A dual method of constructing hereditarily indecomposable Banach spaces
aa r X i v : . [ m a t h . F A ] A p r A DUAL METHOD OF CONSTRUCTING HEREDITARILYINDECOMPOSABLE BANACH SPACES
SPIROS A. ARGYROS AND PAVLOS MOTAKIS
Abstract.
A new method of defining hereditarily indecomposable Ba-nach spaces is presented. This method provides a unified approach forconstructing reflexive HI spaces and also HI spaces with no reflexivesubspace. All the spaces presented here satisfy the property that thecomposition of any two strictly singular operators is a compact one.This yields the first known example of a Banach space with no reflex-ive subspace such that every operator has a non-trivial closed invariantsubspace. Introduction
Defining a hereditarily indecomposable (HI) Banach space is not an easytask. It requires the definition of a subset W of c ( N ) (the space of realsequences which are eventually zero), which in turn, acting as a set of func-tionals on c ( N ), defines an HI norm. In all classical constructions theresulting space admits the unit vector basis of c ( N ) as a boundedly com-plete Schauder basis. This appears to be an inevitable consequence of thesaturation of the set W under certain operations which yield, for every n in N , a lower bound C n of k P nk =1 x k k , for every sequence of successivenormalized block vectors ( x k ) nk =1 , and lim n C n = ∞ .There are two known types of HI spaces whose basis is not boundedlycomplete. The first one concerns the L ∞ HI space X K which appeared in[AH] and is the result of mixing the Bourgain-Delbaen method ([BD]) ofconstructing L ∞ -spaces and the Gowers-Maurey corresponding one ([GM])of constructing HI spaces. The basis of the space is shrinking but not bound-edly complete. However, this is a consequence of the L ∞ structure and notof the HI property of the space. In particular, every block sequence in thespace has a boundedly complete subsequence, hence the space is reflexivelysaturated.The second type concerns HI spaces with no reflexive subspace. All suchspaces whose norm is induced by a norming set W have a boundedly com-plete Schauder basis. This class includes spaces such as the Gowers Treespace [G] and the spaces which appeared in [AAT]. The predual of one Primary 46B03, 46B06, 46B25, 46B45, 47A15
Key words:
Spreading models, Strictly singular operators, Invariant subspaces, Hered-itarily indecomposable spacesThis research was supported by program APIΣTEIA-1082. of the spaces presented in [AAT] is also an HI space without reflexive sub-spaces. This space admits a shrinking basis and none of its subspaces admitsa boundedly complete basis. This predual is essentially different to a spacewhich is induced by a saturated norming set W . The latter, as we haveexplained, always yields spaces with a boundedly complete basis.The preceding discussion leads to the following question. Does there exista method of defining a norming set W such that the resulting space admits ashrinking Schauder basis and no subspace admits a boundedly complete one?This problem is directly related to the problem of the existence of a L ∞ -space which is HI and has no reflexive subspace. Indeed, any HI L ∞ -spacemust have separable dual ([LS], [P]) and if moreover it does not containreflexive subspaces, then it does not contain a boundedly complete basicsequence. More generally, every Banach space with a boundedly completebasis and separable dual is reflexively saturated ([JR]).The aim of the present paper is to answer the first problem by providinga new method of defining a norming set W , which yields an HI space witha shrinking basis with no boundedly complete basic sequence. We perceivethis method as the dual method of the classical one. This new approachallows us to affirmatively answer the second problem. Namely, there existsa L ∞ HI space with no reflexive subspace. This result will appear in aforthcoming paper. Our goal is to use a more classical setting in order topresent the definition of the norming set and its consequences, some of whichare rather unexpected.The definition of the norming set W uses an unconditional frame, namelythe Tsirelson-like space with constraints T (1 / n , S n , α ) n . Norms which aresaturated under constraints were introduced in [ABM] and [AM1] and arerooted in the earlier work of E.Odell and Th. Schlumprecht ([OS1], [OS2]).The norm of T (1 / n , S n , α ) n is described by the following implicit formula:if x ∈ c ( N ) then(1) k x k = max k x k ∞ , sup 12 n d X q =1 k E q x k m q where the supremum is taken over all n ∈ N , S n -admissible successive subsets( E q ) dq =1 of N and sequences ( m q ) dq =1 of N so that m q > max E q − for q =2 , . . . , d . The m -norms appearing in (1) are defined as follows. For m ∈ N and x ∈ c ( N ): k x k m = 1 m sup m X i =1 k G i x k where the supremum is taken over all successive subsets ( G i ) mi =1 of N .The k · k m norms, m ∈ N , which appear in the definition above, do notcontribute to the norm of the element x , in fact they acts as constraints. Thisresults in the neutralization of the operations (1 / n , S n ) on certain sequencesand thus, c spreading models become abundant. As a consequence, every DUAL METHOD OF CONSTRUCTING HI SPACES 3
Schauder basic sequence in the space admits either an ℓ or a c spreadingmodel and both of them are admitted by every infinite dimensional subspace.This norm and its variants have been recently established as an effective toolfor answering certain problems on the structure of Banach spaces and theirspaces of operators [ABM], [AM1], [AM2], [BFM].The norm on T (1 / n , S n , α ) n is induced by the norming set W α whichis the minimal subset of c ( N ) containing the basis ( e ∗ i ) i , all α -averagesof its elements, i.e. averages of successive elements of W α , and it is closedunder the operations (1 / n , S n , α ) for every n ∈ N . The latter means thatfor every very fast growing family ( α q ) dq =1 of successive α -averages, which is S n -admissible, the functional f = (1 / n ) P dq =1 α q is in W α . Any such f iscalled a weighted functional with w ( f ) = n . Hence, the set W α includes theelements of the basis, α -averages and weighted functionals.The norming set W will be chosen to be a subset of W α and its definitionis based on a tree U , called the universal tree. This tree consists of finitesequences { ( f k , x k ) } dk =1 , where ( f k ) dk =1 is a sequence of successive non-zeroweighted functionals in W α , ( x k ) dk =1 is a sequence of successive non-zerovectors in c ( N ) with rational coefficients and for each 1 < m d theweight of f m is uniquely defined by the sequence { ( f k , x k ) } m − k =1 .We will consider a class of subtrees T of the universal tree U . Eachtree T in this class is either well founded and satisfies certain additionalproperties or T = U . For such a tree T we define the norming set W T . Itis worth pointing out that for a well founded tree T the space X T , inducedby the set W T , is a reflexive HI space, while for T = U the space X U admits a shrinking basis and does not contain a reflexive subspace. It isalso interesting, and rather unexpected, that the reflexive and non-reflexivecases have a unified approach, as it is presented in the rest of the paper.Note that the Gowers Tree type HI spaces with no reflexive subspace ([G],[AAT]) have substantially increased complexity, concerning their definitionas well as their proofs, compared to the corresponding reflexive HI spaces.For a subtree T of the universal tree U , as above, we define the norm ofthe space X T , which is very similar to the norm of the space T (1 / n , S n , α ) n .Namely, the norm of X T is described by the implicit formula (1), the differ-ence lying in the definition of k · k m norms, where k x k m = sup ( m m X i =1 g i ( x ) : 1 m m X i =1 g i is an α c -average ) and α c -averages are α -averages which are inductively defined. In otherwords, to define the norm of X T we impose some further restrictions onthe α -averages used as constraints. Alternatively, the norming set W T isthe minimal subset of c ( N ) containing the basis, the α c -averages and all f = (1 / n ) P dq =1 α q where ( α q ) dq =1 is a very fast growing and S n -admissiblefamily of α c -averages. S. A. ARGYROS AND P. MOTAKIS
Let us observe that in the definition of W T the conditional structure,which yields the HI property of the space X T , is contained in the α c -averages.The space X T satisfies the following property. If T is a well founded subtreeof U , for every block sequence with rational coefficients ( y i ) i in X T thereexist a further finite block sequence ( x k ) dk =1 , with 1 / < k x k k
10, and( f k ) dk =1 in W T , such that { ( f k , x k ) } dk =1 is a maximal element of T and k P dk =1 x k k
27. If T = U , the corresponding result holds in the space X U for a branch { ( f k , x k ) } ∞ k =1 of U such that k P dk =1 x k k <
27, for all d ∈ N .Below we summarize the properties of the space X T , in the case the tree T is well founded. Theorem A. If T is well founded, then the space X T satisfies the followingproperties.(i) The space X T has a bimonotone Schauder basis, it is hereditarilyindecomposable and reflexive.(ii) Every Schauder basic sequence in X T admits either ℓ or c as aspreading model and every infinite dimensional subspace of X T ad-mits both of these types of spreading models.(iii) For every block subspace X of X T and every bounded linear operator T : X → X , there is λ ∈ R so that T − λI is strictly singular.(iv) For every infinite dimensional subspace X of X T the ideal of thestrictly singular operators S ( X ) is non separable.(v) For every subspace X of X T and every strictly singular operators S , T on X , the composition T S is compact.(vi) For every block subspace X of X T , every non-scalar bounded linearoperator T : X → X admits a non-trivial closed hyperinvariantsubspace.The above should be compared to the main theorem from [AM1], where aspace with very similar properties is presented. The key difference betweenthe aforementioned case and the present one is in property (v), namely in[AM1] it is only proved for compositions of three strictly singular operators,and not two. In [AM1] special weighted functionals are used, which imposethe necessity to include β -averages in the definition of the norming set. Theabsence of these two notions in the present construction yields property (v),which is the best possible, as well as simplified proofs, compared to those in[AM1].Below we present the main properties of the space X U . Theorem B. If T = U , then the space X U satisfies the following properties.(i) The space X U has a bimonotone and shrinking Schauder basis, it ishereditarily indecomposable and contains no reflexive subspace.(ii) Every Schauder basic sequence in X U admits either ℓ , either, c orthe summing basis of c as a spreading model and every infinite di-mensional subspace of X U admits all three of these types of spreadingmodels. DUAL METHOD OF CONSTRUCTING HI SPACES 5 (iii) For every block subspace X of X U and every bounded linear operator T : X → X , there is λ ∈ R so that T − λI is weakly compact andhence strictly singular.(iv) For every infinite dimensional subspace X of X U the ideal of thestrictly singular operators S ( X ) is non separable.(v) For every subspace X of X U and every strictly singular operators S , T on X , the composition T S is compact.(vi) For every block subspace X of X U , every non-scalar bounded linearoperator T : X → X admits a non-trivial closed hyperinvariantsubspace.This is the first known example of a Banach space with no reflexive sub-space such that the space generated by every block sequence satisfies theinvariant subspace property.In Theorems A and B property (vi) can be stated for every subspace X of the corresponding space, such that every T in L ( X ) is of the form λI + S ,with S strictly singular. The present construction can also be carried outover the field of complex numbers. The corresponding complex HI spacessatisfy Theorems A and B, in particular property (vi) holds for every closedsubspace ([GM, Theorem 18]).2. The norming set of the space X T This section is devoted to the norming set W T of the space. We begin witha brief presentation and discussion concerning the main ingredients involvedin the definition of W T . As we have mentioned in the introduction we willconsider subtrees of the universal tree U . Each such tree T is downwardsclosed and for every node which is non-maximal in T , all of its immediatesuccessors in U are also included in T . For our needs the tree is eitherwell founded, containing at least all elements { ( f k , x k ) } dk =1 of U such that( f k ) dk =1 is S -admissible, or otherwise T = U .The second ingredient are the α c -averages which are inductively definedand are described as follows.To each weight n we assign a unique weight φ ( n ) that appears in thetree T . Two different weights n and m are comparable, if there exist { ( f , x ) , . . . , ( f k , x k ) } in T and 1 i < j k such φ ( n ) = w ( f i ) and φ ( m ) = w ( f j ). Otherwise n , m are incomparable.We consider the following four types of averages. The first one are aver-ages of the basis ( e ∗ i ) i , called basic averages.The second one are IC -averages, i.e. α -averages of the form (1 /n ) P ni =1 g i with { w ( g i ) } ni =1 pairwise incomparable.The third one are IR -averages, i.e. α -averages of the form (1 /n ) P ni =1 g i such that there exist { ( f , x ) , . . . , ( f m , x m ) } in T and 1 k < · · · < k n m with w ( f k i ) = φ ( w ( g i )) and | g i ( x k i ) | > CO -averages. Those are α -averages of the form (1 /n )( g − g + g − g + · · · + ( − n +1 g n ) such that S. A. ARGYROS AND P. MOTAKIS there exist { ( f , x ) , . . . , ( f m , x m ) } in T and 1 k < · · · < k n n with w ( f k i ) = φ ( w ( g i )) and | g i ( x k i ) − g j ( x k j ) | < / i for 1 i < j n .The third and fourth types of averages explain why we consider in theuniversal tree U families of pairs { ( f k , x k ) } dk =1 , instead of ( f k ) dk =1 which isthe approach used in the classical norming sets. We note that the basicaverages permit to begin the construction of weighted functionals in thenorming set W T . The CO -averages are responsible for the whole conditionalstructure in the space X T . The remaining two types of averages are necessaryto exclude the presence of c in the space.2.1. The Schreier families.
