [%,?(~,J if q=-.

and

&,;(q)=(lli2)Qx~(b'2)) 5 i = 2 , 3, ...

Q & J / b ( l ) ) = E Since

and Q,(q)= (lliz) &,(b(") Qx1(q/b( ')) i = 2 , 3, ...

we have that f(lPr,,.)c d, f being also a one-to-one linear map. On the other hand, if q E d then 6 = QnI(v/b( ')) E Zp,,q, and f ( ( ) =q, therefore f ( Z p , , q t ) =A.

Let us now take ,LA E 0, then the series

Q , p ) P ) + 2 (W) Q,,(b'%) 2=2

converges in lp,q to a certain element, say t=(z,). Moreover, .El&., in the case q=,:

Let v=(vn)€Z1 with q n > O for all nEN. Given any &=-O, there exists an NIEN such that i f n s N l then Ilq-r+Pll(p, . .)~~/2, and, since Q is a MONTEL space, there l.L*

212 Math. Nachr. 130 (1985)

also exists an N z € N such that

h

Let y = (yn) =Qn,(b(')p) + (W) Qni(b(2)p) and let N=max {NI, I f2 ) . For every i = 2

n S N and every j P 1, we can choose k EN such that the sequence y associated to k satisfies

[zm-ymI<qm%m=n++, ..., n+j . Thus we have

l lzn+~-qp,- ) ~llYn+i-pinll(p,~) +llYn+i-Ynll(p,-) 4 l Y -Ynll(P,-)

k + c ( l / i Z ) I l b'2'~-((b(2)p)N~ljCp,,,) 5.5 .

i = 2

Therefore [ / z - z " ~ \ ~ p , ~ ~ ~ e if nzAr , that is to say, .El&, . Then, if q-=m (resp. a=-) we obtain that

f : ~ p ~ , * m ( l p , q ) l -QX[%(Q)l b p . I ~ p 4 w : , = . J l - Q X C % ( Q ) l )

is a continuous map, because for every t€Zpt,qt we have

(f(0, p)=(t, Qni(b( l )p) ) + . j (t, (W) Q,(b'"A)=(t, 7) 1=2

where p and t are taken as before. But i(Zp,,q,) = A is closed in QX[S,(Q)], therefore it follows from [S], Prop. 3.17.17 that f is an isomorphism from Z p s , q t [ % 8 ( Z p , q ) ] (resp. Zp,,l[%s(l!&)] if q=-) onto O[%,(Q)], which proves (b) .

Consequently, if f is the subspace of,Q orthogonal to A we have that (c) the space Qlf is isornorphic'to lp3q if q<=oo and to li,* if q=-. Now the theorem follows from (a) and (c).

4.2. Theorem. Let l - c p e m and 1 S q - c m . If Zp,g[am,n] is not a SCHWARTZ

space, then it has a quotient isom,orphic to lp,g .'

Proof. By Theorem 4.1 it is sufficient to consider the case where Zp,p[am,n] is not a MOWTEL space. Then, using Proposition 2.3 , we have that lp,g[am,n] has a sectional subspace isomorphic to tp,g, and has therefore a quotient isomorphic

4.3. Corollary. Assume that 1 <p< - and q = 1 ,a. The space lp...[am,n] is totally reflexive if and only if it is a SCHWARTZ space.

Proof. If Zp,Jam,n] (resp. ZP,Ja,,J) is not a SCHWARTZ space, then, accord- ing to Proposition 2.3 and Theorem 4.1 (resp. Theorem 4.2), i t has a quotient

*

to lP& . =e

Cobos, A New Class of Perfect Frkhet Spaces 21 3

isomorphic to either lp,- or li,* (resp. I,,,), whence it has a quotient which is not reflexive.

Reciprocally, if lp,g[am,n] is a SCHWARTZ space, then every separated quotient of Zp,g[am,n] is a MONTEL space and therefore a reflexive space.

With a parallel reasoning it is possible to establish the following characteriza- tion.

4.4. Corollary. Assume that l<p-=w and l-=q-=-. T h e space lp,g[am,n] is totally MONTEL if and only if it is a SCHWARTZ space.

