We construct a weak Hilbert Banach space such that for every block subspace
$Y$ every bounded linear operator on Y is of the form D+S where S is a strictly
singular operator and D is a diagonal operator. We show that this yields a weak
Hilbert space whose block subspaces are not isomorphic to any of their proper
subspaces.Comment: 32 page

Argyros, Spiros A.

Beanland, Kevin

Raikoftsalis, Theocharis

Comptes Rendus Mathematique

English

arXiv

Argyros, S

Beanland, K

Raikoftsalis, T

DSpace at NTUA

A weak Hilbert space with few symmetries

Spiros A. Argyros

Kevin Beanland

Theocharis Raikoftsalis

Crossref

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1293–1296Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comFunctional AnalysisA weak Hilbert space with few symmetriesUn espace failble de Hilbert avec peu de symétriesSpiros A. Argyros a, Kevin Beanland b, Theocharis Raikoftsalis aa Department of Mathematics, Zografou Campus, National Technical University, Athens 15780, Greeceb Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United Statesa r t i c l e i n f o a b s t r a c tArticle history:Received 8 September 2010Accepted after revision 25 October 2010Available online 18 November 2010Presented by Gilles PisierWe construct a separable Banach space Xwh with an unconditional basis that is a weakHilbert space and no block subspace is linearly isomorphic to any of its proper subspaces.We prove that the space Xwh satisfies these properties by showing it is stronglyasymptotic 2 and that every bounded linear operator on Xwh is a strictly singularperturbation of a diagonal operator with respect to the unit vector basis.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous construisons un space de Banach Xwh qui est un espace failble de Hilbertet n’admettant aucune sous-espace bloc isomorphe linéaire à une sous-espace. Nousdémontrons les propriétés de Xwh par démontrons que Xwh est fortement asymptotique 2et tout opérateur borné de Xwh soit une variation strictment singulière d’un opérateurdiagonal par rappert à la base.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionA Banach space X is called a weak Hilbert space [10] if there are positive constants δ and C such that every finitedimensional subspace E of X contains a subspace F such that dim F  δ dim E , the Banach–Mazur distance between F anddim F2 is at most equal to C and there is a projection from X to F with norm at most C . It has been shown by G. Pisier [11]that the Fredholm theory, as developed by Grothendieck, works in weak Hilbert spaces. W.B. Johnson [8] showed that the2-convexification of Tsirelson’s space is a weak Hilbert space; thus exhibiting a weak Hilbert space not containing 2. Otherconstructions related to the higher order modified Tsirelson spaces could also provide more exotic weak Hilbert spaces.In this Note we construct an example of a weak Hilbert space Xwh with an unconditional basis that has a quite asym-metric structure. In particular, every operator on a block subspace Y of Xwh is a strictly singular perturbation of a diagonaloperator on Xwh restricted to Y . This implies that no block subspace is linearly isomorphic to any of its proper subspaces.In addition, for disjointly supported block subspaces Y and Z of Xwh every operator T : Y → Z is strictly singular. Moreover,Xwh does not contain a quasi minimal subspace and (using the terminology from [6]) its basis is tight by support. In thisannouncement we will outline the argument required to show that every operator on a block subspace has the desireddecomposition.E-mail addresses: sargyros@math.ntua.gr (S.A. Argyros), kbeanland@vcu.edu (K. Beanland), th-raik@hotmail.com (T. Raikoftsalis).1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.10.0321294 S.A. Argyros