On 2-Homogeneous Polynomials on Some Non-stable Banach and Fréchet Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

206, 322]331 Ž1997.

AY975200

NOTE

On 2-Homogeneous Polynomials on Some Non-stable Banach and Frechet Spaces ´ Juan Carlos Dıaz* ´ Dept. de Matematicas, E.T.S.I.A.M., Uni¨ ersidad de Cordoba, 14004 Cordoba, Spain ´ ´ ´ Submitted by Richard M. Aron Received September 28, 1995

Let F be a Banach or a nuclear Frechet space isomorphic to its square. Then ´ P Ž 2 F ., the space of 2-homogeneous polynomials on F, is isomorphic to the space of continuous linear operators LŽ F, F9., both of them endowed with the topology of uniform convergence on bounded sets. In this note we prove that the isomorphism can fail if F is not stable by studying two kind of examples: First, for Banach spaces, we consider James spaces J p constructed with the l p-norm, with p ) 2; second, we treat nuclear power spaces of finite or infinite type. Q 1997 Academic Press

ˆp F the Given a Žreal or complex. Frechet space F we denote by F m ´ completed projective tensor product of F with itself. The symmetric ˆps F, is the closed subspace of projective tensor product, denoted by F m ˆp F spanned by the elements of the form x m y q y m x with x, y g F. Fm ŽWe restrict ourselves to the case of the 2-fold tensor product, e.g., see w6, 9x for details and interesting results about n-fold symmetric tensors.. Symmetric tensors arose in Functional Analysis in the Ph.D. Thesis of Ryan w16x in relation with the study of homogeneous polynomials on Banach spaces. Indeed, if F has the ŽBB.-property Žsee w6x., in particular if ˆps F is the F is a Banach or a nuclear space, then the strong dual of F m 2 space P Ž F . of 2-homogeneous polynomials Ži.e., the space of the restrictions to the diagonal of the symmetric 2-linear forms defined on F = F . *The research of the author has been partially supported by the DGICYT project PB94-0441. E-mail address: [email protected]. 322 0022-247Xr97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

323

NOTE

endowed with the topology of uniform convergence on bounded sets of F, ˆp F is the space of all as well as the fact that the strong dual of F m continuous linear operators LŽ F, F9. endowed with the topology of uniform convergence on bounded sets of F. ˆps F is canonically complemented in F m ˆp F. ŽThe comThe space F m plement is the subspace of antisymmetric tensor products, spanned by the elements of the form x m y y y m x.. On the other hand it follows from w3, ˆp F is complemented in F 2 m ˆps F 2 and the compleLemma 8x that F m s s ˆp F . [ Ž F m ˆp F ., so the following result is a ment is isomorphic to Ž F m straightforward consequence of Pelcynski’s decomposition method:

ˆp F is PROPOSITION 1. If F is stable Ži.e., isomorphic to F 2 . then F m ˆps F. In particular P Ž 2 F . is isomorphic to LŽ F, F9.. isomorphic to F m It has not been checked so far if P Ž 2 F . is not isomorphic to LŽ F, F9. for some Frechet space F. Our aim in this note is to study this question and to ´ provide examples, both of Banach ŽCorollary 7. and of nuclear spaces ŽTheorem 10.. In the first case we deal with quasireflexive James spaces J p , constructed with the l p-norm, p ) 2 and in the second case we consider non-stable power series spaces Žof finite or infinite type.. Our notation is standard. We refer the reader to w13, 5, 7, 11, 15x for results concerning bases, polynomials, and nuclear and general locally convex spaces, respectively. Given 1 - p - ` we denote as usual p9 s prŽ p y 1..