*

5. Closed Subspaces of (Zp,q)(N)

Let us consider the following one-to-one maps

n l ( n ) = 2 n - 1 n=l , 2, ... ~ ~ ( n ) = 2 ~ - ' n ~ ( n ) n = 1 , 2, ... i = 2 , 3, ...

ca

Clearly N= u ni(N), being n i ( N ) f l n , ( N ) = O when i+j. Now, for each mEN, i = i

let (a,,,) be the sequence a('") defined by

&%(a('")) = ( ( n+1)") Qni(a(')) = (i + 1)

m = i , 2, ,.. i = 2 , 3 , ...

and if m = 2 , 3 , ... ( i f l ) ' " for 2 5 i ~ m , I s n S i

Q,,(a("")=(p,,) where pn= ( n f l ) " for 2 5 i ~ m , n = i + l , i + 2 , ... ( (i+l)'" for i = m + l , m + 2 ,..., n = i , 2 ,... The matrix (am,,,) constructed in this way, is strongljr increasing but it

does not satisfy the condition (ii) of Theorem 3.2 (see [14], Chap. 11, $ 5, 6) there- fore, according to Corollary 2.2 and Theorem 3.2, if 1 c p - c ~ and 1 S ~ S - the space lB,g[am,n] is a MONTEL space which is not a SCHWARTZ space. Then, using the same techniques as in [14], Chap. 11, $ 5 , 7, one can derive from Theorem 4.1 the following three results.

5.1. Theorem. Let l-=p-cw. There is in (Zp,m)(N) a closed subspace L and a linear functional o n L which is sequentially continuous but not continuous.

5.2. Theorem. Let 1 -=p< and 1-= q-= m. There is a closed subspace L in ( I ? ~ , ~ ) ( ~ ) sa€isfying;

(i) T h e topology of L is not the MACKEY topology. (ii) T h e subspace L with its MACEEY topology is am (LB)-space.

5.3. Theorem. Let 1 -=p-=w. There i s in (lp,l)(w a closed subspace L and a linear functional on L which is sequentially continuom but not continuous. iMoreover the topology of L is not the M ~ C K E Y topology.

214 Math. Nachr. 120 (1985)

Let us see another way of proving these results: Assume that 1 s q < p and that ~ = ( n ( ~ ' ~ ) - ' ) . The space lp,r is equal to the

space d(w, q ) in the terminology of [S], Def. 4.e.1, both spaces having equivalent norms. Then, by [8], Prop. 4.e.3, there exists a continuous linear operator P : l,,, - -lp,q such that Po P = P and P(lP,,) is isomorphic to I,. Therefore, for every l c p - z - . and every 1 sqs-., we have that ZP,, contains a subspace which is iso- morphic to I,, and consequently the space ( Z p , q ) ( w contains a closed subspace isomorphic to (&)". This proves Theorems 5.1, 5.2 and 5.3 taking into account the results of A. GROTHENDIECK and 81. VALDMA concerning the spaces (Zq)@) (see [14], Chap. 11, 3 5, 7) .

Proceeding in this way we also obtain as a consequence of [14], Chap. 11, 3 5 ,

5.4. Theorem. Let 1 -=p-=-- TheTe is a closed szcbspace L in (Zp,J(N) satisfying; (i) The topology of L is not the MACEEY topology. (ii) The subspace L with its MACEEY topology i s an (LB)-space.

7.(4).

6. Nucfear Spaces Zp,q[am,n].

We end the paper by establishing the following characterizations.

6.1. Theorem. Let l-cp-= - and 1 sq s-. T h e n a necessary and sufficient condi- tion for Zp,p[am,n] to be a nzcclear space i s that f O T each m EN there is a k E N for which

Proof. The condition is necessary : Suppose that Zp,q[am,n] is nuclear. Applying Theorem 2.5 and remembering

the prevjous remark regarding Theorem 2.4, we have that (en) is a basis of the nuclear FRBCHET space Zp,q[am,n]. Then it follows from the theorem of DYYIN- MITIAGN ([9], Thm. 102.1) that the basis (en) is absolute. Let bm,n=vm(e,z)

and let T : lp,q[am,n] +Zl[bm,n] be the operator defined by T 2 (,,en = (tn)- Using

[9], Thm. 10.1.4, we obtain that T is an isomorphism. But ~ , . , = C U ~ . ~ where LL 1

if q<-= and C = 1 if q =a, therefore Z1[bm,*] coincides with

Zl[a,,,], both spaces having equivalent sequences of norms. So then ll[am,,]is a nu- clear space, whence the condition is satisfied by [14], Chap. I T , § 2, 3. (16).

