· · · > αj ≥ 0. Definition 3.11. If (αi )j1 are countable ordinals and (ni )j1 ⊆ N, by ((Sα1 )n1 , . . . , (Sαj )nj ) we denote the class of subsets of N that can be written in the form E11 ∪ · · · ∪ En1 1 ∪ E12 ∪ · · · ∪ En2 2 ∪ · · · ∪ E1j ∪ · · · ∪ Enj j where E11 < E21 < · · · < Enj j and Eik ∈ Sαk for all k ≤ j and i ≤ nk . Proposition 3.12. Let F be a regular set of finite subsets of N with I(F) = ω α1 · n1 + · · · + ω αk−1 · nk−1 + ω αk · nk , in Cantor normal form. Then there exists a subsequence M of N so that we have F(M ) ⊆ ((Sα1 )n1 , . . . , (Sαk )nk ). Remark 3.13. The conclusion of the proposition holds even if I(F) < ω α1 ·n1 + · · · + ω αk · nk . Indeed this follows from the fact that if α < β and F(N ) ⊆ Sα , and N = (ni ), then there exists r ∈ N so that F((ni )i≥r ) ⊆ Sβ (by Proposition 3.2(a)).
26
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES6
Proof of Proposition 3.12. We induct on I(F). If I(F) = 1 then F contains only singletons {n} and so F(N) ⊆ ((S0 )1 ). Assume the proposition holds for all classes with index ≤ β, and let I(F) = β + 1. Let β = ω α1 · n1 + · · · + ω αk−1 · nk−1 + nk (nk ≥ 0). Then D(F) is a regular class of sets with I(D(F)) = β and so there exists M with D(F)(M ) ⊆ ((Sα1 )n1 , . . . , (Sαk−1 )nk−1 , (S0 )nk ) where (S0 )0 ≡ ∅). Note that if F ∈ F \ D(F) then if k = max F and G = F \ {k} we have that G ∈ D(F). Indeed since F is regular, for n ≥ k, G ∪ {n} ∈ F and G ∪ {n} → F . It follows that F(M ) ⊆ ((Sα1 )n1 , . . . , (S0 )nk +1 ). If β is a limit ordinal and the proposition holds for regular classes with index < β we proceed as follows. Let β = ω α1 · n1 + · · · + ω αk · nk where αk > 0 and nk > 0. For j ∈ N set Fj = {F ∈ F : F > j and {j} ∪ F ∈ F}. Then clearly each Fj is regular. Also I(Fj ) < β for if I(Fj ) = β then Dβ (Fj ) = {∅}. By the definition of Fj this implies that {j} ∈ Dβ (F) which contradicts I(F) = β. For each j set Gj = {{j} ∪ F : F ∈ Fj } ∪ Fj . S Each Gj is regular, I(Gj ) ≤ I(Fj ) + 1 < β and F = j Gj . If αk = (αk − 1) + 1 (i.e., is a successor ordinal), choose pj ↑ ∞ so that I(Gj ) < ω α1 · n1 + · · · + ω αk · (nk − 1) + ω αk −1 · pj . By Remark 3.13 and the inductive hypothesis, there exists Mj ⊆ N with Gj (Mj ) ⊆ ((Sα1 )n1 , . . . , (Sαk )nk−1 , (Sαk −1 )pj ) . Without loss of generality we can choose the Mj ’s so that M1 ⊇ M2 ⊇ · · · . Let 2 1 6 F ∈ F then F ∈ Gj for Mj = (mji )∞ i=1 . Let M = (mp1 , mp2 , . . .) ≡ (mi ). If ∅ = j = min F . Since (mj , mj+1 , . . .) ⊆ (mjj , mjj+1 , . . .) and Gj is spreading, we have that (mi )i∈F ∈ ((Sα1 )n1 , . . . , (Sαk )nk −1 , (Sαk −1 )pj ). Also mj = mjpj ≥ pj and so (mi )i∈F ∈ ((Sα1 )n1 , . . . , (Sαk )nk ) . If αk is a limit ordinal, let ηj ↑ αk be the sequence of ordinals defining Sαk . Choose `(j) ↑ ∞ so that I(Gj ) ≤ ω α1 · n1 + · · · + ω αk · (nk − 1) + ω η`(j) . As above we choose M1 ⊇ M2 ⊇ · · · so that Gj (Mj ) ⊆ ((Sα1 )n1 , . . . , (Sαk )nk −1 , (Sη`(j) )) . As in Remark 3.13 we may choose rj ↑ ∞ so that when considering Gj ((mji )i≥rj ) the last set we get is not only in Sη`(j) but also in Sαk . Thus if M = (mj ) = (mjrj )∞ j=1 it follows that F(M ) ⊆ ((Sα1 )n1 , . . . , (Sαk )nk ). 2 Corollary 3.14. Let F be a pointwise closed class of finite subsets of N. Then there exist α < ω1 and a subsequence M of N so that F(M ) ⊆ Sα . Proof. Let R be the regular hull of F; that is, R = {G : there exists F = (n1 , . . . , nk ) ∈ F with G ⊆ (mi )k1 for some m1 < · · · < mk with mi ≥ ni for i ≤ k}. Clearly, R is hereditary and spreading. We check that it is also pointwise closed, and hence the corollary follows from Proposition 3.10. Let Gn → G pointwise for some (Gn ) ⊆ R. If |G| < ∞ then G is an initial segment of Gn for large n and so G ∈ R. It remains to note that |G| = ∞ is impossible. If G = (n1 , n2 , . . .) then for
4. ASYMPTOTIC CONSTANTS AND ∆(X)
27
all k, (n1 , . . . , nk ) is a subset of some spreading of some set Fk ∈ F. In particular |{n ∈ Fk : n ≤ nj }| ≥ j for 1 ≤ j ≤ k. Thus any limit point of (Fk )∞ k=1 is infinite which contradicts the hypotheses that F is pointwise closed and consists of finite sets. For some other interesting properties of the Schreier classes we refer the reader to [ArMeTs] and [AnO]. 4. Asymptotic constants and ∆(X) Asymptotic constants considered in this paper will be determined by the Schreier families Sα ; nevertheless it should be noted that they can be introduced for a very general class of families of finite subsets of N. Definition 4.1. If F is a regular set of finite subsets of N, a sequence of sets E1 < · · · < Ek is F-admissible if (min(Ei ))ki=1 ∈ F. If (xi ) is a basic sequence in a Banach space and (yi )k1 ≺ (xi ), then (yi )k1 is F-admissible (w.r.t. (xi )) if (supp(yi ))k1 is F-admissible, where supp(yi ) is taken w.r.t. (xi ). We use a short form α-admissible to mean Sα -admissible. The next definition was first introduced in [To1] for asymptotic `p spaces with 1 ≤ p < ∞. Definition 4.2. Let F be a regular set of finite subsets of N. For a basic sequence (xi ) in a Banach space X we define δF (xi ) to be the supremum of δ ≥ 0 such that whenever (yi )k1 ≺ (xi ) is F-admissible w.r.t. (xi ) then k
k X i=1
yi k ≥ δ
k X
kyi k .
i=1
If X is a Banach space with a basis (ei ) we write δF (X) for δF (ei ). For α < ω1 , we set δα (xi ) = δSα (xi ) and δα (X) = δSα (X). 0 Remark 4.3. P Note that δF (xi ) is equal to the supremum of all δ ≥ 0 such 0 that kyk ≥ δ kEi yk, for S all y ∈ hxi i and all adjacent F-admissible intervals E1 < · · · < Ek such that Ei ⊇ supp(y). Here the support of y and restrictions 0 Ei y are understood to be w.r.t. (xi ). Indeed, P clearly sup δ ≥ δF (xi ). Conversely, k given (yi )1 ≺ (xi ) F-admissible we set y = yi and we let (E1 , . . . , Ek ) be adjacent intervals such that Ei ⊇ supp(yi ) and min Ei = min supp(yi ) for all i.
In as much as distortion problems involve passing to block subspaces and renormings, it is natural to make two more definitions. Definition 4.4. Let F be a regular set of finite subsets of N and let (ei ) be a basis for X. δ˙F (X) = δ˙F (ei ) = sup{δF (xi ) : (xi ) ≺ (ei )} and δ¨F (X) = δ¨F (ei ) = sup{δ˙F ((ei ), | · |) : | · | is an equivalent norm on X} . We write δ˙S (X) = δ˙α (X) and δ¨S (X) = δ¨α (X). α
α
The asymptotic constants provide a measurement of closeness of block subspaces of X to `1 . Clearly X is asymptotic `1 w.r.t. (ei ) if and only if δ1 (X) > 0. The asymptotic `1 constant of X is then equal to δ1 (X)−1 . On the other hand we also have
28
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES7
Proposition 4.5. X contains a subspace isomorphic to `1 if and only if δ˙α (X) > 0 for all α < ω1 . Proof. This follows from Bourgain’s `1 index of a Banach space X which we recall now. For 0 < c < 1, T (X, c) is the tree of all finite normalized sePk Pk quences (xi )k1 ⊆ X satisfying k 1 ai xi k ≥ c 1 |ai | for (ai )k1 ⊆ R. The order on the tree is (xi )k1 ≤ (yi )n1 if k ≤ n and xi = yi for i ≤ k. For ordinals β < ω1 we define Dβ (T (X, c)) inductively by D1 (T (X, c)) = {(xi )k1 ∈ T (X, c) : k β+1 (x (T (X, c)) = D1 (Dβ (T (X, c))) and Dβ (T (X, c)) = T i )1 is γnot maximal}. D γ 0 iff (ei /kei k) is equivalent to the unit vector basis of `1 ; d): δ¨F (ei ) = sup(xi )≺(ei ) sup{δ˙F ((xi ), |·|) : |·| is an equivalent norm on [xi ]i∈N }; e): δF [G] (xi ) ≥ δF (xi )δG (xi ). Proof. a) and b) are immediate; the first part of a) uses that F(M ) ⊆ F. S∞ c) follows from the fact that n=1 Sn contains all finite subsets of {2, 3, . . .}. d) is true because if Y ⊆ X and | · | is an equivalent norm on Y then | · | can be extended to an equivalent norm on X. For e) notice that if (yi )ki is F[G]-admissible w.r.t. (xi ), then it can be blocked in a F-admissible way into successive blocks each of which consists of G-admissible vectors (w.r.t. (xi )). This directly implies the inequality. The most important situation for the study of the constants δα is when the whole sequence (δα )α 0 choose k with |ak /k − a| < ε. For n > k, an /n − a < an /n − ak /k + ε. Setting n = pk + r, 0 ≤ r < k and using apk ≤ pak we obtain an ak − +ε n k
≤ ≤
apk + ar ak pak ar ak − +ε≤ + − +ε n k pk + r n k pak ak ar ar − + +ε= +ε . pk k n n
The first part of the lemma follows. The moreover part can be easily proved by contradiction. We have an immediate corollary.
4. ASYMPTOTIC CONSTANTS AND ∆(X)
31
Corollary 4.14. Setting γ bα = limn (γα·n )1/n for α < ω1 we have that for n every 0 < ξ < γ bα , ξ γα·n ≤ γ bαn , for all α < ω1 and n ∈ N. 1/n Setting δb = limn (δn (xi )) , for a basic sequence (xi ), we have that for every b ξ n δn (xi ) ≤ δbn for all n ∈ N. 0 < ξ < δ, There is an interesting connection between the constants δ¨α (X) which allow for renormings of a given space X, and the supermultiplicative behavior of γ ∈ ∆(X), in particular of γ bα , which involved the original norm only. Proposition 4.15. Let X have a basis (ei ) and let γ ∈ ∆(X). Then there exists (yi ) ≺ (ei ) so that (yi ) ∆-stabilizes γ and so that for all α < ω1 , δ¨α (yi ) = limn (γα·n )1/n ≡ γ bα . The argument is based on the following renorming result which we shall use again. Proposition 4.16. Let Y be a Banach space with a bimonotone basis (yi ). Let α < ω1 and n ∈ N. Then there exists an equivalent bimonotone norm ||| · ||| on Y 1/n . with δα ((yi ), ||| · |||) ≥ δ[Sα ]n (yi ) Proof. Denote the original norm on Y by | · | and set θ = δ[Sα ]n (yi ). For 0 ≤ j ≤ n define a norm | · |j on Y by X ` |y|j = sup θj |Ei y| :
(Ei y)`1 is [Sα ]j -admissible w.r.t. (yi )
1
and E1 < · · · < E` are adjacent intervals
.