The Schreier families is an increasing se-quence of families of finite subsets of the natural numbers, which first ap-peared in [AA], and is inductively defined in the following manner. Set S = (cid:8) { n } : n ∈ N (cid:9) and S = { F ⊂ N : F min F } . Suppose that S n has been defined and set S n +1 = n F ⊂ N : F = ∪ kj =1 F j , where F < · · · < F k ∈ S n and k min F o . For each n , S n is a regular family. This means that it is hereditary, i.e.if F ∈ S n and G ⊂ F then G ∈ S n , it is spreading, i.e. if F = { i < · · ·
The norming set of the space X T is asubset of W (1 / n , S n ,α ) n , a version of the norming set of Tsirelson space,defined with saturation under constraints.We denote by c ( N ) the space of all real valued sequences ( c i ) i withfinitely many non-zero terms. We denote by ( e i ) i the unit vector basis of c ( N ), while in some cases we shall denote it as ( e ∗ i ) i . For x = ( c i ) i ∈ c ( N ),the support of x is the set supp x = { i ∈ N : c i = 0 } and the range of x ,denoted by ran x , is the smallest interval of N containing supp x . We say thatthe vectors x , . . . , x k in c ( N ) are successive if max supp x i < min supp x i +1 for i = 1 , . . . , k −
1. In this case we write x < · · · < x k . A sequence ofsuccessive vectors in c ( N ) is called a block sequence. Notation.
We remind some notation and terminology which is used con-stantly throughout this paper.
DUAL METHOD OF CONSTRUCTING HI SPACES 7 (i) A sequence of vectors x < · · · < x k in c ( N ) is said to be S n -admissible, for given n ∈ N , if { min supp x i : i = 1 , . . . , k } ∈ S n .(ii) Let G ⊂ c ( N ). A vector α ∈ c ( N ) is called an α -average of G ofsize s ( α ) = n , if there exist f < · · · < f d ∈ G , where d n , suchthat α = 1 n ( f + · · · + f d ) . (iii) A sequence of successive α -averages of G ( α q ) q is called very fastgrowing if s ( α q ) > max supp α q − for q > Definition 2.1.
We define W α = W (1 / n , S n ,α ) n to be the smallest subset of c ( N ) satisfying the following properties:(i) for every i ∈ N , e ∗ i ∈ W α and the set W α is symmetric,(ii) the set W α contains all α -averages of W α ,(iii) for every n ∈ N and every very fast growing and S n -admissible se-quence of α -averages of W α ( α q ) dq =1 , the vector f = (1 / n ) P dq =1 α q is also in W α .We note that, as it is usually the case in this type of constructions, the sizeof an average and the weight of a weighted functional may not be uniquelydefined. However, this does not cause any problems. Remark 2.2.
The set W α satisfies the properties mentioned below. Notethat properties (i), (ii) and (iii) follow readily from property (iv).(i) Every f ∈ W α is either of the form f = ± e ∗ i , either an α -averageof W α or f = (1 / n ) P dq =1 α q , where ( α q ) dq =1 is a very fast growingand S n -admissible sequence of α -averages of W α . In the last case weshall say that f is a weighted functional of W α of weight w ( f ) = n .(ii) For every f ∈ W α and subset of the natural numbers E , the func-tional Ef , i.e. the restriction of f onto E , is also in W α .(iii) The coefficients of every f ∈ W α are rational numbers. In particular, W α is a countable set.(iv) The set W α can be constructed recursively to be the union of anincreasing sequence of sets ( W αm ) ∞ m =0 , where W α = {± e ∗ i : i ∈ N } and if W αm has been defined, then W m +1 is the set of all α -averagesof W αm , W m +1 is the set of all weighted functionals constructed onvery fast growing sequences of elements of W m +1 and W αm +1 = W αm ∪ W m +1 ∪ W m +1 .2.3. The universal tree U . We denote by Q the set of all finite sequences { ( f , x ) , . . . , ( f k , x k ) } satisfying the following:(i) the f , . . . , f k are successive non-zero weighted functionals of W α and(ii) the x , . . . , x k are successive non-zero vectors in c ( N , Q ) (i.e. theyare vectors in c ( N ) with rational coefficients).Note that Q is a subset of ∪ n ( W α × c ( N , Q )) n and hence countable.Choose an infinite subset L ′ = { ℓ k : k ∈ N } of N satisfying: S. A. ARGYROS AND P. MOTAKIS (i) min L ′ > k ∈ N , ℓ k +1 > ℓ k .Define a partition of L ′ into two infinite subsets L and L ′ and choose aone-to-one function σ : Q → L ′ , called the coding function, so that for every { ( f , x ) , . . . , ( f k , x k ) } ∈ Q ,(2) σ ( { ( f , x ) , . . . , ( f k , x k ) } ) > k f k k − ∞ max supp x k . A finite sequence { ( f k , x k ) } dk =1 ∈ Q is called a special sequence if:(i) w ( f ) ∈ L and(ii) if d > w ( f k ) = σ ( { ( f , x ) , . . . , ( f k − , x k − ) } ) for k = 2 , . . . , d . Remark 2.3.
Note that if { ( f k , x k ) } dk =1 is a special sequence, then (2) and(ii) imply that w ( f ) < · · · < w ( f d ).Note that if { ( f k , x k ) } dk =1 is a special sequence and 1 p d , then { ( f k , x k ) } pk =1 is a special sequence as well, hence if we define U to be the setof all special sequences, then U is a tree endowed with the natural ordering“ ⊑ ” of initial segments. Note that the tree U is ill founded, more preciselyevery maximal chain of U is infinite. We shall call the tree U , the universaltree associated with the coding function σ .2.4. Subtrees of U . We fix a subtree T of U which satisfies the followingproperties:(i) for every { ( f k , x k ) } dk =1 in T and 1 p d { ( f k , x k ) } pk =1 is also in T , i.e. T is a downwards closed subtree of U ,(ii) if { ( f k , x k ) } dk =1 is a non-maximal node in T , then for every element( f d +1 , x d +1 ) so that { ( f k , x k ) } d +1 k =1 is in U , { ( f k , x k ) } d +1 k =1 is also in T and(iii) for every { ( f k , x k ) } dk =1 in U with ( f k ) dk =1 being S -admissible, wehave that { ( f k , x k ) } dk =1 is in T . Definition 2.4.
We define L = σ ( T ), which is a subset of L ′ , and L = L ∪ L . Define φ : { i ∈ N : i > min L } → L with φ ( i ) = max { ℓ ∈ L : ℓ i } .Observe that the function φ is non-decreasing, φ ( i ) i for all i ∈ N andlim i φ ( i ) = ∞ . Definition 2.5.
Two natural numbers i and j , both greater than or equalto min L , are called incomparable if one of the following holds:(i) φ ( i ) and φ ( j ) are both in L and φ ( i ) = φ ( j ) or(ii) φ ( i ) and φ ( j ) are both in L and σ − ( φ ( i )), σ − ( φ ( j )) are incompa-rable, in the ordering of T .If i , j are not incomparable they will be called comparable. DUAL METHOD OF CONSTRUCTING HI SPACES 9 α c -averages. We shall define very specific types of averages, based onthe tree T and the notion of comparability of natural numbers from Defini-tion 2.5. Alongside averages of elements of the basis ( e ∗ i ) i , in the definitionof the norming set W T we shall only consider these types of averages. Definition 2.6.
Let g < · · · < g d be weighted functionals in a subset G of W α , all of which have weight greater than or equal to min L , satisfying φ ( w ( g )) < · · · < φ ( w ( g d )).(i) The sequence ( g i ) di =1 is called incomparable, if the natural numbers w ( g i ), i = 1 , . . . , d are pairwise incomparable, in the sense of Defini-tion 2.5. In this case, if n ∈ N with d n we call the average α = 1 n d X i =1 g i an IC -average of G .(ii) The sequence ( g i ) di =1 is called comparable, if there exist m ∈ N with d m , { ( f , x ) , . . . , ( f m , x m ) } ∈ T and 1 k < · · · < k d m sothat the following are satisfied:(a) w ( f k i ) = φ ( w ( g i )),(b) if d > | g i ( x k i ) |
10 for i = 2 , . . . , d − d > | g i ( x k i ) − g j ( x k j ) | < / i for 2 i < j d − n ∈ N with d n and ( ε i ) di =1 is a sequence of alter-nating signs in {− , } we call the average α = 1 n d X i =1 ε i g i a CO -average of G .(iii) The sequence ( g i ) di =1 is called irrelevant, if there exist m ∈ N with d m , { ( f , x ) , . . . , ( f m , x m ) } ∈ T and 1 k < · · · < k d m sothat the following are satisfied:(a) w ( f k i ) = φ ( w ( g i )) and(b) if d > | g i ( x k i ) | >
10 for i = 2 , . . . , d − n ∈ N with d n we call the average α = 1 n d X i =1 g i an IR -average of G .Any average which is of one of the forms defined above, shall be called an α c -average of G . Basic averages will be referred to as α c -averages as well,where a basic average is a functional of the form α = (1 /n ) P di =1 ε i e ∗ j i where d , n , j < · · · < j d ∈ N with d n and ( ε i ) di =1 are any signs in {− , } . Remark 2.7.
The class of α c -averages, is a much more restricted version ofthe one of α -averages and, with the exception of basic averages, α c -averagesare determined using the coding function σ , more precisely the tree T . Remark 2.8.
If ( g i ) di =1 is a sequence in W α which is of one of the threetypes described in Definition 2.6, then any subsequence of it is of the sametype. Moreover, if E is an interval of N and i = min { i : E ∩ ran g i = ∅ } and i = max { i : E ∩ ran g i = ∅ } , then the sequence Eg i , Eg i i +1 , . . . , Eg i is ofthe same type as ( g i ) di =1 . This last part in particular implies that whenever α is an average which is of one of the three types described in Definition 2.6and E is an interval of N , then Eα is an average of the same type.2.6. The norming set W T of the space X T .Definition 2.9. We define W T to be the smallest subset of W α which sat-isfies the following properties.(i) For every i ∈ N , e ∗ i ∈ W T and the set W T is symmetric.(ii) The set W T contains all α c -averages of W T , i.e. it contains all basicaverages and all IC , CO and IR -averages of W T .(iii) For every n ∈ N and every S n -admissible and very fast growingsequence of α c -averages ( α q ) dq =1 of W T , f = (1 / n ) P dq =1 α q is alsoin W T . Remark 2.10.
The set W T satisfies the properties mentioned below. Notethat property (ii) follows from an inductive argument using Remark 2.8 andproperty (iii).(i) Every f ∈ W T is either of the form f = ± e ∗ i , either an α c -averageof W T or a weighted functional f = (1 / n ) P dq =1 α q , where ( α q ) dq =1 is a very fast growing and S n -admissible sequence of α c -averages of W T .(ii) For every f ∈ W T and interval of the natural numbers E , the func-tional Ef , i.e. the restriction of f onto E , is also in W T .(iii) The set W T can be recursively constructed to be the union of anincreasing sequence of sets ( W m ) ∞ m =0 , where W = {± e ∗ i : i ∈ N } and if W m has been defined, then W α c m +1 is the set of all α c -averagesof W m , W wm +1 is the set of all weighted functionals constructed onvery fast growing sequences of elements of W α c m +1 , and W m +1 = W m ∪ W α c m +1 ∪ W wm +1 .The norm of the space X T is the one induced by the set W T , i.e. for every x ∈ c ( N ) we set k x k = sup { f ( x ) : f ∈ W T } and we define X T to be thecompletion of c ( N ) with respect to this norm. By Remark 2.10 the unitvector basis of c ( N ) forms a bimonotone Schauder basis for X T . Remark 2.11.