et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1293–12962. Description of XwhThe definition of the space Xwh uses a type of modified mixed Tsirelson saturation method. The space Xwh is definedas the completion of c00(N) under a norm induced by a norming set of functionals denoted by D wh and described asfollows.For each n ∈ N let Sn denote the Schreier family of order n. See [1,2] for precise definitions. We note that these familiescontain only finite subsets of N and satisfy Sk ⊂ Sn for k < n. A sequence (Ei)di=1 of finite subsets of N is Sn-admissible ifE1 < E2 < · · · < Ed and (min Ei)di=1 ∈ Sn . A sequence (Ei)di=1 of finite sets of N is Sn-allowable if (Ei)di=1 are pairwise disjointand (min Ei)di=1 ∈ Sn . A sequence of vectors (xi)di=1 is Sn-allowable (resp. admissible) if (supp xi)di=1 is Sn-allowable (resp.admissible).The definition of Xwh requires that we fix two increasing sequences of positive integers (ni)∞i=0 and (mi)∞i=0 satisfyingcertain growth conditions. Let m0 = m1 = 2, n0 = 1 and for j  2 let m j+1  m3j ,  j = 3 log2(m j) + 1; n j is chosen such that j(n j−1 + 1) < n j . Now fix infinite disjoint subsets N1 and N2 such that N = N1 ∪ N2. Let,Σ = {(Ei,2 ji)ni=1: Ei ∩ E j = ∅ and j1 < j2 < · · · < jn with j1 ∈ N1 and ji ∈ N2 for i > 1}.Let σ : Σ → N2 be an injection satisfying the following growth condition:m2σ ((E1,2 j1),...,(Ei+1,2 ji+1)) > m2σ ((E1,2 j1),...,(Ei ,2 ji)) · (maxsupp Ei)2.We need the following two definitions:Definition 1. Let D ⊂ c00(N), m > 1 and n ∈ N. We say that D is closed in the modified 2 − (1/m, Sn) operation if for every( f i)di=1 ⊂ D such that ( f i)di=1 is Sn-allowable and (λi)di=1 ∈ Ba(2) the vector1md∑i=1λi f i ∈ D.Let ω( f ) = m (weight of f ) whenever f is the result of the above operation.Definition 2 (σ -special sequences).(1) A sequence (Ei,2 ji)∞i=1 is σ -special if j1 ∈ N1 and for each i  1,σ((E1,2 j1), . . . , (Ei,2 ji)) = ji+1.(2) A σ -special sequence (Ei,2 ji)pi=1 is a Sn2 j+1 σ -special sequence if (min Ei)pi=1 ∈ Sn2 j+1 and j1 > j + 1.(3) ( f i)pi=1 ⊂ c00(N) is a σ -special sequence of functionals (resp. Sn2 j+1 σ -special sequence of functionals) if there exists aσ -special sequence (Ei,2 ji)pi=1 (resp. Sn2 j+1 σ -special sequence) such that supp f i ⊂ Ei and ω( f i) = m2 ji for each1  i  p.We are now ready to define D wh as follows:Definition 3. The norming set D wh is the minimal subset of c00(N) such that(1) {±e∗n: n ∈ N} ⊂ D wh .(2) D wh is closed under 2 − (1/m2 j, Sn2 j ) operations for all j ∈ N.(3) D wh is closed under 2 − (1/m2 j+1, Sn2 j+1) operations for all j ∈ N on Sn2 j+1 σ -special sequences of functionals.The space Xwh is the completion of c00(N) under the norm induced by D wh . Namely, for x ∈ c00(N) let‖x‖ = sup{∣∣ f (x)∣∣: f ∈ D wh}.Finally, note that the norm satisfies‖x‖ = max{sup{‖x‖ j: j ∈ N ∪ {0}},‖x‖∞}where for each j ∈ N, ‖ · ‖ j satisfies the following implicit formulas:S.A. Argyros et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1293–1296 1295‖x‖2 j = sup{1m2 j(k∑i=1‖Ei x‖2) 12: (Ei)ki=1 is Sn2 j -allowable}, (1)‖x‖2 j+1 = sup{1m2 j+1(k∑i=1‖Ei x‖22 ji) 12: (Ei,2 ji)ki=1 is a Sn2 j+1 σ -special sequence}. (2)From the above, it is easy to see that the unit vector basis of Xwh is unconditional.3. Properties of XwhIn order to verify that Xwh is a weak Hilbert space we will use a sufficient condition due to N.J. Nielsen and N. Tomczak-Jaegermann [9]. By applying results of Johnson, they showed that a space with a basis is weak Hilbert whenever there is aC > 0 such that every sequence of n vectors that is disjointly supported after n is C-equivalent to the unit vector basis ofn2. This property has been called strongly asymptotic 2 and has been studied by several authors in more general settings[5,6,12]. The next proposition