Joint spreading models and uniform approximation of bounded operators
aa r X i v : . [ m a t h . F A ] M a r JOINT SPREADING MODELS AND UNIFORMAPPROXIMATION OF BOUNDED OPERATORS
S. A. ARGYROS, A. GEORGIOU, A.-R. LAGOS, AND PAVLOS MOTAKIS
Abstract.
We investigate the following property for Banach spaces. A Ba-nach space X satisfies the Uniform Approximation on Large Subspaces (UALS)if there exists C > A ∈ L ( X ) and convexcompact subset W of L ( X ) for which there exists ε > x ∈ X there exists B ∈ W with k A ( x ) − B ( x ) k ≤ ε k x k , there exists a subspace Y of X of finite codimension and a B ∈ W with k ( A − B ) | Y k L ( Y,X ) ≤ Cε . Weprove that a class of separable Banach spaces including ℓ p , for 1 ≤ p < ∞ , and C ( K ), for K countable and compact, satisfy the UALS. On the other handevery L p [0 , ≤ p ≤ ∞ and p = 2, fails the property and the same holdsfor C ( K ), where K is an uncountable metrizable compact space. Our sufficientconditions for UALS are based on joint spreading models, a multidimensionalextension of the classical concept of spreading model, introduced and studiedin the present paper. introduction This paper is devoted to the study of the Uniform Approximation on Large Sub-spaces (UALS) for an infinite dimensional Banach space X . This concept concernsa special case of the following general question. Find conditions such that the ε -pointwise approximation of a function f by the elements of a family of functions W yields that there exists a g ∈ W which uniformly ε ′ -approximates the function f .One of the best results in this frame is the well known consequence of Hahn-Banachtheorem. If X is a Banach space, it may be viewed as a subspace of X ∗∗ throughthe natural embedding. If x ∈ X , W is a closed convex subset of X and ε > x ∗ ∈ X ∗ there exists x ∈ W with | x ∗ ( x ) − x ∗ ( x ) | ≤ ε k x ∗ k ,then we have that for every ε ′ > ε there exists y ∈ W so that k x − y k ≤ ε ′ .It is natural to ask how the previous is extended to the space of bounded linearoperators L ( X ). The UALS property is an attempt to provide an answer. Noticethat in the definition of UALS there are two differences from the above result. Thefirst one is that the set W is norm compact and this is necessary since the normal-ized operators of rank one ε -pointwise approximate the identity for every ε > X with dim X ≥ C > A ∈ L ( X ) and a convex compact W ⊂ L ( X ) such that, for every x in the unit ball of X , there is a B in W with k A ( x ) − B ( x ) k = 0, whereas the norm distance of A from W is greater than C . Thereare two classes of Banach spaces satisfying the UALS. The first are spaces with thescalar-plus-compact property [AH], [Ar et al. ] and the second one includes spaceswith strong asymptotic homogeneity. The latter concerns the uniform uniquenessof l -joint spreading models, an extension of the classical spreading models [BS].The paper is organized in five sections. In the first section we introduce thenotion of plegma spreading sequences. These are finite collections of Schauder basic Primary 46B03, 46B06, 46B25, 46B28, 46B45.The fourth named author was supported by the National Science Foundation under GrantNumbers DMS-1600600 and DMS-1912897. sequences in a Banach space that interact with one another in a spreading waywhen indexed by plegma families, a notion which first appeared in [AKT].The second section is motivated by the definition of plegma spreading sequencesand concerns the problems of whether or not finite collections of Schauder (uncon-ditional) basic sequences contain subsequences that form, under a suitable order,a common Schauder (unconditional) basic sequence. For Schauder basic sequenceswe provide a complete characterization given in the following.
Theorem I.
Let ( x n ) n , . . . , ( x ln ) n be seminormalized sequences in a Banach space X such that each one is either weakly null, equivalent to the basis of ℓ , or non-trivialweak-Cauchy. Let I ⊂ { , . . . , l } be such that ( x in ) n is a non-trivial weak-Cauchysequence with w ∗ - lim x in = x ∗∗ i for every i ∈ I and set F = span { x ∗∗ i } i ∈ I . Thenthere exist M , . . . , M l infinite subsets of N such that ∪ li =1 { x in } n ∈ M i is a Schauderbasic sequence, under a suitable enumeration, if and only if X ∩ F = { } .For unconditional sequences the following holds. Theorem II.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence such that each ( e in ) n is unconditional. Then ( e in ) li =1 ,n ∈ N is also an unconditional sequence.We also provide a variant of the classical B. Maurey - H. P. Rosenthal example[MR] of two unconditional sequences ( e n ) n , ( e n ) n in a space X such that, for every M, L infinite subsets of N , the sequence ( e n ) n ∈ M ∪ ( e n ) n ∈ L is not unconditional. Thisshows that the assumption of a plegma spreading sequence in the above theorem isnecessary. Further, it is well known that the space generated by two unconditionalsequences is not necessarily unconditionally saturated.In section three we define joint spreading models, which as we have mentioned al-ready, are a multidimensional extension of the classical Brunel-Sucheston spreadingmodels. We also present some of their basic properties.The fourth section concerns spaces that admit uniformly unique joint spreadingmodels with respect to certain families of Schauder basic sequences. Examples ofsuch spaces are ℓ p (Γ), for 1 ≤ p < ∞ , c (Γ) and, as we show, all Asymptotic ℓ p spaces in the sense of [MMT]. We also prove that the James Tree space admits auniformly unique l -joint spreading model with respect to the family of all normalizedweakly null Schauder basic sequences in JT . Each l -joint spreading model generatedby a sequence from this family is √ ℓ andthis is the best constant [HB], [Be]. Our proof is a variant of the well known resultdue to I. Ameniya and T. Ito [AI] that every normalized weakly null sequence in JT has a subsequence equivalent to the basis of ℓ .The fifth section is devoted to the study of spaces satisfying the UALS and toclassical spaces where this property fails. In the first part we study the propertyfor spaces with very few operators, namely spaces with the scalar-plus-compactproperty [AH], [Ar et al. ]. We prove the following. Theorem III.
Every Banach space with the scalar-plus-compact property satisfiesthe UALS.The basic result for UALS concerns spaces which admit uniformly unique jointspreading models with respect to families of Schauder basic sequences that have cer-tain stability properties. For this we first introduce the class of difference-includingfamilies (see Definition 5.9) and we prove the following.
Theorem IV.
Let X be a Banach space and assume that for every separable sub-space Z of X we have a difference-including collection F Z of normalized Schauderbasic sequences in Z . If there exists a uniform K ≥ Z admitsa K -uniformly unique l -joint spreading model with respect to F Z , then X satisfiesthe UALS property. OINT SPREADING MODELS AND UNIFORM APPROXIMATION 3
A key ingedient of the proof is Kakutani’s Fixed Point theorem for multivaluedmappings [BK], [Ka]. This argument has appeared in a work of W. T. Gowers andB. Maurey, which is related to the theorem (see Lemma 9 of [GM]), and was themotivation for defining the UALS property. As a consequence of the above theorem,the following spaces and all of their subspaces satisfy the UALS. The space ℓ p (Γ),for 1 ≤ p < ∞ , c (Γ), the James Tree space and all Asymptotic ℓ p spaces for1 ≤ p ≤ ∞ . The UALS property behaves quite well in duality. In particular thefollowing hold. Theorem V.
Let X be a reflexive Banach space with an FDD. Then X satisfiesthe UALS if and only if X ∗ does. Theorem VI.
Let X be a Banach space with an FDD. Assume that there exista uniform constant C > Z of X ∗ , a difference-including family F Z of normalized sequences in X ∗ such that Z admits a C -uniformly unique l -joint spreading model with respect to F Z . Then X satisfiesthe UALS property.As a consequence of the above, L ∞ -spaces with separable dual and their quo-tients with an FDD satisfy the UALS. Thus the spaces C ( K ), for K countablecompact, have the property. We also provide an example of a reflexive Banachspace that admits a uniformly unique spreading model and fails the UALS prop-erty. This example shows that if a space admits a uniformly unique spreadingmodel, this does not necessarily imply that it admits a uniformly unique l -jointspreading model for every l ∈ N , and that the assumption in Theorems IV and VIof a uniformly unique l -joint spreading model cannot be weakened by assuming auniformly unique spreading model. Finally, we prove that the spaces L p [0 , ≤ p ≤ ∞ and p = 2, and C ( K ), for any uncountable and metrizable compactspace K , fail the UALS. Acknowledgement.
We express our thanks to I. Gasparis, W. B. Johnson andB. Sari for their comments and remarks that allowed us to improve the content ofthe paper. 1.
Plegma Spreading Sequences
We recall the notion of plegma families which first appeared in [AKT] and wereused to define higher order spreading models. Interestingly, they were used in arather different way there and we slightly modify their definition. We shall referto the notion from [AKT] as strict plegma families. We use them to introduce thenotion of plegma spreading sequences. These are finite collections of sequences thatinteract with one another in a spreading way when indexed by plegma families. Westart with some notation we will use throughout the paper.
Notation. By N = { , , . . . } we denote the set of all positive integers. We willuse capital letters as L, M, N, . . . (resp. lower case letters as s, t, u, . . . ) to denoteinfinite subsets (resp. finite subsets) of N . For every infinite subset L of N , thenotation [ L ] ∞ (resp. [ L ] < ∞ ) stands for the set of all infinite (resp. finite) subsetsof L . For every s ∈ [ N ] < ∞ , by | s | we denote the cardinality of s . For L ∈ [ N ] ∞ and k ∈ N , [ L ] k (resp. [ L ] ≤ k ) is the set of all s ∈ [ L ] < ∞ with | s | = k (resp. | s | ≤ k ). Forevery s, t ∈ [ N ] < ∞ , we write s < t if either at least one of them is the empty set, ormax s < min t . Also for ∅ 6 = s ∈ [ N ] ∞ and n ∈ N we write n < s if n < min s .We shall identify strictly increasing sequences in N with their correspondingrange, i.e. we view every strictly increasing sequence in N as a subset of N andconversely every subset of N as the sequence resulting from the increasing order ofits elements. Thus, for an infinite subset L = { l < l < . . . } of N and i ∈ N , we set S. A. ARGYROS, A. GEORGIOU, A.-R. LAGOS, AND P. MOTAKIS L i = l i and similarly, for a finite subset s = { n , . . . , n k } of N and for 1 ≤ i ≤ k , weset s ( i ) = n i .Given a Banach space X with a Schauder basis ( e n ) n , then for every x ∈ X with x = P n a n e n we write supp( x ) to denote the support of x , i.e. supp( x ) = { n ∈ N : a n = 0 } . Generally, we follow [LT] for standard notation and terminologyconcerning Banach space theory. Definition 1.1.