Here we take [Sα ]0 = S0 so that |y|0 = |y|. For 0 ≤ j ≤ n we have |y|j ≥ θj |y| and |y| ≥ θn−j |y|j . The former inequality follows trivially from the definition of | · |j and the latter from the fact that any [Sα ]j -admissible family is [Sα ]n -admissible and the definitionPof θ. n−1 Set |||y||| = n1 0 |y|j for y ∈ Y . Then ||| · ||| is an equivalent Pr norm on PrY . Let (xs )r1 be α-admissible w.r.t. (yi ). First observe that | 1 xs | ≥ θ 1 |xs |n−1 . Indeed, arbitrary [Sα ]n−1 -admissible decompositions for each xs can be put together Pr to give a [Sα ]n -admissible decomposition for 1 xs , thus the estimate follows from the definition of | · |n−1 and the fact that δ[Sα ]n (yi ) = θn . To be more precise, for k(s) s 1 ≤ s ≤ r choose adjacent intervals of integers E1s < · · · < Ek(s) so that (Ejs )1 is [Sα ]n−1 admissible and k(s)
|xs |n−1 = θn−1
X
|Ejs xs | .
j=1 s s r Let Fjs = Ejs if j < k(s) and Fk(s) = [min Ek(s) , min E1s+1 ) if s < r and Fk(r) = r 1 r n Ek(r) . Then F1 < · · · < Fk(1) < · · · < Fk(r) are [Sα ] -admissible adjacent intervals
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES9
32
of N and so r k(s) r X X X Fjs ≥ θ xl
r X xl
n
s=1 j=1
l=1
≥ θ
r X
l=1 k(s)
θn−1
s=1
X
|Ejs (xs )| = θ
j=1
r X
|xs |n−1 ,
s=1
s Pr = |F s (xs + xs+1 )| ≥ |E s (xs )| if s < r, using that the (since Fk(s) l=1 xl k(s) k(s) norm is monotone). Pr Pr Similarly, | 1 xs |j+1 ≥ θ 1 |xs |j for j = 1, 2, . . . , n − 2, by the definitions of | · |j+1 and | · |j . Thus n−2 r r n−1 r r X 1 X X θX 1 X X xs = xs ≥ |xs |n−1 + θ |xs |j . n j=0 s=1 n s=1 n j=0 s=1 j 1 Thus |||
Pr
1
xs ||| ≥ θ
Pr
1
|||xs |||.
Proposition 4.17. Let X be an asymptotic `1 space and let γ ∈ ∆(X). If (ei ) ≺ X ∆-stabilizes γ then for all εi ↓ 0 there exists (xi ) ≺ (ei ) and an equivalent norm | · | on [xi ] satisfying a): For all n and x ∈ hxi i∞ n we have kx|| ≤ |x| ≤ (2 + εn )kxk . b): (xi ) is bimonotone for | · |. c): (xi ) ∆-stabilizes γ¯ ∈ ∆(X, | · |) with γ¯α ≥ γα for all α < ω1 . Proof. We may assume that [ei ] does not contain `1 . Thus by Rosenthal’s theorem [R] there exists (xi ) ≺ (ei ) which is normalized and weakly null. By passing (xi ) we may assume that for all n < m and (ai )m 1 ⊆ R, Pn to a subsequence Pof m k 1 ai xi k ≤ (1 + ε¯n )k 1 ai xi k, where ε¯n = εn /2. Define the norm | · | for x ∈ X by |x| = sup{kExk : E is an interval} . Passing to a block P basis of (xi ) we may assume that (xi ) ∆-stabilizes some γ¯ ∈ m ∆(X, | · |). For x = i=n ai xi with |x| = kF xk we have kxk ≤ |x| ≤ k
max XF i=n
ai xi k + k
min F −1 X
ai xi k ≤ 2(1 + ε¯n )kxk = (2 + εn )kxk .
i=n
Thus a) holds and b) is immediate. It remains to check c). Fix α < ω1 and m ∈ N. ` ∞ Let xm < y1 < · · · < y` (w.r.t. (xi )∞ m ) where (yi )1 is α-admissible w.r.t. (xi )m ∞ and hence w.r.t. (ei )m . Choose intervals E1 < · · · < E` such that |yi | = kEi yi k for i ≤ ` and Ei ⊆ [min supp(yi ), max supp(yi )]. Define (Fi )`1 to be adjacent intervals so that min Fi = min Ei . Thus Fi = [min Ei , min Ei+1 ) ⊆ N for i < ` and F` = E` .
4. ASYMPTOTIC CONSTANTS AND ∆(X)
Let F =
S`
` X yi 1
1
33
Fi . Then, by Remark 4.3, ` ` `
X X X
) yi ) ≥ δα ((ei )∞ kF ( yi )k ≥ F ( j m 1
1
j=1
≥ δα ((ei )∞ ¯m )−1 m )(1 + ε
` X
kFj yj k = δα ((ei )∞ ¯m )−1 m )(1 + ε
j=1
` X
|yj | .
j=1
∞ It follows that δα ((xi )∞ ¯m )−1 . Letting m → ∞ we obtain m , | · |) ≥ δα ((ei )m )(1 + ε γ¯α ≥ γα .
Remark 4.18. It is worth noting the following. Let (ei ) be a basic sequence in X ∆-stabilizing γ ∈ ∆(X). Then there exists (xi ) ≺ (ei ) and an equivalent monotone norm | · | on [xi ] so that (xi ) ∆-stabilizes γ ∈ ∆(X, | · |). Furthermore |x| − kxk < εn for x ∈ hxi i∞ n and some εn ↓ 0. Assuming as we may that [ei ] does not contain `1 , this is accomplished by taking i ) to be a suitable weakly null P P(x n block basis of (ei ) and setting | ai xi | = supn k 1 ai xi k. A similar argument yields Proposition 4.19. Let F be a regular set of finite subsets of N and let (ei ) be a basis for X. Given ε > 0 and εi ↓ 0 there exists an equivalent norm | · | on some block subspace [xi ] ⊆ X satisfying a) and b) of Proposition 4.17 and δF ((xi ), | · |) ≥ δF (ei ) − ε. As a corollary to these propositions we obtain Theorem 4.20. Let Y be a Banach space with a basis (yi ). Let α < ω1 , n ∈ N, ε > 0 and θn = δ[Sα ]n (yi ). Then there exists an equivalent norm ||| · ||| on X = [xi ] ≺ Y with δα ((xi ), ||| · |||) ≥ θ − ε. Proof of Proposition 4.15. Let (xi ) ≺ (ei ) ∆-stabilize γ (for the original norm k · k). We may assume that X 0 = [xi ] does not contain `1 . It follows that there exists α0 < ω1 so that δ¨β (X 0 ) = 0 = γβ for all β > α0 . Also from Lemma 4.6, ¨ δ¨α (zi ) ≤ δ¨α (wi ) if (zi ) ≺ (wi ) ≺ (ei ); moreover, δ¨α ((zi )∞ n ) = δα (zi ) for all n ∈ N. ¨ We can therefore stabilize the δα ’s (as in the proof of Proposition 4.10) to find (yi ) ≺ (xi ) so that for all α ≤ α0 , δ¨α (yi ) = δ¨α (zi ) if (zi ) ≺ (yi ). Of course (yi ) still ∆-stabilizes γ. We shall prove that δ¨α (yi ) = limn (γα·n )1/n . Note that if | · | is an equivalent norm on [yi ] and γ¯ ∈ ∆((yi ), | · |) then limn (¯ γα·n )1/n = limn (γα·n )1/n . Indeed if (zi ) ≺ (yi ) ∆-stabilizes γ¯ in | · | then since (zi ) ∆-stabilizes γ in k · k and the norms are equivalent, we obtain c¯ γβ ≤ γβ ≤ d¯ γβ for all β < ω1 and for some constants c, d > 0. Thus γ¯α ≤ sup(¯ γα·n )1/n = lim(γα·n )1/n . n
n
By Proposition 4.11 we obtain that δ¨α (yi ) ≤ limn (γα·n )1/n . Fix θ < limn (γα·n )1/n . Thus there exists n0 with θn0 < γα·n0 . Choose (zi ) ≺ (yi ) with θn0 < δα·n0 (zi ). By Corollary 3.4 there exists M so that [Sα ]n0 (M ) ⊆ Sα·n0 , which yields δα·n0 (zi ) ≤ δ[Sα ]n0 ((zi )M ). So letting (wi ) = (zi )M we have δ[Sα ]n0 (wi ) > θn0 . By Theorem 4.20 there exists an equivalent norm ||| · ||| on [wi0 ]N ,
34
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES10
for some (wi0 ) ≺ (wi ) with δα ((wi0 ), ||| · |||) > θ. The reverse inequality, δ¨α (yi ) ≥ limn (γα·n )1/n , follows. 2 As we will see in later sections, some further regularity properties of sequences γ ∈ ∆(X) are closely related to distortion properties of the space X, and they may or may not hold in general. In contrast, the sequences (δ¨α ) which allow for renorming display a complete power type behavior. In fact, we will give a comprehensive description of behavior of such sequences in Theorem 4.23 below. In the result that follows we shall be particularly interested in part c). Proposition 4.21. Let X have a basis (ei ). Let α < ω1 and n ∈ N. a): δ¨[Sα ]n (X) = (δ¨α (X))n b): δ¨[Sα ]n (X) = δ¨α·n (X) c): δ¨α·n (X) = (δ¨α (X))n Proof. c) will follow from a) and b). a) Since for any equivalent norm |·| on X we have δ[Sα ]n ((yi ), |·|) ≥ (δα ((yi ), | · |))n (Lemma 4.6, e)), the inequality δ¨[Sα ]n (X) ≥ (δ¨α (X))n follows from g) of proposition 4.11. To see the reverse inequality let | · | be an equivalent norm on X, and let (yi ) ≺ (ei ) and θ > 0 satisfy δ[Sα ]n ((yi ), | · |) > θn . By Theorem 4.20 there exist (xi ) ≺ (yi ) and an equivalent norm ||| · ||| on [xi ]i∈N such that δα ((xi ), ||| · |||) > θ. This completes the proof. b) As we have shown earlier, whenever (yi ) ≺ (ei ) and | · | is an equivalent norm, by Corollary 3.4 there exists a subsequence M such that δ[Sα ]n ((yi )M , | · |) ≥ δα·n ((yi ), | · |). It follows that δ¨[Sα ]n (X) ≥ δ¨α·n (X). The reverse inequality follows by choosing N with Sα·n (N ) ⊆ [Sα ]n . Let us introduce the following natural and convenient definition. Definition 4.22. Let X be an asymptotic `1 space. The spectral index of X, I∆ (X), is defined to be I∆ (X) = inf{α < ω : δ¨α (X) < 1} . Theorem 4.23. If X is an asymptotic `1 space not containing `1 , then I∆ (X) = ω α for some α < ω1 . If I∆ (X) = α0 and δ¨α0 (X) = θ then δ¨α0 ·n+β (X) = θn for all n ∈ N and β < α0 . Finally, δ¨β (X) = 0 for all α0 · ω ≤ β < ω1 . Proof. For the proof of the first statement, it suffices to show that if β < I∆ (X) then for all n ∈ N, β·n < I∆ (X) ([Mo], Thm. 15.5). But by Proposition 4.21, δ¨β·n (X) = (δ¨β (X))n = 1, so β · n < I∆ (X). Now let α0 = ω α for some α and assume that δ¨α0 (X) = θ for some 0 < θ < 1. Fix β < α0 . We first show that for any ε > 0 we can find (yi ) ≺ X and an equivalent norm ||| · ||| on [yi ]N with δβ ((yi ), ||| · |||) > 1 − ε and δα0 ((yi ), ||| · |||) > θ − ε. Indeed, let θ0 = θ − ε and choose by Proposition 4.17 (xi ) ≺ X and an equivalent 0 bimonotone norm | · | on X so thathδα0 ((x i i ), | · |) > θ . Given m ∈ N we can choose j a subsequence N of N so that Sα0 [Sβ ] (N ) ⊆ Sβ·j+α0 = Sα0 for j = 0, 1, . . . , m; this follows from Proposition 3.2, Corollary 3.4 and the fact that β · m + ω α = ω α . Let (yi ) = (xi )N and am ≡ δ[Sβ ]m ((yi ), | · |). Note that since [Sβ ]m (N ) ⊆ Sα0 then
5. EXAMPLES–TSIRELSON SPACES
35
am ≥ θ0 and so a ≥ (θ0 )1/m . For y ∈ [yi ]N and 0 ≤ j ≤ m set |y|j = sup{aj
` X
|Ei y| :
(Ei y)`1 is [Sβ ]j -admissible w.r.t. (yi )
1
