Multilinear Forms, Subsymmetric Polynomials, and Spreading Models on Banach Spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 202, 379᎐397Ž. 1996 ARTICLE NO. 0322

Raquel GonzaloU

Departamento de Analisis´´ Matematico, Facultad de Matematicas, ´ Uni¨ersidad Complutense, Madrid, 28040, Spain

View metadata, citation and similar papers at core.ac.uk brought to you by CORE Submitted by William F. Ames provided by Elsevier - Publisher Connector Received May 15, 1995

Some applications of Ramsey theory to the study of multilinear forms and

polynomials on Banach spaces are given, related to the existence of lower lq-estimates of sequences. We give an explicit representation of subsymmetric polynomials in Banach spaces with subsymmetric basis. Finally we apply our results to get approximation of polynomials by subsymmetric polynomials, and to the problem of stabilization of a certain chain of algebras generated by polynomials. ᮊ 1996 Academic Press, Inc.

INTRODUCTION

The aim of this paper is to give some applications of the Ramsey theory to the study of polynomials and multilinear forms on Banach spaces. It will be first applied to the connection between the weak sequential continuity of polynomials, and more generally multilinear forms, and the existence of certain estimates in the sequences. This was first studied by Pelczynskiwx 21 and later on inwx 2, 5, 14 with results along the same lines. Recently, the theory of ‘‘spreading models’’ has been introducedwx 11, 12 in the study of weak sequential continuity of polynomials, with some applications to polynomial reflexivity on Banach spaces. By using Ramsey theory it is possible to obtain for a given polynomial a new polynomial now defined on the ‘‘spreading model’’ and which preserves some properties of the original one. This construction was done inwx 15 . We here generalize this result to multilinear formsŽ. Theorem 1.4 . An interesting consequence of the results

U Partially supported by DGYCIT Grant PB 93-0452. E-mail address: raquel26@eucmax. sim.ucm.es.

379

0022-247Xr96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 380 RAQUEL GONZALO of the first section is Corollary 1.6 which shows that a condition to ensure that all polynomials until a certain degree N are weakly sequentially continuous is the non-existence of lower estimates of power strictly bigger than N in the sequences of the space. An analogous result to this one was obtained inwx 3, 11 with estimates not in the space but in the ‘‘spreading model,’’ which is less general. The second section is totally devoted to the study of subsymmetric polynomials. Nemirovski and Semenovwx 19 introduced these polynomials Ž.they called them ‘‘standard’’ polynomials on l p-spaces. We here investi- gate this kind of polynomials in a more general context: Banach spaces with subsymmetric basis. The main result in this section is Theorem 2.1 that gives the explicit representation of subsymmetric polynomials as linear combinations of some elementary subsymmetric polynomials that are described. Since there is a finite amount of elementary subsymmetric polynomials, in particular, the vector space of subsymmetric polynomials of a certain degree is finite dimensional. In the proof of this result we use the Generalized Rademacher Functions that were introduced inwx 4 and also used inwx 5 . An easy proof of the characterization of Banach spaces which are Cϱ-smoothwx 8, 15 is given as a consequence of the representation theorem of subsymmetric polynomials. We also give an approximation result of polynomials by subsymmetric polynomials that generalizes the analogous result given inwx 19 , and more recently in wx 16 , for l p-spaces. In the last section we apply the results obtained before to one problem of stabilization of algebras of polynomials that appears inwx 19 . There, they proved that in l p-spaces the chain of algebras generated by polynomials of a fixed degree does not stabilize. We extend this result to a wider class of Banach spaces which includes all superreflexive spaces.

1. MULTILINEAR MAPPINGS AND LOWER ESTIMATES

Let X be a Banach space over ދ Ž.where ދ s ޒ or ރ and let 1-q-ϱ. Recall that a sequence Ä4un in X is said to have a lower q-estimate if there exists a constant C ) 0 such that

kk1rq <

for every x1,..., xk gދ and every k g ގ. Now if X1,..., XN are Banach spaces over ދ we say that a sequence Ä4 ÄŽ1 N.4 unnnsu,...,u in X1= иии = XNhas a lower estimate if there exists POLYNOMIALS AND SPREADING MODELS 381 a constant C ) 0 such that

kkk 1N11 NN ÝÝÝ<

1 N for all xnn,..., x gދ; ns1,...,k; kgގ. Note that from the Holder¨ inequality it follows that if each sequence Ä i 4 Ž. uniihas a lower p -estimate in Xis1,...,N and 1rp1q иии q1rpNs Ä4 ÄŽ1 N.4 1, then the sequence unnns u ,...,u has a lower estimate. On the ÄŽ.4 other hand, if X1 s иии s XNnns X, then the sequence u ,...,u has a N lower estimate in X if and only if the sequence Ä4un has a lower N-estimate in X. Next we give a multilinear version of Proposition 1.9 inwx 15 .