The conditional structure of the space X T is only imposedby the CO -averages in the norming set W T , which are merely averages. Inthis sense, the conditionality appearing in the space X T is not as strict asin other HI constructions. DUAL METHOD OF CONSTRUCTING HI SPACES 11 Special convex combinations and evaluation of their norm
We first remind the notion of the ( n, ε ) special convex combinations, (see[AD],[AGR],[AT]) which is one of the main tools used in the sequel. Wethen include, without proof, some estimates from [AM1], which also applyto the present case.
Definition 3.1.
Let x = P k ∈ F c k e k be a vector in c ( N ) and n ∈ N , ε > x is called a ( n, ε )-basic special convex combination (or a ( n, ε )-basics.c.c.) if the following are satisfied:(i) F ∈ S n , c k > k ∈ F and P k ∈ F c k = 1,(ii) for any G ⊂ F , with G ∈ S n − , we have that P k ∈ G c k < ε . Remark 3.2.
We note for later use the following easy fact. If x = P i ∈ F c i e i is a ( n, ε )-basic s.c.c. with 0 < ε < / i ∈ F \ { min F } we set c ′ i = c i / ( P j ∈ F \{ min F } c j ) then y = P i ∈ F \{ min F } c ′ i e i is a ( n, ε )-basic s.c.c.The next result is from [AMT]. For a proof see [AT, Chapter 2, Proposi-tion 2.3]. Proposition 3.3.
For every infinite subset of the natural numbers M , any n ∈ N and ε >
0, there exist F ⊂ M and non-negative real numbers ( c k ) k ∈ F ,such that the vector x = P k ∈ F c k e k is a ( n, ε )-basic s.c.c. Definition 3.4.
Let x < · · · < x m be vectors in c ( N ) and ψ ( k ) =min supp x k , for k = 1 , . . . , m . If the vector P mk =1 c k e ψ ( k ) is a ( n, ε )-basics.c.c., for some n ∈ N and ε >
0, then the vector x = P mk =1 c k x k is called a( n, ε )-special convex combination (or ( n, ε )-s.c.c.).By T we denote Tsirelson space and by k·k T its norm, as they were definedin [FJ]. This space is actually the dual of Tsirelson’s original Banach spacedefined in [T]. The proof of the following result can be found in [AM1,Proposition 2.5]. Proposition 3.5.
Let n ∈ N , ε > x = P k ∈ F c k e k be a ( n, ε )-basic s.c.c.and G ⊂ F . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ G c k e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T n X k ∈ G c k + ε. The next result can also be found in [AM1, Corollary 2.8]. A number ofsteps are required in order to reach this estimate, however the argumentsused there also work in the present case unchanged and therefore we omitthe proof.
Proposition 3.6.
Let ( x k ) k be a block sequence in X T with k x k k k ∈ N , ( c k ) k be a sequence of real numbers and φ ( k ) = max supp x k forall k . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k c k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k c k e φ ( k ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T . The next crucial estimate follows from Propositions 3.5 and 3.6. A proofcan be found in [AM1, Corollary 2.9].
Corollary 3.7.
Let n ∈ N , ε > x = P mk =1 c k x k be a ( n, ε )-s.c.c. in X T , such that k x k k
1, for k = 1 , . . . , m . If F is subset of { , . . . , m } then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k ∈ F c k x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k ∈ F c k + 12 ε. In particular, k x k / n + 12 ε .Using Propositions 3.3 and 3.7 one can easily derive the next result. Fora proof see [AM1, Corollary 2.10]. Proposition 3.8.
The basis of X T is shrinking. In particular, the dual of X T is separable.We now give some definitions which will be crucial in the next sections,where we prove the properties of the space X T . Rapidly increasing sequencesare defined exactly as in [AM1, Definition 2.13]. Definition 3.9.
Let C > n k ) k be a strictly increasing sequence ofnatural numbers. A block sequence ( x k ) k is called a ( C, ( n k ) k )-rapidly in-creasing sequence (or ( C, ( n k ) k )-RIS) if k x k k C for all k and the followinghold:(i) for every k and every weighted functional f in W T with w ( f ) = j C, θ, n )-exact vector is definedidentically as in [AM1, Definition 2.15]. Definition 3.10. Let n ∈ N , C > θ > 0. A vector x ∈ X T is calleda ( C, θ, n )-vector if there exist 0 < ε < / (36 C n ) and a block sequence( x k ) mk =1 with k x k k C for k = 1 , . . . , m such that:(i) min supp x > C n ,(ii) there exist non-negative real numbers ( c k ) mk =1 so that the vector P mk =1 c k x k is a ( n, ε )-s.c.c.,(iii) x = 2 n P mk =1 c k x k and k x k > θ .If moreover there exists a strictly increasing sequence of natural numbers( n k ) mk =1 with n > n so that ( x k ) mk =1 is a ( C, ( n k ) mk =1 )-RIS, then x is calleda ( C, θ, n )-exact vector. Remark 3.11. Let x be a ( C, θ, n )-vector in X T . Then, using Corollary 3.7we conclude that k x k < C . Remark 3.12. Let x be a ( C, θ, n )-vector in X T . By the choice of ε and k x k k C for k = 1 , . . . , m , we obtain k x k ∞ < / (2 n DUAL METHOD OF CONSTRUCTING HI SPACES 13 The α -index In all recent constructions involving saturation under constraints ([AM1],[AM2], [ABM], [BFM]), the α -index has been used to help determine thespreading models admitted by block sequences. In contrast to the HI con-structions [AM1] and [AM2], where the α -index is not sufficient to fully char-acterize the spreading models of block sequences, the present case resemblesmore closely the unconditional example from [ABM], where the α -index isthe only necessary tool to study spreading models admitted by the space.This is due to the fact that only α -averages, more precisely α c -averages, arethe only ingredient used to construct weighted functionals. The definitionof the α -index of a block sequence given below is identical to the one from[AM1] and [AM2]. Definition 4.1. Let ( x k ) k be a block sequence in X T that satisfies thefollowing: for every n ∈ N , for every very fast growing sequence of α c -averages of W T ( α q ) q , for every increasing sequence of subsets of the naturalnumbers ( F m ) m , such that ( α q ) q ∈ F m is S n -admissible for all m ∈ N and forevery subsequence ( x k m ) m of ( x k ) k , we have thatlim k X q ∈ F m | α q ( x k m ) | = 0 . Then we say that the α -index of ( x k ) k is zero and write α (( x k ) k ) = 0.Otherwise we write α (( x k ) k ) > α -index zero,and its proof can be found in [AM1, Proposition 3.3]. Although here it isformulated slightly differently, the two versions are easily seen to be equiv-alent. Proposition 4.2. Let ( x k ) k be a block sequence in X T . The followingassertions are equivalent.(i) The α -index of ( x k ) k is zero.(ii) For every ε > j ∈ N such that for every n ∈ N thereexists k n ∈ N such that for every k > k n and for every very fastgrowing and S n -admissible sequence of α c -averages ( α q ) dq =1 , with s ( α q ) > j for q = 1 , . . . , d , we have that P dq =1 | α q ( x k ) | < ε .The next result is proved in [AM1, Proposition 3.5]. Proposition 4.3. Let ( x k ) k be a seminormalized block sequence in X T with α (( x k ) k ) > 0. Then there exist θ > x k m ) m of ( x k ) k that generates an ℓ n spreading model with a lower constant θ/ n , for all n ∈ N . More precisely, for every n ∈ N , subset of the natural numbers F , sothat ( x k m ) m ∈ F is S n -admissible, and real numbers ( c m ) m ∈ F we have that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X m ∈ F c m x k m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > θ n X m ∈ F | c m | . In particular, for all k , n ∈ N , there exists a finite subset of the natu-ral numbers F with min F > k and non-negative real numbers ( c m ) m ∈ F ,such that the vector x = 2 n P m ∈ F c m x k m is a ( C, θ, n )-vector, where C =sup {k x k k : k ∈ N } .We now prove that block sequences with α -index zero admit only c asa spreading model and that Schreier sums of them define rapidly increasingsequences. Proposition 4.4. Let ( x k ) k be a normalized block sequence in X T with α (( x k ) k ) = 0. Then ( x k ) k has a subsequence, which we also denote by( x k ) k , that generates a spreading model which is isometric to the unit vectorbasis of c . Moreover, there exists a strictly increasing sequence of naturalnumbers ( j k ) k so that for every natural numbers n k < · · · < k n , realnumbers ( c i ) ni =1 and weighted functional f of W T with w ( f ) = j < j n , wehave (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f n X i =1 c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < / j max i n | c i | . Proof. Using Proposition 4.2, we pass to a subsequence of ( x k ) k , again de-noted by ( x k ) k , and choose a strictly increasing sequence of natural numbersso that the following are satisfied:(i) for every k ∈ N , 1 / j k +1 max supp x k < / k and(ii) for every k , k ∈ N with k > k and every very fast growingand S j k -admissible sequence of α c -averages ( α q ) nq =1 with s ( α q ) > max supp x k we have d X q =1 | α q ( x k ) | < / ( k k ) . We claim that ( x k ) k generates a spreading model isometric to c . Usingthe third assertion of Remark 2.10 we shall inductively prove the following:for every f ∈ W m , natural numbers n k < · · · < k n and real numbers c , . . . , c n in [ − , 1] we have(3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f n X i =1 c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < n . If moreover f is a weighted functional with w ( f ) = j < j n , then(4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f n X i =1 c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < / n j . The desired conclusion clearly follows from the above and the fact that thebasis of X T is bimonotone, omitting if necessary a finite number of terms ofthe sequence ( x k ) k .We now proceed to the proof of the inductive step. The case m = 0 is animmediate consequence of the fact that the sequence ( x k ) k is normalized and DUAL METHOD OF CONSTRUCTING HI SPACES 15 W = {± e ∗ i : i ∈ N } . Assume now that m is such that the conclusion holdsfor every functional in W m and let f ∈ W m +1 . If f is an α c -average of W m ,then by the inductive assumption we conclude that (3) holds. Otherwise, f is a weighted functional of weight w ( f ) = j , i.e. there is a very fastgrowing and S j admissible sequence of α c -averages of W m ( α q ) dq =1 so that f = (1 / j P dq =1 α q . Assuming that f ( P ni =1 c i x k i ) = 0, set q = min { q :max supp α q > min supp x k } . Omitting, if it is necessary, the first q − q = 1. the We distinguish three casesconcerning the weight of f . Case 1: j < j k . Since the sequence ( α q ) dq =1 is very fast growing, for q > s ( α q ) > max supp α > min supp x k . Also, since ( α q ) dq =2 is S j admissible with j < j k , by (ii) we conclude:(5) d X q =2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α q n X i =1 c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < n k k n . Moreover, by the inductive assumption we obtain | α ( P ni =1 c i x k i ) | < / n . Combining this with (5):(6) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f n X i =1 c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < / n j . This concludes the proof of the first case and also (4) of the inductive as-sumption. Case 2: there is 1 i < n so that j k i j < j k i . Arguing in anidentical manner as in the previous case, we obtain(7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f X i>i c i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < / k i j ki n . Also, if i > 1, by (i) we have that 1 / j max supp x k i − < / k i − andhence:(8) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f X i j k n . Using that | f ( x k n ) | | f ( P ni =1 c i x k i ) | < / n and this concludes the proof. (cid:3) Propositions 4.3 and 4.4 yield the following result, which characterizesthe spreading models admitted by a given block sequence. Corollary 4.5. Let ( x k ) k be a normalized block sequence in X T . Then ( x k ) k has a subsequence that generates either an isometric c spreading model oran ℓ n spreading model for every n ∈ N . More precisely, the assertions statedbelow hold.