1 For any 1 - p - ` the James space J p is defined as

Jp

¡ ~ [ Ž x . ; 5Ž x . 5 [ ¢ i

i

ky1

sup 0sn 0-n 1- ??? -n k

p

n jq1

 ž /0 Ý

js0

Ý

isn jq1

xi

1rp

¦¥ -` . §

The structure of J p has been mostly studied for p s 2 Žsee w12x. but the results can be extended word by word to the general case. We denote by e i the sequence taking the value 1 in the ith place and 0 elsewhere, e i shall be also denoted by eUi when considered as an element of the dual Ž J p .9. Then Ž e i . is a Schauder basis of J p and Ž eUi . is a sequence of associated

324

NOTE

biorthogonal functionals in Ž J p .9. The following result can be proved in the same way as w4, Lemma 2.ax. LEMMA 2. Let 0 s r 0 - r 1 - . . . be an increasing sequence of natural nq 1 numbers and let yn s Ý risr a i e i be a bounded block basic sequence in J p n q1 nq 1 with 5 yi 5 ) « for some « ) 0 and e¨ ery i g N. Let us assume that Ý risr ai n q1 s 0 for e¨ ery n g N. Then Ž yn . is equi¨ alent to the unit ¨ ector basis of l p . PROPOSITION 3.

Let 1 - p - `, then:

Ža. E¨ ery weakly null and normalized sequence in J p has a subsequence equi¨ alent to the unit ¨ ector basis of l p . Žb. Gi¨ en a weakly null sequence Ž z n . in Ž J p .9, with sup n 5 z n 5 ) 0 there exist a subsequence Ž zs Ž n. . and a quotient map Q : Ž J p .9 ª l p9 such that QŽ zs Ž n. . s e n , n g N. Proof. Ža. The proof is quite similar to the one of w4, Lemma 3x but some new ideas are necessary so we include it for the sake of correctness. Note that for every r g N the element Ý`isr eUi belongs to Ž J p .9 where the series converges in the topology s ŽŽ J p .9, J p .. Now let Ž x n ., with x n s Ý`is1 x n, i e i , be a normalized and weakly null sequence in J p . By induction we shall construct a subsequence Ž xs Ž n. . equivalent to a sequence Ž yn . that satisfies the hypotheses of Lemma 2. We fix « ) 0, since Ž x n . weakly converges to 0 there exists s Ž1. g N such that 5 xs Ž1., 1 e1 5 s

«

`

2

Ý eUi :

² xs Ž1. ,

and

is2

`

Ý xs Ž1., i

-

is2

« 2

.

Then we choose r 1 ) 1 such that r1

Ý xs Ž1., i is2

-

« 2

`

and

Ý

isr 1q1

xs Ž1., i e i -

1 1 and set y 1 [ yŽÝ ris2 xs Ž1., i . e1 q Ý ris2 xs Ž1., i e i . Therefore

5 y 1 y x 1 5 F 5 xs Ž1., 1 e1 5 q

r1

Ý xs Ž1., i e1 is2

q

`

Ý

isr 1q1

xs Ž1., i e i -

3« 2

.

« 2

,

325

NOTE

Assume that we have already chosen  xs Ž j. ; j F n y 14 , r 0 s 0 - r 1 - ??? - r Ž n y 1. and  y j ; j F n y 14 , satisfying the hypotheses of Lemma 2 and such that 5 xs Ž j. y y j 5 - 3 «r2 j. Once more we have that Ž ny1.q1 U :. Ž² x m , Ý ris1 e i m and Ž² x m , Ý`isr Ž ny1.q2 eUi :. m converge to 0, so we can choose s Ž n. ) s Ž n y 1. such that r Ž ny1 .q1

Ý

`

xs Ž n., i e i - «r2 n

and

Ý isr Ž ny1 .q2

is1

xs Ž n., i - «r2 n .

Now we select r Ž n. ) r Ž n y 1. satisfying `

r Ž n.

Ý isr Ž ny1 .q2

xs Ž n., i - «r2 n ,

Ý isr Ž n .q1

xs Ž n., i e i - «r2 n .

Let us define yn [ y

ž

r Ž n.

Ý isr Ž ny1 .q2

r Ž n.

/

xs Ž n., i e r Ž ny1.q1 q

Ý isr Ž ny1 .q2

xs Ž n., i e i .