LEMMA 1.1. Let X1,..., XN be Banach spaces with unconditional bases Ä 14ÄN4 that will be denoted by enn,..., e , respecti¨ely. Then the following condi- tions are equi¨alent

Ž.1 There exists a bounded N-linear form A on X1 = иии = XN such that, Ž 1 N . for all integer n, we ha¨eAe< nn,...,e < G1. 1 N Ž.2 The sequenceÄŽ enn,...,e.4 has a lower estimate.

Proof. To proveŽ. 2 implies Ž. 1 we only have to define the N-linear form

nn n Axe11,..., xeNN x1иии x N. ž/ÝÝii i i s Ýi i is1 is1 is1

Now to proveŽ. 1 implies Ž. 2 we consider the case that ދ s ރ. Otherwise we work, as it is done inwx 15 , with the complexified spaces and with the extension of the multilinear mapping. As inwx 5 , we will use the N-Gener- Ä Ž.4 alized Rademacher Functions rtn for N G 2. These are complex-valued functions defined on the unit interval and satisfying

1 1, if i1 s иии s iN HrtiiŽ.иии rtdt Ž. s 1 N ½ 0 0, otherwise.

Ž 1 N. Now we choose ⑀ng ރ with <<⑀ns 1 and A ⑀nne ,...,⑀ nne s 382 RAQUEL GONZALO

1 N < AeŽ nn,...,e .<. Then

kk 1N 11 NN ÝÝ<

The above lemma can be used to obtain isomorphic copies of lϱ in the space of multilinear mappings, as we see in the following corollary. The same ideas of Theorem 3 inwx 10 give, in fact, this more general result.

COROLLARY 1.2. Let X1,..., XN be Banach spaces which haë uncondi- Ä 14ÄN4 tional bases enn,..., e , respectiëly. Assume that there exists a bounded 1N multilinear form A on X1 = иии = XNnn such thatÄ AŽ e ,...,e.4 does not ŽN. conërge to zero. Then, the space L X1 = иии = XN of bounded N-linear forms on X1 = иии = XN contains an isomorphic copy of lϱ. Proof. It follows by Lemma 1.1 that some subsequence Äe1 4Ä,..., eN4 nnii has a lower estimate. We may then consider the isomorphism from lϱ into ŽN .Ž. LX1 =иии = XNbgiven by TbsA, where

ϱϱ ϱ 11 NN 1 N AxebiiiiinnÝÝ,..., xe s Ýbx иии x ž/ii is1 is1 is1

Ž. for b s bi g lϱ. In the polynomial case we have the following result.

COROLLARY 1.3. Let X be a Banach space with unconditional basisÄ4 en Ä Ž.4 such that there is an N-homogeneous polynomial P on X ¨erifying that P en does not con¨erge to zero. Then

N lϱ; PŽ.X . POLYNOMIALS AND SPREADING MODELS 383

In next theorem we extend multilinear continuous mappings to the ‘‘spreading models’’ associated to certain sequences, by using Ramsey theory. Recall that if Ä4un is weakly null and a normalized sequence in X the ‘‘spreading model’’ built over Ä4un Ž.or a subsequenceŽ seewx 6. is a Banach space F with subsymmetric basis Ä4en , called the fundamental sequence of the spreading model, and verifying that if x1,..., xk gދ, kgގ, kk

ÝÝxeii s lim xui n . n-иии -n i is1 F 1 k is1

1 N THEOREM 1.4. Let X1 = иии = XNnn be Banach spaces andÄ u 4Ä,..., u 4 weakly null normalized sequences in X1,..., XN , respecti¨ely, which admit 1 N spreading models F1,...,FNnn with unconditional basisÄ e 4Ä,..., e 4, respecti¨ely. Let A be an N-linear continuous form from X1 = иии = XN into ދ. U Then, there is an N-linear continuous form A from F1 = иии = FN and an infinite set of integers H such that

kk k k AxeU11,..., xeNN lim Axu11,..., xuNN ÝÝii i i s Ýi nii Ýi n ž/ž/n1-иии -nk is1 is1 is1 is1 nigH