and E1 < · · · < Ek are adjacent intervals} . It can be checked by a straightforward calculation, using the choice of N and that (yi ) is monotone for |·|, that δα0 ((xi ), |·|j ) ≥ δα0 ((yi ), |·|) > θ0 for j = 0, . . . , m. Pm−1 1 0 For y ∈ [yi ]N set |||y||| = m j=0 |y|j . Then δα0 ((yi ), ||| · |||) > θ and from the proof of Proposition 4.16, δβ ((yi ), ||| · |||) ≥ a > (θ0 )1/m . Taking m such that (θ0 )1/m ≥ 1−ε we get what we wanted. Now by Proposition 3.2 there exists a subsequence M of N with Sα0 +β (M ) ⊆ Sβ [Sα0 ]. It follows that δα0 +β ((yi )M , ||| · |||) > (1 − ε)θ0 = (1 − ε)(θ − ε) . Hence δ¨α0 +β (X) = θ. The case of general n is proved similarly, replacing α0 by α0 · n above and recalling (Proposition 4.21) that δ¨α0 ·n (X) = (δ¨α0 (X))n . The last statement is obvious. 5. Examples–Tsirelson Spaces Our primary source of examples of asymptotic `1 spaces with various behaviors of asymptotic constants is the class of mixed Tsirelson spaces introduced by Argyros and Deliyanni in [ArD]. Definition 5.1. Let I ⊆ N and for n ∈ I let Fn be a regular family of finite subsets of N. Let (θn )n∈I ⊆ (0, 1) satisfy supn∈I θn < 1. The mixed Tsirelson space T (Fn , θn )n∈I is the completion of c00 under the implicit norm X ! k k kxk = max kxk∞ , sup sup θn kEi xk : (Ei )i=1 is Fn -admissible . n∈I
i=1
It is shown in [ArD] that such a norm exists. It is also proved that if I is finite or if θn → 0, then T (Fn , θn )n∈I is a reflexive Banach space, in which the standard unit vectors (ei ) form a 1-unconditional basis. In [ArD] it is proved that for an appropriate choice of θn and Fn the space T (Fn , θn )n∈N is arbitrarily distortable. Deliyanni and Kutzarova [DKut] proved a result that illustrates the possible complexity these spaces can possess. They proved that a mixed Tsirelson space may uniformly contain `n∞ ’s in all subspaces. Notice that the Tsirelson space T satisfies T = T (Sn , 2−n )n∈N = T (S1 , 2−1 ). For 0 < θ < 1 we denote the θTsirelson space by Tθ = T (S1 , θ). Theorem 5.2. Let (ei ) denote the unit vector basis for T . a): If (xi ) ≺ (ei ) then for all n, δn (xi ) = 2−n and δ¨n (xi ) = 2−n . b): For all γ ∈ ∆(T ), γn = 2−n for n ∈ N and γα = 0 for α ≥ ω. ¨ ), γn ≤ 2−n for n ∈ N. c): For all γ ∈ ∆(T d): I∆ (X) = 1 for all X ≺ T . Remark 5.3. Condition a) immediately implies that for an arbitrary equivalent norm | · | on T and (xi ) ≺ (ei ), we have δ1 ((xi ), | · |) ≤ 1/2. Since the asymptotic
36
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES11
`1 constant is equal to δ1−1 , this improves the constant in Proposition 2.8 from to 2.
√
2
Remark 5.4. For Tθ we have δn (Tθ ) = δ¨n (Tθ ) = θn for n ∈ N; and all other equalities and inequalities from Theorem 5.2 hold with appropriate modifications. Also, clearly, I∆ (Tθ ) = 1. Proof of Theorem 5.2. a) By definition of the norm k · k for T , δn (ei ) ≥ 2−n and so if (xi ) ≺ (ei ) then δn (xi ) ≥ 2−n as well. We next show that there exists C < ∞ so that δm (xi ) ≤ C2−m for all m. This will yield the equality for δn . Indeed if for some n, δn (xi ) = A/2n where A > 1 then since δnk (xi ) ≥ (δn (xi ))k (Remark 4.12), we would have that C2−nk ≥ δnk (xi ) ≥ Ak 2−nk for all k, which is impossible. First we consider the case P(xi ) = (ei )i∈M where M is a subsequence of N. Let ε > 0, n ∈ N and let x = i∈F ai ei be an (n, n − 1, ε)-average of (ei )i∈M (see Proposition 3.6 and Notation 3.7). kxk ≥ 2−n . Iterating the definition of the Pn Thus P −i n norm in T yields that kxk = i=1 2 j∈Fi aj where (Fi )i=1 partitions F into −n sets with Fi ∈ Si for i ≤ n. Thus if ε < 2 , kxk = k
X i∈F
ai ei k ≤
n−1 X i=1
2−i ε + 2−n
X
aj ≤ 2/2n = 2/2n
j∈Fn
X
kai ei k .
i∈F
Hence δn ((ei )M ) ≤ 2/2n . If (xi ) is normalized with (xi ) ≺ (ei ) then by [CJoTz] (see also [CSh]), there exists a subsequence M such that (xi ) is D-equivalent to (ei )i∈M , where D is an absolute constant (we let mi = min supp(xi ), and then M = (mi )). Thus δn (xi ) ≤ Dδn ((ei )M ) ≤ 2D/2n . To get the equality for δ¨n we first observe that for any equivalent norm | · | on T there is a constant C 0 (depending on | · |) such that δn ((xi ), | · |) ≤ C 0 δn (xi ), and then we follow the previous argument. b) is immediate from the first part of a); and c) and d) follow from the second part of a). 2 Remark 5.5. For the subsequence M = (mi ) above one could take any mi ∈ supp(xi ) for all i. In the space Tθ , any normalized block basis is D-equivalent to (ei )M as well, with the equivalence constant D = cθ−1 , where c is an absolute constant. The choice of a subsequence M is the same as indicated above (for θ = 1/2). The next example illustrates Theorem 4.23. Example 5.6. Let α < ω1 and let X = T (Sωα , θ). Then a): δ¨ωα ·n (X) = θn for n ∈ N b): I∆ (X) = ω α Proof. a) Let (xi ) ≺ X be a normalized block basis that ∆-stabilizes γ ∈ n α ∆(X). Let n ∈ N and let ε > 0. Choose N by Corollary P3.4 so that [Sω ] α⊇ n−1 Sωα ·n (N ) and also [Sωα ] (N ) ⊆ Sωα ·(n−1) . Choose x = F ai xi to be an (ω · n, ω α · (n − 1), ε) average of (xi )N w.r.t. (ei ), the unit vector basis of X. Clearly kxk ≥ θn . As in T , kxk is calculated by a tree of sets where the first level of sets is Sωα -admissible, the second level is [Sωα ]2 -admissible and so on.
5. EXAMPLES–TSIRELSON SPACES
37
If we stop this tree after n − 1 levels, discarding sets which stopped before then and shrinking those sets which split the support of some xi we obtain for some (Ei x)`1 being ω α · (n − 1)-admissible, kxk ≤ θn−1
` X
kEi xk + ε .
1
The next level of splitting may indeed split the supports of some of the xj ’s. However since those xj ’s have not yet been split the contribution of aj xj to the next level of sets is at most aj θ−1 . Thus we obtain X kxk ≤ θn aj θ−1 + ε = θn−1 + ε . It follows that γωα ·n ≤ θn−1 = θ1 (θn ). Thus, just as in the case of T , γωα ·n = θn . Indeed, if γωα ·n0 > θn0 then 1 n0 k θ θ for large enough k (Proposition 4.11), which is a contradiction. ¨ Similarly if γ ∈ ∆(X) then for some C, γωα ·n ≤ Cθn and so γωα ·n ≤ θn for all ¨ n. This yields that δωα ·n (X) = θn . b) The argument in Proposition 4.23(b) yields this result: for β < ω α and ε > 0 there exists (xi ) and ||| · ||| with δβ ((xi ), ||| · |||) > 1 − ε. γωα ·n0 k ≥ (γωα ·n0 )k >
Before we pass to further examples, let us note a fundamental and useful connection between the spectrum ∆(X) and a lower estimate for the norm on some block subspace. Proposition 5.7. Let X be an asymptotic `1 space and let (zi ) ≺ X be a normalized bimonotone block basis ∆-stabilizing some γ ∈ ∆(X) with 0 < γ1 < 1. Let (ei ) be the unit vector basis of Tγ1 ≡ T (S1 , γ1 ). Then for all ε > 0 there exists a subsequence (xi ) of (zi ) satisfying for all (ai ) ⊆ R X X k ai xi k ≥ (1 − ε)k ai ei kTγ1 . Proof. We shall prove the proposition in the case where γ1 = 1/2 (and so Tγ1 = T ). We shall describe below the argument in a general case, but the reader is advised to first test the special case when δ1 (zi ) = 1/2 (when Pε∞n = 0 for all n and the mi ’s can be omitted.) Choose integers mi ↑ ∞ so that 1 2−mi < ε and then choose εn ↓ 0 to satisfy, for all k ∈ N, k Y 1 1
(5)
2
− εn(i)
> (1 − ε)2−k whenever (n(i))ki=1 ⊆ N satisfy for every j, |{i : n(i) = j}| ≤ mj .
Let (xi ) be a subsequence of (zi ) which Pn Pnsatisfies: for all n, if xn ≤ y1 < · · · < yn w.r.t. (xi ) then k 1 yi k > ( 12 − εn ) 1 kyi k. Such a sequence exists since (zi ) ∆-stabilizes γ with γ1 = 1/2. P` P` Let x = P 1 ai xi and assume that k 1 ai ei kT = 1. We shall show that kxk > (1 − ε)2 . If k ai ei kT = |aP then kxk =P1. Otherwise for some 1j | for some j P n admissible family of sets, k ai ei kT = 21 j=1 kEj ( ai ei )kT . Accordingly we
38
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES12
have that (here is where the bimonotone assumption is used) X n 1 kxk > − εi kEj xk 2 j=1 where i = min(supp E1 x). We then repeat the step above for each Ej x. Ultimately we obtain for some J ⊆ N, X X 1=k ai ei kT = 2−`(i) |ai | i∈J
where `(i) = the number of splittings before we stop at |ai |. We follow the same tree of splittings in getting a lower estimate for kxk with one additional proviso. Each splitting of Ex in hxi i will introduce a factor of ( 12 − εn ) for some n. A given factor ( 21 − εn ) may be repeated a number of times. If any ( 12 − εn ) is repeated mn times we shall discard the corresponding set kExk at that instant. P By virtue of (5) we thus obtain that kxk ≥ (1 − ε)2−`(i) |ai | where I ⊆ J i∈I and ai xi belonged to a discarded set for i ∈ JP\ I. However the contribution P ∞ −mn of the discarded sets to k ai ei kT is at most < ε since from our n=1 2 1 construction for any given n (where ( 2 − εn ) is repeated mn times) we will discard at most one set, something of the form 2−k kExkT where k ≥ mn . It follows that kxk > (1 − ε)(kxkT − ε) = (1 − ε)2 . The proof also yields the following block result. Corollary 5.8. Let (zi ) be a bimonotone basic sequence in a Banach space X which ∆-stabilizes γ ∈ ∆(X) where 0 < γ1 < 1. Let (ei ) be the unit vector basis of Tγ1 . Then for all ε > 0 there exists a subsequence (xi ) of (zi ) satisfying for all (yj )k1 ≺ (xi ) if mj = min(supp(yi )) w.r.t. (xi ) then k k
X
X
kyj kemj yi ≥ (1 − ε)
1
1
.