Let M ∈ [ N ] ∞ and F be either [ M ] k for some k ∈ N or [ M ] ∞ .A plegma (resp. strict plegma ) family in F is a finite sequence ( s i ) li =1 in F satisfyingthe following properties.(i) s i ( j ) < s i ( j ) for every 1 ≤ j < j ≤ k or j < j ∈ N and 1 ≤ i , i ≤ l .(ii) s i ( j ) ≤ s i ( j ) (cid:0) resp. s i ( j ) < s i ( j ) (cid:1) for every 1 ≤ i < i ≤ l and every1 ≤ j ≤ k or j ∈ N .For each l ∈ N , the set of all sequences ( s i ) li =1 which are plegma families in F willbe denoted by P lm l ( F ) and that of the strict plegma ones by S - P lm l ( F ).The following is a consequence of Ramsey’s Theorem [Ra]. Theorem 1.2 ([AKT]) . Let M be an infinite subset of N and k, l ∈ N . Then forevery finite partition S - P lm l ([ M ] k ) = ∪ ni =1 P i , there exist L ∈ [ M ] ∞ and 1 ≤ i ≤ n such that S - P lm l ([ L ] k ) ⊂ P i . Definition 1.3.
Let π = { , . . . , l } × { . . . , k } , s = ( s i ) li =1 be a plegma family in[ N ] k and ( e in ) li =1 ,n ∈ N be a sequence in a linear space E .(i) The plegma shift of π with respect to the plegma family s is the set s ( π ) = { ( i, s i ( j )) : ( i, j ) ∈ π } and for a subset A of π , the plegma shift of A with respect to s is the set s ( A ) = { ( i, s i ( j )) : ( i, j ) ∈ A } .(ii) Let x ∈ E with x = P ( i,j ) ∈ F a ij e ij and F ⊂ π . The plegma shift of x withrespect to s is the vector s ( x ) = P ( i,j ) ∈ F a ij e is i ( j ) .Recall that a sequence ( e n ) n in a seminormed space E is called spreading if k P ni =1 a i e i k = k P ni =1 a i e k i k for every n ∈ N , k < . . . < k n and a , . . . , a n ∈ R .Then, under Definition 1.3, we have the following reformulation: ( e n ) n is spread-ing if k P ni =1 a i e i k = k s ( P ni =1 a i e i ) k for every n ∈ N , a , . . . , a n ∈ R and everyplegma family s ∈ P lm ([ N ] n ). Next we introduce the notion of plegma spreadingsequences, which are an extension of the above. Definition 1.4.
A sequence ( e in ) li =1 ,n ∈ N in a Banach space E will be called plegmaspreading if each ( e in ) n is a normalized Schauder basic sequence and, for every x ∈ span { e in } li =1 ,n ∈ N , we have that k x k = k s ( x ) k for all plegma shifts s ( x ) of x . Remark 1.5.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence.(i) For every I ⊂ { , . . . , l } the sequence ( e in ) i ∈ I,n ∈ N is also plegma spreadingand in particular the sequence ( e in ) n is spreading for every 1 ≤ i ≤ l .(ii) The set { e in } li =1 ,n ∈ N is linearly independent.(iii) For every ( s i ) li =1 ∈ P lm l ([ N ] ∞ ), the sequence ( e is i ( n ) ) li =1 ,n ∈ N is isometric to( e in ) li =1 ,n ∈ N under the natural mapping T ( e in ) = e is i ( n ) .(iv) For every k ∈ N , x ∈ span { e in } l,ki =1 ,n =1 and s n = ( s ni ) li =1 ∈ P lm l ([ N ] k ), suchthat s ml ( k ) < s n (1) for every m < n , we have that the sequence ( x n ) n , with x n = s n ( x ), is spreading.2. Finite Families of Sequences in Banach Spaces
In this section we study in which cases l -tuples of Schauder (unconditional) basicsequences in a given Banach space have subsequences indexed by plegma familiesthat form a common Schauder (unconditional) basic sequence with a natural order. OINT SPREADING MODELS AND UNIFORM APPROXIMATION 5
As it turns out this is related to the w ∗ limits of these sequences in the second dual.The case in which some of the sequences are equivalent to the unit vector basis of ℓ is the interesting one and we use ultrafilters to deduce the desired conclusion.2.1. Finite Families of Schauder Basic Sequences.
We first treat non-trivialweak-Cauchy sequences and sequences equivalent to the unit vector basis of ℓ andeventually we consider weakly null sequences as well. We include a proof of thefollowing well known lemma for completeness. Lemma 2.1.
Let ( x n ) n be a normalized sequence in a Banach space X and x ∗ , . . . , x ∗ k ∈ X ∗ with lim x ∗ i ( x n ) = 0 for all 1 ≤ i ≤ k . For every δ >
0, thereexists n ∈ N such that d ( x n , ∩ ki =1 ker x ∗ i ) < δ for every n ≥ n . Proof.
Let Y = span { x ∗ , . . . , x ∗ k } and F be a finite δ/ S Y . Then there exists n ∈ N such that f ( x n ) < δ/ f ∈ F and n ≥ n . Pick any n ≥ n . Then,if d ( x n , ∩ ki =1 ker x ∗ i ) ≥ δ , we may find x ∗ ∈ X ∗ with k x ∗ k = 1 such that x ∗ ( x n ) ≥ δ and ∩ ki =1 ker x ∗ i ⊂ ker x ∗ . Hence x ∗ ∈ Y and there exists f ∈ F with k x ∗ − f k < δ/ x ∗ ( x n ) ≥ δ since f ( x n ) < δ/ (cid:3) The following is a variation of Mazur’s method [LT, Theorem 1.a.5] for findingSchauder basic sequences in infinite dimensional Banach spaces.
Proposition 2.2.
Let ( e n ) n , . . . , ( e ln ) n be seminormalized sequences in a Banachspace X and let E denote the closed linear span of { e in } li =1 ,n ∈ N and S E the unitsphere of E . Assume that there exist ε > { K F : F ⊂ S E finite } of finite subsets of X ∗ such that(i) for every finite F ⊂ S E and x ∈ F there exists x ∗ ∈ K F with k x ∗ k = 1 and x ∗ ( x ) ≥ ε ,(ii) for every finite F ⊂ S E and 1 ≤ i ≤ l , there exists L ∈ [ N ] ∞ such thatlim n ∈ L x ∗ ( e in ) = 0 for all x ∗ ∈ K F , and(iii) for every finite F ′ ⊂ F ⊂ S E , we have K F ′ ⊂ K F .Then there exist M , . . . , M l ∈ [ N ] ∞ and a suitable enumeration under which ∪ li =1 { e in } n ∈ M i is a Schauder basic sequence. Proof.
We may assume that the sequences ( e in ) n , 1 ≤ i ≤ l , are normalized. Indeed,if we normalize the given sequences then conditions (i), (ii), and (iii) will not beaffected. If we obtain the result for the normalized versions of the given sequences,then we can revert to subsequences of the initial ones. Let ( ε n ) n be a sequence in(0 , /
2) such that P ∞ n =1 ε n < ε/
5. We will construct, by induction on N , a Schauderbasic sequence ( x k ) k with x k = e i k n k , where i k = ( k − l ) + 1 and n k +1 > n k .Hence the sets M i = { n k : i k = i } , for 1 ≤ i ≤ l , and the lexicographic order on N × { , . . . , l } yield the desired result.We set x = y = e and F = { x / k x k , − x / k x k} . Assume that x , . . . , x k , y , . . . , y k , and F ⊂ F ⊂ · · · ⊂ F k have been chosen, for some k ∈ N , suchthat the following are satisfied: for 1 ≤ m ≤ k each x m is of the form e i m n m , each F m is an ε m / X m = span { x , y , . . . , x m , y m } , and for m > y m is in Y m = ∩{ E ∩ ker x ∗ : x ∗ ∈ K F m − } with k x m − y m k < ε m .We describe the next inductive step. By property (ii) there is L ∈ [ N ] ∞ suchthat for all x ∗ ∈ K F k we have lim n ∈ L x ∗ ( e i k +1 n ) = 0. Apply Lemma 2.1 to thesequence ( e i k +1 n ) n and the subset { x ∗ | E : x ∗ ∈ F k } of E ∗ to find x k +1 = e i k +1 n k +1 such that d ( x k +1 , ∩{ Y ∩ ker x ∗ : x ∗ ∈ K F k } ) < ε k +1 /
2. Then we may choose a y k +1 ∈ ∩{ Y ∩ ker x ∗ : x ∗ ∈ K F k } with k x k +1 − y k +1 k < ε k +1 . Finally, pick an ε k +1 -net F k +1 of the unit sphere of X k +1 = span { x , y , . . . , x k +1 , y m +1 } .Note that for each k ∈ N and x ∈ X m there exists x ∗ ∈ K F m with k x ∗ k = 1 and x ∗ ( x ) ≥ ( ε − ε m / k x k ≥ (9 ε/ k x k . Also, because for m ≤ n we have K F m ⊂ K F n we have y n +1 ∈ Y n +1 ⊂ Y m and hence for x ∗ ∈ K F m we deduce x ∗ ( y n +1 ) = 0. We S. A. ARGYROS, A. GEORGIOU, A.-R. LAGOS, AND P. MOTAKIS use these facts to first observe that ( y k ) k is K -Schauder basic, for K = (10 / (9 ε )).Indeed, if m ≤ n and a , . . . , a n ∈ R then x = P mi =1 a i y i ∈ X m and for some x ∗ ∈ F m with k x ∗ k = 1 (cid:13)(cid:13)(cid:13) m X i =1 a i y i (cid:13)(cid:13)(cid:13) ≤ ε x ∗ (cid:16) m X i =1 a i y i (cid:17) = 109 ε x ∗ (cid:16) n X i =1 a i y i (cid:17) ≤ ε (cid:13)(cid:13)(cid:13) n X i =1 a i y i (cid:13)(cid:13)(cid:13) . By the principle of small perturbations, for ( x k ) k to be Schauder basic it suffices toshow 2 K P k k x k − y k k / k y k k <
1. This follows from P k k x k − y k k / k y k k < P n ε k < ε/ (cid:3) Remark 2.3.
Let us observe that ( M i ) li =1 , as constructed in the previous proof,is a plegma family in [ N ] ∞ . In general, for every M , . . . , M l ∈ [ N ] ∞ , there exists aplegma family ( s i ) li =1 in [ N ] ∞ with s i ⊂ M i .Moreover, for any plegma family ( s i ) li =1 ∈ P lm l ([ N ] ∞ ), we associate a naturalorder on { s i ( n ) } li =1 ,n ∈ N and that is the lexicographic order on [ N ] × { , . . . , l } .The following lemma is an immediate consequence of the principle of local re-flexivity [LR], however we also give the following easy proof. Lemma 2.4.