j for all xi g ދ, j s 1,...,N; is1,...,k. Proof. In order to prove it we proceed as inwx 15, Theorem 1.8 . For any 1 1 N N integer k let x1,..., xk ,..., x1 ,..., xk gދ be fixed and consider the function

kk 11 NN ⌿Ž.n1,...,nkininsAxuÝÝ,..., xu if n1- иии - nk. ž/ii is1 is1 Since ⌿ is a bounded function on the set of k-tuples of different integers, by using Ramsey theorem we get an infinite set of integers H such that

lim ⌿Ž.n1,...,nk n1-иии -nk nigH does exist. Now by a reiterative applying of Ramsey theorem and after some diagonalization procedures as it is done inwx 15 we have that there is an infinite set of integers H such that

kk lim Axu11,..., xuNN ÝÝiniii n n1-иии -nk ž/ is1 is1 nigH 384 RAQUEL GONZALO

1 1 N N does exist for any x1,..., xk ,..., x1 ,..., xk gދ and this number will be U ŽÝk 11 ÝkNN. denoted by A is1 xeii,..., is1xeii. Besides, it is easy to prove that U Ais an N-linear continuous form on F1 = иии = FN . Ä4 Remark 1.5. Suppose that X1 s иии s XNns X and u is a weakly null sequence in X which admits a spreading model F with fundamental sequence Ä4en . Given an N-homogeneous polynomial P on X with associated N-linear form A, from Theorem 1.4 we obtain an N-linear form AU, an N-homogeneous polynomial PU on F, and an infinite set H of integers, such that

kk PxeUlim Pxu. ÝÝii s i ni ž/n1-иии -nk ž / is1 is1 nigH

COROLLARY 1.6. Suppose that in a Banach space X there is no weakly null normalized sequence with a lower q-estimate. Then if N - qe¨ery polynomial of degree at most N on X is weakly sequentially continuous. Proof. Assume that there is an N-homogeneous polynomial P which is not weakly sequentially continuous and N - q. We may assume without loss of generality that P is not weakly sequentially continuous at zero. Let

Ä4xn be a weakly null sequence on which P is bounded away from zero at modulus. Let F be a spreading model built over the sequence Ä4xn Žor a subsequence if necessary. with fundamental sequence Ä4en . Then by Re- mark 1.5 it is possible to construct a polynomial PU on F which is bounded away from zero at modulus. Now by Lemma 1.1 we have that Ä4en has a lower N-estimate. It follows from Theorem 1.4 inwx 14 that there is a subsequence of Ä4xn with a lower q-estimate for all q ) N. Remark 1.7. The above corollary improves the results ofwx 3, 11 where the power 2 N for the lower estimate in the fundamental sequence of the ‘‘spreading model’’ is given. On the other hand, the power N obtained here is sharp: consider for instance the space lN where N is an even integer. Besides, if X has property WUŽ that is, every weakly null normalized sequence has an unconditional subsequence, seewx 20. , the power q s N in Corollary 1.6 can be easily obtained from the above results.

2. SUBSYMMETRIC POLYNOMIALS

Through this section X will denote a Banach space with subsymmetric basis Ä4en , i.e., an unconditional basis in X which is equivalent to each of POLYNOMIALS AND SPREADING MODELS 385 its subsequences. It is knownŽ seewx 17. that there is an equivalent norm on Xverifying that if x1,..., xniigދand n is an integer and ⑀ g ދ, <<⑀ F 1, then

nn ⑀xe xe if k - иии - k . ÝÝiiki s ii 1 n is1 is1

We always consider this norm on X and Ä4en a normalized and subsymmetric basis. An N-homogeneous polynomial on X, P, is said to be a subsymmetric polynomial if for all x1,..., xn gދ and all integers n we have

PxeÝÝii sPxeik if k1- иии - kn. ž/ž/i is1 is1 Note that this definition coincides with the definition of standard polynomials introduced by Nemirovski and Semenov inwx 19 . In order to give the explicit representation of subsymmetric polynomials it is essential to introduce an index which is associated to the basis and that we call the polynomial degree of Ä4en defined by

degÄ4enns minÄ4p g ގ: Ä4e has a lower p-estimate

Ž.where the minimum of the empty set is infinite ; the reason to denote this number by polynomial degree is that bywx 13 it can be easily proved that

degÄ4enns minÄ4m g ގ; Ä4e is not Pm-null .