Tγ
Remark 5.9. We can remove the bimonotone assumption on the norm if we Pm Pm have that for some εn ↓ 0, ky0 + 1 yi k ≥ (γ1 − εn ) 1 kyi k, whenever zn ≤ y0 ≤ zm < y1 < · · · < ym . Without either this assumption or the bimonotone property we obtain a slightly weaker result. Theorem 5.10. Let X be an asymptotic `1 space and let (zi ) ≺ X be a basic sequence ∆-stabilizing some γ ∈ ∆(X), with 0 < γ1 < 1. Then for all ε > 0 there exists a normalized (xi ) ≺ (zi ) satisfying for all (ai ) ⊆ R X X 1 k ai xi k ≥ (1 − ε)k ai ei kTγ1 . 2 Moreover if (yi )k1 ≺ (xi ) with mj = min(supp(yi )) w.r.t. (xi ) then one has k
k X 1
yi k ≥
k
X
1
(1 − ε) kyi kemi . 2 Tγ1 1
Proof. By Proposition 4.17 there exists a k · k-normalized (xi ) ≺ (zi ) and a bimonotone norm | · | on [xi ] with kxk ≤ |x| ≤ (2 + ε)kxk for x ∈ [xi ] and such that (xi ) ∆-stabilizes γ¯ ∈ ∆(X, |·|) with γ¯1 ≥ γ1 . We may thus assume that (xi ) satisfies
5. EXAMPLES–TSIRELSON SPACES
39
the conclusion of Corollary 5.8 for | · | and ε0 such that (1 − ε0 )/(2 + ε0 ) = 12 (1 − ε). Thus if (yi )k1 is as in the statement of the theorem, k
X
yi ≥
1
k k k
X
1 1 X 1 − ε0
X
≥ ≥ y |y |e . (1 − ε) ky ke
i i m i m i i 2 + ε0 1 2 + ε0 1 2 Tγ¯1 Tγ1 1
The following can be proved by an argument similar to that in Proposition 5.7. Proposition 5.11. Let X be an asymptotic `1 space and let (zi ) ≺ X be a normalized bimonotone block basis ∆-stabilizing γ ∈ ∆(X). Let α < ω1 with 0 < γα < 1 and let ε > 0. Then there exists a subsequence (xi ) of (zi ) satisfying the following: if (yi )k1 ≺ (zi ) with min(supp(yi )) = mi (w.r.t. (xi )) then k k
X
X
kyi kemi yi ≥ (1 − ε)
1
1
.
T (Sα ,γα )
The next example is a space X for which the sequences of asymptotic constants (δα (X)) and (δ¨α (X)) are “essentially” the same as for Tsirelson’s space T ; still, X and T have no common subspaces–no subspace of X is isomorphic to a subspace of T . It is worth noting that X also has the property that the sequence δ¨ = (δ¨α (X)) ¨ does not belong to ∆(X). Example 5.12. Let 0 < c < 1 and let X = T (Sn , c2−n )n∈N . Then a): δ¨n (X) = 2−n for all n ¨ b): For all γ ∈ ∆(X), γn < 2−n for all n. c): No subspace of X embeds isomorphically into T . Before verifying these assertions we first require some observations. The norm of x ∈ X, if not equal to kxk∞ , is computed by a tree of sets, the first level being (Ei )`1 where for some j, (Ei )`1 is j-admissible and kxk =
` c X kEi xk . 2j i=1
For each i, if kEi xk does not equal kEi xk∞ , then we split kEi xk into a second level of sets mi -admissible for some mi , and so on. If every set keeps splitting then after k steps we obtain an expression of the form (6)
ck
r X
2−n(s) kFs xk .
s=1
Of course some sets may stop splitting, in which case if we carry on for k-steps, we only obtain a lower estimate for kxk. Consider the case where (xi ) ≺ X and x ∈ hxi i. We set kxkTk ,(xi ) to be the largest of the expressions of the form (6) obtained by splitting k-times (a k level tree of sets, where (Fs )r1 is the k th -level), subject to the additional constraint that for all i and s, Fs does not split xi . Thus Fs xi is either xi or 0. Lemma 5.13. Let (xi ) ≺ X, ε > 0 and k ∈ N. Then there exists x ∈ hxi i with kxk = 1 such that kxkTk ,(xi ) > 1 − ε.
40
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES13
Proof. Assume without loss of generality that kxi k =P1 for i ∈ N. P We call x ∈ [xi ] an P (n, ε)-normalized average (of (xi ) w.r.t. (ei )) if x = i∈F ai xi /k i∈F ai xi k, where i∈F ai xi is an (n, n − 1, cε/2n )-average of (xi ) w.r.t. (ei ). Thus (xi )i∈F is n-admissible w.r.t. (eP i ) and if G ⊆ F satisfies (xi )i∈G is (n − 1)-admissible then P n a < cε/2 . Also i G i∈F ai = 1 and ai > 0 for i ∈ F . (We can always find such vectors by Proposition 3.6.) Note that if (xi )i∈G is (n−1)-admissible and if we write P P x in theP form x = i∈F bi xi (for some bi > 0), then G bi < (cε/2n )(2n /c) = ε (since k i∈F ai xi k ≥ c/2n ). We first indicate how to find x satisfying kxk = 1 and kxkT1 ,(xi ) > 1 − ε. Let P∞ εi = 2−(i+1) ε so that 1 εi = ε/2. Let ` n o X k · kn = sup c2−n kEj xk : (Ej x)`j=1 is n-admissible . j=1
and observe that for all x, limn kxkn = 0. Let n1 = 1 and choose (yi1 ) ≺ (xi ) and nj ↑ ∞ by induction so that each yj1 is an (nj , εj )-normalized average of (xi ) and Pj for all j, k i=1 yi1 km < εj+1 if m ≥ nj+1 . Then we choose y 2 to be an (n, ε/2)normalized average of (yi1 ) where n ∈ N is not important but we may assume that P 2 y = F bi yi1 where n < nmin F . We have 1 = ky 2 k and so by the definition of the norm in X, there exists j such P` that 1 = ky 2 kj = c/2j s=1 kEs (y 2 )k where (Es y 2 )`1 is j-admissible. We claim that by somewhat altering the Es ’s we can ensure, by losing no more than ε, that the sets Es do not split any of the xi ’s. Indeed P if 1 ≤ j < n, then G = {i ∈ F : Es splits yi1 for some s} ∈ Sj . Since j < n, s∈G bs < ε/2 and thus by shrinking the offending sets Es to avoid splitting yi ’s we obtain the desired sets. If n ≤ j < nmin F then if we fix i ∈ F and consider Gi = {r : Es splits one or more of the xr ’s in the support of yi } we get that, by similarly shrinking the offending Es ’s so as to not es be the new sets, that split such an xr , and letting E X X c es y 2 k > 1 − k E bi ε i > 1 − ε . 2j i∈F
Finally if F = (k1 , . . . , kr ) and nkp ≤ j < nkp+1 then
X
bi yi < εkp and bkp < ε/2
i∈F,i 1 − ε . j 2 i∈F,i>kp
This proves the lemma in the case k = 1. For the general case we continue as above letting (yi2 ) be (n2i , εi )-normalized averages of (yi1 ), etc. If x = y1k+1 then x satisfies the lemma for k. We omit the tedious calculations. Proof of the assertions in Example 5.12. By Proposition 4.16, since δn (X) ≥ c2−n , ¨ we have δ¨1 (X) ≥ 2−1 . If there exists γ ∈ ∆(X) with γ1 ≥ 2−1 then by Theorem 5.10
5. EXAMPLES–TSIRELSON SPACES
41
there exists (xi ) ≺ (ei ) and d > 0 so that for all (yj )`1 ≺ (xi ) if mj = min(supp(yj )) w.r.t. xi , then ` `
X
X
kyj kemj . (7) yj ≥ d
1
T
1
Fix an arbitrary k. By Lemma 5.13 there exists x ∈ hxi i with kxk = 1 and kxkTk ,(xi ) > 1/2. Thus there exists a k-level tree of sets whose final level Pr is (E1 , . . . , Er ) so that ck s=1 2−n(s) kEs xk > 1/2. Following the same partition scheme in T and using (7) for ys = Es x we get (with ms = min(supp(Es x))), r r
X X 1
2−n(s) kEs xk > (c−k ) . kEs xkems ≥ d−1 = d−1 kxk ≥ 2 T s=1 s=1 Since c < 1, this is impossible for large enough k. This proves b) for n = 1 and that δ¨1 (X) = 2−1 . Then Proposition 4.21 yields δ¨n (X) = 2−n for all n. The remainder of b) easily follows from the proof of Proposition 4.16. Indeed ¨ assume that some γ ∈ ∆(X) satisfies γn = 2−n , for some n > 1. By Proposition 4.17 there is (yi ) ≺ X and an equivalent bimonotone norm | · | on [yi ] such that (yi ) ¨ ∆-stabilizes γ¯ ∈ ∆(X, | · |) and γ¯n = 2−n . By passing to a subsequence we may assume that for some sequence εn ↓ 0, for all m, k k X X xi ≥ 2−n (1 − εm ) |xi | 1
1
and (xi ) is n-admissible w.r.t. (yi )∞ if ≺ 1 . Let ||| · ||| be the norm constructed in the proof of Proposition 4.16 for α = 1 and θ = 1/2. If yr ≤ x1 < · · · < xr then r r X X xs ≥ (1/2)(1 − εr ) |xs |n−1 . (xi )k1
(yi )∞ m
1
1
The remaining estimates remain true and, as in the proof of Proposition 4.16, we obtain r r X X |||xs ||| . xs ||| ≥ (1/2)(1 − εr ) ||| 1
1
Thus γ1 = 1/2 which is impossible. If c) were not true, then, by Theorem 5.2 b), a subspace Y of X isomorphic to a subspace of T would admit a renorming for which γ1 (Y ) = 1/2, in contradiction to b). 2 Remark 5.14. The above example X yields the following. There exists (xi ) ≺ (ei ) and a sequence of equivalent norms ||| · |||j so that for all k on [xi ]∞ k , kxk ≥ |||x|||j ≥ c2 kxk if j ≥ k and furthermore δ1 (||| · |||j , (xi )) > 21 − εj for some εj → 0. ¨ Yet γ1 < 12 for all γ ∈ ∆(X). To see this one needs only choose (xi ) so that on [xi ]∞ k , kxk = sup`≥k kxk` . This can be accomplished by taking each xj to be an iterated Pj j +1-normalized average of (ei ) (as in lemma 5.13). Then set |||x|||j = (1/j) 1 kxki . 2 Since kxk ≥ kxki ≥ ckxkj ≥ c2 kxk on hxs i∞ j , kxk ≥ |||x|||j ≥ c kxk. We mention one other example, taken from [AnO]. First suppose that X = T (Sn , θn )n∈N where 1 > supn θn and limn→∞ θn = 0. We shall call (θn ) regular if
42
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES14
for all n, m ∈ N, θn+m ≥ θn θm . It is easy to verify that every such X has a regular representation, i.e., for some regular sequence (θn ) we have X = T (Sn , θn )N . Thus 1/n limn θn exists by Lemma 4.13. Example 5.15. Let X = T (Sn , θn )N where 1 > supn θn , θn → 0 and (θn ) is 1/n regular. Let θ = limn θn . Then a): For all Y ≺ X we have δ¨1 (Y ) = θ. b): For all Y ≺ X and for alln ∈ N, δ¨n (Y ) = θn and δ¨ω = 0. ω if θ = 1 c): For all Y ≺ X, I∆ (Y ) = 1 if θ < 1 d): For all Y ≺ X and j ∈ N we have δj (Y ) ≤ θj supn≥j θn θ−n ∨ θj /θ1 . In particular, if θn θ−n → 0 then X is arbitrarily distortable. 6. Renormings of T , and spaces of bounded distortion Definition 6.1. The distortion constant of a space X is defined by D(X) = sup d(X, | · |) . |·|∼k·k
So X is distortable iff D(X) > 1. Similarly, X is arbitrarily distortable iff D(X) = ∞. Finally, X is of bounded distortion iff there is D < ∞ such that D(Y ) ≤ D for every subspace Y ⊆ X. As we saw in Proposition 2.7, Tsirelson’s space T satisfies D(T ) ≥ 2. Similarly one can show that D(Tθ ) ≥ θ−1 . However, not much more is known about distorting T . It is unknown if T is arbitrarily distortable, or at least whether it contains an arbitrarily distortable subspace; and, if not, what is D(T ) or at least a reasonable upper estimate for it. The interest in these questions lies in the fact that, as already mentioned, no examples are yet known of distortable spaces which are of bounded distortion. ¿From techniques developed earlier in this paper we easily get some information on asymptotic constants of equivalent norms on Tsirelson space. This should be compared with Theorem 5.2 where the constants for the original norm were established. Surprisingly, it is not known if there exists (xi ) ≺ T and an equivalent norm | · | on [xi ] with δ1 ((xi ), | · |) < 1/2. Our next result shows that the class of equivalent norms for which δ1 = 1/2 cannot arbitrarily distort T . Theorem 6.2. There exists an absolute constant D with the following property. Let X ≺ T and let |·| be an equivalent norm on X such that for some γ ∈ ∆(X, |·|), γ1 = 1/2. Then d(X, | · |) ≤ D. Proof. Let (zi ) be a basic sequence in X ∆-stabilizing γ under | · | where γ1 = 1/2. Let ε > 0. By passing to a block basis of (zi ) and multiplying | · | by a constant if necessary we may assume that k · kT ≥ | · | on [zi ] and for all (wi ) ≺ (zi ) there exists w ∈ hwi i with 1 + ε > kwkT ≥ |w| = 1. Choose a normalized block basis (wi ) of (zi ) satisfying 1 + ε ≥ kwi kT ≥ |wi | = 1 for all i. Theorem 5.10 allows us to also assume that X X | ai wi | ≥ (1/2 − ε)k ai ei kT . There exists an absolute constant D1 so that (wi /kwi kT ) is D1 -equivalent to (emi ) in k · kT , where mi = min supp(wi ) w.r.t. (ei ), for each i [CJoTz]. Thus we have,
6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION
43
for all (ai ) ⊆ R, (8) (1 + ε)D1 k
X
ai emi kT ≥ k
X
ai wi kT ≥ |
X
ai wi | ≥ (1/2 − ε)k
X
ai ei kT .