Let X be a Banach space and F be a linear subspace of X ∗∗ withfinite dimension and X ∩ F = { } . Then, for every δ > x ∈ X with k x k = 1,there exists x ∗ ∈ X ∗ with k x ∗ k ≤ δ such that x ∗ ( x ) ≥ ε = d ( S X , F ) and x ∗∗ ( x ∗ ) = 0 for every x ∗∗ ∈ F . Proof.
Let x ∈ X with k x k = 1 and Y = span { F ∪ { x }} . Then there exists x ∗∗∗ ∈ S Y ∗ such that x ∗∗∗ ( x ) ≥ ε and x ∗∗∗ ( x ∗ ) = 0 for every x ∗∗ ∈ F . Then weconsider the identity map I : Y → X ∗∗ and recall that its conjugate I ∗ : X ∗∗∗ → Y ∗ is w ∗ - w ∗ -continuous. Since B X ∗ is a w ∗ -dense subset of B X ∗∗∗ and Y is of finitedimension, it follows that I ∗ ( B X ∗ ) is norm dense in B Y ∗ and hence, for every δ > B Y ∗ ⊂ I ∗ ((1 + δ ) B X ∗ ), which yields the desired result. (cid:3) Recall that a sequence ( x n ) n in a Banach space X is called non-trivial weak-Cauchy if there exists x ∗∗ ∈ X ∗∗ \ X such that w ∗ - lim x n = x ∗∗ . The next proposi-tion provides a complete characterization to the aforementioned problem for finitecollections of such sequences. Proposition 2.5.
Let ( e n ) n , . . . , ( e ln ) n be seminormalized non-trivial weak-Cauchysequences in a Banach space X and F = span { e ∗∗ i } li =1 , where w ∗ - lim e in = e ∗∗ i .Then there exists an ( s i ) li =1 ∈ P lm l ([ N ] ∞ ) such that { e is i ( n ) } li =1 ,n ∈ N is a Schauderbasic sequence, enumerated by the natural plegma order, if and only if X ∩ F = { } . Proof.
Let X ∩ F = { } , hence there exists x = P li =1 a i e ∗∗ i ∈ X with x = 0.If there exists a plegma family ( s i ) li =1 ∈ P lm l ([ N ] ∞ ) such that ∪ li =1 { e in } n ∈ M i is aSchauder basic sequence, we consider the sequence ( x n ) n with x n = P li =1 a i e is i ( n ) .Notice that ( x n ) n is a Schauder basic sequence with w - lim x k = x , which is acontradiction since x = 0.Suppose now that X ∩ F = { } and let ε = d ( S E , F ). Then for every x ∈ S E ,by Lemma 2.4, there exists an f x ∈ E ∗ with k f x k = 1 such that f x ( x ) ≥ ε/ x ∗∗ ( f x ) = 0 for every x ∗∗ ∈ F and hence lim x ∗ ( e in ) = 0 for every 1 ≤ i ≤ l . Finally,applying Proposition 2.2 (setting K F = { f x : x ∈ F } ) and Remark 2.3 the proof iscomplete. (cid:3) Next we give an example of a plegma spreading sequence, formed by two non-trivial Weak-Cauchy sequences, that is not Schauder basic.
OINT SPREADING MODELS AND UNIFORM APPROXIMATION 7
Definition 2.6 ([J1]) . On the space c ( N ) we define the following norm k x k J = sup n X i =1 (cid:18) X k ∈ I i x ( k ) (cid:19) ! , where the supremum is taken over all finite collections I , . . . , I n of disjoint intervalsof natural numbers. The James’ space , denoted by J , is the completion of c ( N )with respect to k · k J . Example 2.7.
Let ( e n ) n denote the standard basis of James’ space and recall thatit is a non-trivial weak-Cauchy sequence. We consider the sequences ( e n ) n and( e n ) n in J , with e n = e n + e and e n = e n +1 − e , which are also non-trivial weak-Cauchy and denote by e ∗∗ , e ∗∗ their w ∗ -limits. Notice that the sequence ( e in ) i =1 ,n ∈ N is a plegma spreading sequence in J . Moreover, since e ∈ J ∩ span { e ∗∗ , e ∗∗ } and T ( e in ) = e is i ( n ) is an isometry for every ( s i ) i =1 ∈ P lm l ([ N ] ∞ ), then the samearguments as in Proposition 2.5 yield that ( e in ) i =1 ,n ∈ N is not Schauder basic.We pass now to study the case of finite families of ℓ sequences in a Banach space.As is well known, β N denotes the Stone- ˇCech compactification of N and therefore ℓ ∞ ( N ) is isometric to C ( β N ). It is also known that the elements of β N are theultrafilters on N . The identification of ℓ ∞ ( N ) with C ( β N ) yields that the conjugatespace of ℓ ∞ ( N ) is isometric to M ( β N ), the set of all regular measures on β N .For f ∈ ℓ ∞ ( N ) and p an ultrafilter on N , the evaluation of the Dirac measure δ p on the function f is given as δ p ( f ) = lim p f ( n ), where lim p f ( n ) is the unique limitof ( f ( n )) n with respect to the ultrafilter p . Let us also observe that if T : ℓ → X is an isomorphic embedding, then T ∗∗ : M ( β N ) → X ∗∗ and for every p ∈ β N and x ∗ ∈ X ∗ , we have that T ∗∗ δ p ( x ∗ ) = lim p x ∗ ( T e n ). For further information onultrafilters we refer to [CN]. Lemma 2.8.
Let X be a Banach space and T : ℓ → X be an isomorphic em-bedding. Let α ∈ R , x ∗ , . . . , x ∗ k ∈ X ∗ and p be a non-principal ultrafilter on N such that T ∗∗ δ p ( x ∗ i ) = α for all 1 ≤ i ≤ k . Then there exists M ∈ [ N ] ∞ such thatlim n ∈ M x ∗ i ( T e n ) = α for every 1 ≤ i ≤ k . Proof.
Notice that T ∗∗ δ p ( x ∗ i ) = lim p x ∗ i ( T e n ) and also that, for any n ∈ N , the set M n = { m ∈ N : | x ∗ i ( T e m ) − α | < /n, for all 1 ≤ i ≤ l } is in p and is not finite,since p is a non-principal ultrafilter. Let M be a diagonalization of ( M n ) n , i.e. M ( n ) ∈ M n for all n ∈ N , then ( T e n ) n ∈ M is the desired subsequence. (cid:3) Lemma 2.9.
Let X be a separable Banach space and F be a finite dimensionalsubspace of X ∗∗ with X ∩ F = { } . Let also ( e n ) n , . . . , ( e ln ) n be sequences in X suchthat each one is equivalent to the basis of ℓ and denote by T i the correspondingembedding. Then there exist non-principal ultrafilters p , . . . , p l on N such that(i) The set F ∪ { T ∗∗ i δ p i } li =1 is linearly independent.(ii) X ∩ span { F ∪ { T ∗∗ i δ p i } li =1 } = { } . Proof.
Let us observe that the cardinality of β N is 2 c whereas that of any separableBanach space is less or equal to c . We also remind that the family { δ p : p ∈ β N } is equivalent to the basis of ℓ (2 c ) and hence linearly independent. The same re-mains valid for a fixed 1 ≤ i ≤ l and the family { T ∗∗ i δ p : p ∈ β N } , since T ∗∗ i is anisomorphism. We consider the linear space X ∗∗ /X and we denote by Q the naturalquotient map Q : X ∗∗ → X ∗∗ /X . Claim : For every 1 ≤ i ≤ l , there exists an uncountable subset A i of β N suchthat the family { QT ∗∗ i δ p : p ∈ A i } is linearly independent. S. A. ARGYROS, A. GEORGIOU, A.-R. LAGOS, AND P. MOTAKIS
Proof of Claim.
If not, there would exist a countable subset A i of β N such that { QT ∗∗ i δ p : p ∈ A i } is a maximal independent subfamily of { QT ∗∗ i δ p : p ∈ β N } , forsome 1 ≤ i ≤ l . Then { T ∗∗ i δ p : p ∈ β N } ⊂ span { X ∪ { T ∗∗ i δ p : p ∈ A i }} , which yieldsa contradiction, since the algebraic dimension of X is less or equal to c .Since F satisfies X ∩ F = { } , it follows that Q | F is an isomorphism and byinduction we will choose, for every 1 ≤ i ≤ l , an ultrafilter p i on N such that p i ∈ A i and QT ∗∗ i δ p i / ∈ span { Q [ F ] ∪ { QT ∗∗ j δ p j } j
Let ( e n ) n , . . . , ( e ln ) n be sequences in a separable Banach space X such that each one is equivalent to the basis of ℓ and denote by T i the correspondingembedding. Then, for every k ∈ N and every 1 ≤ i ≤ l and 1 ≤ j ≤ k , there existsa non-principal ultrafilter p ij on N such that(i) The set { T ∗∗ i δ p ij } l,ki =1 ,j =1 is linearly independent.(ii) X ∩ span { T ∗∗ i δ p ij } l,ki =1 ,j =1 = { } .The following lemma is an immediate consequence of the above and it will usedin the next subsection. Lemma 2.11.
Let ( e n ) n , . . . , ( e ln ) n be sequences in a separable Banach space X such that each one is equivalent the basis of ℓ and denote by T i the correspondingembedding. Then there exist x ∗ . . . , x ∗ l ∈ X ∗ such that the following hold.(i) For every 1 ≤ i ≤ l , there exists M i ∈ [ N ] ∞ such that lim n ∈ M i x ∗ i ( e in ) = 1.(ii) For every 1 ≤ i, j ≤ l , there exists M ij ∈ [ N ] ∞ such that lim n ∈ M ij x ∗ i ( e jn ) = 0. Proof.