Thus r Ž ny1 .q1

5 yn y xs Ž n. 5 F

xs Ž n., i e i

Ý is1

q q

ž

r Ž n.

Ý isr Ž ny1 .q2 `

Ý isr Ž n .q1

/

xs Ž n., i e r Ž ny1.q1

xs Ž n., i e i - 3 «r2 n .

Therefore with a suitable choice of « we get a sequence Ž xs Ž n. . equivalent to the basic sequence Ž yn . which, by Lemma 2, turns out to be equivalent to the unit vector basis of l p . Žb. The dual space Ž J p .9 has a Schauder basis given by Ž eUi . i G 0 where U e0 [ Ý`is1 eUi , the convergence of the series in the weak* topology. Let Ž z n . be a weakly null sequence with sup n 5 z n 5 ) 0. By a classical perturbation argument, due to Bessaga and Pelczynski, we choose a subsequence ´ U nq 1 Ž z r Ž n. . and a block basis sequence ynU s Ý risr a q1 i e i , where 0 - r 0 - r 1 n U . . . and 5 yn 5 ) « for some « ) 0 and every n, such that `

Ý 5 f n 5 5 z r Ž n. y ynU 5 - 1, ns1

326

NOTE

where Ž f n . n is a sequence of biorthogonal functionals associated to Ž ynU . n Že.g., see w10, 30.13x or w13, 1.a.12x and the proof of w13, 1.a.11x.. Then there exists an isomorphism T : Ž J p .9 ª Ž J p .9 mapping z r Ž n. into ynU , n g N Žsee the proof of w10, 30.11x.. By induction we construct a bounded block sequence Ž yn . in Ž J p . biorthogonal to Ž ynU .. Since J p does not contain l 1 we can assume, by taking subsequences if necessary, that Ž yn . is weak-Cauchy, hence Ž y 2 n y y 2 nq1 . is weakly null. By using part Ža. there is a subsequence Ž y 2 s Ž n. y y 2 s Ž n.q1 . equivalent to the unit vector basis of l p . Let i : l p ª J p be an isomorphic embedding with iŽ e n . s y 2 s Ž n. y y 2 s Ž n.q1 , n g N. Then i* : Ž J p .9 ª l p9 is a quotient map and note that ² i* Ž yU2 s Ž n. . , e m : s ² yU2 s Ž n. , i Ž e m . : s ² yU2 s Ž n. , y 2 s Ž m. y y 2 s Ž m.q1 : s dn , m , whence i*Ž yU2 s Ž n. . s e n , n g N. Therefore Q [ i* ? T : Ž J p .9 ª l p9 is the announced quotient map. COROLLARY 4. E¨ ery continuous linear operator from X into Y is compact in the following cases: Ža. X s Jq , Y s J p , with 1 - p - q - `. Žb. X s J p and Y s Ž Jq .9 with p ) q9, in particular if X s J p and Y s X 9 with p ) 2. Proof. We prove Žb., the other case is similar. Let T : X ª Y be a continuous linear operator. By contradiction if T is not compact Žand since X does not contain l 1 . there is a weakly null and normalized sequence Ž x n . in X such that 5 T Ž x n .5 ) « for some « ) 0. By using Proposition 3 twice there exist a subsequence Ž xs Ž n. ., an imbedding i : l p ª J p , and a quotient Q : Ž Jq .9 ª l q9 such that Q ? T ? iŽ e n . s QŽT Ž xs Ž n. .. s e n , n g N, hence Q ? T ? i is a non-compact mapping from l p into l q9 , p ) q9, a contradiction with Pitt’s theorem. Remark 5. The result above cannot be obtained just from the well known fact that Jr is hereditarily l r Ži.e., every closed subspace of Jr contains a copy of l r .. Indeed, Neidinger w14, Theorem 2.2x proved that for every 1 - p - q - r - ` the canonical inclusion from l p into l q , which is not compact, factorizes through an hereditarily l r space. As a consequence of Corollary 4 we can readily compute the dual and ˆp J p and J p m ˆps J p if p ) 2. bidual of J p m