shows that Xwh has this property.Proposition 4. The basis (ei)i∈N of Xwh is strongly asymptotic 2 . In particular, for every sequence of disjointly supported vectors(xk)dk=1 such that d  supp xi for i ∈ {1, . . . ,d},1m2(d∑k=1‖xk‖2) 12∥∥∥∥∥d∑k=1xk∥∥∥∥∥ (d∑k=1‖xk‖2) 12.The lower inequality follows from Eq. (1) for j = 1. The proof of the upper inequality relies on induction on the heightof the tree analysis of a functional. This notion is central in constructions related to saturation methods (see, for example,[3]) and it essentially describes the “history” of the functional, namely how it is built through the application of the 2-operations. The key point in this proof is that the coding function is chosen with respect to the sets ‘Ei ’ and not the specificfunctionals ‘ f i .’As in all constructions of this type, there are several technical obstacles to overcome in order to verify that the space ofoperators on Xwh has the desired properties. In the present case, however, the need to evaluate norms with 2 structureand the fact that the norming functionals are built from others with disjoint and not successive supports requires newtechniques for calculating norms. The methods developed for evaluating norms in this modified mixed Tsirelson setting maybe of independent interest. It is also worth mentioning that in this context the standard and powerful tool for reducing thecomplexity of calculations (i.e. the basic inequality) is not, and perhaps cannot, be used.We will state the main proposition required to verify that the operators on block subspaces have the desired decompo-sition. To do so we recall a few more definitions.The crucial notion needed is that of a rapidly increasing sequence (RIS) of vectors. In our case, the vectors that form a RISare seminormalized 2-(ε,n) averages for ε > 0 and n ∈ N. These averages have been defined previously in the papers [2,4].Each RIS (xn)n is associated with a constant C and an increasing sequence ( jk), where jk+1 is chosen inductively withrespect to the support of xk . It is routine to show that every block subspace contains a RIS. Another element we need is thenotion of a diagonal free operator which is given in the following:Definition 5. Let X be a Banach space with a Schauder basis (en)n and T : X → X be a bounded linear operator. T is calleddiagonal free if e∗n(T en) = 0, for all n ∈ N.The final step in our work is to show that every diagonal free operator T ∈ L(Y ), where Y is a block subspace of Xwh isstrictly singular. This actually follows from the fact that for every RIS (xn)n in Y , limn T xn = 0.In order to show that the diagonal free operators converge in norm to 0 on RIS’s, we use the following.Given a (C, (is))-RIS (yn) one can inductively build a further block sequence (xn) that satisfies the following:(a) For j ∈ N with j + 1 < j1 ∈ N1 there is a sequence (Ei)∞i=1 such that (Ei,2 ji)∞i=1 is σ -special with (⋃i∈N Ei) ∩(⋃i∈N supp xi) = ∅.(b) The sequence (xk) is seminormalized and for each k ∈ N, xk is a 2-(1/m32 jk ,n2 jk ) average.Such a sequence (xk) is called a (0, C,2 j + 1) dependent sequence with respect to (Ei,2 ji)∞i=1.The critical and most technically demanding inequality that is used for proving the decomposition property of operatorson block subspaces and its asymmetrical structure is the following:Proposition 6. Let j ∈ N, C > 0, and (xk)dk=1 be a (0, C,2 j + 1) dependent sequence and let∑dk=1 bkxk be a (1/m22 j+2,n2 j+1)average. Then,1296 S.A. Argyros et al. / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1293–1296∥∥∥∥∥d∑k=1bkxk∥∥∥∥∥  Cm22 j+1 .Granting the above, one can show that a diagonal free operator goes to zero along an RIS. The proof essentially reduces(using a technical counting argument from [7]) to showing that for every RIS (xn)n