From Corollary 2.10, there exist p , . . . , p l , q , . . . , q l non-principal ultrafilterson N such that the set { T ∗∗ i δ p i } li =1 ∪ { T ∗∗ i δ q i } li =1 is linearly independent. Then,for each 1 ≤ i ≤ l , we choose x ∗∗∗ i ∈ X ∗∗∗ such that x ∗∗∗ i ( T ∗∗ j δ p j ) = δ ij and x ∗∗∗ i ( T ∗∗ j δ q j ) = 0 for every 1 ≤ j ≤ l . The principle of local reflexivity then yieldsan x ∗ i ∈ X ∗ such that T ∗∗ j δ p j ( x ∗ i ) = δ ij and T ∗∗ j δ q j ( x ∗ i ) = 0 for every 1 ≤ j ≤ l .Finally, applying Lemma 2.8 we obtain the desired subsequences. (cid:3) Next we give a characterization to the problem in the general case. Recall that asfollows from Rosenthal’s ℓ theorem [Ro] and the theory of Schauder bases, if ( x n ) n is a Schauder basic sequence in a Banach space X , then it contains a subsequencewhich is either weakly null or equivalent to a basis of ℓ or non-trivial weak-Cauchy. Theorem 2.12.
Let ( e n ) n , . . . , ( e ln ) n be seminormalized sequences in a Banachspace X such that each one is either weakly null, equivalent to the basis of ℓ , ornon-trivial weak-Cauchy. Let I ⊂ { , . . . , l } be such that ( e in ) n is a non-trivial weak-Cauchy sequence with w ∗ - lim e in = e ∗∗ i for every i ∈ I and set F = span { e ∗∗ i } i ∈ I .Then there exists ( s i ) li =1 ∈ P lm l ([ N ] ∞ ) such that { e is i ( n ) } li =1 ,n ∈ N is a Schauderbasic sequence, enumerated by the natural plegma order, if and only if X ∩ F = { } . Proof.
Let J ⊂ { , . . . , l } be such that ( e in ) n is equivalent to the basis of ℓ foreach i ∈ J and denote by T i the corresponding embedding. Then Lemma 2.9 yieldsfor every i ∈ J a non-principal ultrafilter p i on N such that X ∩ Y = { } , where Y = span { F ∪ { T ∗∗ i δ p i } i ∈ J } . For ε = d ( S X , Y ) it follows from Lemma 2.4 that,for every x ∈ X with k x k = 1, there exists f x ∈ X ∗ with k f x k = 1 such that f x ( x ) ≥ ε/ x ∗∗ ( f x ) = 0 for every x ∗∗ ∈ Y . For every finite F ⊂ S X , we set K F = { f x : x ∈ F } . Note that lim x ∗ ( e in ) = 0 for every i ∈ I , every finite F ⊂ S X and every x ∗ ∈ K F . Also, from Lemma 2.8, for each i ∈ J , there exists M i ∈ [ N ] ∞ OINT SPREADING MODELS AND UNIFORM APPROXIMATION 9 with lim n ∈ M i x ∗ ( e in ) = 0 for all x ∗ ∈ K f . Applying Proposition 2.2, we derive thedesired result. (cid:3) More specifically, for a plegma spreading sequence ( e in ) li =1 , Rosenthal’s theoremyields that each ( e in ) is either weakly null, equivalent to the unit vector basis of ℓ , or non-trivial Weak-Cauchy, since it is spreading. Taking also into account thebehavior of plegma spreading sequences we give a corollary of the above theorem. Corollary 2.13.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence in a Banach space X and I ⊂ { , . . . , l } be such that ( e in ) n is a non-trivial weak-Cauchy sequence with w ∗ - lim e in = e ∗∗ i for every i ∈ I and set F = span { e ∗∗ i } i ∈ I . Then ( e in ) li =1 ,n ∈ N is aSchauder basic sequence, enumerated by the lexicographic order on [ N ] × { , . . . , l } ,if and only if X ∩ F = { } . Proof.
Theorem 2.12 yields a plegma family ( s i ) li =1 in [ N ] ∞ such that ( e is i ( n ) ) li =1 ,n ∈ N is a Schauder basic sequence if and only if X ∩ F = { } , and since T ( e in ) = e is i ( n ) is an isometry, then the same holds for ( e in ) li =1 ,n ∈ N . (cid:3) Finite Families of Unconditional Sequences.
We now study the case ofunconditional sequences. We start with plegma spreading sequences. Recall thatevery weakly null spreading sequence in a Banach space is unconditional. The fol-lowing proposition extends this result to plegma spreading sequences, using similararguments as in the classical case.
Proposition 2.14.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence such that each( e in ) n is weakly null. Then ( e in ) li =1 ,n ∈ N is an unconditional sequence. Proof.
Let π = { , . . . , l } × { , . . . , k } and x = P ( i,j ) ∈ π a ij e ij . Since each ( e in ) n is weakly null, then for every ε > i , j ) ∈ π , there exist s ( x ) , . . . , s m ( x )plegma shifts of x and a convex combination P mt =1 λ t s t ( x ) such that(i) s t ( e ij ) = s t ( e ij ) for every ( i, j ) ∈ π ′ = π \ { ( i , j ) } and 1 ≤ t , t , ≤ m .(ii) k P nt =1 λ t s t ( e i j ) k < ε k x k /a i j .(iii) P mt =1 λ t s t ( x ) = P ( i,j ) ∈ π ′ a ij s ( e i j ) + P nt =1 λ t a i j s t ( e i j ).Then since ( e in ) li =1 ,n ∈ N is plegma spreading we have that k s t ( x ) k = k x k for all1 ≤ t ≤ m and also k P ( i,j ) ∈ π ′ a ij s ( e ij ) k = k P ( i,j ) ∈ π ′ a ij e ij k and hence (cid:13)(cid:13)(cid:13) X ( i,j ) ∈ π ′ a ij e ij (cid:13)(cid:13)(cid:13) ≤ (1 + ε ) (cid:13)(cid:13)(cid:13) X ( i,j ) ∈ π a ij e ij (cid:13)(cid:13)(cid:13) . Finally, applying iteration we show that for every ε > F ⊂ π , (cid:13)(cid:13)(cid:13) X ( i,j ) ∈ F a ij e ij (cid:13)(cid:13)(cid:13) ≤ (1 + ε ) (cid:13)(cid:13)(cid:13) X ( i,j ) ∈ π a ij e ij (cid:13)(cid:13)(cid:13) . (cid:3) Proposition 2.15.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence. If each ( e in ) n is equivalent to the basis of ℓ , then the same holds for ( e in ) li =1 ,n ∈ N . Proof.
Let 0 < ε <
1, 0 < δ < (1 − ε ) / l , and x = P kj =1 P li =1 a ij e ij with P kj =1 P li =1 | a ij | = 1. Then either for x or − x there exists 1 ≤ i ≤ l suchthat P j ∈ J + i a i j ≥ / l , where J + i = { j : a i j > } . Moreover, from Lemma2.11, for each 1 ≤ i, j ≤ l , there exist x i ∈ X ∗ and M i , M ij in [ N ] ∞ such thatlim n ∈ M i x ∗ i ( e in ) = 1 and lim n ∈ M ij x ∗ i ( e jn ) = 0. We set M = max {k x ∗ i k : i = 1 , . . . , l } and choose a plegma family ( s i ) li =1 in [ N ] k such that(i) s i ( j ) ∈ M i and x ∗ ( e is i ( j ) ) > − ε for every j ∈ J + i . (ii) s i ( j ) ∈ M i i for every j ∈ J − i = { j : a i j < } .(ii) s j ⊂ M i j for every 1 ≤ j ≤ l with j = i .(iii) x ∗ i ( P kj =1 P li =1 i = i a ij e is i ( j ) + P j ∈ J − i a i j e i s i ( j ) ) < δ .Hence x ∗ i ( x ) ≥ − ε l − δ and therefore k x k ≥ ( − ε l − δ ) /M which yields the desiredresult. (cid:3) Combining the two previous propositions we have the following final result.
Theorem 2.16.
Let ( e in ) li =1 ,n ∈ N be a plegma spreading sequence such that each( e in ) n is unconditional. Then ( e in ) li =1 ,n ∈ N is also an unconditional sequence. Proof.
Let I ⊂ { , . . . , l } be such that the sequence ( e in ) n is weakly null for every i ∈ I and denote its complement by J . We denote by E the closed linear spanof { e in } i ∈ I,n ∈ N and by E that of { e in } i ∈ J,n ∈ N . Then, for any x ∈ E + E with x = P kj =1 ( P i ∈ I a ij e ij + P i ∈ J b ij e ij ), using similar arguments as in the proof ofProposition 2.14, we have that for every ε > (cid:13)(cid:13)(cid:13) k X j =1 X i ∈ J b ij e ij (cid:13)(cid:13)(cid:13) ≤ (1 + ε ) (cid:13)(cid:13)(cid:13) k X j =1 (cid:16) X i ∈ I a ij e ij + X i ∈ J b ij e ij (cid:17)(cid:13)(cid:13)(cid:13) . Hence E + E = E ⊕ E and since both ( e in ) i ∈ I,n ∈ N and ( e in ) i ∈ J,n ∈ N are uncondi-tional sequences, as follows from the two previous propositions, then the same holdsfor their union. (cid:3) Unconditional Sequences in Singular Position.
The following is a variantof the Maurey-Rosenthal classical example [MR]. As Theorem 2.16 asserts, thestrong assumption of being plegma spreading yields that l -tuples of unconditionalsequences are jointly unconditional. The purpose of this example is to demonstratethat this strong condition is in fact necessary. Proposition 2.17.
Let N , N be a partition of N into two infinite sets. Thereexists a Banach space X with a Schauder basis ( e n ) n such that the following hold.(i) The sequences ( e n ) n ∈ N and ( e n ) n ∈ N are unconditional.(ii) For every M ⊂ N such that M ∩ N and M ∩ N are both infinite sets, thesequence ( e n ) n ∈ M is not unconditional.We fix a strictly increasing sequence of natural numbers ( µ i ) i such that ∞ X i =1 X j>i √ µ i √ µ j ≤ P the collection of all finite sequences ( E k ) nk =1 of successive non-empty fi-nite subsets of N . Take an injection σ : P → N such that σ (( E k ) nk =1 ) > max { E k } nk =1 ,for every ( E k ) nk =1 ∈ P , and finally fix a partition of N into two infinite subsets N and N . Definition 2.18.
A sequence ( E ik ) ,ni =1 ,k =1 of non-empty finite subsets of N is calleda special sequence if the following hold.(i) E k ⊂ N and E k ⊂ N for every 1 ≤ k ≤ n .(ii) The sets E k and E k are successive for every 1 ≤ k ≤ n .(iii) The sets E k and E k +1 are successive for every 1 ≤ k < n .(iv) E = E = µ j , for some j ∈ N .(v) E k = E k = µ j k , where j k = σ (( E ik ) ,k − i =1 ,k =1 ) for every 1 < k ≤ n . Remark 2.19.