ˆp J p .0 ' Ž J p .0 m ˆp Ž J p .0 and Ž J p m ˆps J p .0 COROLLARY 6. If p ) 2, Ž J p m s ˆp Ž Jp .0. ' Ž J p .0 m

327

NOTE

Proof. By Corollary 4 and since J p has the approximation property Žin ˆp J p is K Ž J p , Ž J p .9. ' Ž J p .9 m ˆ« fact it has a Schauder basis. the dual of J p m Ž J p .9 w11, 18, 3x, moreover Ž J p .0, being isomorphic to J p , has the Radon Nikodym property and the approximation property hence the dual of ˆ« Ž Jp .9 is Ž J p .0 m ˆp Ž J p .0. The second statement readily follows since Ž J p .9 m the symmetric projective tensor product is canonically complemented in the projective tensor product and by duality arguments; see w1x.

ˆp J p is not COROLLARY 7. If p ) 2 the projecti¨ e tensor product J p m ˆps Jp . Equi¨ alently LŽ J p , Ž J p .9. isomorphic to a complemented subspace of J p m is not a complemented subspace of P Ž 2 J p .. Proof. We first prove

žJ

p

ˆp J p 0r J p m ˆp J p ( J p [ J p , m

/ ž

žJ

/

p

ˆps J p 0r J p m ˆps J p ( J p . Ž 1 . m

/ ž

/

Indeed Ž J p .0 can be written as J p [ K where the copy of J p is just the canonical embedded of this space in its bidual. Then by Corollary 6

žJ

p

ˆp J p 0 ' Ž J p [ K . m ˆp Ž J p [ K . ' J p m ˆp J p [ J p [ J p [ K, Ž 2 . m

/

ž

/

where the first component in the right hand side expression corresponds to ˆ J p in the bidual, thus the first assertion the canonical embedding of J p m of Ž1. follows since J p [ K ( J p . The second one is analogous; one just needs to use the following isomorphism, which is obtained by restricting the equivalence in Ž2. to the subspace of symmetric tensors,

ˆps Ž J p [ K . ' ž J p m ˆps Ž Jp [ K. m

Jp [ Jp [ K

/

Ž x, a . m Ž y, b . q Ž y, b . m Ž x, a . ª Ž x m y q y m x, a y q b x, ab . . Therefore Ž1. has been established. Now if X and Y are Banach spaces such that X is a complemented subspace of Y then X 0rX is also ˆp J p is a complemented subspace of complemented in Y 0rY ; thus if J p m ˆps J p we conclude by Ž1. that J p [ J p is complemented in J p , what Jp m cannot happen since J p has codimension 1 in its bidual and J p [ J p has codimension 2. To show the second assertion, since Ž J p .0 is isomorphic to J p we get that ˆp Ž J p .0 is not a complemented subspace of Ž J p .0 m ˆps Ž J p .0 and Ž J p .0 m Ž Ž . . consequently L J p , J p 9 cannot be isomorphic to a complemented subspace of P Ž 2 J p .. 2 In the second part of this note we consider nuclear power series spaces ˆp L t Ž a . is L t Ž a ., t s 1 or `, and, as a main result, prove that L t Ž a . m

328

NOTE

ˆps L t Ž a . if and only if L t Ž a . is stable isomorphic to a subspace of L t Ž a . m Ži.e., isomorphic to its square.. We recall some definitions. Let a s Ž a n . be an increasing sequence of positive numbers such that sup nŽlog nra n . ` Žrespectively lim nŽlog nra n . s 0., and the infinite Žrespectively finite. type power series space L`Ž a . Žrespectively L 1Ž a .. is defined as the Kothe ¨ echelon space,

½

L `Ž a . [ Ž x i . ; 5 Ž x i . 5 k s

`

Ý < x i
5

,

respectively

½

L 1Ž a . [ Ž x i . ; 5 Ž x i . 5 k s

`

Ý < x i
5

.