Consider the subsequence (pi ) of N defined by induction by p1 = 1 and pi+1 = mpi , for i ≥ 1. There is a universal constant D2 so that (epi ) is D2 -equivalent to (epi+1 ) in k · kT [CJoTz]. Also, on the subspace [wpi ] we have, by (8), X X X (1 + ε)D1 k ai epi+1 kT ≥ | ai wpi | ≥ (1/2 − ε)k ai epi kT . Thus the conclusion follows with D = 2D1 D2 .
A natural question in light of the above results is whether one can quantify the distortion d(X, | · |) of an equivalent norm | · | on X ≺ T in terms of ∆(X, | · |). Problem 6.3. Let | · | be an equivalent norm on T and let (xi ) ≺ T (∆, | · |)stabilize γ. Thus for some c > 0, c2−n ≤ γn ≤ 2−n for all n. Does there exist a function f (c) so that d(X, | · |) ≤ f (c)? We shall give a suggestive partial answer to a weaker problem. First we note the following proposition. Proposition 6.4. For n ∈ N define the equivalent norm k · kn on T by kxkn = P` sup{2−n 1 kEi xk : (Ei x)`1 is n-admissible}. Given X ≺ T and εn ↓ 0 there exists (xi ) ≺ X so that for all n if x ∈ hxi i∞ n then kxk − kxkn < εn kxk. Proof. First note that if nX o k · kSn = sup |x(i)| : E ∈ Sn i∈E
then for all x ∈ T we have kxkn ≤ kxk ≤ kxkn +kxkSn . Indeed, if kxk = 6 kxk∞ then kxk = x∗ (x) for some functional x∗ (with kx∗ k = 1) determined by the successive iterations of the implicit equation of the norm in T ; in particular, x∗ (ei ) = ±2−n(i) for all i. We may write x∗ = y ∗ + z ∗ where z ∗ (ei ) = ±2−n(i) if n(i) ≤ n and 0 otherwise. Thus, since the support of z ∗ is n-admissible, |z ∗ (x)| ≤ (1/2)kxkSn and |y ∗ (x)| ≤ kxkn . Furthermore, kxkSn ≤ 2n kxk. Since the Schreier space Sn is n isomorphic to a subspace of C(ω ω ) (Remark 3.5), , it is c0 -saturated, i.e., every infinite-dimensional subspace contains a copy of c0 , and thus k · kSn cannot be equivalent to k · k on any infinite-dimensional subspace of T . In particular we can chose (xi ) ≺ X so that for all x ∈ hxi i∞ n , kxkSn ≤ εn kxk. The conclusion follows. Problem 6.5. Let | · | be an equivalent norm on X = [xi ] ≺ T . Let (yi ) ≺ (xi ), −1 C < ∞ and suppose that for all n, if y ∈ [yi ]∞ |y|n ≤ |y| ≤ C|y|n , where n then C P` ` −n |E y| : (E y) is n-admissible w.r.t. (x |y|n = sup{2 i )}. Does there exist a i i 1 1 function F (C) so that d(Y, | · |) ≤ F (C)? Proposition [xi ]. Suppose that defined as above). on [yi ]∞ n0 such that
6.6. Let (yi ) ≺ (xi ) ≺ T and let | · | be an equivalent norm on −1 for all n and y ∈ [yi ]∞ |y|n ≤ |y| ≤ C|y|n (where | · |n is n , C Then for all ε > 0 there exists n0 and an equivalent norm ||| · ||| 1 ∞ C −1 |||y||| ≤ |y| ≤ C|||y||| for y ∈ [yi ]∞ n0 and δ1 ((yi )n0 , ||| · |||) > 2 − ε.
44
3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES15
Pn0 1 Proof. Choose n0 so that C 2 /n0 < ε. On [yi ]∞ n0 define |||y||| = n0 1 |y|j . Clearly the inequality between the norms hold. Let p ∈ N and let (zi )p1 ≺ [yi ]∞ n0 Pp satisfy yn0 +p ≤ z < · < z . Let z = z . Then (see the proof of Proposition 4.16) 1 p i 1 Pp |z|j+1 ≥ 12 i=1 |zi |j for j = 1, . . . , n0 − 1. Hence |||z||| ≥
p p p n0 −1 1 X 1X 1 X 1X |zi |j = |||zi ||| − |zi |n0 . n0 j=1 2 i=1 2 i=1 2n0 i=1
Now |zi |n0 ≤ C|zi | ≤ C 2 |||zi ||| and so p p 1X C2 1X |||z||| ≥ |||zi ||| 1 − > (1 − ε) |||zi ||| , 2 i=1 2n0 2 i=1 completing the proof.
Finally, let us recall the following known [CJoTz] property of T . There exists an absolute constant D1 so that if x1 < y1 < x2 < y2 < · · · are normalized in T then (xi ) is D1 -equivalent to (yi ). It turns out that equivalent norms on T that satisfy this property (with a fixed constant) cannot arbitrarily distort T . The result, in fact, holds in any space having this subsequence property. Proposition 6.7. There exists a function f (D) satisfying the following. If | · | is an equivalent norm on [xi ]N ≺ T so that (yi ) is D-equivalent to (zi ) whenever y1 < z1 < y2 < · · · is a normalized block basis of (xi ), then d(X, | · |) ≤ f (D). Proof. By passing to a block basis of (xi ) and scaling the norm | · | we may assume that there exists d > 1 so that for all x ∈ [xi ], d−1 kxk ≤ |x| ≤ kxk; furthermore, in any block subspace Y of (xi ) there exist y, z ∈ Y with |y| = |z| = 1 and kyk ≤ 2 and kzk > d/2. Choose a | · |-normalized block basis of P(xi ), y1 < z1 < y2 < · · · with kzi k > d/2 and kyi k ≤ 2 for all i. There exists w = Pai zi satisfying |w| = 1 and kwk < 2. Since (zi ) and (yi ) are D-equivalent for | · |, | ai yi | > D−1 . Also (zi /kzi kT ) and (yi /kyi kT ) are D1 -equivalent in T . Thus X X X k ai yi kT ≤ 2D1 k ai zi /kzi kT kT ≤ 4D1 /d k ai zi kT ≤ 8D1 /d . Thus D−1 ≤ 8D1 /d and so d ≤ 8D1 D ≡ f (D).
We now turn to some results about spaces of bounded distortion. Theorem 6.8. Let X be an asymptotic `1 space. Let γ ∈ ∆(X) and let (yi ) ≺ X ∆-stabilize γ. If Y = [yi ] is of D-bounded distortion then for any α < ω1 and n, m ∈ N, a): D−1 (δ¨α (Y ))n ≤ γα·n ≤ (δ¨α (Y ))n b): γα·n γα·m ≤ γα·(n+m) ≤ D2 γα·n γα·m . ¨ ). Choose an equivalent norm | · | on Y and Proof. a) Let γ = ( γ α ) ∈ ∆(Y (wi ) ≺ (yi ) which (∆, | · |)-stabilizes γ. Let ε > 0. By passing to a block basis of (wi ) and scaling | · | we may suppose that |w| ≤ kwk ≤ (D + ε)|w| for w ∈ [wi ] .
6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION
45
Let α < ω1 and n ∈ N. We may assume that δα·n ((wi ), | · |) > γ α·n − ε. Thus if (xs )r1 is α · n-admissible w.r.t. (wi ), r r r r
X
X X γ −εX
xs ≥ ( γ α·n − ε) xs ≥ kxs k . |xs | ≥ α·n
D+ε 1 1 1 1 It follows that γα·n ≥ γ α·n /D and so γ α·n ≤ Dγα·n . Passing to the supremum over all γ α·n and using Proposition 4.11 g), we get δ¨α·n (Y ) ≤ Dγα·n . Hence by Proposition 4.21, D−1 (δ¨α (Y ))n = D−1 δ¨α·n (Y ) ≤ γα·n ≤ δ¨α·n (Y ) = (δ¨α (Y ))n . b) Using part a) and Proposition 4.11 d), γα·n γα·m ≤ γα·(n+m) ≤ (δ¨α (Y ))n+m = completing the proof.
(δ¨α (Y ))n (δ¨α (Y ))m ≤ D2 γα·n γα·m ,
Combining the proposition with Theorem 4.23 we get a complete description, up to equivalence, of sequences γ from ∆(X), in spaces of D-bounded distortion. We leave the details to the reader. Recall the notation γ bα = limk (γα·k )1/k , for α < ω1 (Corollary 4.14). If Y ≺ X ∆-stabilizes γ, we may write γ bα (Y ) to emphasize the subspace Y . By Proposi¨ tion 4.15, γ bα (Y ) = δα (Y ). Therefore, by Proposition 2.5, we have an important sufficient condition for an asymptotic `1 space to contain an arbitrary distortable subspace. Corollary 6.9. Let X be an asymptotic `1 space. Let γ ∈ ∆(X) and let (yi ) ≺ X ∆-stabilize γ. If there exists α < ω1 such that γα > 0 and limn γα·n γ bα (Y )−n = 0, then Y contains an arbitrarily distortable subspace. Let us present an alternative approach to Corollary 6.9, taken from [To1], which is of independent interest. It is based on a construction of certain asymptotic sets in a general asymptotic `1 space. An alternative proof of Corollary 6.9. (Sketch) Let γ ∈ ∆(X), let Y = [yi ] ≺ X ∆-stabilize γ and let (yi∗ ) be the biorthogonal functionals in Y ∗ . Suppose that Y is of D-bounded distortion. Fix an arbitrary α < ω1 . We shall show that (1/3D)(b γα (Y ))n ≤ γα·(n−1) . By Proposition 4.15, this is slightly weaker than Theorem 6.8, but sufficient to imply Corollary 6.9. Fix n ∈ N. First we shall show that for all ε > 0, all normalized blocks (xi ) ≺ (yi ), and all 0 < λ < 1, there is an (α · n, α · (n − 1), ε) average x of (xi ) w.r.t. (yi ) such that kxk ≥ λ(b γα (Y ))n ≡ λ0 . This is done by blocking, in the spirit of James [J]. Fix m sufficiently large and pick N ⊆ N such that [Sα·n ]m (N ) ⊆ Sα·(n m) (Corollary 3.4) and that λγα·(n m) ≤ δα·(n m) ((xi )N ) (this is possible by the Definition 4.7 of the ∆-spectrum). Pick (1) (1) (zi ) ≺ (xi )N such that for all i, zi is an (α · n, α · (n − 1), ε) average of (xi )N (1) (2) (1) w.r.t. (yi ). If for all i, kzi k < λ0 , then pick (zi ) ≺ (zi ) such that for all i, (2) (1) (1) zi is an (α · n, α · (n − 1), ε) average of (zi /kzi k) w.r.t. (yi ). And keep going. (k) Assume that after m steps we still had that kzi k < λ0 for all i and all k ≤ m. P (m) Write z1 = j∈N bj xj ; then bj ≥ 0 and let J be the set of all j ∈ N such that
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3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES16
bj > 0. It is easily seen that (xj )j∈J is [Sα·n ]m (N )-admissible w.r.t. (yi ), hence also (α · (n m))–admissible w.r.t. (yi ). Moreover, our assumption on the norms of P (k) the zi ’s easily yields that bj > (1/λ0 )m−1 . Thus (1/λ0 )m−1 λγα·(n m) ≤ λγα·(n m)
X
kbj xj k
j∈J
≤ δα·(n m) ((xi )N )
X j∈J
kbj xj k ≤ k
X
(m)
bj xj k = kz1
k < λ0 .