Let ( E ik ) ,ni =1 ,k =1 and ( F ik ) ,mi =1 ,k =1 be special sequences and set k =min { k : E k = F k } . Then since σ is an injection, notice that if k > OINT SPREADING MODELS AND UNIFORM APPROXIMATION 11 (i) If k > E k = F k and E k = F k for every 1 ≤ k < k − E k − = F k − and E k − = F k − or E k − = F k − .(iii) E k = F k for every k ≤ k ≤ min { n, m } .Let ( e ∗ i ) i as well as ( e i ) i denote the unit vector basis of c ( N ) and for every f, x ∈ c ( N ) with f = P ni =1 a i e ∗ i and x = P mi =1 b i e i set f ( x ) = P min { n,m } i =1 a i b i . Finally,for f, g ∈ c ( N ) with f = P nk =1 a i k e ∗ i k and g = P nk =1 b j k e ∗ j k , where a i k , b j k = 0, wewill say that f and g are consistent if sgn( a i k ) = sgn( b j k ) for every 1 ≤ k ≤ n . Definition 2.20.
Consider the following subsets of c ( N ). W = { } ∪ {± e ∗ i : i ∈ N } ,W = ( √ µ j X i ∈ E ε i e ∗ i : E ⊂ N or E ⊂ N , E = µ j , ε i ∈ {− , } for i ∈ E ) ,W = ( X i =1 n X k =1 f ik : f ik ∈ W ∪ W , (supp( f ik )) ,ni =1 ,k =1 is a special sequence ,f k and f k are consistent for every 1 ≤ k ≤ n ) and set W = { P E ( f ) : f ∈ W ∪ W ∪ W , E interval of N } . Define a norm on c ( N ) by setting k x k = sup { f ( x ) : f ∈ W } and let X (2) MR denote its completionwith respect to this norm. Remark 2.21.
For any f = P ni =1 a i e ∗ i in W and 1 ≤ k < l ≤ n , it follows that P li = k a i e ∗ i is in W as well. That is, the sequence ( e i ) i forms a normalized andbimonotone Schauder basis for X (2) MR . Remark 2.22.
For any f = P ni =1 a i e ∗ i in W and any choice of signs ( ε i ) i ∈ N (or ( ε i ) i ∈ N ), it follows that there exist ( b i ) ni =1 such that b i = ε i a i for all i ∈ N ∩ { , . . . , n } (or i ∈ N ∩ { , . . . , n } ) and g = P ni =1 b i e ∗ i is in W . Hence, thesequences ( e i ) i ∈ N and ( e i ) i ∈ N are 1-unconditional. Definition 2.23.
We will call an x in X (2) MR a weighted vector if x = √ µ ℓ P i ∈ E ε i e i with E = µ ℓ and ε i ∈ {− , } . We also define the weight of x as w ( x ) = µ ℓ .Moreover, any f = √ µ ℓ P i ∈ E ε i e ∗ i in W with E = µ ℓ and ε i ∈ {− , } , will becalled a weighted functional and we define the weight of f as w ( f ) = µ ℓ . Lemma 2.24.
Let x , . . . , x n be successive weighted vectors with increasing weightsand f , . . . , f m be successive functionals with increasing weights. If w ( x i ) = w ( f j )for every 1 ≤ i ≤ n and 1 ≤ j ≤ m , then P mi =1 P ni =1 | f j ( x i ) | ≤ . Proof.
Let us observe that for x, f weighted with w ( x ) = µ ℓ and w ( f ) = µ k suchthat µ ℓ = µ k , it holds that | f ( x ) | ≤ min {√ µ ℓ , √ µ k } max {√ µ ℓ , √ µ k } and hence | f ( x ) | ≤ √ µ ℓ √ µ k if µ ℓ < µ k and | f ( x ) | ≤ √ µ k √ µ ℓ if µ k < µ ℓ . Let w ( x i ) = µ ℓ i and w ( f j ) = µ k j . Then, for eachpair ( i, j ), we have that | f j ( x i ) | ≤ min n √ µ ℓi √ µ kj , √ µ kj √ µ ℓi o and since each pair ( µ ℓ i , µ k j )appears only once, we have that m X j =1 n X i =1 | f j ( x i ) | ≤ ∞ X j =1 X j>i √ µ i √ µ j ≤ . (cid:3) Proposition 2.25.
Let ( E ik ) ,ni =1 ,k =1 be a special sequence and define the vector x ik = (1 / p E ik ) P i ∈ E ik e i for 1 ≤ k ≤ n and i = 1 ,
2. Then k P i =1 P nk =1 x ik k ≥ n whereas k P i =1 P nk =1 ( − i x ik k ≤ Proof.
Set µ j k = E k for 1 ≤ k ≤ n . The first part follows easily from thefact that f = P i =1 P nk =1 (1 / p E ik ) P j ∈ E ik e ∗ j is in W . For the second part set y = P i =1 P nk =1 ( − i x ik and let g = P i =1 P mk =1 1 √ µ jk P j ∈ F ik e ∗ j in W and also k = min { k : E k = F k } , making the convention min ∅ = m + 1. If k = 1, thenthe previous lemma yields that | g ( y ) | ≤ . Otherwise, by Remark 2.19 and Lemma2.24 the following hold.(i) If k >
2, then g k ( y ) = − g k ( y ) for every 1 ≤ k < k − | g ik − ( y ) | ≤ P mk = k | g ik ( y ) | ≤ for i = 1 , | g ( y ) | ≤
3. Finally, in the general case that g ∈ W , using similar argumentswe conclude that | g ( y ) | ≤ (cid:3) Proposition 2.26.
Let M ⊂ N be such that M ∩ N and M ∩ N are both infinitesets. Then the sequence ( e i ) i ∈ M is not unconditional. Proof.
We may, choose for each n ∈ N , a special sequence ( E ik ) ,ni =1 ,k =1 such that E k ⊂ M ∩ N and E k ⊂ M ∩ N for every 1 ≤ k ≤ n , and apply Proposition 2.25to conclude thatsup ((cid:13)(cid:13)(cid:13) X i ∈ M ε i a i e i (cid:13)(cid:13)(cid:13) : ε i ∈ {− , } , (cid:13)(cid:13)(cid:13) X i ∈ M ε i a i e i (cid:13)(cid:13)(cid:13) ≤ ) ≥ n . Since n is arbitrary, it follows that ( e i ) i ∈ M is not unconditional. (cid:3) The following problem is however open.
Problem 1.
Let ( e n ) n and ( e n ) n be subsymmetric sequences in a Banach space,i.e. spreading and unconditional. Do there exist M, L infinite subsets of N suchthat the sequence { e n } n ∈ M ∪ { e n } n ∈ L is unconditional?Despite the fact that for ( e n ) n ∈ N and ( e n ) n ∈ N any subsequences fail to forma common unconditional sequence, the following more general result shows that wemay find further block subsequences which satisfy this property. Proposition 2.27.
Let ( x n ) n and ( y n ) n be unconditional sequences in a Banachspace X . There exist block sequences ( z n ) n and ( w n ) n , of ( x n ) n and ( y n ) n respec-tively, such that { z n } n ∈ N ∪ { w n } n ∈ N is an unconditional sequence. Proof.
Assume that there exist two subsequences ( x n ) n ∈ M and ( y n ) n ∈ M such that d ( S Z , S Y ) >
0, where Z = span { x n } n ∈ M and Y = span { y n } n ∈ M . Then Y + Z isclosed. Hence by the Closed Graph Theorem we have that Y + Z = Y ⊕ Z and thisyields that { x n } n ∈ M ∪ { y n } n ∈ M is unconditional.Otherwise, we choose by induction ( z n ) n and ( w n ) n which are normalized blocksof ( x n ) n and ( y n ) n respectively with P ∞ n =1 k z n − w n k < / (2 C ), where C is thebasis constant of ( x n ) n , and hence also of ( z n ) n . Then, by the principle of smallperturbations, it follows that the sequence { z n } n ∈ N ∪ { w n − } n ∈ N is equivalent to( z n ) n , which is unconditional. (cid:3) Remark 2.28.
A natural question arising from the previous proposition is whetherevery space generated by two unconditional sequences is unconditionally saturated.The answer is negative and this follows from a well known more general result. Let X be a Banach space with a Schauder basis ( x n ) n , Y be a separable Banach spaceand ( d n ) n be a dense subset of the unit ball of Y . Then the sequences ( x n ) n and( y n ) n with y n = x n + d n / n are equivalent and generate the space X ⊕ Y . Hence OINT SPREADING MODELS AND UNIFORM APPROXIMATION 13 if ( x n ) n is unconditional and Y contains no unconditional sequence we obtain thedesired result. We thank Bill Johnson for bringing to our attention this classicalargument. 3. Joint Spreading Models
We introduce the notion of l -joint spreading models which is the central conceptof this paper. It describes the joint asymptotic behavior of a finite collection ofsequences. As it is demonstrated in [AM1], in certain spaces this behavior may beradically more rich than the one of usual spreading models. It is worth pointingout that spreading models have been tied to the study of bounded linear operators[AM2] and the present paper clarifies that joint spreading models are no exception. Definition 3.1.
Let l ∈ N , ( x n ) n , . . . , ( x ln ) n be Schauder basic sequences in aBanach space ( X, k · k ) and ( e in ) li =1 ,n ∈ N be a sequence in a Banach space ( E, k · k ∗ ).Let M ∈ [ N ] ∞ . We will say that the l -tuple (( x in ) n ∈ M ) li =1 generates ( e in ) li =1 ,n ∈ N as an l-joint spreading model if the following is satisfied. There exists a null sequence( δ n ) n of positive reals such that for every k ∈ N , ( a ij ) l,ki =1 ,j =1 ⊂ [ − ,
1] and everystrict-plegma family ( s i ) li =1 ∈ S - P lm l ([ M ] k ) with M ( k ) ≤ s (1), we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij e ij (cid:13)(cid:13)(cid:13) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ k . We will also say that (( x in ) n ) li =1 admits ( e in ) li =1 ,n ∈ N as an l -joint spreading model if there exists M ∈ [ N ] ∞ such that (( x in ) n ∈ M ) li =1 generates ( e in ) li =1 ,n ∈ N .Finally, for a subset A of X , we will say that A admits ( e in ) li =1 ,n ∈ N as an l -jointspreading model if there exists an l -tuple (( x in ) n ) li =1 of sequences in A which admits( e in ) li =1 ,n ∈ N as an l -joint spreading model .Notice that for l = 1, the previous definition recovers the classical Brunel-Sucheston spreading models. Remark 3.2.