Recall that a sequence Ä4enNis said to be P -nullŽ seewx 13, 15. if for all ÄŽ.4 M-homogeneous polynomials P with M F N the sequence Pen converges to zero. If two bases are equivalent, then both have the same polynomial degree. Of course, the reciprocal is not true; however, the polynomial degree equal to 1 implies that the basis is equivalent to the unit vector basis of l1. Next we see what the polynomial degree is in some of the most well-known spaces with subsymmetric basis: Ž. 1 For the spaces l p where 1 F p - ϱ the polynomial degree of the basis coincides with

p if p is integer N s ½ wxpq1 otherwise, where wxp is the greatest integer less than p. 386 RAQUEL GONZALO

Ž. 2 For the space c0 , the polynomial degree of the basis is r s ϱ since the unit vector basis in this space has all the upper estimates and so it has no lower estimates.

Ž.3 In the case of Orlicz sequence spaces lM associated to an Orlicz function M, the polynomial degree of the basis can be obtained fromwx 13 ; so, we have that

MtŽ. degÄ4en s min N g ގ; inf N ) 0. ½5t t gwx0,1

Ž.4 For Lorentz sequence spaces dwŽ.,p with 1 - p - ϱ, associated to a sequence Ä4wn we do not know the exact polynomial degree of the basis; however, a lower bound for this number is found inwx 13 . There, it is proved that if

qsinfÄ4s G 1; Ä4wnsg l

Ä4 UU then deg en G qpwhere p is the conjugate of p. So, in the particular Ž. case that q s 1 for instance if wn s 1rn , the polynomial degree of the basis is infinite. Once we have introduced the polynomial degree we can prove the main result of this section that is the following theorem:

THEOREM 2.1. Let X be a Banach space with subsymmetric basisÄ4 en Ä4 which has polynomial degree r s deg en . If N G r then e¨ery N-homogeneous subsymmetric polynomial can be expressed as a linear combination of elementary N-homogeneous subsymmetric polynomials on X that are

ϱ ii1 s Pxei,...,innÝÝsx nnиии x , 1sž/ 1 s ns1 n1-иии -ns

where i1 q иии qi sjs N and each i G r. If N - r then there is no non-tri¨ial N-homogeneous subsymmetric polynomials. Proof. The proof of this theorem will be done in the complex case; for the real case, we consider the complexified space of X and the polynomial extension to this complex space, and we proceed in the same way. POLYNOMIALS AND SPREADING MODELS 387

First note that for each i1,...,is gގ, verifying that i1 q иии qi s s N Ä4 and that all i jnG deg e s r the following expressions

ϱ ii1 s Pxei,...,innÝÝsx nnиии x 1sž/ 1 s ns1 n1-иии -ns

Ýϱ are well defined; indeed, for x s ns1 xennwe have

ϱϱϱ ii1 s Pxei,...,innnÝÝÝF<

ϱϱi1rrisrr ϱN <

Aei1,...,eis Aei1,...,eis . Ž.1snns Ž.1 s

If we write the expression of P in terms of the coordinates, then if x1,..., xk gދ we have that

k N ! Pxe xiiii1иии xAes 1,...,e s ÝÝjj s Ýn1 ns Ž. n1 ns ž/j 1 i иии i Nni1 !иии i s! -иии -n k s 1q q ss 1 sF N! ii1 s ii1 s sÝÝAeŽ.1,...,exsnnиии x . i!иии i ! 1s i1q иии qissNn1 s 1-иии -nsFk Ž.1

Ži1 is. The only thing that we have to prove now is that Ae1 ,...,es s0 Ä4Ä4 Ä4 whenever min i1,...,isn-deg e . Assume that i11s min i ,...,is, other- Ži1 is. wise we proceed in the same way. If

i1 Ž i1 i s . ⑀ g ރ such that ⑀ Ae1 ,...,es s␣. Then

kk ii ␣ <<11 << ii12 is ÝÝxjjjs xAeŽ.,ekq2,...,ekqs js1js1 k ii <<1⑀ 1ii2 s sÝxAjjkž/Ž.e ,eq2,...,ekqs js1

i k 1 1 ⑀ii2 s sHAÝ xrjjŽ. t e j,e kq2,...,edtkqs 0ž/j1 ž/s

ki1

FCA55Ý xejj , js1 where ÄrtnŽ.4is the sequence if i1-Generalized Rademacher Functions if i1 G 2 and C is the unconditional constant of the basis; note that in the case of i1 s 1 the same inequality holds by using the linearity of Ž i2 i s . Ä4 A., ekq2 ,...,ekqsj. Therefore e would have a lower i1-estimate and since i1 - degÄ4en that is not possible. Now the result follows fromŽ. 1 , since for all k g ގ k N ! ii1 s PxeÝÝjj s AeŽ.1,...,es ž/j 1 i иии i N ; i r i1 !иии i s! s 1q q sjs G