(i) The sequence ( x k ) k admits only c as a spreading model if and onlyif α (( x k ) k ) = 0.(ii) The sequence ( x k ) k has a subsequence that generates an ℓ n spreadingmodel for every n ∈ N if and only if α (( x k ) k ) > Estimations on exact vectors In this section we provide estimations on exact vectors whose sums definenon-trivial weakly Cauchy sequences in X U and in the general case pro-vide the fact that the space X T is hereditarily indecomposable. We givethe definitions of exact vectors and exact sequences and several technicalintermediate steps are presented in order to achieve the main estimate.The next estimate uses Proposition 3.6 and the properties of special con-vex combinations. It is proved in [AM1, Lemma 3.8] and identical argumentsalso apply in this case. Lemma 5.1. Let x be a ( C, θ, n )-vector in X T . Let also ( a q ) dq =1 be a veryfast growing and S j -admissible sequence of α c -averages, with j < n . Then d X q =1 | α q ( x ) | < Cs ( α ) + 12 n . These next two results follows readily form Lemma 5.1 and Proposition4.4. Their proof can also be found in [AM1, Propositions 3.9 and 3.10] Proposition 5.2. Let C > θ > 0. If ( x k ) k is a block sequence in X T so that each x k is a ( C, θ, n k )-vector, with ( n k ) k a strictly increasingsequence of natural numbers, then α (( x k ) k ) = 0 and hence, every spreadingmodel admitted by ( x k ) k is isometric, up to scaling, to the unit vector basisof c . Proposition 5.3. Let x be a ( C, θ, n )-vector in X T . Then for any weightedfunctional f in W T such that w ( f ) = j < n we have | f ( x ) | < C j . We now give the definition of an exact pair and a dependent sequence. Definition 5.4. A pair ( f, x ) where x is a ( C, θ, n )-exact vector in X T and f is a weighted functional in W T with w ( f ) = n , ran f ⊂ ran x and f ( x ) = θ is called a ( C, θ, n )-exact pair. Definition 5.5. Let C > θ > 0. A sequence of pairs { ( f k , x k ) } ℓk =1 ,where f k ∈ W T and x k is a vector with rational coefficients in X T for k =1 , . . . , ℓ , is called a ( C, θ )-dependent sequence if the following are satisfied: DUAL METHOD OF CONSTRUCTING HI SPACES 17 (i) ( f k , x k ) is a ( C, θ, w ( f k ))-exact pair for k = 1 , . . . , ℓ and(ii) { ( f k , x k ) } ℓk =1 is in T ,We introduce some notation baring similarities to the one used in [AM1,Subsection 3.2] and [AM2]. Notation. Let x = 2 n P mk =1 c k x k be a ( C, θ, n )-exact vector, with ( x k ) mk =1 a( C, ( n k ) mk =1 )-RIS. Let also g < · · · < g d be weighted functionals in W T , allof which have weight greater than or equal to min L satisfying φ ( w ( g )) < · · · < φ ( w ( g d )) (see Definition 2.4). We define the following subsets of N : I ( x, ( g i ) di =1 ) = { j : n w ( g j ) < n } ,I ( x, ( g i ) di =1 ) = { j : w ( g j ) < n } and I ( x, ( g i ) di =1 ) = { j : 2 n w ( g j ) } . Remark 5.6. Let x be a ( C, θ, n )-exact vector and g < · · · < g d beweighted functionals in W T , all of which have weight greater than or equalto min L satisfying φ ( w ( g )) < · · · < φ ( w ( g d )).(i) If n ∈ L , then the set I ( x, ( g i ) di =1 ) is either empty or a singleton.Indeed, by the choice of L ′ , the fact that L ⊂ L ′ and the definitionof φ it is straightforward to check that if j ∈ I ( x, ( g i ) di =1 ), then φ ( w ( g j )) = n and clearly at most one j can satisfy this condition.(ii) Also, the sets I ( x, ( g i ) di =1 ), I ( x, ( g i ) di =1 ) are successive intervalsof { , . . . , d } , which clearly follows from the fact that φ is non-decreasing. Lemma 5.7. Let n > x be a ( C, θ, n )-exact vector in X T and g < · · · < g d be weighted functionals in W T , all of which have weight greaterthan or equal to min L satisfying φ ( w ( g )) < · · · < φ ( w ( g d )). If we set I ( x ) = I ( x, ( g i ) di =1 ), then X j ∈ I ( x ) | g j ( x ) | < d C n . Proof. We will actually show that if g is a weighted functional in W T with w ( g ) > n , then | g ( x ) | < C/ n . If x = 2 n P mk =1 c k x k , with ( x k ) mk =1 a( C, ( n k ) mk =1 )-RIS, recall that according to Definition 3.10 we have that 2 n Let 1 C / θ > { ( f k , x k ) } ℓk =1 be a ( C, θ )-dependentsequence and 1 n m ℓ be natural numbers. Let also ( g j ) dj =1 be asequence of weighted functionals in W T and ( ε j ) dj =1 be a sequence of signsin {− , } , so that one of the following is satisfied:(i) the sequence ( g j ) dj =1 is comparable and the signs ( ε j ) dj =1 are alter-nating or(ii) the sequence ( g j ) dj =1 is either incomparable or irrelevant.If for j = 1 , . . . , d we define D j = { n k m : w ( g j ) < w ( f k ) } , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X j =1 ε j g j m X k = n x k ! − d X j =1 ε j g j m X k ∈ D j x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C + d C w ( f n ) . Proof. Recall that each x k is a ( C, θ, w ( f k ))-exact vector and for all 1 k ℓ define A k = I ( x k , ( g j ) dj =1 ) and B k = I ( x k , ( g j ) dj =1 ). Observe that d X j =1 ε j g j m X k ∈ D j x k = m X k =1 X j ∈ B k ε j g j ( x k ) . Therefore, if we define C k = I ( x k , ( g j ) dj =1 ) we conclude (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X j =1 ε j g j m X k = n x k ! − d X j =1 ε j g j m X k ∈ D j x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = DUAL METHOD OF CONSTRUCTING HI SPACES 19 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k = n X j ∈ A k ε j g j ( x k ) + X j ∈ C k ε j g j ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k = n X j ∈ A k ε j g j ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + m X k = n d C w ( f k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k = n X j ∈ A k ε j g j ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + d C w ( f n ) where the first inequality follows from Lemma 5.7 while the second onefollows from the fact that the w ( f j )’s are strictly increasing (see Remark2.3).We will show that | P mk =1 P j ∈ A k ε j g j ( x k ) | C , which will conclude theproof. We remind that by Remark 3.11, k x k k < C for all 1 k ℓ . Wealso remind that by Remark 5.6 each set A k is either empty or a singletonand in particular, we note the following: if j ∈ A k then φ ( w ( g j )) = w ( f k ).Moreover, the assumptions yield the φ ( w ( g j ))’s are strictly increasing. If thesets A k are all empty there is nothing to prove. Otherwise, let k < · · · < k s be all the k ’s in { , . . . , ℓ } satisfying A k i = ∅ . Let also 1 j < · · · < j s d be so that for each i , j i is the unique element of A k i , and hence φ ( w ( g j i )) = w ( f k i ) for i = 1 , . . . , s .If s k x k k < C for all 1 k ℓ . Otherwise, s > g i ) di =1 is notincomparable, i.e. there are 1 i < i ′ d so that w ( g j ) and w ( g j ) arenot incomparable in the sense of Definition 2.5. Indeed, since { ( f k , x k ) } mk =1 is in T we have that σ − ( φ ( w ( g j ))) = σ − ( w ( f k )) = { ( f k , x k ) } k − k =1 ⊑ { ( f k , x k ) } k − k =1 = σ − ( w ( f k )) = σ − ( φ ( w ( g j )))which means that w ( g j ) and w ( g j ) are comparable.We conclude that the sequence ( g j ) dj =1 is either comparable, or irrelevantand therefore there exists m ′ ∈ N with d m ′ , natural numbers 1 k ′ < · · · < k ′ d m ′ and { ( h k , y k ) } m ′ k =1 in T , so that φ ( w ( g j )) = w ( h k ′ j ) for j =1 , . . . , d . Observe the following:(12) { ( h k , y k ) } k ′ js − k =1 = σ − ( φ ( w ( g j s ))) = σ − ( w ( f k s )) = { ( f k , x k ) } k s − k =1 . The above implies that { j , . . . , j s } is an initial interval of { , . . . , d } , inparticular:(a) j i = i for i = 1 , . . . , s and(b) k ′ i = k i for i = 1 , . . . , s .Indeed, if 1 t < j s then φ ( w ( g t )) = w ( h k ′ t ) = w ( f k ′ t ) and hence j ∈ A k ′ t .This yields that there is 1 i < s so that t = j i and k i = k ′ t . A simplecardinality argument yields that { j , . . . , j s } = { , . . . , s } and for 1 i < sk ′ i = k i . Also, since j s = s , (12) clearly yields that k s = k ′ s .Observe that the sequence ( g j ) dj =1 is not irrelevant. Indeed, the oppositewould imply that 10 < | g ( y k ′ ) | = | g ( x k ) | C 10, a contradiction. In the last remaining case, the sequence ( g j ) dj =1 is comparable. Define E = { i : k i ∈ { n, . . . , m }} , observe that E is an interval of { , . . . , s } andchoose successive two-point intervals E , . . . , E p of E \ { max E, min E } , sothat E \ ∪ pi =1 E i has at most three elements. The fact that the sequence( g j ) dj =1 is comparable and (b) yield that | g i ( x k i ) − g j ( x k j ) | < / i for all2 i < j s − ε i ) di =1 are alternating, if foreach i we write E i = { r i , r i + 1 } then we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ E i ε j g j ( x k j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) g r ( x k ri ) − g r +1 ( x k ri +1 ) (cid:12)(cid:12)(cid:12) < r i i for i = 1 , . . . , p and hence (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k = n X j ∈ A k ε j g j ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ E ε i g i ( x k i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C + p X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ E i ε i g i ( x k i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C. (cid:3) The result below is the main one of this section and it is used later toprove the main properties of the space X T and its operators. Proposition 5.9. Let 1 C / { ( f k , x k ) } ℓk =1 be a ( C, θ )-dependentsequence and f be a weighted functional in W T . If for some natural numbers1 n m ℓ we set D = { k ∈ { n, . . . , m } : w ( f ) < w ( f k ) } , then: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f X k ∈ D x k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C w ( f ) . In particular, for every natural numbers 1 n m ℓ , k P mk = n x k k C . Proof. We first assume that the first statement holds to prove the fact thatfor 1 n m ℓ , k P mk = n x k k C . Let f ∈ W T . We may as-sume that f is either an element of the basis, or a weighted functional.In the first case, | f ( P mk = n x k ) | max {k x k k ∞ : n k m } < C byRemark 3.12. If on the other hand f is a weighted functional, we distin-guish three cases regarding the weight of f . If w ( f ) < w ( f n ), then thefirst statement yields that | f ( P mk = n x k ) | < C/ w ( f ) < C . If there is n k < m with w ( f k ) w ( f ) < w ( f k +1 ), then as before we obtain that | f ( P k>k x k ) | C/ w ( f ) C/ w ( f k ) < C (recall that w ( f k ) ∈ L andmin L > | f ( x k ) | C while (2) and Remark3.12 yield that | f ( P k 8. Combining (14)with the above, (13) follows.Let now f = (1 / j ) P dq =1 α q be a weighted functional in W p +1 , with( α q ) dq =1 a very fast growing and S j -admissible sequence of α c -averages of W p , and let also 1 n m ℓ be natural numbers. Define D = { k ∈{ n, . . . , m } : j < w ( f k ) } and also for k ∈ D set M k = { q : ran α q ∩ ran x k = ∅ } and N k = n q ∈ M k : s ( α q ) > C w ( f k ) o . Lemma 5.1 yields that for k ∈ D , X q ∈ N k | α q ( x k ) | < w ( f k )2 S. A. ARGYROS AND P. MOTAKIS and therefore: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X q =1 α q X k ∈ D x k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ D X q ∈ M k \ N k α q ( x k ) + X k ∈ D X q ∈ N k α q ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ D X q ∈ M k \ N k α q ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X k ∈ D w ( f k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ D X q ∈ M k \ N k α q ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 42 w ( f n ) (15)where we used that, according to