Let ( x n ) n , . . . , ( x ln ) n be Schauder basic sequences in a Banach space( X, k · k ). Let also M ∈ [ N ] ∞ be such that the l -tuple (( x in ) n ∈ M ) li =1 generates thesequence ( e in ) li =1 ,n ∈ N as an l -joint spreading model. Then the following hold.(i) For every 1 ≤ i ≤ l , the sequence ( e in ) n is the spreading model admitted by( x in ) n .(ii) The sequence ( e in ) li =1 ,n ∈ N is plegma spreading. Although l -joint spreadingmodels are defined using strict plegma families, these sequences behave ina spreading way that involves plegma families.(iii) For every M ′ ∈ [ M ] ∞ we have that (( x in ) n ∈ M ′ ) li =1 generates ( e in ) li =1 ,n ∈ N asan l -joint spreading model.(iv) For every ( δ n ) n null sequence of positive reals there exists M ′ ∈ [ M ] ∞ suchthat (( x in ) n ∈ M ′ ) li =1 generates ( e in ) li =1 ,n ∈ N as an l -joint spreading model withrespect to ( δ n ) n .(v) If |k · k| is an equivalent norm on X , then every l -joint spreading modeladmitted by ( X, k · k ) is equivalent to an l -joint spreading model admittedby ( X, |k · k| ).Next we prove a Brunel-Sucheston type result for l -joint spreading models. Theorem 3.3.
Let l ∈ N and X be a Banach space. Then every l -tuple of Schauderbasic sequences in X admits an l -joint spreading model.First we describe the following combinatorial lemma which will yield the theorem. Lemma 3.4.
Let ( x n ) n , . . . , ( x ln ) n be bounded sequences in a Banach space X and ( δ n ) n be a decreasing null sequence of positive real numbers. Then for every M ∈ [ N ] ∞ , there exists L ∈ [ M ] ∞ such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x it i ( j ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ k for every k ∈ N , ( a ij ) l,ki =1 ,j =1 ⊂ [ − ,
1] and ( s i ) li =1 , ( t i ) li =1 ∈ S - P lm l ([ L ] k ) with s (1) , t (1) ≥ L ( k ). Proof.
Let
C > k x in k < C for all i = 1 , . . . , l and n ∈ N , and set L = M . We will construct, by induction, a decreasing sequence ( L k ) k ≥ such thatfor every k ∈ N , ( a ij ) l,ki =1 ,j =1 ⊂ [ − ,
1] and ( s i ) li =1 , ( t i ) li =1 ∈ S - P lm l ([ L k ] k ), (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x it i ( j ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ k . Suppose that L , . . . , L k − have been chosen, for some k ∈ N . Let A be a finite δ k klC -net of [ − ,
1] and B be a partition of [0 , lC ] consisting of disjoint intervals withlength less than δ k . We set F = { f : A kl → B } and for f ∈ F P f = n ( s i ) li =1 ∈ S - P lm l ([ L k − ] k ) : (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) ∈ f ( a ) , for all a = (( a ij ) kj =1 ) li =1 ∈ A kl o . Then S - P lm l ([ L k − ] k ) = ∪ f ∈F P f and by Theorem 1.2 there exist L k ∈ [ N ] ∞ and f ∈ F such that S - P lm l ([ L k ] k ) ⊂ P f . Hence for every ( a ij ) l,ki =1 ,j =1 ⊂ A and( s i ) li =1 , ( t i ) li =1 ∈ S - P lm l ([ L k ] k ), we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) n X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) n X j =1 l X i =1 a ij x it i ( j ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < δ k . Since A is a net of [ − , L k is as desired. Finally, choosing L to be a diagonalization of ( L k ) k completes the proof. (cid:3) Proof of Theorem 3.3.
Let ( x n ) n , . . . , ( x ln ) n be Schauder basic sequences. Observethat the previous lemma yields an infinite subset L of N such that for every k ∈ N and ( a ij ) l,ki =1 ,j =1 ⊂ [ − ,
1] and every sequence (( s ni ) li =1 ) n of strict plegma familiesin [ L ] k with lim s n (1) = ∞ , the sequence ( k P k,lj =1 ,i =1 a ij x is ni ( j ) k ) n is Cauchy, withthe limit independent from the choice of the sequence (( s ni ) li =1 ) n .Denote by ( e n ) n the usual basis of c ( N ) and for every i = 1 , . . . , l and n ∈ N ,set e in = e k ( i,n ) , where k ( i, n ) = ( n − l + in . Using the above observation, wedefine a seminorm k · k ∗ on c ( N ) as follows: (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij e ij (cid:13)(cid:13)(cid:13) ∗ = lim n (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is ni ( j ) (cid:13)(cid:13)(cid:13) where (( s ni ) li =1 ) n ⊂ P lm l ([ L ] k ) with s n (1) → ∞ and ( a ij ) l,ki =1 ,j =1 ∈ [ − , x in ) n is a Schauder basic sequence and hence does not contain any normconvergent subsequences, a modification of [BL, Proposition 1.B.2] yields that k · k ∗ is a norm. Denote by E the completion of c ( N ) with respect to this norm andnotice that the l -tuple (( x in ) n ∈ L ) li =1 generates the sequence ( e in ) li =1 ,n ∈ N in E as an l -joint spreading model. (cid:3) OINT SPREADING MODELS AND UNIFORM APPROXIMATION 15
The following proposition is an immediate consequence of the definition of l -jointspreading models and Theorem 2.12. Proposition 3.5.
Let ( x n ) n , . . . , ( x ln ) n be Schauder basic sequences in a Banachspace X such that each one is either weakly null, equivalent to the basis of ℓ , or non-trivial weak-Cauchy and the l -tuple ( x in ) li =1 ,n ∈ N generates the sequence ( e in ) li =1 ,n ∈ N as an l -joint spreading model. Let I ⊂ { , . . . , l } be such that ( x in ) n is a non-trivial weak-Cauchy sequence with w ∗ - lim x in = x ∗∗ i for every i ∈ I and set F =span { x ∗∗ i } i ∈ I . If X ∩ F = { } , then ( e in ) li =1 ,n ∈ N is a Schauder basic sequence.The next example demonstrates that the opposite statement of the above is notalways true, that is, ( e in ) li =1 ,n ∈ N may be Schauder basic whereas X ∩ F = { } . Example 3.6.
Define the norm on c ( N ) given by k x k = sup P ni =1 | P k ∈ I i x ( k ) | ,where the supremum is taken over all finite collections I , . . . , I n of successive in-tervals of natural numbers with n ≤ min I . Denote by X the completion of c ( N )with respect to this norm. Then the usual basis ( e n ) n is a non-trivial weak-Cauchysequence that generates a spreading model equivalent to the unit vector basis of ℓ . Consider the sequences e n = e n − − e and e n = e n + e . As follows fromProposition 2.5, none of their subsequences form a common Schauder basic sequencewhereas the l -joint spreading model admitted by ( e in ) i =1 ,n ∈ N is equivalent to theunit vector basis of ℓ .Recall that spreading models generated by weakly null sequences are uncondi-tional. This is extended to joint spreading models by an easy modification of theclassical case [BL, Proposition 5.1]. Proposition 3.7.
Let ( x n ) n , . . . , ( x ln ) n be weakly null Schauder basic sequencesin a Banach space X that admit ( e in ) li =1 ,n ∈ N as an l -joint spreading model. Then( e in ) li =1 ,n ∈ N is 1-suppression unconditional and hence for every ε > k ∈ N there exists n ∈ N such that for every ( s i ) li =1 ∈ S - P lm l ([ N ] k ) with n ≤ s (1), thesequence ( x is i ( j ) ) l,ki =1 ,j =1 is (1 + ε )-suppression unconditional. Remark 3.8.
The notion of l -joint spreading models can be naturally extended,by a diagonalization argument, to ω -joint spreading models which are ω -plegmaspreading sequences and are generated by countably many Schauder basic sequences.4. spaces with unique joint spreading model In this section we study spaces that admit a uniformly unique joint spreadingmodel with respect to certain families of sequences. In the first part we prove theuniform uniqueness of l -joint spreading models for the classical ℓ p and c spaces.Then we pass to Asymptotic ℓ p spaces [MMT] and in the last part we study thisproblem for the James Tree space. Definition 4.1.
Let F be a family of normalized sequences in a Banach space X .We will say that X admits a uniformly unique l -joint spreading model with respectto F if there exists K > l ∈ N , any two l -joint spreadingmodels generated by sequences from F are K -equivalent. Remark 4.2.
Let F be a family of normalized sequences in a Banach space X such that, for some l ∈ N , there exists K l > l -joint spreadingmodels generated by sequences from F are K l -equivalent.(i) For every l ′ < l , there exists K l ′ ≤ K l such that any two l ′ -joint spreadingmodels generated by sequences from F are K l ′ -equivalent.(ii) The space X may fail to admit a uniformly unique l -joint spreading modelwith respect to F . For examples of such spaces see [AM1] and the space inDefinition 5.28. Notation 4.3.
For a Banach space X , we will denote by F ( X ) the set of allnormalized Schauder basic sequences in X , by F ( X ) its subset consisting of thesequences that are weakly null and by F C ( X ) the set of all normalized C -Schauderbasic sequences in X . Finally, if X has a Schauder basis, we shall denote by F b ( X )the set of all normalized block sequences in X .Next we present some examples of spaces admitting a uniformly unique l -jointspreading model. We start with the classical sequence spaces ℓ p and c . Proposition 4.4.
Each of the following spaces admits a uniformly unique l -jointspreading model which is in fact equivalent to its unit vector basis.(i) The spaces ℓ p , for 1 < p < ∞ , with respect to F ( ℓ p ).(ii) The space ℓ with respect to F b ( ℓ ), but not with respect to F ( ℓ ).(iii) The space c with respect to F ( c ), but not with respect to F ( c ).It is immediate to see that the above remain valid for the spaces ℓ p (Γ), for1 ≤ p < ∞ , and c (Γ), for any infinite set Γ. Remark 4.5.