j∈J
1
bα (Y ), a conIt follows that (λγα·(n m) ) m < λ0 , hence (γα·(n m) )1/n m < λ1/n−1/mn γ tradiction, if m is large enough. Now we shall define asymptotic sets A, B ⊆ S(Y ) and a set A∗ in the unit ball of Y ∗ such that A∗ 2-norms A and the action of A∗ on B is small. By passing to a tail subspace of Y if necessary, we may assume without loss of generality that 7 3 of this 4 γα·(n−1) ≤ 8 δα·(n−1) (Y ). Fix ε > 0, quite small as determined at the end P proof. Let A∗ consist of all functionals in Y ∗ of the form y ∗ = 34 γα·(n−1) k∈K wk∗ , where (wk∗ )K ≺ (yi∗ ) is (α · (n − 1))-admissible (w.r.t. (yi∗ )); and let A consist of all y ∈ S(Y ) that are 2-normed by A∗ . The set A is asymptotic by the definition of the ∆-stabilization. Since Y ∆-stabilizes γ, it is not difficult to see that A is asymptotic in Y and that functionals from A∗ have the norm not exceeding 1. Then B consists of all vectors of the form x/kxk, where x is an (α · n, α · (n − 1), ε) average w.r.t. (yi ) of some normalized (xi ) ≺ (yi ), such that kxk ≥ (1 − ε)(b γα (Y ))n . By the first part of this proof, B is asymptotic in Y . We will show that if y ∗ ∈ A∗ and z ∈ B, then |y ∗ (z)| ≤ 43 γ bα (Y )−n (γα·(n−1) + 76 ε)/(1 − ε) ≡ η. This is a direct consequence of the following estimate. If x is an (α · n, α · (n − 1), ε) average as above, and if (Ek ) ∈ Sα·(n−1) , and Ek x denotes the restriction of P x whose support w.r.t. (y ) is E ; then kE k k xk ≤ 1 + 7ε/6γα·(n−1) . To see this, Pi write x in the form x = i∈F ai xi where (xi )i∈F is αP · n-admissible w.r.t. P (yi ) and if J ⊆ F satisfies (xi )J is α · (n − 1)-admissible then G ai < ε. Also i∈F ai = 1 and ai > 0 for i ∈ F . Set I = {i : Ek ∩ supp(xi ) 6= ∅ for at most one k} and J = F \ I; and for i ∈ J let Ki = {k : Ek ∩ supp(xi ) 6= ∅}. Then it can be checked that (xi )J is α · (n − 1)-admissible, hence X k
kEk xk ≤
X i∈I
ai kxi k+
X i∈J
ai
X
kEk xi k ≤ 1+ε/δα·(n−1) (Y ) ≤ 1+7ε/6γα·(n−1) .
k∈Ki
P Now, if y ∗ = 43 γα·(n−1) k∈K wk∗ ∈ A∗ then letting Ek = supp(wk ) for all k we get |y ∗ (z)| ≤ η, as required. As mentioned in Section 2, Y is (1/2+1/4η)-distortable. Hence the assumption of D-bounded distortion implies 1/2 + 1/4η ≤ D. Substituting the definition of η and taking ε > 0 sufficiently small we get the inequality (1/3D)(b γα (Y ))n ≤ γα·(n−1) , as promised. 2 As we remarked earlier, the assumption of bounded distortion implies the existence of certain subspaces with a nice structure ([MiTo], [M], [To2]). We would like to identify more such regular subspaces in the class of asymptotic `1 spaces of bounded distortion.
6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION
47
Recall (Proposition 6.4) that in Tsirelson’s space T = Tθ , for all εn ↓ 0 there exists (xi ) ≺ T so that for all n and all x ∈ hxi i∞ n we have (1 + εn )−1 kxkT ≤ sup{θn
` X
kEi xkT : (Ei )`1 is n-admissible} ≤ kxkT .
1
In any asymptotic `1 space with bounded distortion one can find a block basis that displays an isomorphic version of this phenomenon. Theorem 6.10. Let X be an asymptotic `1 space of D-bounded distortion not containing `1 . There exist (wi ) ≺ X, α = ω β0 , 0 < θ < 1, and (zi ) ≺ (wi ) such that for every k ∈ N we have, for z ∈ [zi ]∞ k , n X o (1/4 D) sup sup θn kEi zk : (Ei ) is α · n-admissible ≤ kzk 1≤n≤k
n X o ≤ 4 D inf sup θn kEi zk : (Ei ) is α · n-admissible . 1≤n≤k
(Here, for an interval P E of N and z = w.r.t. (wi ), i.e., Ez = i∈E ai wi .)
P
ai wi ∈ [wi ], Ez denotes the restriction
Proof. By Proposition 4.5, δ¨β (X) > 0 for at most countably many β’s; write this set as (βm ). For an arbitrary β < ω1 , it follows from Lemma 4.6 that if ¨ (yi ) ≺ (ei ) then δ¨β ((yi )∞ n; and that δ¨β (zi ) ≤ δ¨β (yi ) whenever n ) = δβ (yi ) for all P −m ¨ (zi ) ≺ (yi ). Letting, for example, f (yi ) = 2 δβm (yi ), by a standard induction argument, similar to that in Proposition 4.10, we can stabilize f (yi ). That is, we can find (yi ) ≺ X such that f (zi ) = f (yi ) for all (zi ) ≺ (yi ). Since δ¨β (X) = 0 implies δ¨β (zi ) = 0 for all (zi ) ≺ X, the stabilization of f implies that we have, for all (zi ) ≺ (yi ), δ¨β (zi ) = δ¨β (yi )
for all β < ω1 .
Let α = I∆ (yi ); by Theorem 4.23, α = ω β0 for some β0 < ω1 . Let θ = δ¨α (yi ). Then δ¨α·n (yi ) = θn for n ∈ N, by Proposition 4.21. By an inductive construction followed by a diagonal argument, using Proposition 4.17, we can find (wi ) ≺ (yi ) ∞ and equivalent bimonotone norms | · |n on [wi ]∞ n such that for all (zi ) ≺ (wi )n and n ∈ N, (9)
−1/n δα ([zi ]∞ θ. n , | · |n ) ≥ 2
Notice that (9) is preserved if the norms involved are multiplied by constants. Therefore by scaling and the assumption of bounded distortion we may additionally ensure that kwk ≤ |w|n ≤ 2 Dkwk for w ∈ [wi ]∞ n and all n ∈ N. Now, given any α-admissible family of intervals (Fi )k1 of N, let (Gi )k1 be adjacent intervals such that min Fi = min Gi for i < k and let Gk = Fk . Since the norms | · |n are bimonotone, |Fi w|n ≤ |Gi w|n for w ∈ [wi ]∞ n and all n ∈ N. In particular, by Remark 4.3 for n ∈ N and w ∈ [wi ]∞ we get n k k X X |w|n ≥ δα [wi ]∞ |Gi w|n ≥ δα [wi ]∞ |Fi w|n . n , | · |n n , | · |n i=1
i=1
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3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES17
Using this and the assumption (9) on δα ’s we easily get, for n ∈ N and w ∈ [wi ]∞ n , 2 Dkwk ≥ |w|n
k n X o ≥ sup δαn |Ei w|n : (Ei ) is [Sα ]n -admissible i=1 k n X o ≥ (1/2) sup θn kEi wk : (Ei ) is [Sα ]n -admissible , i=1
where we have abbreviated δα [wi ]∞ n , |·|n to δα . Finally, using Corollary 3.4 and a diagonal argument, construct a subsequence M = (mi ) of N such that setting Mn = n (mi )∞ n we get Sα·n (Mn ) ⊆ [Sα ] for all n. Thus for w ∈ [wi ]i∈Mn replacing the supremum in the last formula by the supremum over Sα·n (Mn )-admissible families and relabeling the subsequence by (wi0 ) we get, for n ∈ N and w ∈ [wi0 ]∞ n , n X o kwk ≥ (1/4 D) sup θn kEi wk : (Ei ) is α · n-admissible . It should be noted that in this last estimate, the admissibility condition is understood with respect to the above subsequence (wi0 ) of (wi ) which indeed corresponds to the subsequence M of N. n P We relabel once more, denoting (wi0 ) simply by (wi ). Set |||w|||n = sup θn kEi wk : o (Ei ) is α · n-admissible , for w ∈ [wi ]∞ n and n ∈ N. These are equivalent norms on the subspaces where they are defined. Therefore stabilizing all norms ||| · |||n on a nested sequence of block subspaces, using the assumption of bounded distortion, and passing to a diagonal subspace we get (zi ) ≺ (wi ) and An such that ∞ ∞ [zi ]∞ n ≺ [wi ]n and An |||z|||n ≤ kzk ≤ 2 DAn |||z|||n for z ∈ [zi ]n . Since for all (zi ) ≺ (wi ) we have δα·n ([zi ], k · k) ≤ δ¨α·n (zi ) = θn < 2θn , then for all (zi ) ≺ (wi ) and all n ∈ N, there exists vn ∈ [zi ]∞ n such that kvn k ≤ 1 and |||vn |||n ≥ 1/2. Hence An ≤ 2, thus kzk ≤ 4 D|||z|||n on [zi ]∞ n . We have shown that for all k ∈ N, kzk ≥ (1/4 D) sup1≤n≤k |||z|||n on [wi ]∞ k ∞ [zi ]∞ ; and kzk ≤ 4 D inf |||z||| on [z ] . 1≤n≤k i k k n We would like to directly relate the norm of an asymptotic `1 space of bounded distortion with a norm in some Tsirelson space. While we were unable to obtain two-sided estimates we did obtain the following lower estimate. Proposition 6.11. Let X be an asymptotic `1 space of D-bounded distortion, α ≺ ω1 and suppose that δ¨α (Y ) = θ ∈ (0, 1) for all Y ≺ X. Let εn ↓ 0. ∞ There exist P (wi ) ≺ X so that for all n if w ∈ [wi ]n then kwk ≥ (1 − εn )(D + −1 εn ) k kEi wkepi kT (Sα ,θ−εn ) , whenever E1 < E2 < · are adjacent intervals, Ei w denotes the restriction of w w.r.t. (wi ) and pi = min Ei . Note that the first paragraph of the proof of Theorem 6.10 shows how to choose a subspace X satisfying the above hypothesis in an asymptotic `1 space of bounded distortion. Proof. Choose (zi ) ≺ X so that for all n there exists an equivalent bimonotone ∞ norm | · |n on [zi ]∞ n with δα ((zi )n , | · |n ) > θ − εn . This can be done by Propo¨ sition 4.17 using that δα (Z) = θ for all Z ≺ X. Hence by a diagonal argument,
6. RENORMINGS OF T , AND SPACES OF BOUNDED DISTORTION
49
0 0 applying Corollary 5.8, we may assume also that if z ∈ hzi i∞ n and E1 < · · · < E` are adjacent intervals then `
X
|z|n ≥ (1 − εn ) |Ei0 z|n eri 1
,
T (Sα ,θ−εn )
where ri = min Ei0 and Ei0 z is the restriction of z w.r.t. (zi ). Using that X is of D-bounded distortion and scaling | · |n we may obtain (wi ) ≺ (zi ) so that for all n 1 ∞ and w ∈ hwi i∞ n , kwk ≥ |w|n ≥ D+εn kwk. We thus obtain for w ∈ hwi in ,
X
kwk ≥ (1 − εn ) |Ei0 w|n epi T (Sα ,θ−εn )
X 1 − εn
≥ kEi0 wkeri .