Power series spaces were introduced by Grothendieck in w8x; more details can be seen in w7x. The conditions relating Ž a n . with Žlog n. are equivalent to the nuclearity of L t Ž a .. Moreover the fact that Ž a n . is increasing is equivalent to the regularity of the canonical basis Ži.e., 5 e nq1 5 kq 1r5 e nq1 5 k G 5 e n 5 kq 1r5 e n 5 k for every k, n g N. and this is important in order to get Lemma 9 below. LEMMA 8.

Let t be equal to 1 or to `, then:

ˆp L t Ž a . is isomorphic to L t Ž s . where s s Ž sm . is an Ža. L t Ž a . m increasing rearrangement of the double sequence Ž a i q a j . i, j g N . Moreo¨ er Ž sm . can be chosen in such a way that when the element sm coincides with 2 a n for some n then m G n2 . ˆps L t Ž a . is isomorphic to L t Žg . where g s Žgm . is an Žb. L t Ž a . m increasing rearrangement of the family Ž a i q a j . i F j . Moreo¨ er Žgm . can be constructed in such a way that when the element gm coincides with a 1 q a n for some n G 2 then m F n2r2. ˆp L t Ž a . is Proof. Ža. It is quite known Žsee w2, part 2x. that L t Ž a . m isomorphic to the space of all double sequences Ž x i, j . such that 5Ž x i , j . 5 k [

`

Ý

< x i , j < a k Ž i , j . - `,

;k g N,

i , js1

where a k Ž i, j . s expŽ k a i .expŽ k a j . if t s ` and a k Ž i, j . s expŽya irk .expŽya jrk . if t s 1. Therefore it is isomorphic to the power series space associated to the increasing rearrangement of Ž a i q a j . i, j . Now, when ordering Ž a i q a j . i, j increasingly we observe that 2 a n G a i q

NOTE

329

a j , for every i, j F n hence the index m such that sm s 2 a n can be taken to be greater than or equal to n2 . ˆps L t Ž a . is the subspace of L t Ž a . m ˆp L t Ž a . Žb. The space L t Ž a . m spanned by the elements Ž e i m e j q e j m e i . iF j and this is readily checked to be isomorphic to the space of generalized sequences Ž x i, j . iF j such that Ý iF j < x i, j < a k Ž i, j . - `, thus it is isomorphic to the power series space associated to the increasing rearrangement of Ž a i q a j . iF j . In this case, given n g N, all the elements preceding a 1 q a n are selected only from the set of elements with the form a i q a j with i F j F n y 1, hence if gm s a 1 q a n with n G 2 we have m F nŽ n y 1.r2 q 1 F n2r2. The following lemma is well known. Part Ža. follows from the properties and estimates of the diameters of Kolmogorov Žsee w7, I, 6.2.2, 6.3.2x.. Part Žb. is a direct consequence of the property of quasi-equivalence of regular bases of nuclear Frechet spaces w7, III, 2x. ´ LEMMA 9.

Let t be equal to 1 or `.

Ža. If L t Ž b . is isomorphic to a subspace of L t Ž a ., then sup n a nr bn - `. Žb. L t Ž a . is stable if and only if sup n a 2 nra n - `. Now we come to our main result. THEOREM 10. Let L t Ž a . be a power series space with t s 1 or `. The following conditions are equi¨ alent: Ž1. L t Ž a . is stable. ˆp L t Ž a . is isomorphic to L t Ž a . m ˆps L t Ž a .. Ž Equi¨ alently Ž2. L t Ž a . m 2 the space of 2-homogeneous polynomials P Ž L t Ž a .. is isomorphic to LŽ L t Ž a ., Ž L t Ž a ..9... Ž3. Ž4.