For C ≥
1, the space ℓ admits a uniformly unique l -joint spreadingmodel with respect to F C ( ℓ ). To see this, let ( x n ) n be an arbitrary normalized C -Schauder basic sequence in ℓ . Passing, if it necessary, to a subsequence ( x n ) n has a point-wise limit x in ℓ . That is, lim n e ∗ i ( x n ) = e ∗ i ( x ) for all i ∈ N . Also,if we set z n = x n − x , then lim n k z n k = λ may be assumed to exist. It followsthat k x k + λ = 1 and that 0 < λ ≤
1. We can also assume that ( λ − z n ) n ≥ n is(1 + 1 /n )-equivalent to the usual basis of ℓ . We conclude that for any M ∈ N : M (1 − λ ) = M k x k ≤ lim k (cid:13)(cid:13)(cid:13) M X n =1 x n + k (cid:13)(cid:13)(cid:13) ≤ C lim k (cid:13)(cid:13)(cid:13) M X n =1 x n + k − M X n = M +1 x n + k (cid:13)(cid:13)(cid:13) = C lim k (cid:13)(cid:13)(cid:13) M X n =1 z n + k − M X n = M +1 z n + k (cid:13)(cid:13)(cid:13) = C M λ.
Therefore, λ ≥ / (2 C + 1). Hence if ( x in ) n , 1 ≤ i ≤ l , is an l -tuple of C -Schauderbasic sequences we may pass to subsequences so that ( x in ) n converges point-wise tosome x i for 1 ≤ i ≤ l and if we set ( z in ) n = ( x in − x i ) for 1 ≤ i ≤ l then thesesequences are point-wise null and they are all bounded bellow from 1 / (2 C + 1). Wemay then conclude that for any ε >
0, passing to appropriate subsequences, ( x in ) n ,1 ≤ i ≤ m , is jointly (2 C + 1 + ε )-equivalent to the unit vector basis of ℓ . Remark 4.6.
Although for any C ≥ ℓ admits a uniformly unique l -joint spreading model with respect to F C ( ℓ ), this is no longer true for spaces withthe Schur property. For example, define for each n ∈ N the norm k·k n on ℓ given by k x k n = max {k x k ℓ , n − k x k ℓ } . Set X = ( P n ⊕ X n ) ℓ , where X n = ( ℓ , k·k n ), whichhas the Schur property. Although every spreading model of this space is equivalentto the unit vector basis of ℓ , this does not happen for a uniform constant.Another example of spaces admitting a uniformly unique joint spreading modelare asymptotic ℓ p spaces. We start with their definition. Definition 4.7 ([MT]) . A Banach space X with a normalized Schauder basis is asymptotic ℓ p (resp. asymptotic c ) if there exists C > x i ) ni =1 of normalized vectors in X with n < supp( x ) < · · · < supp( x n )is C -equivalent to the standard basis of ℓ np (resp. c n ), for 1 ≤ p < ∞ .The classical examples of asymptotic ℓ p spaces are Tsirelson’s original space [T]and its p-convexifications [FJ]. The next proposition follows easily from the abovedefinition and the fact that an asymptotic ℓ p space, for 1 < p < ∞ , is reflexive. OINT SPREADING MODELS AND UNIFORM APPROXIMATION 17
Proposition 4.8.
Every asymptotic ℓ p or asymptotic c space X admits a uni-formly unique l -joint spreading model with respect to F b ( X ). Moreover, everyasymptotic ℓ p space, for 1 < p < ∞ , admits a uniformly unique l -joint spreadingmodel with respect to F ( X ).The following proposition concerns spaces with uniformly unique joint spreadingmodels with respect to families that have certain stability properties. The jointspreading models of such spaces are unconditional and sometimes even equivalentto some ℓ p or c . Families with such properties play an important role in the studyof the UALS in the next section. Proposition 4.9.
Let X be a Banach space that admits a K -uniformly unique l -joint spreading model with respect to a family of normalized Schauder basic se-quences F . Assume that F is such that:(a) If ( x j ) j in F then any subsequence of ( x j ) j is in F .(b) If ( x j ) j is in F then there exists an infinite subset L = { l i : i ∈ N } of N such that if z i = k x l i − − x l i k − ( x l i − − x l i ), for i ∈ N , then the sequence( z i ) i is in F .(c) If ( x j ) j is in F and ( λ i ) Ni =1 is a finite sequence of scalars, not all of whichare zero, then there exists an infinite subset L = { l i : i ∈ N } of N such thatif for n ∈ N : z n = (cid:13)(cid:13)(cid:13) N X i =1 λ i x l N ( n − i (cid:13)(cid:13)(cid:13) − N X i =1 λ i x l N ( n − i ! , then the sequence ( z i ) i is in F .The following statements hold.(i) If F satisfies (a) then every l -joint spreading model admitted by an l -tuple of sequences in F is spreading when enumerated with the naturalplegma order and it is K -equivalent to the spreading model generated byany sequence in F .(ii) If F satisfies (a) and (b) then every l -joint spreading model admitted bysequences in F is K -suppression unconditional.(iii) If F satisfies (a) and (c) then every l -joint spreading model admitted bysequences in F is K -equivalent to the unit vector basis of ℓ p (for p = ∞ wemean the unit vector basis of c ). Proof.
The first item follows by taking an arbitrary sequence ( x j ) j in F , passing toa subsequence that generates some spreading model ( e i ) i , and then taking disjointlyindexed subsequences ( x i ) i , . . . , ( x li ) i , which by assumption are all in F . Clearly,they generate an l -joint spreading model that is isometrically equivalent to ( e i ) i .We conclude that any l -joint spreading model generated by an l -tuple of sequencesin F is K -equivalent to ( e i ) i , when endowed with the natural plegma order.For the second item it is sufficient, by (i), to show that any spreading modeladmitted by a sequence in F has the desired property. Pick an arbitrary sequence( x j ) j in F which by (a) may be chosen to generate some spreading model ( e j ) j .Applying (b) to ( x j ) j we can deduce that there is a sequence in F that gener-ates as a spreading model the sequence ( k e j − − e j k − ( e j − − e j )) j , which by[BL, Proposition 4.3] is 1-suppression unconditional. Observe that any sequencethat is K -equivalent to a 1-suppression unconditional sequence is K -suppressionunconditional.Assume now that (a) and (c) hold. Clearly, (a) and (c) together imply (b) sowe may pick up where we left off, namely having at hand a sequence ( x j ) j in F that generates a spreading model that is 1-suppression unconditional. By [MMT,Paragraph 1.6.3], as a direct application of Krivine’s Theorem [Kr] [L], for any m ∈ N and ε >
0, we may choose scalars λ , . . . , λ N such that any m terms of theresulting sequence ( z n ) n are (1 + ε )-equivalent to the unit vector basis of ℓ mp , forsome 1 ≤ p ≤ ∞ . This means that there exists a constant K such that, for any m ∈ N , there exists 1 ≤ p m ≤ ∞ such that the first m terms of any spreading modelgenerated by a sequence from F are K -equivalent to the unit vector basis of ℓ p m .Taking a limit point of ( p m ) m yields the conclusion. (cid:3) Coordinate-free asymptotic ℓ p spaces (Asymptotic ℓ p spaces). Noticethat Definition 4.7 of an asymptotic ℓ p space from [MT] depends on the Schauderbasis of X and not only on X . A coordinate-free version of this definition canbe found in [MMT, Subsection 1.7] and it is based on a game of two players (S)and (V). In each turn of the game player (S) chooses a closed finite codimensionalsubspace Y of X and player (V) chooses a normalized vector y ∈ Y . A Banachspace X is called Asymptotic ℓ p if there exists a constant C such that, for every n ∈ N , player (S) has a wining strategy in the game G ( p, n, C ), that is to forcein n steps player (V) to choose a sequence ( y i ) ni =1 that is C -equivalent to the unitvector basis of ℓ np (or c n for p = ∞ ). We point out that the original formulation ofthis property is different. The equivalence of the original definition with this moreconvenient version follows from [MMT, Subsection 1.5].Next we show that for a separable Asymptotic ℓ p space X , for 1 ≤ p ≤ ∞ ,there exists a certain family of sequences in X , described in Proposition 4.11, withrespect to which X admits a uniformly unique l -joint spreading model. This familyhas certain properties that F ( X ) fails when X contains ℓ and this result will beused in the next section to prove that an Asymptotic ℓ space satisfies the UALS.We start with the following lemma. Lemma 4.10.
Let X be a separable C -Asymptotic ℓ p space, for 1 ≤ p ≤ ∞ . Thenthere exists a countable collection Y of finite codimensional subspaces of X suchthat, for every ε > n ∈ N , player (S) has a winning strategy in the game G ( p, n, C + ε ) when choosing finite codimensional subspaces from Y . Proof. If X is C -asymptotic ℓ p in the sense described in the paragraph above weshall, for fixed n ∈ N , assume the role of player (V) and let player (S) follow awinning strategy during a multitude of outcomes in a game of G ( p, n, C ). Moreaccurately, we will describe how to define a collection of vectors of X of the form { x nF : ∅ 6 = F ∈ [ N ] ≤ n } and a collection of closed finite codimensional subspaces of X of the form { Y nF : F ∈ [ N ] ≤ n − } that satisfy:(i) for all F ∈ [ N ] ≤ n − , the norm-closure of { x nF ∪{ i } : i > max( F ) } is the unitsphere of Y nF (here, max( ∅ ) = 0) and(ii) for every { k , . . . , k m } , in [ N ] ≤ n we have that (cid:16) Y n ∅ , x n { k } (cid:17) , (cid:16) Y n { k } , x n { k ,k } (cid:17) , . . . , (cid:16) Y n { k ,...,k m − } , x n { k ,...,k m } (cid:17) is the outcome of a game of G ( p, n, C ) after m rounds in which player (S)has followed a winning strategy.Player (S) initiates and he chooses a finite codimensional subspace Y n ∅ . As player(V), we choose a dense subset { x n { i } : i ∈ N } of the unit sphere of Y n ∅ . If for some1 ≤ m < n we have chosen { x nF : ∅ 6 = F ∈ [ N ] ≤ m } and { Y nF : F ∈ [ N ] ≤ m − } ,we complete the inductive step as follows: for every F = { k < · · · < k m } , byassumption (ii), player (S) may continue following a winning strategy and choosea closed finite codimensional subspace Y such that for every unit vector y ∈ Y thesequence ( x nk i ) mi =1 ⌢ ( y ) is C -equivalent to the unit vector basis of ℓ m +1 p . Set Y nF = Y and then, for all i > max( F ), choose a unit vector x nF ∪{ i } in Y F such that the set { x nF ∪{ i } : i > max( F ) } is dense in the unit sphere of Y nF . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 19
Set Y = { Y nF : n ∈ N , F ∈ [ N ] ≤ n − } and fix ε > n ∈ N . Let us alsotake ˜ ε > G ( p, n, C + ε ), choosing finite codimensional subspaces from Y .Player (S) initiates the game and chooses the subspace Y = Y n ∅ and player (V)chooses an arbitrary normalized vector y from Y . Before the next turn, player(S) also chooses k ∈ N such that k y − x nk k < ˜ ε/n . Let ( Y , y ) , . . . ( Y m , y m ) bethe outcome in the first m turns of the game, for 1 ≤ m < n , while player (S)has also chosen k , . . . , k m ∈ N with k y i − x nk i k < ˜ ε/n , for 1 ≤ i ≤ m . In thenext turn of the game, player (S) chooses the subspace Y m +1 = Y n { k ,...,k m } and k m +1 ∈ N such that k y m +1 − x nk m +1 k < ˜ ε/n , where y m +1 is the vector player (V)choose from Y m +1 . Hence if ( Y , y ) , . . . ( Y n , y n ) is the final outcome of the game,notice that the sequence ( x nk i ) ni =1 is C -equivalent to the unit vector basis of ℓ np and k y i − x nk i k < ˜ ε/n for all 1 ≤ i ≤ n . If we take 1 ≤ A, B with AB ≤ C suchthat (1 /A ) ≤ ( P ni =1 | a i | p ) /p ≤ k P ni =1 a i x nk i k ≤ B ( P ni =1 | a i | p ) /p then we concludethat ( y i ) ni =1 is C (1 + ˜ ε ) / (1 − ˜ εC ) equivalent to the unit vector basis of ℓ np . For ˜ ε sufficiently small we deduce the conclusion. (cid:3) Proposition 4.11.