k =Pxe i1,...,ijjsÝ ž/j1 s then P can be expressed as a linear combination of the polynomials P as we required. i1,...,is Note that from this representation it follows that the vector space of N-homogeneous subsymmetric polynomials is finite dimensional. As a consequence of the above theorem we can give an easy proof of the characterization of real Banach spaces with subsymmetric basis which have a separating polynomial that was given inwx 15 . Recall that a polynomial P is separating if it verifies that PŽ.0 s 0 and Px Ž .G1if 55x s1. COROLLARY 2.2. Let X be a real Banach space with subsymmetric basis

Ä4en which has a separating polynomial. Then X is isomorphic to lr , where Ä4 rsdeg eisanen ¨en integer.

Proof. First, if r s 1 then the space X is isomorphic to l1 which does not have a separating polynomial, hence r ) 1 and Ä4en is weakly null. We POLYNOMIALS AND SPREADING MODELS 389 may assume that X has a 2k-homogeneous separating polynomial P Žsee, e.g.,wx 9, p. 209. . From Remark 1.5, since the spreading model of a weakly null subsymmetric basis coincides with itself, it easily follows that there is a U 2k-homogeneous subsymmetric separating polynomial P on X. Since Ä4en is weakly null, we have that r F 2k and now we can apply Theorem 2.1 obtaining that

ϱ U ii1 s PxeÝÝÝnn s ␣iиии innxиии x . ž/ 1sž/1s ns1 i1q иии qiss2 k n1-иии -ns ijGr

Then,

ϱϱ2 k U ii1 s ÝÝÝÝxenn FPxenn F <<␣iиии inn <<<

ϱϱ ϱ2 krr Cx<

since the basis has a lower r-estimate. Therefore, Ä4en has also an upper r-estimate and is equivalent to the unit vector basis in lr where r must be an even integerŽ see, e.g.,wx 7. as we required. Next, combining Ramsey theory and Theorem 2.1, we give a generaliza- tion of the result obtained by Nemirovskiwx 19 and also by wx 16 about the approximation of polynomials by subsymmetric polynomials on l p-spaces for 1 F p - ϱ.

THEOREM 2.3. Let X be a Banach space with subsymmetric basisÄ4 en . If PisanN-homogeneous polynomial on X, there is an N-homogeneous subsymmetric polynomial PU such that for each ⑀ ) 0 there exists an infinite set of integers H such that

5 U 5 P y P X H F ⑀ ,

Ä4 where XHn is the closed subspace generated by e ; n g H . Proof. Let A be the N-linear symmetric form associated to P and consider PU the subsymmetric polynomial associated to P given by Re- mark 1.5 and AU its associated multilinear form. Then, there is an infinite 390 RAQUEL GONZALO set of integers H such that

kk k k AxeU1,..., xeNlim Axe,..., xe . ÝÝii i i s Ýinii Ýin ž/ž/n1-иии -nk is1 is1 is1 is1 nigH In order to prove the theorem we need the following fact: Fact. Let N be an integer, P an N-homogeneous polynomial with associated N-linear symmetric form A. Then, for each infinite set of X integers H and ⑀ ) 0 there exists an infinite set H ; H such that

xiiii1иии xAes 1,...,e s -⑀, Ý nnnn1 sŽ.1 s n-иии -n 1 s XH X Ä4 whenever i1,...,is gގ verify that i1 q иии qi s s N and min i1,...,is -r. To prove this fact we use an argument of induction on the degree of the polynomial. In the case that N s 1, a 1-homogeneous polynomial is a U U linear form x g X . Then, for r s 1 there is nothing to prove and if r ) 1, since the basis is weakly null it is possible to choose an infinite set of X integers H ; H such that

U ÝxeŽ.n -⑀. X ngH

Assume now that the result holds for any integer M F N y 1 and let P, A, ⑀, and H be fixed, and consider i1,...,is gގ verifying that i1 Ä4 qиии qi s s N and min i1,...,is -r. Then, two cases may be given.