Remark 2.3, the w ( f k )’s are strictly in-creasing.Define A = ∪ k ∈ D M k \ N k , for q ∈ A set D q = { k ∈ D : q ∈ M k \ N k } andobserve the following:(16) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X k ∈ D X q ∈ M k \ N k α q ( x k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X q ∈ A α q X k ∈ D q x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . We will show that the D q ’s are disjoint intervals of { n, . . . , m } . Indeed, let q ∈ A and k , k ∈ D q . If k < k < k , we will show that k ∈ D q . The factthat q ∈ M k ∩ M k means that ran α q ∩ ran x k = ∅ and ran α q ∩ ran x k = ∅ which, of course, yields that ran α q ∩ ran x k = ∅ , i.e. q ∈ M k . Also, q ∈ M k \ N k means that s ( α q ) C w ( f k ) < C w ( f k ) , in other words q / ∈ N k and hence k ∈ D q . We now show that the D q ’s are pairwise disjoint.Let q < q be in A and assume that k ∈ D q ∩ D q . By the fact thatran α q ∩ ran x k = ∅ and Definition 3.10 we obtain8 C w ( f k ) min supp x k max supp α q and since the sequence ( α q ) dq =1 is very fast growing, we obtain that s ( α q ) > max supp α q > C w ( f k ) which means that q ∈ N k , which contradicts k ∈ D q .If we set n q = min D q , then the n q ’s are strictly increasing and since the D q ’s are intervals, by (13)(17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α q X k ∈ D q x k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Cs ( α q ) + 2 C w ( f nq ) for all q ∈ A . Combining (15), (16) and (17): (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d X q =1 α q X k ∈ D x k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X q ∈ A Cs ( α q ) + X q ∈ A C w ( f nq ) + 42 w ( f n ) < C DUAL METHOD OF CONSTRUCTING HI SPACES 23 where we used that, as implied by the definition of very fast growing se-quences, P q (1 /s ( α q )) < 2, that the w ( f n q )’s are strictly increasing ele-ments of L and that min L > 8. Finally, we conclude that | f ( P k ∈ D ) x k | < C/ j . (cid:3) Non-trivial weakly Cauchy sequences and the HI propertyof the space X T In this section we prove that in every block subspace of X T one can find aseminormalized block sequence ( x k ) k and a sequence of weighted functionals( f k ) k so that { ( f k , x k ) } k forms a maximal chain in T . We conclude that X T is hereditarily indecomposable. We also show that in the case T is wellfounded, then the space X T reflexive. On the other hand, if T = U , thenwe show that X U contains no reflexive subspace. Lemma 6.1. Let ( f k ) k be an infinite sequence of non-averages in W T sothat for each n ∈ N the set of all k ’s, so that f k is a weighted functionalof weight w ( f k ) = n , is finite. Then there exists a subsequence of ( f k ) k ,again denoted by ( f k ) k , so that for every natural numbers k < · · · < k d and alternating signs ( ε i ) di =1 in {− , } , the functional α = (1 /d ) P di =1 ε i f k i is an α c -average in W T . Proof. By passing to a subsequence, either all f k ’s are weighted functionals,or they are all of the form f k = ε k e ∗ i k where ε k ∈ {− , } . Assume thatthe second case holds and recall that for al natural numbers d n , i < · · · < i d and for any choice of signs ε j , j = 1 , . . . , d the functional α =(1 /n ) P dj =1 ε j e ∗ i j is an α c -average. The result easily follows.Assume now that the f k ’s are all weighted functionals. Then lim k w ( f k ) = ∞ and so we may pass to a subsequence so that the sequence φ ( w ( f k )) isstrictly increasing. By Ramsey’s theorem [Ra, Theorem A], by passing to afurther subsequence, the φ ( w ( f k ))’s are either all pairwise incomparable, orall pairwise comparable, in the sense of Definition 2.5. If the first one holds,then for any natural numbers d n , k < · · · < k d and for any choice ofsigns ε j , j = 1 , . . . , d the sequence of functionals ( ε j f k j ) dj =1 is incomparable,and hence α = (1 /n ) P dj =1 ε j f k j is an IC -average. This easily implies thedesired result.We assume now that the φ ( w ( f k ))’s are pairwise comparable in the senseof Definition 2.5. Observe first that for at most one k ∈ N we have that φ ( w ( f k )) ∈ L and hence we may assume that φ ( w ( f k )) ∈ L for all k ∈ N .This further implies that ( σ − ( φ ( w ( f k )))) k is a chain in T and hence, thereexist sequences ( h i ) i in W α and ( y i ) i in c ( N , Q ), so that { ( h i , y i ) } ni =1 isin T for all n ∈ N and there is a strictly increasing sequence of naturalnumbers ( m k ) k , so that w ( h m k ) = φ ( w ( f k )) for all k ∈ N . By passing oncemore to a subsequence, we may assume that either | f k ( y m k ) | > 10 for all k ∈ N , or | f k ( y m k ) | 10 for all k ∈ N . If the first one holds, then forany natural numbers d n , k < · · · < k d and for any choice of signs ε j , j = 1 , . . . , d the sequence of functionals ( ε j f k j ) dj =1 is irrelevant, and hence α = (1 /n ) P dj =1 ε j f k j is an IR -average, which implies the desired result.Otherwise, we pass to an even further subsequence so that for every naturalnumbers k < n we have that | f k ( y m k ) − f n ( y m n ) | < / k . This means that forany natural numbers d n , k < · · · < k d sequence of functionals ( f k j ) dj =1 iscomparable and therefore for alternating signs ( ε j ) dj =1 , α = (1 /n ) P dj =1 ε j f k j is a CO -average. The conclusion follows easily. (cid:3) If we assume that the tree T is well founded, then there does not exista strictly increasing sequence of natural numbers which are pairwise com-parable in the sense of Definition 2.5. In this case, the proof of Lemma 6.1yields the following. Lemma 6.2. Assume that the tree T is well founded and let ( f k ) k be aninfinite sequence of non-averages in W T so that for each n ∈ N the setof k ’s, so that f k is a weighted functional of weight w ( f k ) = n , is finite.Then there exists a subsequence of ( f k ) k , again denoted by ( f k ) k , so thatfor every natural numbers k < · · · < k d the functional α = (1 /d ) P di =1 f k i is an α c -average in W T . Lemma 6.3. Let ( x k ) k be a block sequence in X T and assume that there isa constant C > k P ℓk =1 x k k C for all ℓ ∈ N . Then α (( x k ) k ) = 0. Proof. Assume that this is not the case. Then there exist ε > m ∈ N ,a very fast growing sequence of α c -averages ( α q ) q , a sequence of succes-sive subsets ( F n ) n of N , with ( α q ) q ∈ F n S m -admissible for all n ∈ N and asubsequence ( x k n ) n of ( x k ) k so that X q ∈ F n α q ( x k n ) > ε. We may also assume that ran α q ⊂ ran x k n for all q ∈ F n and n ∈ N , hence: X q ∈ F n α q ( x k ′ ) = 0 for k ′ = k n . Choose n > m +1 C/ε and observe that the functional f = 12 m +1 2 n − X n = n X q ∈ F n α q is a weighted functional in W T of weight w ( f ) = m + 1. Also observe thatby (6) and (6) C > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k n − X k =1 x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > f k n − X k =1 x k > m +1 n ε > C which is absurd. (cid:3) DUAL METHOD OF CONSTRUCTING HI SPACES 25 Lemma 6.4. Let ( x k ) k be a seminormalized block sequence in X T with α (( x k ) k ) = 0. Let also ( f k ) k be a sequence of non-zero functionals in W T ,so that f k ( x k ) > (3 / k x k k for all k ∈ N . Then for each n ∈ N the set of k ’s, so that f k is a weighted functional of weight w ( f k ) = n , is finite. Proof. Assume that, passing to a subsequence, there is m ∈ N so that f k isa weighted functional with w ( f k ) = m for all k ∈ N . Proposition 4.4 yieldsthat, passing to a further subsequence, there is k ∈ N so that f k ( x k ) (1 / m )(8 / k x k k < (3 / k x k k for all k > k , which is absurd. (cid:3) Lemmas 6.3 and 6.4 immediately yield the following. Lemma 6.5. Let ( x k ) k be a seminormalized block sequence in X T andassume that there is a constant C > k P ℓk =1 x k k C for all ℓ ∈ N . Let also ( f k ) k be a sequence of non-averages in W T , so that f k ( x k ) > (3 / k x k k for all k ∈ N . Then for each n ∈ N the set of k ’s, so that f k is aweighted functional of weight w ( f k ) = n , is finite.We obtain the first result that depends on the properties of the tree T . Proposition 6.6. If the tree T is well founded, then the space X T is re-flexive. Proof. We will show that the basis of X T is boundedly complete, whichin conjunction with Proposition 3.8 and James’ well known theorem [J,Theorem 1] yield the desired result. Let us assume that this is not the case,i.e. there is a seminormalized block sequence ( x k ) k and a constant C with k P mk =1 x k k C for all m ∈ N . For each k ∈ N choose a functional in W T ,which is not an average, so that ran f k ⊂ ran x k and f k ( x k ) > (3 / k x k k .Lemmas 6.5 and 6.2 yield that there is an infinite subset of the naturalnumbers M , so that for every finite subset F of M the functional α F =(1 / F ) P k ∈ F f k is an α c -average of W T . Note that for m > max F we have α F m X k =1 x k ! = 1 F X k ∈ F f k ( x k ) > 34 inf k k x k k Choose a natural number d > C/ (4 inf k x k k ) and F < · · · < F d so thatthe sequence ( α F q ) dq =1 is S -admissible and very fast growing. Then f =(1 / P dq =1 α F q is in W T and if m = max F d we obtain f ( P mk =1 x k ) > C ,which is absurd. (cid:3) The next result is one of the main features of saturation under constraintsand it plays an important role in deducing the properties of the space. Proposition 6.7. Every block subspace X of X T contains a block sequencegenerating an ℓ spreading model, as well as a block sequence generating a c spreading model. Proof. By Proposition 4.5, it suffices to find, given a block sequence gen-erating an ℓ spreading model, a further block sequence with α -index zero and, given a block sequence generating a c spreading model, a further blocksequence with α -index positive. Assume that ( x k ) k is a block sequence gen-erating an ℓ spreading model, i.e. α (( x k ) k ) > 0. By Proposition 4.3 wemay find C > θ > y k ) k so that each y k isa ( C, θ, n k )-vector, with ( n k ) k strictly increasing. Proposition 5.2 yields thedesired result. Assume now that ( x k ) k is a normalized block generating a c spreading model, i.e. α (( x k ) k ) = 0. Choose a sequence ( f k ) k of non-averagesin W T so that for each k , ran f k ⊂ ran x k and f k ( x k ) > / 4. By Lemma6.4 and Lemma 6.1 we may pass to a further subsequence so that for every k < · · · < k d and alternating signs ( ε i ) di =1 , the functional (1 /d ) P di =1 ε i f k i is an α c -average of W T . Choose a sequence ( F n ) n of successive subsets of N with F n min F n for all n ∈ N and lim n F n = ∞ . Also choose sequencesof alternating signs ( ε i ) i ∈ F n and set y n = P i ∈ F n ε i x i , α n = (1 / F n ) P ε i f i for all n ∈ N . Since ( x k ) k generates a c spreading model we conclude that( y n ) n is bounded. Furthermore for each n , α n is an α c -average of size F n so that α n ( y n ) > / 4. It easily follows that α (( y n ) n ) > (cid:3) Lemma 6.8. Let ( x k ) k be a block sequence in the unit ball of X T generatinga c spreading model and ( f k ) k be a sequence of functionals in W T so thatthe following are satisfied:(a) f k is not an α c -average, ran f k ⊂ ran x k for all k ∈ N and(b) there is a θ > 0, so that (3 / k x k k < f k ( x k ) = θ for all k ∈ N .Then for every n ∈ N there are successive finite subsets of the naturalnumbers ( F k ) mk =1 , sequences of signs ( ε i ) i ∈ F k , k = 1 , . . . , m and a sequenceof non-negative real numbers ( c k ) mk =1 so that the following are satisfied:(i) the vector x = 2 n P mk =1 c k ( P i ∈ F k ε i x i ) is a (9 / , θ, n )-exact vector,(ii) the functional α k = (1 / F k ) P i ∈ F k ε i f i is an α c -average of W T for k = 1 , . . . , m and(iii) the sequence ( α k ) dk =1 is S n -admissible and very fast growing. Inparticular, f = (1 / n ) P mk =1 α k is a weighted functional in W T with,ran f ⊂ ran x , w ( f ) = n and f ( x ) = θ . Proof. By Corollary 4.5 α (( x k ) k ) = 0 and by Lemma 6.4 we obtain that,passing to a subsequence, ( f k ) k satisfies the conclusion of Lemma 6.1, i.e.