and every partition Cn, Bn of supp xn ,limn Cn T Bnxn = 0. This argument relies heavily on Proposition 6; we give an outline of the proof below.Proposition 7. Let Y be a block subspace of Xwh generated by (yn)n and T : Y → Y be a diagonal free operator with respect to (yn)n.Let also (xn)n be a RIS in Y , then T xn → 0.Proof. Suppose, toward a contradiction, that the conclusion fails. Then, from the preceding discussion by passing to asubsequence if necessary, there exists ε > 0 such that ‖Cn T Bnxn‖ > ε for every n ∈ N. It is easy to see that (Bnxn) is a(C, (2in))-RIS. For each n ∈ N let fn ∈ D wh such that fn(Cn T Bnxn) > ε and supp fn ⊂ Cn . Choose j ∈ N such that 1m2 j+1 <ε‖T ‖C . Next we construct sequences (zk)∞k=1, (gk)∞k=1 and (Ei,2 ji)∞i=1 such that,(1) (zk)∞k=1 is a block subsequence of (xn).(2) (zk)∞k=1 is a (0, C,2 j + 1) dependent sequence with respect to (Ei,2 ji)∞i=1.(3) (gk)∞k=1 ⊂ D wh is σ -special with respect to (Ei,2 ji)∞i=1.(4) gk(T zk)  ε for all k ∈ N.Given this, we arrive at a contraction in the following way: Find d ∈ N such that (min Ei)di=1 is a maximal element ofSn2 j+1 . There is a sequence (bk)dk=1 ∈ Ba(2) such that∑dk=1 bk zk is a (1/m22 j+2,n2 j+1) average. Using the conditions on thesequences and Proposition 6, the contradiction to our choice of j is as follows,εm2 j+1<1m2 j+1d∑k=1bk gk(d∑k=1bk T zk)∥∥∥∥∥T(d∑k=1bkzk)∥∥∥∥∥  C‖T ‖m22 j+1 . The above proposition yields that every diagonal free operator is strictly singular. This fact combined with the uncondi-tionality of the basis of Xwh yield the following main result of the Note:Theorem 8. Let Y be a block subspace of Xwh generated by (yn)n and T : Y → Y be a bounded linear operator. Then T can be writtenas T = D + S, where D is a diagonal operator with respect to the basis (yn)n of Y and S is a strictly singular operator. Moreover, everyblock subspace of Xwh cannot be isomorphic to any of its proper subspaces.It is easy to see that every bounded linear operator T : Y → Y is the sum of a diagonal operator D and a diagonal freeoperator S and from Proposition 7 we conclude that S is strictly singular. The last part of Theorem 8 follows from thedecomposition of T and Fredholm theory [7].References[1] D.E. Alspach, S.A. Argyros, Complexity of weakly null sequences, Dissertationes Math. (Rozprawy Mat.) 321 (1992) 44.[2] G. Androulakis, K. Beanland, A hereditarily indecomposable asymptotic l2 Banach space, Glasg. Math. J. 48 (3) (2006) 503–532.[3] S.A. Argyros, S. Todorcevic, Ramsey Methods in Analysis, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2005.[4] I. Deliyanni, A. Manoussakis, Asymptotic lp hereditarily indecomposable Banach spaces, Illinois J. Math. 51 (3) (2007) 767–803.[5] S.J. Dilworth, V. Ferenczi, D. Kutzarova, E. Odell, On strongly asymptotic lp spaces and minimality, J. Lond. Math. Soc. (2) 75 (2) (2007) 409–419.[6] V. Ferenczi, C. Rosendal, Banach spaces without minimal subspaces, J. Funct. Anal. 257 (1) (2009) 149–193.[7] W.T. Gowers, B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (4) (1997) 543–568.[8] W.B. Johnson, A reflexive Banach space which is not sufficiently Euclidean, Studia Math. 55 (2) (1976) 201–205.[9] N.J. Nielsen, N. Tomczak-Jaegermann, Banach lattices with property (H) and weak Hilbert spaces, Illinois J. Math. 36 (3) (1992) 345–371.[10] G. Pisier, Weak Hilbert spaces, Proc. London Math. Soc. (3) 56 (3) (1988) 547–579.[11] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, vol. 94, Cambridge University Press, Cambridge,1989.[12] A. Tcaciuc, On the existence of asymptotic-lp structures in Banach spaces, Canad. Math. Bull. 50 (4) (2007) 619–631.

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A weak Hilbert space with few symmetries

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