First Case. minÄ4i2 ,...,is -r. Then, if we consider Q1 to be the homogeneous polynomial given by the M-linear form defined by

i1 Au11Ž.,...,uMsAeŽ.11,u,...,uM, where M s i2 q иии qi s, by using the hypothesis of induction we can find Ä 14 1 an increasing sequence of integers H1 s ␣n ; H such that p11s ␣ ) 1 and

Aeiiii2,...,exs2иии x s- ⑀ 2. Ý 1Ž.nnnn2 s2 s r n-иии -n X 2 s H 1

We consider now the polynomial Q with associated multilinear form p1

AuŽ.,...,u Aei1,u,...,u p111Mps Ž.1M POLYNOMIALS AND SPREADING MODELS 391 and proceeding in the same way as above we can again find an increasing Ä 24 2 sequence of integers H2 s ␣n ; H111with ␣ ) p and such that

Aeiiii2,...,exs2иии x s- ⑀ 2.2 Ý pn12Ž. ns n2 ns r n-иии -n X 2 s H 2

2 Then, we choose p22s ␣ . So, inductively we construct increasing se- Ä␣ k4 ␣ kk␣ quences Hkns ; H ky1such that pkks ) 1) pky1and verifying

iiii2 s2 s k Ý AepnŽ.,...,ex nnnиии x - ⑀ 2; ky12 s 2 s r n-иии -n X 2 s H k Ä4 for any integer k. We consider now the diagonal set H s pk . Then, if xgXH and 55x F 1 we have that

Aeii12,e ,...,ex is i1иии x is Ý Ž.nn12 ns n1 ns n1-иии -ns nigH

ϱ i Aeii12,e ,...,ex iis 2иии xx is <<1 FÝÝŽ.pnk 2 ns n2 nsk p k 1 ž/pk-n2-иии -ns s nigH

ϱ i <

Ä4 Second Case. min i2 ,...,is Gr and i11- r or i s N - r; in the last N case since the basis is PNn-null we have that ÄAeŽ.4converges to zero and X by choosing H ; H such that

N ÝAeŽ.n -⑀ X ngH Ä4 we have the result. Otherwise if min i2 ,...,is Gr and i1 - r, then by the U Ž i1 i s . proof of Theorem 2.1 we have that Ae1 ,...,es s0 and then

lim Aei1,ei2,...,eis 0. Ž.nn2 nss n-n2-иии -ns nigH 392 RAQUEL GONZALO

We now proceed in the following way: for ⑀ ) 0 there exists N1 g H such that if n2 - иии - ns in H and n21) N , then

Aeii12,e ,...,e is ⑀; Ž.Nn12 nsF

again, for ⑀r2 there exists N21) N such that if n2- иии - nsin H and n22)Nthen

Aeii12,e ,...,e is ⑀. Ž.Nn22 nsF

In this way we construct an increasing sequence of integers Ä4Nk such that if n2 - иии - ns and n2 ) Nk we have ␧ ii12 is AeŽ.Nn,e ,...,e nF . k 2 s 2k

X Ä4 X Therefore, if H s Nkkand x g XH, x s Ý xen nof norm one we have

xiiii1иии xAes 1иии e s Ý nnnn1 sŽ.1 s n1-иии -ns X nigH ϱ ii12isi12iis FÝÝxxNnиии xAenŽ.N,en,...,en kž/2sk2 s ks1Nk-n2-иии -ns ϱϱ⑀ ϱ ii12 is F55AxÝÝ

where C ) 0 is the constant of the lower r-estimate of the basis. So, the fact is proved. Now, the polynomial P can be written as

ϱϱϱ PxePxePxe, ž/ž/ž/ÝÝÝnn s0 nn q1 nn ns1 ns0 ns0 where

ϱ N ! iiii1 s 1 s Pxe1ÝÝnn s Ýxnиии xAe nŽ. n,...,e n , ž/ i!иии i ! 1 s 1 s ns1 i1q иии qissN 1 s n1- иии -ns ijGr POLYNOMIALS AND SPREADING MODELS 393

Ä is an N-homogeneous polynomial, whenever the set i1,...,is gގ; i1 Ä4 4 qиии qi sjs N, min i G r is not empty; and

N! iiii1 s 1 s Pxe0Ž.ÝÝnns Ýx nиии xAe nŽ. n,...,e n . i!иии i ! 1 s 1 s i1q иии qissN 1 s n1- иии -ns minÄ4ij -r

By using the Fact, we have that there exists an infinite set of integers X H ; H such that 55 P0XHX-⑀.

