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

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Multilinear Forms, Subsymmetric Polynomials, and Spreading Models on Banach Spaces JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 202, 379]397Ž. 1996 ARTICLE NO. 0322 Multilinear Forms, Subsymmetric Polynomials, and Spreading Models on Banach Spaces Raquel GonzaloU Departamento de AnalisisÂÂ Matematico, Facultad de Matematicas, Â 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-esti- mates of sequences. We give an explicit representation of subsymmetric polynomi- als 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. Q 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 Q 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 C`-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 K Ž.where K s R or C and let 1-q-`. 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 <<xqCxu ž/ÝÝnnF n ns1 ns1 for every x1,..., xk gK and every k g N. Now if X1,..., XN are Banach spaces over K we say that a sequence Ä4 ÄŽ1 N.4 unnnsu,...,u in X1= ??? = XNhas a lower estimate if there exists POLYNOMIALS AND SPREADING MODELS 381 a constant C ) 0 such that kkk 1N11 NN ÝÝÝ<<xnn??? x F Cxu nn??? xunn ns1 ns1 XX1ns1 N 1 N for all xnn,..., x gK; ns1,...,k; kgN. Note that from the HolderÈ inequality it follows that if each sequence Ä i 4 Ž. uniihas a lower p -estimate in Xis1,...,N and 1rp1q ??? 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 ??? 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 = ??? = 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 x1??? x N. ž/ÝÝii i i s Ýi i is1 is1 is1 Now to proveŽ. 1 implies Ž. 2 we consider the case that K s C. 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 ??? s iN HrtiiŽ.??? rtdt Ž. s 1 N ½ 0 0, otherwise. Ž 1 N. Now we choose eng C with <<ens 1 and A enne ,...,e nne s 382 RAQUEL GONZALO 1 N < AeŽ nn,...,e .<. Then kk 1N 11 NN ÝÝ<<xnn??? x F AŽ.e nnnnnn <<<<xe,...,e xe ns1 ns1 kk 111 NN sHAennnnnnnnÝÝ<<xertŽ.,...,e <xert < Ž. dt 0ž/ ns1 ns1 kk 111 NN FH55AxrteÝÝ <nn <Ž. n ??? <xrten < nŽ. n dt 0 ns1 XX1ns1 N kk 11 NN FCxeÝÝnn ??? xen n , ns1 XX1ns1 N where C only depends on 55A and on the unconditional constants of the bases. The above lemma can be used to obtain isomorphic copies of l` 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 ha¨e uncondi- Ä 14ÄN4 tional bases enn,..., e , respecti¨ely. Assume that there exists a bounded 1N multilinear form A on X1 = ??? = XNnn such thatÄ AŽ e ,...,e.4 does not ŽN. con¨erge to zero. Then, the space L X1 = ??? = XN of bounded N-linear forms on X1 = ??? = XN contains an isomorphic copy of l`. 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 l` into ŽN .Ž. LX1 =??? = XNbgiven by TbsA, where `` ` 11 NN 1 N AxebiiiiinnÝÝ,..., xe s Ýbx ??? x ž/ii is1 is1 is1 Ž. for b s bi g l`. 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 l`; 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 gK, kgN, kk ÝÝxeii s lim xui n . n-??? -n i is1 F 1 k is1 1 N THEOREM 1.4. Let X1 = ??? = 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, respec- ti¨ely. Let A be an N-linear continuous form from X1 = ??? = XN into K. U Then, there is an N-linear continuous form A from F1 = ??? = 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-??? -nk is1 is1 is1 is1 nigH j for all xi g K, 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 gK be fixed and consider the function kk 11 NN CŽ.n1,...,nkininsAxuÝÝ,..., xu if n1- ??? - nk.
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