D + εn T (Sα ,θ−εn ) Now given adjacent intervals E1 < E2 < · · · , take intervals E10 < E20 < · · · such that for all w ∈ hwi i∞ n , and all i, the restriction Ei w w.r.t. (wi ) coincides with the restriction Ei0 w with respect to (zi ). Then we have ri = min Ei0 ≥ P pi = min Ei for all iP and since Sα is invariant under spreading we easily get that k ai eri kT (Sα ,θ0 ) ≥ k ai epi kT (Sα ,θ0 ) for all (ai ) and all 0 < θ0 < 1. Thus the final lower estimate follows. The following proposition generalizes the fact that for the Tsirelson space Tθ , D(Tθ ) ≥ θ. Proposition 6.12. Let X be an asymptotic `1 space. Then sup{D(Y ) : Y ≺ X} ≥ sup{γ1−1 : γ ∈ ∆(X)}. Proof. Let γ ∈ ∆(X) and let (xi ) ≺ X ∆-stabilize γ. Thus for some εn ↓ 0, all n and all (yi ) ≺ (xi )∞ n , γ1 ≥ δ1 (yi ) ≥ γ1 − εn . For n ∈ N and (yi ) ≺ (xi ) define n n X δ1 (n)(yi ) = sup δ : kyk ≥ δ kEi yk : y ∈ [yi ], Ey is a restriction w.r.t.(yi ), 1
E1 < · · · < En0 are adjacent intervals with
[
o Ei = supp(y) .
Now observe that given ε > 0 there exists n0 ∈ N and (yi ) ≺ (xi ) so that δ1 (n0 )(wi ) < γ1 + ε for all (wi ) ≺ (yi ). Indeed, if not, we could, by a diagonal argument, produce (yi ) ≺ (xi ) with δ1 (yi ) ≥ γ1 + ε. On [yi ] define the norm X n0 |y| = sup kEi yk : E1 y < · · · < En0 y w.r.t. (yi ) and 1
E1 < · · · < En0 are adjacent intervals with
[
Ei = supp(y) .
Thus, by the choice of (yi ), for all W ≺ Y = [yi ]N , there exists w ∈ W , kwk = 1 and |w| > γ11+ε . Also by considering long `k1 -averages (see the proof of Proposition 2.7) there exists x ∈ W , kxk = 1 and |x| < 1 + ε. Thus D(Y ) ≥ d(X, | · |) ≥ (1 + ε)/(γ1 + ε).
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3. PROXIMITY TO `1 AND DISTORTION IN ASYMPTOTIC `1 SPACES18
More generally, we have Proposition 6.13. Let X be asymptotic `1 and suppose that I∆ (X) = α0 . then sup{D(Y ) : Y ≺ X} ≥ sup{γα−1 : γ ∈ ∆(X)} . 0 Proof. We may assume α0 > 1 by Proposition 6.12. Thus by Theorem 4.23, α0 is a limit ordinal. Let αn ↑ α0 be the ordinal sequence used in defining Sα0 . Let γ ∈ ∆(X), ε > 0. Then for some n0 , γαn0 < γα0 + ε. Let (xi ) ∆-stabilize γ. Choose (yi ) ≺ (xi ) and an equivalent norm |·| on [yi ] with δαn0 ((yi ), |·|) > 1−ε. By passing to a block basis of (yi ) and scaling | · | if necessary we may assume that for some D we have k · k ≤ | · | ≤ Dk · k on [yi ], and for all W = [wi ] ≺ Y there exists w ∈ W , kwk = 1 and |w| < 1 + ε. Since γαn0 < γα0 + ε there exists z ∈ W with kzk = 1 P` P` and 1 kzi k ≥ 1/(γα0 + ε), for some decomposition z = P 1 zi where (zi )`1 is αn0 P admissible w.r.t. (wi ). Hence |z| ≥ (1−ε) |zi | ≥ (1−ε) kzi k ≥ (1−ε)/(γα0 +ε). Comparing the norms |z| and kzk we get D(Y, | · |) > (1 − ε)(1 + ε)/(γα0 + ε). We have a simple corollary. Corollary 6.14. Let X be asymptotic `1 with I∆ (X) = I∆ (Y ) = α0 for all Y ≺ X. If δ¨α0 (X) = 0 then no subspace of X is of bounded distortion.
CHAPTER 4
Asymptotic Versions of Operators and Operator Ideals 1
1. Preliminaries The goal of this note is to introduce new classes of operator ideals, and, moreover, a new way of constructing such classes through employing the asymptotic structure recently introduced in [MMiTo]. 1.1. K¨ onig’s Unendlichkeitslemma . An added technicality relative to [MMiTo] responsible for the use of the following lemma is that for our proofs we need to be able to find asymptotic versions of spaces and operators, not only in the entire space, but also inside any set which large enough to be asymptotic. Indeed, suppose we know we can extract asymptotic subsets approximating some fixed asymptotic version arbitrarily well from any collection of arbitrarily long asymptotic set. If we have that the set of sequences with a certain property is large enough to be asymptotic, we immediately know that we can find an asymptotic version of the space or of the operator with the same property. The proof of the said combinatorial lemma is an elementary exercise, and can be found in [K¨ o]. A rooted tree is a connected tree with some vertex labeled as ’root’. Lemma 1.1. A rooted tree has an infinite branch emanating from the root if: (1) There are vertices arbitrarily far from the root, and (2) The set of vertices with any fixed distance to the root is finite. 1.2. Extracting an asymptotic version from asymptotic sets . We will now prove the extraction theorem for asymptotic versions of operators. We use the following technical terminology. Definition 1.2. A truncation (of length k) of a given collection, Σ ⊆ S(X)n< , is the collection of sequences of the leading k blocks from sequences in Σ. A truncation of an asymptotic set is obviously an asymptotic set. Theorem 1.3. Let X be a space with a shrinking basis. For every operator T ∈ L(X) and every sequence, {Φn }∞ n=1 , of asymptotic sets with increasing lengths there exists T˜ ∈ {T }∞ approximated arbitrarily well by asymptotic sets which are truncations of asymptotic subsets of Φn ’s. Proof. The proof will split into three parts. First we will extract asymptotic subsets of block sequences, whose normalized images under T are closely equivalent to asymptotic spaces of X (this is the only part where we use the shrinking 1
Joint work with Professor V. Milman. To appear in ??? 51
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property of the basis). Then we will use a compactness argument and Lemma 4.4 (chapter 1) to extract asymptotic subsets of block sequences, where the norm of linear combinations, the image under T and the action of T are stabilized. Finally we will pipe such asymptotic sets of different lengths together by means of lemma 1.1, to approximate an asymptotic version of T . First step: For every δ > 0 and for every asymptotic set of length n there is an asymptotic subset of sequences, whose normalized images under T are (1 + δ)equivalent to elements of {X}n . Proof of first step: We want to show that the set Φ of sequences mapped (up to δ) to sequences in the collection Σn,ε (X) from Remark 6.3 (chapter 1) is asymptotic. To do that, we must show that V has a winning strategy in the game for Φ. To produce a sequence from Φ, in the first step of the game the vector player must pick a block whose image is essentially (up to a norm δ perturbation) supported far enough to fit into a sequence from Σn,ε (X). Suppose this cannot be achieved. Specifically, suppose that for a sequence, {xk1 }∞ k=1 , of normalized vectors supported arbitrarily far, and for some natural n1 and δ > 0, we have kPn1 (T (xk1 ))k ≥ δ. This means that one of the bounded functionals P{ei } (T (x)), 1 ≤ i ≤ n1 , does not go to zero when applied to some sequence of norm bounded vectors supported arbitrarily far. This contradicts the shrinking property. So, the vector player can choose a vector x1 with kPn1 (T (x1 ))k < δ, for arbitrary n1 and δ, and therefore insure that that it fits essentially as the first vector from a sequence in Σn,ε (X). The same reasoning allows the vector player to choose x2 with image which is essentially supported far enough to fit as the second vector of a Σn,ε (X)-admissible sequence beginning with a slight perturbation of T (x1 ). Repeating this argument, we achieve a sequence of essentially-consecutive vectors, arbitrarily-well equivalent to a Σn,ε (X)-admissible sequence (with equivalence constant going to 1 as δ and ε go to zero), and we are through. Second step: Consider two copies of the Minkowski compactum of order n, M and N . Consider finite coverings of M and N , {Vi }i and {Wj }j respectively. Consider a finite covering of the cube [0, kT k]n , {Ik }k . For every asymptotic set of length n, Φ, there is an asymptotic subset, Φ0 , of block-sequences with the following additional properties: (1) The sequences in Φ0 are contained in some fixed Vi0 . (2) The normalized images under T of sequences from Φ0 are contained in some fixed Wj0 . (3) For all {xi }ni=1 ∈ Φ0 , the sequences {kT (xi )k}ni=1 are contained in some fixed Ik0 . Proof of second step: This is an easy application of Lemma 4.4 (chapter 1) and the fact that the covering is finite. Split the sequences in the asymptotic subset from step 1 into the collections: Φi,j,k = {{xi }ni=1 |[xi ]ni=1 ∈ Vi ,
T (xi ) n ∈ Wj , {kT (xi )k}ni=1 ∈ Ik }. kT (xi )k i=1
By Lemma 4.4 (chapter 1) one of those collections must be an asymptotic set.
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Third step: For every operator T ∈ L(X) and every collection, {Φn }∞ n=1 , of asymptotic sets of increasing lengths there exists a formally diagonal operator, T˜ ∈ {T }∞ , approximated arbitrarily well by truncated asymptotic subsets of {Φn }n Proof of the third step: Fix a positive sequence converging to zero, {εn }n . Choose inductively open finite coverings by cells of diameter less than εn of the compact product of the two copies of the Minkowski compactum of order n and the cube [0, kT k]n (as in step 2 above), such that the projection of the covering of the order n product space onto the order n − 1 product space refines the covering of the order n − 1 product space. The cells of this covering form a tree in an obvious way. Take the collection {Φn }n of asymptotic sets. Applying steps 1 and 2, we get that for every k ≤ n, on the k-th level of the tree some vertex (i.e. covering cell) contains an asymptotic subset of k-truncations of Φn (note that we take not only an asymptotic subset of each Φn , but also an all truncations of these subsets, to guarantee we indeed have a subtree). The subtree spanned by such vertices has properties 1 and 2 from Lemma 1.1. Therefore it has an infinite branch. Let {Φ0n }n be the truncated asymptotic subsets contained in the vertices of the infinite branch. For every n, the truncated asymptotic subsets {Φ0n }n of {Φn }n have the following properties: (1) The sequences of blocks from {Φ0n }n are (1 + εn )-equivalent to the basis of [Y ]n for some fixed Y ∈ {X}∞ . (2) The normalized images under T of all sequences in {Φ0n }n are (1 + εn )equivalent to the basis of [Z]n for some fixed Z ∈ {X}∞ . (3) The norm of the image of a block from a sequence in {Φ0n }n depends (up to εn ) only on the place of this vector in the sequence. Therefore, the result of this process is a sequence of asymptotic sets approximating arbitrarily well a formally diagonal asymptotic version. Remark 1.4. Note that since we may start with any collection of asymptotic sets with increasing lengths, we may choose to extract subsets approximating an asymptotic version of T arbitrarily well from asymptotic sets approximating a given asymptotic version of X. We thus have for any operator T ∈ L(X) and for any ˜ ∈ {X}∞ an asymptotic version of T whose domain is X. ˜ X 2. Asymptotic versions of operator ideals 2.1. Compact operators. Proposition 2.1. If T is compact then {T }∞ = {0}. If T is non compact then not all operators in {T }∞ are compact. Proof. If T is compact, take asymptotic sets, {Φn }n , approximating an asymptotic version of the operator. Let V play his winning strategy for Φn , and let S play tail subspaces of [X]>n with n such that kT k[X]>n < ε. The block-sequence resulting from this game will still be an approximation of the same asymptotic version. This shows that any asymptotic version of T can be approximated arbitrarily well by operators with norm smaller than any positive ε. Therefore the only asymptotic version of T is zero.