ˆp L t Ž a . is isomorphic to a subspace of L t Ž a . m ˆps L t Ž a .. Lt Ž a . m ˆp L t Ž a . is stable. Lt Ž a . m

Proof. Part Ž1. implies Ž2. was observed in the Introduction. ŽThe equivalence stated in Ž2. is straightforward.. Part Ž2. implies Ž3. as well as Ž1. implies Ž4. are obvious, so only Ž3. implies Ž1. and Ž4. implies Ž1. need a proof. Both cases are quite similar. Ž3. « Ž1.. By hypothesis and by using the notation of Lemma 8 we have that L t Ž s . is a subspace of L t Žg .. From Lemma 9Ža. there exists M ) 0 such that gm rsm - M for every m g N. We shall prove that sup n a 2 nra n - ` and then apply Lemma 9Žb.. Along this proof given any s g N we denote by I Ž'2 s . the integer part of '2 s. Given any n g N, n G 2, we

330

NOTE

take m such that sm s 2 a n . By Lemma 8Ža. we have that m G n2 . Now we take m 0 such that gm 0 s a 1 q a IŽ '2 n. . By 8Žb. 2

m 0 F Ž I Ž '2 n . . r2 F n2 F m, whence gm G gm 0 . Then

a IŽ'2 n . an

F2

gm 0 sm

F2

gm sm

F 2 M,

for every n g N. Now observe that 2 n F I Ž'2 I Ž'2 I Ž'2 n... for n G 3, thus

a2 n an

F

a IŽ'2 n . a IŽ'2 I Ž'2 n .. a IŽ'2 I Ž'2 Ž I Ž'2 n ... F 8 M 3, an a IŽ'2 n . a IŽ'2 I Ž'2 n ..

n G 3.

To check that Ž4. implies Ž1. we use that s 2 m rsm is bounded Žby Lemma 9Žb.. and that the index m such that sm s a 1 q a n can be selected to satisfy m F n2 . Then given n g N we take m and m 0 such that sm s 2 a n and sm 0 s a 1 q a 2 n . Then m 0 F Ž2 n. 2 F 4 m. Therefore

a2 n an

F2

sm 0 sm

F2

s4 m sm

s2

s 2 m s4 m sm s 2 m

.

This finishes the proof.

ACKNOWLEDGMENTS Part of this note was obtained during a stay of the author at the University of Trier Žpartially supported by the Junta de Andalucıa, ´ Spain.. He gratefully thanks S. Dierolf for her kind hospitality and for many hours of fruitful conversations.

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331

6. S. Dineen, Holomorphic functions and the ŽBB.-property, Math. Scand. 74 Ž1994., 215]236. 7. E. Dubinsky, The structure of nuclear Frechet spaces, in ‘‘Lecture Notes in Mathemat´ ics,’’ Vol. 720, Springer-Verlag, BerlinrHeidelbergrNew York, 1979. 8. A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Mem. Amer. Math. Soc. 16 Ž1955.. 9. H. Huser, ‘‘Lokalkonvexe Topologien auf Raumen n-linearer Abbildungen und n-ho¨ ¨ mogener Polynome,’’ Dissertation, Trier, September 1994. 10. G. J. O. Jameson, ‘‘Topology and Normed Spaces,’’ Chapman & Hall, London, 1974. 11. H. Jarchow, ‘‘Locally Convex Spaces,’’ Teubner, Stuttgart, 1981. 12. J. Lindenstrauss and C. Stegall, Examples of separable spaces which do not contain l 1 and whose duals are non-separable, Studia Math. 54 Ž1975., 81]105. 13. J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces, I,’’ Springer-Verlag, BerlinrHeidelbergrNew York, 1977. 14. R. D. Neidinger, Factoring operators through hereditarily-l p spaces, in ‘‘Banach Spaces, Proceedings, Missouri, 1984’’ ŽN. Kalton and E. Saab, Eds.., pp. 116]128, Lecture Notes in Math., Vol. 1166, Springer-Verlag, BerlinrHeidelbergrNew YorkrTokyo, 1985. 15. P. Perez Carreras and J. Bonet, Barrelled locally convex spaces, in ‘‘North-Holland ´ Math. Studies,’’ Vol. 131, North-Holland, Amsterdam, 1987. 16. R. Ryan, ‘‘Applications of Topological Tensor Products to Infinite Dimensional Holomorphy,’’ Thesis, Trinity College, Dublin, 1980.