We may assume without loss of generality that H X also verifies that if X n1 - иии - nsi, n g H , then

U X Aeii1,...,e sAe ii1,...,e s⑀ Ž.2 Ž.nn1 sy Ž.1sF for a convenient choice of ⑀ X in order to get that

5 U 5 P y P1 X HX

N ! s Ý i ! иии i ! i1q иии qissN 1 s ijGr

U = xii1иии xAes i1,...,e isAe i1,...,e is ⑀. Ýnn1sž/Ž.1 snny Ž.1 sF n1-иии -ns

So, the proof is concluded, since

5 U 5555U 5 PyPXXHHXXFP01qPyP XHX as we required. From Theorems 2.1 and 2.3 the following result follows.

COROLLARY 2.4. Let X be a Banach space with subsymmetric basisÄ4 en . Then, if ⑀ ) 0 and P is an N-homogeneous polynomial on X with N - degÄ4en there exists an infinite set of integers H such that

55 PXHF⑀. 394 RAQUEL GONZALO

3. APPLICATIONS TO SOME ALGEBRAS OF POLYNOMIALS

Let Ak be the smallest uniformly closed algebra of functions on X containing all real polynomials of degree at most k. It is clear that

иии иии A1 ; Akk; A q1; . In the next theorem we give a general condition for Banach spaces to ensure that the above chain does not stabilize. This result generalizes the analogous one obtained by Nemirovski inwx 19 in the case of l p for 1 - p - ϱ.

THEOREM 3.1. Let X be a Banach space which contains a weakly null Ä4 sequence un ¨erifying that there exists a continuous linear operator Ž. ÄŽ.4 T:Xªlpnp1-p-ϱsuch that T u is the canonical basis of l . Then the chain of algebras Ä4Ak does not stabilize.

Proof. If N G p and ⌿Npis the N-homogeneous polynomial on l defined by

ϱϱ ⌿xe Ž.x N, Niiiž/ÝÝs is1 is1 we consider the N-homogeneous polynomial on X, PNNs ⌿ (T. Without loss of generality we may assume that Ä4un is a basic sequence U in X that admits a spreading model F with subsymmetric basis Ä4enN.If P is the polynomial associated to PN given in Remark 1.5, it is easy to see Ýϱ that if x s is1 xeiithen

ϱϱ PxeU Ž.xN. Niiiž/ÝÝs is1 is1

Assume that the chain Ä4Ak does stabilize, then for some integer k we have that

Akks A ql for all integers l. In other words, every polynomial of degree greater than or equal to k can be uniformly approximated by polynomials of degree at Ä4 most k. Let ⑀ ) 0 and let N be fixed, where N G N0 s max p, k . Then, there exist polynomials of degree at most k, that we denote by

Q12,Q,...,Qnand a real polynomial of n variables RtŽ.1,...,tnsuch that

PNyRQŽ.1,...,Qn-⑀. POLYNOMIALS AND SPREADING MODELS 395

Now, by Remark 1.5, there can be found an infinite set of integers H and U U subsymmetric polynomials Q1 ,...,Qn on the spreading model F such that for any x1,..., xm gދ, mgގ, we have

mm QxeUlim Qxu jiiÝÝs jini ž/n1-иии -nm ž / is1 is1 nigH for j s 1,...,n. Then, by passing to the limit we have

UUU ⑀ PNyRQŽ.1,...,QnFH-2 ,

Ä4 where FHnis the closed subspace on F generated by e , n g H . Having in U U U mind that PN , Q1 ,...,Qn are subsymmetric polynomials the above inequality holds on F, i.e.,

UUU PNyRQŽ.1,...,Qn-2⑀.3Ž.

U To conclude, we proceed as inwx 19 : we have that all the polynomials Pn on F can be uniformly approximated by algebraic combinations of subsymmetric polynomials of degree at most k on F. By using the finite dimen- sionality of subsymmetric polynomials of degree at most k on F ŽTheorem 2.1. we can find two points in F that cannot be separated by subsymmetric U polynomials of degree at most k but can be separated by some PN for N large enough. Then, the inequalityŽ. 3 does not hold for sufficiently small ⑀, which proves the theorem. Remark 3.2. The class of Banach spaces which verify the hypothesis in Theorem 3.1 is very large and includes Ž.1 All the superreflexive spaces.