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If T is non compact, there is an ε > 0 such that for every n one can choose an asymptotic set of length n, Φn , with every {x1 , . . . , xn } ∈ Φn having kT (xi )k ≥ ε for all 1 ≤ i ≤ n. Using theorem 1.3, extract asymptotic subsets approximating an asymptotic version of T arbitrarily well. The norm of this asymptotic version will not be smaller than ε on any element of the basis of its domain, and will therefore be non-compact. Asymptotic versions induce a seminorm on operators, through the formula: |||T ||| = sup kT˜k where the supremum is taken over all asymptotic versions of T and the double-bar norm is the usual operator norm. It is interesting to note that this gives a way of looking at the Calkin algebras L(X)/K(X) (c.f. [CaPfY]). Proposition 2.2. Suppose X is a Banach space with a shrinking basis such that the norm of all tail projections is exactly 1 (this can always be achieved by renorming, see [LTz]). Then the norm of the image of an operator T in the Calkin algebra is equal to |||T |||. Proof. One direction is clear. If K is a compact operator on X, the norm of T + K is at least the supremum of norms of asymptotic versions of T + K. The latter, by the proof of Proposition 2.1, are the same as asymptotic versions of T . For the other direction we will show that for every T there exist compact operators K such that the norm of T + K is almost achieved by asymptotic versions of T + K, which, again, are the same as asymptotic versions of T . We will perturb T by a compact operator K, such that the set of normalized blocks mapped by T + K to vectors of norm greater than kT + Kk − ε is asymptotic as a set of sequences of length 1. If we manage to do that, then an asymptotic version of T + K, which can be approximated arbitrarily well by subsets extracted from asymptotic sets composed of sequences of the above blocks, will almost achieve the norm of T + K, as required. Take λ to be (up to ε) the largest such that {x ∈ S(X)1< ; kT (x)k ≥ λ} is asymptotic. By this we mean that the set {x ∈ S(X)1< ; kT (x)k ≥ λ + ε} does not have elements in some tail subspace, [X]>m . Consider the compact perturbation of T , T 0 = T − T ◦ Pm . By our assumption on the basis kT 0 k ≤ λ + ε, and the set {x ∈ S(X)1< ; kT 0 (x)k ≥ λ} is still asymptotic. The proof is now complete.
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2.2. Finitely singular and asymptotically finitely singular operators. Definition 2.3. An operator T on a sequence space X is asymptotically finitely singular if for every ε there exists n(ε, T ) and an admission set Σn of length n, such that T takes some normalized block from the span of each sequence in Σ to a vector with norm less than ε. T is called finitely singular if it satisfies the above definition with Σn = S(X)n< . In other words, an operator T is asymptotically finitely singular, if, when restricted to the span of a sequence from Σ, T −1 is either not defined or has norm larger than 1ε An operator ideal close to the ideal of finitely singular operators was defined in [Mi], and called σ0 . The difference is that the original definition referred to all n dimensional subspaces, rather than just block subspaces, as we read here. Proposition 2.4. Operators on a Banach-space X, which are asymptotically finitely singular with respect to a given basis, form a Banach space with the usual operator norm and a two sided ideal of L(X). Proof. Let T be asymptotically finitely singular, and let S be bounded. ST ε , T ), and the is obviously asymptotically finitely singular. Indeed, n(ε, ST ) ≤ n( kSk admission sets for ST are the same as those for T . To see that T S is asymptotically finitely singular we use the proof of step 1 in Theorem 1.3, and find admission sets Σ0 , whose normalized image under S is essentially contained in the admission sets Σ used to define T as an asymptotically finitely singular operator. T S will take some normalized block from the span of any sequence from Σ0 to a vector with arbitrarily small norm. Indeed, S takes (essentially) Σ0 to Σ, where T has an ’almost kernel’. To show that the sum of two asymptotically finitely singular operators is also asymptotically finitely singular, we need to use the following: Claim: If T is asymptotically finitely singular then for every ε and every k there exists an N (k, ε, T ), and an admission set Σ of length N , such that every sequence from Σ has a k-dimensional block subspace on which T has norm less than ε. This claim is standardly proved by taking concatenations of asymptotic sets from the definition of T as asymptotically finitely singular (it is easier to think here of the finitely singular case: if T has an ’almost kernel’ on every n dimensional block subspace, then it has a k dimensional ’almost kernel’ in every N (k) dimensional block subspace). To complete the proof of the proposition, fix ε > 0 and consider asymptotically finitely singular operators, T and S. By definition of asymptotically finitely singular produce an admission set, Σ, of length n( 2ε , S), such that S takes some normalized block in the span of any Σ-admissible sequence to a vector with norm less than 2ε . Take the admission set Ψ of length N (n, 2ε , T ) from the above claim. It is possible to extract an admission subset Ψ0 ⊆ Ψ, such that any n consecutive blocks of a sequence in Ψ0 are also in Σ (similarly to Remark 4.2). We therefore have that in any block sequence in Ψ0 there is an n-dimensional block subspace where T has norm less than 2ε , and inside this subspace a normalized vector, whose image under S has norm less than 2ε . This means that T + S is asymptotically finitely singular. The fact that asymptotically finitely singular operators form a closed subspace of L(X) is straightforward.
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Remark 2.5. (1) From Gowers’ combinatorial lemma (in [G5], see also chapter 2) it follows that for every asymptotically finitely singular operator T , there is a block subspace Y where: for every ε there exists an n, such that in the span of every sequence in S(Y )n< supported after the n-th basic element, there is a normalized vector whose image under T has norm less than ε. It is easy to see that on this block subspace T is finitely singular; indeed, every sequence of n blocks contains a sequence of [ n2 ] blocks supported after the [ n2 ]-th basic element. (2) It is not true, however, that the ideals of asymptotically finitely singular and finitely singular operators coincide. Consider the following example: Let X be the `2 sum of increasingly long `n2 ’s, and let Y be their `3 sum. The formal identity from X to Y is not finitely singular, but is asymptotically finitely singular. (3) It is easy to extend Proposition 2.4 and show that asymptotically finitely singular operators form a two sided ideal in the operator norm when restricting our attention to the category of Banach spaces with shrinking bases. Finitely singular operators will only form a left-sided ideal in the (shrinking) basis context. This is because a bounded operator multiplied to the right side of a finitely singular operator does not have to preserve blocks. (4) It is worth noting that in the Gowers-Maurey space (from [GM1]), all operators are finitely singular perturbations of a scalar operator (this follows from Lemma 22 and Lemma 3 in [GM1]). This point is even more interesting in light of Corollary 2.8 below. The following proposition claims a strong dichotomy in the asymptotic structure of operators: either it contains an isomorphism, or it is composed only of finitely singular operators. Proposition 2.6. If T is an asymptotically finitely singular operator then all operators in {T }∞ are finitely singular. If T is not asymptotically finitely singular, {T }∞ contains an isomorphism. Proof. Let T be asymptotically finitely singular. Take asymptotic sets approximating an asymptotic version of T , Φn . Let player V play the winning strategy for {Φn }n while S plays the winning strategy for the admission sets Σ in the definition of asymptotically finitely singular operators. The resulting vector sequences must approximate the same asymptotic version, but must also contain an ’almost kernel’ for T . Therefore any asymptotic version of T is finitely singular. Suppose T is not asymptotically finitely singular. Then for some ε > 0 the sets Φn of all sequences in S(X)n< , on which T is an isomorphism with kT −1 k ≤ 1ε , are asymptotic. Indeed, if they weren’t, by Lemma 4.3 (chapter 1), for some ε and for every n, Φcn would be admission sets, and therefore T would be asymptotically finitely singular, in contradiction. Using Theorem 1.3, extract from Φn asymptotic subsets Φ0n , which approximate arbitrarily well an asymptotic version of T . This asymptotic version is an isomorphism.
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Remark 2.7. Note that the proof shows that if T is asymptotically finitely singular, all its asymptotic versions will be finitely singular operators with the same n(ε) as T . We offer here an application to the theory developed above. Recall that an asymptotic `p space is a space, all whose asymptotic versions are isomorphic to `p . Corollary 2.8. Every asymptotically finitely singular operator from an asymptotic `p space X to itself is compact. Proof. Consider the set of asymptotically finitely singular operators AFS(X) ⊂ L(X). The asymptotic versions of these operators, by proposition 2.6 above, are all finitely singular operators in L(`p ). In particular they are all asymptotically finitely singular, and therefore belong to some proper closed two-sided ideal. It is well known (cf. [GoMaFel]), that the only proper closed two sided operator ideal in L(`p ) is the ideal of compact operators, but let us sketch a very simple proof: All asymptotic versions of T are diagonal finitely singular operators in L(`p ). It is clear, then, that given any ε > 0, only finitely many entries on the diagonal are larger than ε; otherwise, restricting to the span of the basic elements corresponding to the entries larger than ε we get an isomorphism. Therefore the entries on the diagonal go to zero, and the operator is compact. We can now complete the proof of the corollary, using proposition 2.1 once more. AFS(X) = {T ∈ L(X)|{T }∞ ⊆ FS(`p )} = {T ∈ L(X)|{T }∞ ⊆ K(`p )} = K(X). where K(Z) is the set of all compact operators on Z, and FS(Z) is the set of all finitely singular operators on Z. Note that, by this corollary, if an asymptotic-`p space with a shrinking basis has the the property of the Gowers-Maurey space from the last point of Remark 2.5, than all bounded linear operators on this space will be compact perturbations of scalar operators. We do not know whether such space exists. 2.3. A general theorem . We conclude with a theorem which explains that the above phenomena are a part of a more general situation. When referring to operator ideals we invoke the categorical algebraic definition from [P]. An injective operator ideal, J, has the property that if T : X → Y is in J, then the same operator with a revised range, T : X → Im(T ) is also in J. The following theorem states that the ’asymptotic preimage’ in L(X) of an injective operator ideal is an ideal in the algebra L(X). Theorem 2.9. Let J be an injective operator ideal, and let X be a space with a shrinking basis. The set of operators: J 0 = {T ∈ L(X)|{T }∞ ⊆ J} is an operator ideal in L(X). Proof. If we multiply an operator S ∈ J 0 with an operator T ∈ L(X), an asymptotic version of the product will always be a product of asymptotic versions. ˜ of Indeed, take the asymptotic sets approximating an asymptotic version, R, R = ST (or R = T S), and extract (by the proof of Theorem 1.3) asymptotic subsets, which approximate an asymptotic version T˜ of T , and whose normalized images under T approximate an asymptotic version S˜ of S. The product of these
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asymptotic versions, S˜T˜ ∈ J, is also approximated by the same asymptotic subsets. ˜ But these asymptotic subsets must still approximate R. 0 ˜ Therefore R ∈ J, and R must be in J . ˜ Let T and S be in J 0 . Let R = S+T , and find asymptotic sets approximating R, an asymptotic version of R. Extract asymptotic subsets approximating asymptotic ˜ = S˜ + T˜, versions of S and T , S˜ and T˜ respectively. Note that we cannot say that R ˜ ˜ since S and T may have different ranges. However, we do have: (10)
˜ ˜ kR(x)k ≤ kS(x)k + kT˜(x)k, ˜ ∈ J. This is enough in order to prove R Indeed, we can write: ˜ = P ◦ (i1 ◦ S˜ + i2 ◦ T˜), R where ˜:X ˜ →W ˜, R
˜ → Y˜ , S˜ : X
˜ i1 : Y˜ → Y˜ ⊕ Z,
˜ → Z, ˜ T˜ : X
˜ i2 : Z˜ → Y˜ ⊕ Z,
i1 (y) = (y, 0), i2 (z) = (0, z), ˜ ˜ is defined by: P (i1 ◦ S˜ + i2 ◦ T˜)(x) = R(x) ˜ and P : Im(i1 ◦ S + i2 ◦ T˜) → W and by continuity. Inequality (10) assures that P is well defined and continuous. Now, T˜ and S˜ are in J, so i1 ◦ S˜ + i2 ◦ T˜ is also in J. By injectivity, we are ˜ allowed to modify the range as we compose with P , and still get that the result ,R, is in J. Therefore R ∈ J 0 , and we conclude that J 0 is an ideal in L(X).
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