The condition is sufficient : If p-cr-=-., then, as sets,

Uarn,nl c lp,g[am,nl c Br[.~rn,nI

and the inclusion maps are continuous. But, by the hypothesis and the inequality of HOLDER, we also have that c Zl[arn,,] with the inclusion map being

Cobos, A New Class of Perfect Frbchet Spaces 215

continuous. Thus the space Zp,q[am,n] coincides with ll[am,fl], both spaces having equivalent sequences of norms. Therefore, according to [14], Chap. 11, $ 2, 3. (16), the space Zp,q[am,n] is nuclear.

6.2. Theorem. Let l<p-=- and 1 sq Sm. The space Zp,q[um,n] is nuclear i f and only if = Z,,8[amJ (with equivalent sy tems of n o r m ) whenever 1 -= r-= 03, l s s 5 m o r r = s = l .

Proof . If Zp,q[am,n] is nuclear then, applying Theorem 6.1 and using the ine- quality of HOLDER, Z,:q[am,n] = Ir,s[am,n], both spaces having equivalent se- quences of norms.

=I=

Reciprocally, i f we suppose that

Z,s[am,nl= ~ l b m , n l = a,[am,Tal

(l,(am + k , n ) II I Ilz(nm )

-(zi(%l +zb,n) 7 I / lIz1(am +u,n) )

- 4 2 ( ~ r n , ? a ) l I1 II12(am,nJ -

then for each mCN, there exist u, k € N such that the following inclusion maps are continuous

But every continuous linear operator from I, into I , is absolutely summing ([8], Thm. 2.b.6), therefore the inclusion map from (Z2(amCk,J, / / / [12(am+k,n) ) into

( L ( L I ~ , ~ ) , 1 1 I / 1 2 ( a m , d ) is also absolutely summing, from which the result follows taking into account [ l o ] , Thm. 2q.7.2. =k

The fac t that the condition stated in Theorem 6.2 is sufficient for Zp,q[am,n] to be a nuclear space, has been kindly pointed out to us by Professor QLBRECHT

PIETSCH and we should like to thank him for this observation.

Acknowledgements. We would like to thank Professor MA NU EL A. FUGAROLAS for his helpful comments and constant interest during the preparation of our thesis. We would like also to express our indebtedness to Professor MANTJEL VALDIV~~ for suggesting the subject treated in section 4 and for his valuable infor- mation on [l], [2] and [14].

References

[l] G. CROFTS, Concerning Perfect Fr6chet Spaces and Diagonal Transformations. Nath. Ann. 1S2

[2] E. DUBINSEY, Perfect Frkchet Spaces. Math. Ann. 174 (1967) 186-194 [S] C. FE~SEE, E. SCHOCK, Nnclearitat und lokele Konvexitat von Folgenraumen. 3K.lath. Nachr.

[4] D. FREITAG, Reelle Interpolation zwischen kp-RLumen mit Gewichten. Xixth. Wachr. 77

[5] C. GOULAOUIC, Prolongements de foncteurs d'interpolation et applicntions. Ann. Inst. Fou-

(1969) 67-76

45 (1970) 327-335 .

(1977) 101-115

rier Univ. Grenoble 18 (1968) 1-98

216 klath. Nach. 120 (1985)

[6] J. HORVATE, Topological vector spaces and distributions. Addison- Wesley, London 1966 [7] G. ROTHE, Topological Vector Spaces I. Springer, Berlin - Heidelberg - New York 1969 [8] J. ILNDEXSTRAUSS, L. TZAFRIRI, Classical Banach Spaces, Vol. I , Sequence Spaces. Springer,

[9] A. PIETSCH, Nuclear Locally Convex Spaces. Springer, Berlin - Heidelberg - New York 1972 Berlin - Heidelberg - New York 1977

[lo] A. PIETSCH, Operator Ideals. North-Holland, Amsterdam - New York - Oxford 1980 [ill J. PRADA-BUXCO, Local convexity in sequence spaces. Math. Nachr. 95 (1980) 21-26 [12] T. TERZIOGLU, On Schwartz Spaces. Nath. Ann. 1SB (1969) 236-242 [13] H. TRIEBEL, Interpolation Theory, Function Spaces, Differential Operators. North-Holland,

[14] 31, VALDIVIA, Topics in Locally Convex Spaces. North-Holland Xathematics Studies 67, Amsterdam - New York - Oxford 1978

-4msterdam - New York - Oxford 1982

Universidad A u t h m a de Madrid, Facultad de Ciencias Divisidn de Matedt icas Xadrid - 280 49 Spain