(c) for every natural numbers k < · · · k d and alternating signs ( ε i ) di =1 ,the functional α = (1 /d ) P di =1 ε i f k i is an α c -average of W T .Corollary 4.5 yields that α (( x k ) k ) = 0 and so we pass once more to asubsequence and find a strictly increasing sequence of natural numbers ( j k ) k ,so that the conclusion of Proposition 4.4 holds, i.e. for every natural numbers d k < · · · < k d , scalars ( λ i ) di =1 we have(18) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X i =1 λ i x k i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (9 / 8) max i d | λ i | DUAL METHOD OF CONSTRUCTING HI SPACES 27 and for every weighted functional in W T f with w ( f ) = j < j d , we have(19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f d X i =1 λ i x k i !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < / j max i d | λ i | . Inductively choose a sequence of successive intervals of N ( I q ) q so that thefollowing are satisfied:(d) min I q I q for all q ∈ N ,(e) I q +1 > max supp x max Iq for all q ∈ N and(f) 1 / j min Iq +1 max supp x max I q < / j max Iq for all q ∈ N .For each q choose alternating signs ( ε i ) i ∈ I q and define w q = X i ∈ I q ε i x i and α q = 1 I q X i ∈ I q ε i f i . Then (18) and (d) yield k w q k / q ∈ N while by (c) and (e) ( α q ) q is a very fast growing sequence of α c -averages of W T . Since ran α q ⊂ ran w q for all q we easily obtain the following:(g) whenever F ⊂ N is such that ( w q ) q ∈ F is S n -admissible, then f =(1 / n ) P q ∈ F α q is in W T and hence, if ( λ q ) q ∈ F are non-negativescalars with P q ∈ F λ q = 1, then f (2 n P q ∈ F λ q w q ) = θ .Furthermore, by (19) and (f), the sequence ( w q ) q is (9 / , ( j ′ q ) q )-RIS, where j ′ q = j min I q for all q ∈ N .By Proposition 3.3 we may choose q < · · · < q m and non-negative realnumbers ( c k ) mk =1 so that the vector x = 2 n P mk =1 c k w q k satisfies all assump-tions of the definition of a (9 / , θ, n )-exact vector (see Definition 3.10).Therefore, the ( I q k ) mk =1 , ( ε i ) i ∈ I qk for k = 1 , . . . , m and ( c k ) mk =1 satisfy thedesired conclusion. (cid:3) Remark 6.9. Let ( x k ) k , ( f k ) k satisfy the assumptions of Lemma 6.8. As-sume moreover that ( g k ) k is a sequence of successive functionals in W T suchthat for each n ∈ N , the set of k ’s with w ( g k ) = n is finite. The same methodof proof, and an argument involving Proposition 3.3, Remark 3.2 and thespreading property of the Schreier families, yields that we may find ( F k ) mk =1 ,( ε i ) i ∈ F k , k = 1 , . . . , m and ( c k ) mk =1 satisfying the conclusion of Proposition6.8 so that moreover the functional g = (1 / n ) P mk =1 ((1 / ( F k )) P i ∈ F k ε i g i )is a weighted functional of weight w ( g ) = n in W T . Lemma 6.10. Let X be a block subspace of X T and n ∈ N . Then thereexists a (9 / , / , n )-exact pair ( f, x ) so that x is in X . Proof. By Proposition 6.7 there exists a normalized block sequence ( x k ) k in X generating a c spreading model. Choose a sequence of functionals f k in W T so that for each k , f k is not an average, f k ( x ) k > / f k ⊂ ran x k . Define x ′ k = (8 / (9 f k ( x k ))) x k and observe that the assumptions ofLemma 6.8 are satisfied for ( x ′ k ) k , ( f k ) k and θ = 8 / 9. The first and thirdassertions of the conclusion of that proposition yield the desired result. (cid:3) Lemma 6.11. Let X and Y be block subspaces of X T , both generated byvectors with rational coefficients. Then there exists an initial interval E of N (finite or infinite) and a sequence of exact pairs { ( f k , x k ) } k ∈ E so that thefollowing are satisfied:(i) for k odd x k is in X while for k even x k is in Y ,(ii) { ( f k , x k ) } mk =1 is a (9 / , / m ∈ E and(iii) {{ ( f k , x k ) } mk =1 : m ∈ E } is a maximal chain of T . Proof. Using an inductive argument and Lemma 6.10, we choose a sequenceof (9 / , / , n k )-exact pairs { ( f k , x k ) } ∞ k =1 so that (i) of the conclusion holdsand { ( f k , x k ) } mk =1 is in U for all m ∈ N . By property (i) of T from Subsec-tion 2.4 we obtain that { ( f , x ) } is in T . If for all m ∈ N we have that { ( f k , x k ) } mk =1 is in T , then we obtain that for E = N the conclusion is satis-fied. Otherwise, set m = max { m ∈ N : { ( f k , x k ) } mk =1 ∈ T } and by property(ii) of T from Subsection 2.4 we obtain that, setting E = { , . . . , m } , theconclusion holds. (cid:3) Recall that T is a subtree of the universal tree U associated with thecoding function σ . If we take T to be all of U , we obtain the result below. Theorem 6.12. The space X U contains no reflexive subspace. Proof. It is enough to show that any block sequence with rational coefficientsis not boundedly complete. Indeed, let ( z k ) k be such a block sequence andapply Lemma 6.11, for X = Y = [( z k ) k ] to find a sequence of exact pairs { ( f k , x k ) } k ∈ E satisfying the conclusion of that lemma. Recall that everymaximal chain in U is infinite and hence E = N . Finally, k x k k > / k ∈ N while by Proposition 5.9 we have that k P nk =1 x k k 27 for all n ∈ N . (cid:3) Theorem 6.13. The space X T is hereditarily indecomposable. Proof. We will show that for every block subspaces X and Y of X T , bothgenerated by vectors with rational coefficients, and for every n ∈ N thereexists x ∈ X and y ∈ Y so that k x + y k 53 and k x − y k > (4 / n .by passing to further block subspaces, we may assume the X and Y aregenerated by block sequences ( z k ) k and ( w k ) k respectively, so that(i) min supp z > n ,(ii) min supp z k > max supp w k and min supp w k > max supp z k for all k ∈ N .Apply Lemma 6.11 to find sequences ( x k ) k ∈ E and ( f k ) k ∈ E satisfying theconclusion of that lemma. The maximality property of that conclusion inconjunction with property (iii) of T from Subsection 2.4 yield that there is aninitial interval G of E so that the set { min supp f k : k ∈ G } is a maximal S -set. By the definition of S choose a partition of G into successive intervals G , . . . , G d so that:(a) { min supp f min G q : q = 1 , . . . , d } is an S -set and DUAL METHOD OF CONSTRUCTING HI SPACES 29 (b) { min supp f k : k ∈ G q } is an S -set for q = 1 , . . . , d .Then (i) implies that n d while the maximality of { min supp f k : k ∈ G } implies that each { min supp f k : k ∈ G q } is a maximal S -set, i.e. G q =min supp f min G q , for q = 1 , . . . , d .Define G o = { k ∈ G : k odd } and G e = { k ∈ G : k even } . Set x = P k ∈ G o x k and y = P k ∈ G e x k . Then x ∈ X , y ∈ Y and k x + y k / < 53 by Proposition 5.9.The sequence ( f k ) k ∈ G q can easily seen to be comparable and hence, thefunctional α q = (1 / G q ) P k ∈ G q ( − k f k is an α c -average for each q =1 , . . . , n with α q ( P k ∈ G q ( − k x k ) = 8 / 9. Also the sequence ( α q ) dq =1 is S -admissible by (a). Also by (ii), s ( α q +1 ) = min supp f min G q +1 > max supp α q and hence the sequence ( α q ) dq =1 is very fast growing. We conclude that f =(1 / P dq =1 α q is in W T and f ( x − y ) = (1 / P dq =1 α q ( P k ∈ G q ( − k x k ) =(1 / d (8 / > (4 / n which yields the desired result. (cid:3) The spreading models of X T In the case T is well founded, i.e. the space X T is reflexive, Propositions4.5 and 6.7 clarify all types of spreading models admitted by Schauder basicsequences in subspaces of X T . In the case of the space X U non-trivial weaklyCauchy sequences exist in every subspace of the space and this section is de-voted to determining what types of spreading models these sequences admit.We start the section by presenting some simple general facts about spreadingsequences, i.e. sequences which are equivalent to their subsequences. Lemma 7.1. Let ( e k ) k be a conditional and spreading Schauder basic se-quence so that if for all k we set u k = e k − − e k , then ( u k ) k is equivalentto the unit vector basis of c . Then ( e k ) k is equivalent to the summing basisof c . Proof. Define d = e and d k = e k − e k − for k > 2. The fact that ( e k ) k isconditional and spreading implies that ( d k ) k is Schauder basic. We will showthat ( d k ) k is equivalent to the unit vector basis of c , which easily yields theconclusion. Since ( d k ) k is Schauder basic, it dominates the unit vector basisof c so it remains to prove that it is dominated by it as well. Note thatby the spreading property of ( e k ) k , both sequences ( d k ) k and ( d k − ) k areequivalent to ( u k ) k and hence also to the unit vector basis of c . Therefore,if ( a k ) k is a sequence of scalars, finitely many of which are non-zero, then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 a k d k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 a k d k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =2 a k − d k − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) which implies the desired result. (cid:3) Lemma 7.2. Let ( e k ) k be a 1-spreading sequence and ( z k ) k be a convexblock sequence of ( e k ) k . Then ( z k ) k is 1-dominated by ( e k ) k . Proof. Let n ∈ N and ( λ k ) nk =1 be a sequence of scalars. We may assumethat there is d ∈ N and ( c i ) di =1 so that P di =1 c i = 1 and z k = P di =1 c i e p k,i ,where p k,i < p m,j for k < m and p k,i p k,j for i < j . Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 λ k z k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X k =1 c k max i d (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 λ k e p k,i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 λ k e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (cid:3) Lemma 7.3. Let ( e k ) k be a 1-spreading Schauder basis sequence and ( z k ) k be an absolutely convex block sequence of ( e k ) k . Then for every ε > 0, ( z k ) k has a subsequence which is (1 + ε )-dominated by ( e k ) k . Proof. If for each k , z k = P i ∈ F k c i e i with P i ∈ F k | c i | = 1 for all k ∈ N ,define F + k = { i ∈ F k : λ i > } , F − k = F k \ F + k and λ + k = P i ∈ F + k λ i , λ − k = P i ∈ F − k λ i . We fix δ > λ + , λ − with P k | λ + k − λ + | < δ and P k | λ − k − λ − | < δ . Note that | λ + | + | λ − | = 1. We shall assume that λ + = 0 as well as λ − = 0, as theother cases are treated similarly. We may pass to a further subsequenceso that for all k , λ + k = 0 and λ − k = 0 and so we may define the vectors z + k = (1 /λ + k ) P i ∈ F + k λ i e i , z − k = (1 /λ − k ) P i ∈ F − k λ i e i . Observe that ( z + k ) k and( z − k ) k are both convex block sequences of ( e i ) i . Then if ( a k ) nk =1 is a sequenceof scalars: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k z k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k λ + k z + k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k λ − k z − k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) | λ + | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k z + k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | λ − | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k z + k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 2 δ max | a k | | λ + | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | λ − | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 2 δ max | a k | = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 a k e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 2 δ max | a k | where the third inequality follows from Lemma 7.2. For δ < ε/ (4 C ), where C is the basis constant of ( e k ) k , the result follows. (cid:3) Proposition 7.4. Let X be a Banach space and ( x k ) k , ( y k ) k be Schauderbasic sequences in X . If ( x k ) k admits an ℓ spreading model while ( y k ) k does not, then ( x k − y k ) k admits an ℓ spreading model. Proof. We pass to a subsequence so that ( x k ) k , ( y k ) k and ( x k − y k ) k generatespreading models ( z k ), ( w k ) k and ( u k ) k respectively. We will show that ( u k ) k is equivalent to the unit vector basis of ℓ . According to the assumption,( w k ) k is not equivalent to the unit vector basis of ℓ and so we may choose DUAL METHOD OF CONSTRUCTING HI SPACES 31 an absolutely convex block vector w = P pk =1 λ k w k with k w k < c/ 2, where c > z k ) k c -dominates the unit vector basis of ℓ . Since ( w k ) k is 1-spreading, by copying the vector w , we may find successive convex blockvector P k ∈ F n λ k w k , all of which have norm strictly smaller that c/ 2. For n ∈ N define d n = P k ∈ F n λ k u k . Then ( d n ) n is an absolutely convex blocksequence of ( u k ) k and we will show that it is equivalent to the unit vectorbasis of ℓ . Lemma 7.3 will yields that ( u k ) k is equivalent to the unit vectorbasis of ℓ as well, which is the desired result. Indeed, let ( c n ) mn =1 be asequence of scalars. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n d n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = lim i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n X k ∈ F n λ k ( x k + i − y k + i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > lim i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n X k ∈ F n λ k x k + i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − lim i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n X k ∈ F n λ k y k + i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n X k ∈ F n λ k z k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X n =1 c n X k ∈ F n λ k w k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > c m X n =1 | c n | − c m X n =1 | c n | = c m X n =1 | c n | and the proof is complete. (cid:3) Proposition 7.5. Let ( λ i ) i be a sequence of scalars so that if ( e i ) i is thebasis of X T and x k = P ki =1 λ i e i for all k ∈ N , then ( x k ) k is bounded andnon-convergent in the norm topology. Then ( x k ) k admits only the summingbasis of c as a spreading model. Proof. Pass to a subsequence of ( x k ) k that generates a spreading model( z k ) k . The fact that ( x k ) k is non-trivial weakly easily implies that ( z k ) k iseither equivalent to the unit vector basis of ℓ , or a conditional spreadingsequence. Then, if y k = x k − − x k and u k = z k − − z k for all k ∈ N , thesequence ( y k ) k generates ( u k ) k as a spreading model. Lemma 6.3 impliesthat α (( y k ) k ) = 0 and hence by Proposition 4.4, ( u k ) k is equivalent to theunit vector basis of c . Therefore, ( z k ) k is conditional and spreading and byLemma 7.1 we deduce the desired result. (cid:3) Remark 7.6. Note that the summing basis norm is the minimum condi-tional spreading norm, in terms of domination. An argument similar to thatused in the proof of Lemma 7.2 yields the following: if ( x k ) k is a sequencegenerating the summing basis of c as a spreading model, then every convex block sequence of ( x k ) k admits only the summing basis of c as a spreadingmodel as well.The next will be useful in the sequel. Lemma 7.7. Let ( x k ) k be a non-trivial weakly Cauchy sequence in X T .Then there is a convex block sequence ( y k ) k of ( x k ) k that generates thesumming basis of c as a spreading model. Proof. Let x ∗∗ be the w ∗ -limit of ( x k ) k and y k = P ki =1 x ∗∗ ( e ∗ i ) e i . Then byProposition 3.8 ( y k ) k , w ∗ -converges to x ∗∗ . By Lemma 7.5, passing to asubsequence, ( y k ) k generates the summing basis of c as a spreading model.As ( x k − y k ) k is weakly null, by Mazur’s theorem there is a convex blocksequence of ( x k ) k that is equivalent to a convex block sequence of ( y k ) k . ByRemark 7.6 we deduce the desired result. (cid:3) Proposition 7.8. Every non-trivial weakly Cauchy sequence in X T admitsa spreading model which is either equivalent to the summing basis of c or equivalent to the unit vector basis of ℓ . If moreover T = U , then ev-ery infinite dimensional subspace of X U contains non-trivial weakly Cauchysequences admitting both of these types of spreading models. Proof. Let ( x k ) k be a non-trivial weakly Cauchy sequence in X T and x ∗∗ beits w ∗ -limit. If for k ∈ N we set y k = P ki =1 x ∗∗ ( e ∗ i ) e i , By proposition 3.8we obtain that ( y k ) k w ∗ -converges to x ∗∗ and hence, setting z k = y k − x k ,the sequence ( z k ) k is weakly null. By Proposition 7.5 ( y k ) k admits only thesumming basis of c as a spreading model, while ( z k ) k is either norm null,or it is not. If it is norm null then clearly ( x k ) k admits only the summingbasis of c as a spreading model. Otherwise, it follows from Proposition4.5 that ( z k ) k either admits only the unit vector basis of c as a spreadingmodel, or it admits the unit vector basis of ℓ as a spreading model. If thefirst one holds, we conclude that any spreading model admitted by ( x k ) k must be equivalent to the unit vector basis of c and if the second one holds,Proposition 7.4 yields that ( x k ) k admits an ℓ spreading model.The second assertion is proved as follows: by Theorem 6.12, and Propo-sition 7.5 we obtain that every subspace of X U admits the summing basis of c as a spreading model. Combining this with Propositions 6.7 and 7.4 wededuce that there is a non-trivial weakly Cauchy sequence in every subspacegenerating an ℓ spreading model. (cid:3) Remark 7.9. We comment that using the α -index it can be shown thatevery non-trivial weakly Cauchy sequence in X T admitting an ℓ spreadingmodel, has a subsequence that generates an ℓ n spreading model with lowerconstant θ/ n , for all n ∈ N and some θ > X T , depending on the choice of T . Proposition 7.10. Every seminormalized weakly null sequence in X T ad-mits either ℓ or c as a spreading model and every non-trivial weakly Cauchy DUAL METHOD OF CONSTRUCTING HI SPACES 33 sequence in X T admits either ℓ or the summing basis of c as a spreadingmodel. In particular the following hold.(i) If T is well founded (i.e. the space X T is reflexive), then everySchauder basic sequence in X T admits either ℓ or c as a spread-ing model and both of these types are admitted by every infinitedimensional subspace.(ii) If T = U , then every Schauder basic sequence in X T admits either ℓ , either c , or the summing basis of c as a spreading model andall three of these types are admitted by every infinite dimensionalsubspace. 8. Operators on the space X T In this final section we prove the properties of the operators defined onsubspaces of X T . We characterize strictly singular operators with respect totheir action on sequences generating certain types of spreading models. Weconclude that the composition of any pair of singular operators is a compactone. This ought to be compared to [AM1, Theorem 5.19 and Remark 5.20].We also show that all operators defined on block subspaces of X T non-trivialclosed invariant subspaces and that operators defined on X U are strictlysingular if and only if they are weakly compact. Lemma 8.1. Let x , y be non-zero vectors in X T . Then there exist non-averages f , g in W T so that the following hold:(i) ran f ⊂ ran x and ran g ⊂ ran y ,(ii) f ( x ) > (8 / k x k and g ( y ) > (8 / k y k ,(iii) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:18) f ( x ) x (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) / Proof. Choose a non-average g in W T with g ( y ) > (8 / k y k . If | g ( x ) | > (8 / k x k define f = sgn( g ( x )) g | ran x and observe that f , g satisfy the con-clusion. Otherwise g ( x ) (8 / k x k and choose any non-average f in W T with f ( x ) > (8 / k x k and ran f ⊂ ran x . A simple calculation yields that f , g satisfy the conclusion. (cid:3) Lemma 8.2. Let ( f, x ) be an (9 / , / , n )-exact pair in X T and let also ρ in [ − / , / g in W T of weight w ( g ) = n , so that ran g ⊂ ran x and | g ( x ) − ρ | < / n +1 . Proof. By Remark 3.12, we have that k x k ∞ < / (2 n < / n +1 . Thefact that f ( x ) = 8 / E of ran f and ε ∈ {− , } , so that g = εEf is the desired functional. (cid:3) The following result characterizes strictly singular operators, defined onsubspaces of X T , in the following manner: an operator is strictly singu-lar if and only if it does not preserve any type of spreading model. It isworth mentioning this we could neither prove nor disprove the same result in [AM1]. The reason for this difference is the presence of β -averages in thatpaper and their absence in the present one. Proposition 8.3. Let X be an infinite dimensional closed subspace of X T and T : X → X T be a bounded linear operator. The following assertionsare equivalent.(i) The operator T is strictly singular.(ii) There exists a normalized weakly null sequence ( y k ) k in X so that( T y k ) k converges to zero in norm.(iii) For every sequence ( x k ) k in X generating a c spreading model,( T x k ) k converges to zero in norm.(iv) For every sequence ( x k ) k in X generating an ℓ spreading model,( T x k ) k does not admit an ℓ spreading model. Proof. That (i) implies (ii) follows from the fact that ℓ does not embedinto X T and that (iv) implies (i) follows from Proposition 6.7. We shall firstdemonstrate that (iii) implies (iv) and then that (ii) implies (iii).We assume that (ii) is true and towards a contradiction assume that thereis a sequence in ( x k ) k in X , so that both ( x k ) k and ( T x k ) k generate an ℓ spreading model. By taking differences, we may assume that both ( x k ) k and ( T x k ) k are block sequences with α -index positive. By Proposition 4.3we may assume that there is θ > ℓ n spreading model with a lower constant θ/ n for all n ∈ N . Using thesame Proposition, construct a block sequence ( y k ) k of ( x k ) k , so that each y k is a ( C, θ, n k )-vector and k T y k k > θ for all k ∈ N with a ( n k ) k a strictlyincreasing sequence of natural numbers. Proposition 5.2 yields that ( y k ) k admits only c as a spreading model, which contradicts (ii).We shall now prove that (ii) implies (iii). Toward a contradiction assumethat there is normalized weakly null sequence ( y k ) k in X with lim k T y k = 0in norm, as well as a sequence ( x k ) k in X generating a c spreading model, sothat ( T x k ) k does not converge to zero in norm. By perturbing the operator T we may assume that the following are satisfied:(a) ( y k ) k , ( x k ) k and ( T x k ) k are all seminormalized block sequences withrational coefficients and(b) T y k = 0 for all k ∈ N .For each k ∈ N , choose f k and g k so that the conclusion of Lemma 8.1is satisfied, i.e. ran f k ⊂ ran x k , ran g k ⊂ ran T x k , f k ( x k ) > (8 / k x k k , g k ( T x k ) > (8 / k T x k k and | g k ((8 / f k ( x k )) x k ) | / 9. Hence, if for all k weset x ′ k = (8 / f k ( x k )) x k and θ = (8 / inf k k T x k k / sup k k x k k > 0, then forall k ∈ N :(c) ran f k ⊂ ran x ′ k , ran g k ⊂ ran T x ′ k ,(d) f k ( x ′ k ) = 8 / g k ( T x ′ k ) > θ and(e) | g k ( x ′ k ) | / DUAL METHOD OF CONSTRUCTING HI SPACES 35 We note that the boundedness of T yields that ( T x k ) k admits only c as aspreading model, combining this with g k ( T x k ) > (8 / k T x k k for all k ∈ N and Lemma 6.4 we obtain that(f) for each n ∈ N , the set of k ’s so that g k is a weighted functional ofweight w ( g k ) = n is finite.We pass to a subsequence, so that there is ρ in [ − / , / 9] so that(g) | g k ( x ′ k ) − ρ | < / k +1 for all k ∈ N .Let now n ∈ N with n > k T k /θ . We construct a (9 / , / { ( h k , z k ) } mk =1 with the following properties:(h) min supp h > n and ( h k ) mk =1 is S -admissible,(j) There is a partition of N into successive intervals ( G k ) k and succes-sive subsets of the natural numbers ( F j ) j as well as a sequence ofsigns ( ε i ) i so that for k odd: z k = 2 w ( h k ) X j ∈ G k c j X i ∈ F j ε i x ′ i h k = 12 w ( h k ) X j ∈ G k F j X i ∈ F j ε i f i , (k) for k odd the functional φ k = 12 w ( h k ) X j ∈ G k F j X i ∈ F j ε i g i is a weighted functional in W T of weight w ( φ k ) = w ( h k ) and(l) for k even, ran φ k − < ran z k < ran φ k +1 and z k is a linear combina-tion of the ( y k ) k .Note that in the construction for k odd we use Lemma 6.8, (f) and Remark6.9. For k even we just use Lemma 6.10 while the fact that we continue thisprocess until ( h k ) mk =1 is S -admissible follows from properties (ii) and (iii)from Subsection 2.4.Proposition 5.9 yields k P mk =1 z k k 27. We will finish the proof by show-ing that k T ( P mk =1 z k ) k > k T k , which is absurd.For k even, by Lemma 8.2, we may choose φ k in W T with ran φ k ⊂ ran z k and | φ k ( z k ) − ρ | < / w ( h k )+1 / k +1 . Moreover, (g), (j) and (k) yieldthat for k odd, | φ k ( z k ) − ρ | < / k +1 as well. We conclude:(m) | φ k ( z k ) − φ k ′ ( z k ′ ) | < / k for 1 k k ′ m .Since { ( h k , z k ) } mk =1 is in T and φ k is a functional of weight w ( h k ) for k =1 , . . . , m by (l) and (m) we conclude that the sequence ( φ k ) mk =1 is compatible,in the sense of Definition 2.6. Arguing identically as in the proof of Theorem6.13, for the already fixed n we may choose a partition of { , . . . , m } intosuccessive intervals ( E q ) nq =1 so that if α q = (1 / E q ) P k ∈ E q ( − k +1 φ k , thenthe sequence ( α q ) dq =1 is a very fast growing and S -admissible of α c -averages of W T . Define ψ = (1 / P nq =1 α q which is in W T . Then, by (b) and (l) T ( P mk =1 z k ) = P k odd T z k . By (d), (j) and (k) we obtain: (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) T m X k =1 z k !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X k odd T z k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) > ψ X k odd T z k ! = 12 n X q =1 E q X odd k ∈ E q φ k ( T z k ) > θ n > k T k . (cid:3) We remind that in [AM1, Theorem 5.19] it is proved that the compositionof any triple of strictly singular operators, defined on a subspace of X ISP , isa compact one. We were unable to determine whether that result is optimalor if it could be stated for couples of strictly singular operators. As wecommented before Proposition 8.3, the construction of the space X ISP form[AM1] uses β -averages while the present one does not. A direct consequenceof this difference is that in the case of the space X T we can prove thefollowing. Theorem 8.4. Let X be a closed subspace of X T and S , T : X → X bestrictly singular operators. Then the composition T S is a compact operator. Proof. Since ℓ does not embed into X T , it suffices to show that T S mapsweakly null sequences to norm null ones and ( x k ) k be a weakly null sequencein X . If it is norm null then there is nothing more to prove. Otherwise,it either admits a c or an ℓ spreading model. If the first one holds, thenby Proposition 8.3 ( Sx k ) k has a subsequence which is norm null. If on theother hand ( x k ) k admits an ℓ spreading model then, passing to subsequence,( Sx k ) k is either norm null, or it generates a c spreading model and hence,arguing as above, we obtain that ( T Sx k ) k is norm null. (cid:3) Corollary 8.5. Let X be an infinite dimensional closed subspace of X T and S : X → X be a non-zero strictly singular operator. Then S admits anon-trivial closed hyperinvariant subspace. Proof. Assume first that S = 0. Then it is straightforward to check thatker S is a non-trivial closed hyperinvariant subspace of S . Otherwise, if S = 0, then Theorem 8.4 yields that S is compact and non-zero. Since S commutes with its square, by [Si, Theorem 2.1], it is sufficient to checkthat for any α , β ∈ R with β = 0, we have ( αI − S ) + β I = 0 (see also[H, Theorem 2]). The fact that S is strictly singular, easily implies that thiscondition is satisfied. (cid:3) Lemma 8.6. Let ( x k ) k be a seminormalized block sequence in X T with α (( x k ) k ) = 0 and X = [( x k ) k ]. Let T : X → X T be a linear operator andassume that there exist ε > g k ) k in W T satisfying the following:(i) g k ( T x k ) > ε and g k ( x k ) = 0 for all k ∈ N and DUAL METHOD OF CONSTRUCTING HI SPACES 37 (ii) for all n ∈ N the set of k ’s so that g k is a weighted functional ofweight w ( f k ) = n is finite.Then T is unbounded. Proof. Towards a contradiction we assume that T is bounded. We mayassume that the x k ’s have rational coefficients. Choose a sequence of nonaverages in W T so that ran f k ⊂ ran x k and f k ( x k ) > (8 / k x k k for all k ∈ N .For all k ∈ N define x ′ k = (8 / (9 f k ( x k ))) x k and set θ = (8 ε ) / (9 sup k x k k ) > g k ( x ′ k ) = 0 for all k ∈ N and(b) g k ( T x ′ k ) > ε for all k ∈ N .Let now n ∈ N with n > k T k /θ . We construct a (9 / , / { ( h k , z k ) } mk =1 so that min supp h > n , ( h k ) mk =1 is S -admissible,there is a partition of N into successive intervals ( G k ) k and successive subsetsof the natural numbers ( F j ) j as well as a sequence of signs ( ε i ) i so that for k = 1 , . . . , m : z k = 2 w ( h k ) X j ∈ G k c j X i ∈ F j ε i x ′ i h k = 12 w ( h k ) X j ∈ G k F j X i ∈ F j ε i f i , and the functional φ k = 12 w ( h k ) X j ∈ G k F j X i ∈ F j ε i g i is a weighted functional in W T of weight w ( φ k ) = w ( h k ). Note that by (a)(c) φ k ( z k ) = 0 for k = 1 , . . . , m .Proposition 5.9 yields that k P mk =1 z k k 27. We will show that also k T ( P mk =1 z k ) k > k T k , which will complete the proof.Since { ( h k , z k ) } mk =1 is in T and φ k is a functional of weight w ( h k ) for k = 1 , . . . , m by (c) we easily conclude that the sequence (( − k φ k ) mk =1 iscompatible, in the sense of Definition 2.6. Arguing in the proof of Theorem6.13 we choose a partition of { , . . . , m } into successive intervals ( E q ) nq =1 sothat if α q = (1 / E q ) P k ∈ E q φ k , then the sequence ( α q ) dq =1 is a very fastgrowing and S -admissible of α c -averages of W T . An argument similar tothat used in the end of the proof of Proposition 8.3 yields k P mk =1 T z k k >nθ/ > k T k . (cid:3) Remark 8.7. If E is an interval of N , we denote by P E the projection onto E , associated with the Schauder basis ( e i ) i of X T . It easily follows that if( x k ) k , ( y k ) k are block sequences in X T , then(i) if α (( x k ) k ) = 0 and ( E k ) k is a sequence of successive intervals of thenatural numbers, then α (( P E k x k ) k ) = 0. (ii) if α (( x k ) k ) = 0 and α (( y k ) k ) = 0, then α (( x k + y k ) k ) = 0. Lemma 8.8. Let ( x k ) k be a seminormalized block sequence in X T and X = [( x k ) k ]. Let T : X → X T be a bounded linear operator and for each k ∈ N set y k = P ran x k T x k . If the sequence ( y k ) k is norm null, then T isstrictly singular. Proof. By Proposition 8.3 it suffices to find a seminormalized weakly nullsequence ( u k ) k in X so that ( T u k ) k is norm null. For all k define z k = P [1 , min ran x k − T x k and w k = P [max ran x k +1 , ∞ )] T x k . By perturbing T andpassing to a subsequence, we may assume that T x k = z k + w k and z k 4) inf k z k k and for each k choose ( g k ) k with ran g k ⊂ ran z k and g k ( x k ) > (3 / k x k k . By Remark 8.7 we obtain that α (( z k ) k ) = 0 and byLemma 6.4 we conclude that the assumptions of Lemma 8.6 are satisfied,i.e. T is unbounded, which is absurd. Case 2: ( x k ) k admits an ℓ spreading model and ( T x k ) k does not, i.e. itis either norm null, or passing to a subsequence it generates a c spreadingmodel. In the first case we are done, in the second case choose a sequenceof successive S sets ( F k ) k with lim k F k = 0 and for all k define u k =(1 / F k ) P i ∈ F k x i . Then ( u k ) k is the desired sequence. Case 3: by passing to a subsequence, both ( x k ) k and ( T x k ) k generate an ℓ spreading model. Remark 8.7 yields that either α (( z k ) k ) > α (( w k ) k ) > n ∈ N , δ > 0, a very fast growing sequence of α c -averages ( α q ) q of W T and a sequence of successive subsets ( F k ) k of N , so that(a) ( α q ) q ∈ F k is S n admissible for all k ∈ N ,(b) ran α q ⊂ ran z k for all q ∈ F k , k ∈ N and(c) P q ∈ F k α q ( z k ) > δ for all k ∈ N .By Proposition 4.3, there are C > θ > u k ) k so that for each k , u k = 2 n k P j ∈ G k c k x k is a ( C, θ, n k )-vector with ( n k )strictly increasing. Using an argument involving Proposition 3.3, Remark3.2 and the spreading properties of the Schreier families, we may also chosethe sets G k so that ( α q ) q ∈∪ j ∈ Gk F j is S n + n k -admissible and hence, g k =(1 / n + n k ) P q ∈∪ j ∈ Gk F j α q is a weighted functional of weight w ( g k ) = n + n k for all k ∈ N . By (b) we obtain g k ( u k ) = 0 and by(c) g k ( T u k ) > δ/ n forall k ∈ N . Finally, combining these facts with Proposition 5.2 we concludethat ( u k ) k admits a c spreading model, i.e. the assumptions of Lemma 8.6are satisfied. This means that T is unbounded, which is absurd. (cid:3) Theorem 8.9. Let X be a block subspace of X T . Then for every boundedlinear operator T : X → X there is a λ ∈ R so that T − λI is strictlysingular. DUAL METHOD OF CONSTRUCTING HI SPACES 39 Proof. Let ( x k ) k be the normalized block sequence so that X = [( x k ) k ]. Wemay, of course, assume that ( x k ) k is normalized and let Q { n } denote theprojections associated with the basis ( x n ) n of X , i.e. Q { n } x m = δ n,m . Thenfor each k ∈ N , Q { k } T x k = λ k x k for some λ k ∈ R . Choose an accumulationpoint λ of ( λ k ) k and by Lemma 8.8 it easily follows that T − λI is strictlysingular. (cid:3) Remark 8.10. The reason the above result cannot be stated for everyclosed subspace of X T , is that in the definition of the norming set W T it isnot allowed to take α -averages of convex combinations of elements of W T .We note that the construction presented in this paper can also be used toobtain a space X C T defined over the field of complex numbers. In that case, asit was proved in [GM, Theorem 18], every subspace of X C T satisfies the scalarplus strictly singular property. Therefore, compared to Theorem 8.11 whichis stated for block subspaces of X T , every closed subspace of X C T satisfiedthe invariant subspace property. Theorem 8.11. Let X be a block subspace of X T and T : X → X bea non-scalar bounded linear operator. Then T admits a non-trivial closedhyperinvariant subspace. Proof. By Theorem 8.9 there is a λ ∈ R so that the operator S = T − λI isstrictly singular. Note that S = 0, otherwise T would be a scalar operator.Corollary 8.5 yields that S admits a non-trivial closed hyperinvariant sub-space Y . It is straightforward to check that Y is a hyperinvariant subspacefor T . (cid:3) We note that the following property of the strictly singular operators on X U , was also proved for an HI space which appeared in [AAT]. Theorem 8.12. Let X be a closed subspace of X U and T : X → X U be abounded linear operator. The following assertions are equivalent.(i) The operator T is strictly singular.(ii) The operator T is weakly compact. Proof. The implication (ii) ⇒ (i) immediately follows from Theorem 6.12.Assume now that T is strictly singular and not weakly compact, whichimplies that there is a sequence ( x k ) k in X so that both ( x k ) k and ( T x k ) k are non-trivial weakly Cauchy. By Lemma 7.7 we may assume that ( x k ) k generates the summing basis of c as a spreading model. 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