Ž.2 All reflexive spaces whose dual has the Sp-property Žseewx 14. for some 1 - p - ϱ; for instance the dual space T of the Tsirelson original space. Ž.3 Banach spaces with an unconditional basis which is not polynomi- ally nullŽ. i.e., there is an integer N such that the basis is not PN-null . All these spaces have in common the existence of a polynomial which is not weakly continuous on the unit ball. Besides, if every polynomial is weakly continuous on the unit ball, by the Stone᎐Weierstrass theorem we have that all the algebra generated by all polynomials up to a fixed degree coincides with the algebra of all weakly uniformly continuous functions on the unit ball. Therefore it is natural to ask the following question: 396 RAQUEL GONZALO

QUESTION. Will it be true that if there is a polynomial on X which is not weakly continuous on the unit ball then the chain does not stabilize?

ACKNOWLEDGMENTS

I am very grateful to J. A. Jaramillo for several valuable and helpful conversations concerning the work in this paper, and to R. Aron who posed the problem of stabilization of algebras to us. Note added in proof. Corollaries 1.2 and 1.3 should be compared with Corollary in ‘‘Bases in spaces of Multilinear forms over Banach space,’’ by V. Dimaut and I. Zaldmendo, to appear in J. Math. Anal. Appl. and Proposition 6 in ‘‘A Dvoretzky Theorem for Polynomials,’’ by S. Dineen, Proc. Am. Math. Soc. 123 Ž.1995 , 2817᎐2821.

REFERENCES

1. R. Aron, C. Herves,´ and M. Valdivia, Weakly continuous mappings on Banach spaces, J. Funct. Anal. 52 Ž.1983 , 189᎐204. 2. R. Alencar and K. Floret, Weak-strong continuity of multilinear mappings and Pelczyn- ski-Pitt theorem, to appear. 3. R. Alencar and K. Floret, Weak continuity of multilinear mappings on Tsirelson’s space, to appear.

4. R. M. Aron, J. Globevnik, Analytic functions on c0 , Re¨. Mat. Uni¨. Complut. Madrid 2 Ž.1989 , 27᎐33. 5. R. M. Aron, M. La Cruz, R. Ryan, and A. Tonge, The Generalized Rademacher Functions, Note Mat. 12 Ž.1992 , 15᎐25. 6. B. Beauzamy and Lapreste,´`´` ‘‘Modeles etales des espaces de Banach,’’ Hermann, Paris, 1984. 7. N. Bonic and J. Frampton, Smooth functions of Banach manifolds, J. Math. Mech. 15 Ž.1966 , 877᎐899. 8. R. Deville, A characterization of Cϱ-smooth Banach spaces, Bull. London Math. Soc. 22 Ž.1990 , 13᎐17. 9. R. Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, in ‘‘Pitman Monographs and Surveys in Pure and Applied Mathematics,’’ Vol. 64, Longman, Harlow, 1993. 10. V. Dimant and I. Zalduendo, Bases in spaces of multilinear forms over Banach spaces, to appear. 11. J. Farmer, Polynomial reflexivity in Banach spaces, Israel J. Math. 87 Ž.1994 , 257᎐273. 12. J. Farmer and W. B. Johnson, Polynomial Schur and polynomial Dunford Pettis properties, Contemp. Math. 144 Ž.1993 , 95᎐105. 13. R. Gonzalo, Upper and lower estimates in Banach sequence spaces, Comment. Math. Uni¨. Carolin. 36 Ž.1995 , 641᎐653. 14. R. Gonzalo and J. A. Jaramillo, Compact polynomials between Banach spaces, Extracta Math. 8, No. 1Ž. 1993 , 42᎐48. 15. R. Gonzalo and J. A. Jaramillo, Smoothness and estimates of sequences in Banach spaces, Israel J. Math. 89 Ž.1995 , 321᎐341. 16. P. Hajeck and P. Habala, Stabilization of polynomials, C. R. Acad. Sci. Paris 320 Ž.1995 , 821᎐825. POLYNOMIALS AND SPREADING MODELS 397

17. J. Lindenstrauss and L. Tzafriri, ‘‘Classical Banach Spaces, I,’’ Springer-Verlag, Berlin, 1977. 18. L. Nachbin, ‘‘Topology on Spaces of Holomorphic Functions,’’ Springer, New York, 1969. 19. A. S. Nemirovski and S. M. Semenov, On polynomial approximation of functions on Hilbert space, Math. USSR-Sb. 21 Ž.1973 , 255᎐277. 20. E. Odell, On quotients of Banach spaces having shrinking unconditional basis, Illinois J. Math. 36, No. 4Ž. 1992 , 681᎐695. 21. P. Pelczynski, A property of multilinear operations, Studia Math. 16 Ž.1957 , 173᎐182.