Let X be a separable C -Asymptotic ℓ p space, for 1 ≤ p ≤ ∞ .There exists a countable subset A of X ∗ such that, if F , A = n ( x n ) n : ( x n ) n is normalized and lim n f ( x n ) = 0 for all f ∈ A o , then X admits ℓ p as a C -uniformly unique l -joint spreading model with respect tothe family F , A . Proof.
Let Y be as in Lemma 4.10 and, for each Y ∈ Y , choose a finite subset f Y , . . . , f Yk Y of X ∗ such that Y = ∩ k Y i =1 ker( f Yi ) and set A = ∪ Y ∈ Y { f Y , . . . , f Yk Y } which is a countable set. We will show that it is the desired one. To that end, let l ∈ N and ( x n ) n , . . . , ( x ln ) n be sequences in F , A generating an l -joint spreadingmodel ( e in ) li =1 ,n . Let k ∈ N , we will show that ( e ij ) l,ki =1 ,j =1 is C -equivalent to theunit vector basis of ℓ lkp . Set m = lk , fix ε > y , . . . , y m and ( s i ) li =1 in S - P lm l ([ N ] k ) such that(i) ( Y , y ) , . . . , ( Y m , y m ) is the outcome of the game G ( m, p, C + ε ).(ii) Y , . . . , Y m ∈ Y .(iii) For 1 ≤ i ≤ l and 1 ≤ j ≤ k , if we take n ( i, j ) = ( i − k + j then k y n ( i,j ) − x is i ( j ) k ≤ ε/m .It follows that for any scalars ( a ij ) l,ki =1 ,j =1 with | a ij | ≤
1, we have that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij x is i ( j ) (cid:13)(cid:13)(cid:13) − (cid:13)(cid:13)(cid:13) k X j =1 l X i =1 a ij y n ( i,j ) (cid:13)(cid:13)(cid:13)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε. As ( y i ) mi =1 is ( C + ε )-equivalent to the unit vector basis of ℓ mp the conclusion follows. (cid:3) The following is an immediate corollary of the above. In the general case, allAsymptotic ℓ p spaces admit a uniformly unique joint spreading model with respectto the family of normalized weakly null Schauder basic sequences. Corollary 4.12.
Every Asymptotic ℓ p space X , for 1 ≤ p ≤ ∞ , admits a uniformlyunique l -joint spreading model with respect to F ( X ). Moreover, every l -jointspreading model generated by a sequence from this family is equivalent to the unitvector basis of ℓ p (or of c if p = ∞ ). James Tree Space.
We show that the James Tree space JT admits a uni-formly unique joint spreading model with respect to F ( JT ). This is however nottrue for joint spreading models with respect to F ( JT ) or F b ( JT ). Notation 4.13.
We denote by D the dyadic tree , i.e. D = { , } < ∞ , ordered by theinitial part order. We will use S to denote segments of D and B to denote branches.For m < n , the band Q [ m,n ] is the set { s ∈ D : m ≤ | s | ≤ n } . We set c ( D ) to bethe linear space of all eventually zero sequences x : D → R . For a segment S of D we denote by S ∗ the linear functional on c ( D ) defined as S ∗ ( x ) = P s ∈ S x ( s ). Definition 4.14 ([J2]) . On c ( D ) we define the following norm k x k JT = sup n X i =1 (cid:16) X s ∈ S i x ( s ) (cid:17) ! where the supremum is taken over all finite collections S , . . . , S n of pairwise disjointsegments. The James Tree space , denoted by JT , is the completion of c ( D ) withrespect to the above norm. Remarks 4.15.
Let us observe, for later use, that(i) The following set is norming for JT : W = ( n X i =1 b i S ∗ i : n ∈ N , n X i =1 b i ≤ , { S i } ni =1 pairwise disjoint segments ) . (ii) Let ε > x ∈ JT with k x k = 1 and { S i } i ∈ I be pairwise disjoint segmentswith the property that | S ∗ i ( x ) | ≥ ε for every i ∈ I . Then I ≤ /ε .(iii) Let S , . . . , S n be pairwise disjoint segments and b , . . . , b n ∈ R , then (cid:13)(cid:13)(cid:13) n X i =1 b i S ∗ i (cid:13)(cid:13)(cid:13) ≤ n X i =1 b i . We will prove the following theorem.
Theorem 4.16.
The space JT admits a uniformly unique l -joint spreading modelwith respect to F ( JT ) and every l -joint spreading model generated by sequencesfrom this family is √ ℓ .The proof of the theorem is a variant of the well known result due to I. Ameniyaand T. Ito [AI], that every normalized weakly null sequence in JT contains a sub-sequence which is 2-equivalent to the usual basis of ℓ . From this follows that everyspreading model generated by a normalized weakly null sequence is 2-equivalent tothe unit vector basis of ℓ . Our approach yields that every l -joint spreading modelgenerated by sequences from F ( JT ) is equivalent to the unit vector basis of ℓ withequivalence constant √
2, which as mentioned in [HB], [Be] is the best possible.As a consequence of the fact that James space (see Definition 2.6) is isometric toa subspace of JT it follows that J also admits a uniformly unique l -joint spreadingmodel with respect to F ( J ).We break up the proof of the theorem into several lemmas and we start with thefollowing Ramsey type result. Definition 4.17.
Let ( Q [ p n ,q n ] ) n be successive bands in D and ( F n ) n be a sequenceof finite subsets of JT . We will say that ( F n ) n is a weakly null level block family with respect to ( Q [ p n ,q n ] ) n if the following hold.(i) supp( x ) ⊂ Q [ p n ,q n ] and k x k = 1 for every n ∈ N and x ∈ F n .(ii) The sequence ( x n ) n is weakly null for any choice of x n ∈ F n . Lemma 4.18.
Let ( F n ) n be a weakly null level block family with sup n F n < ∞ .Then, for every ε >
0, there exists an L ∈ [ N ] ∞ such that for every initial segment S there exists at most one n ∈ L with the property that | S ∗ ( x ) | ≥ ε for some x ∈ F n . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 21
Proof.
If the conclusion is false, using Ramsey’s theorem from [Ra], we may as-sume that there exists L ∈ [ N ] ∞ such that, for every pair m < n in L , there exist aninitial segment S m,n and x ∈ F m , y ∈ F n such that | S ∗ m,n ( x ) | ≥ ε and | S ∗ m,n ( y ) | ≥ ε . Claim : Set µ = max n F n /ε . Then { S m,n | [0 ,p n ] : m ∈ L, m < n } ≤ µ forevery n ∈ L , where for a segment S and p, q ∈ N we denote S | [ p,q ] = S ∩ Q [ p,q ] . Proof of Claim.
If { S m,n | [0 ,p n ] : m ∈ L, m < n } > µ for some n ∈ L , then usingthe pigeon hole principle, we may find an x ∈ F n and F ⊂ { , . . . , n − } with F > /ε such that | S m,n ( x ) | ≥ ε and the segments S m,n | [ p n ,q n ] are pairwisedisjoint for m ∈ F . This contradicts item (ii) of Remark 4.15.Hence, for every n ∈ L , let { S m,n | [0 ,p n ] : m ∈ L, m < n } = { S n , . . . , S nµ ( n ) } with µ ( n ) ≤ µ and set L ni = { m ∈ L : m < n and S m,n | [0 ,p n ] = S ni } , for 1 ≤ i ≤ µ ( n ),and L ni = ∅ , for µ ( n ) < i ≤ µ . Notice that { m ∈ L : m < n } = ∪ µi =1 L ni for all n ∈ L . Passing to a further subsequence we may assume that, for every 1 ≤ i ≤ µ ,( L ni ) n ∈ L converges point-wise and we denote that limit by L i . Then it is easy tosee that L = ∪ µi =1 L i and hence some L i is an infinite subset of L such that, forevery n ∈ L i , there exists an initial segment S n such that, for all m < n in L i ,we have that | S ∗ n ( x m ) | ≥ ε for some x m ∈ F m . Then there exist M ∈ [ L i ] ∞ anda sequence ( x n ) n ∈ M with x n ∈ F n , such that ( S n ) n ∈ M converges point-wise to abranch B and | S ∗ m ( x n ) | ≥ ε for all m > n in M . Hence | B ( x n ) | ≥ ε for all n ∈ M ,which contradicts item (ii) of Definition 4.17. (cid:3) Lemma 4.19.
Let ε > F n ) n be a weakly null level block family with respectto ( Q [ p n ,q n ] ) n and assume that sup n F n < ∞ . Then there exist an increasingsequence ( n k ) k in N and a decreasing sequence ( ε k ) k of positive reals such that(i) For every k ∈ N and every initial segment S there exists at most one k ′ > k such that | S ∗ ( x ) | ≥ ε k for some x ∈ F n k ′ .(ii) P ∞ k =1 q nk P ∞ i = k ( i + 1) ε i < ε . Proof.
Let ( δ n ) n be a sequence of positive reals such that P ∞ n =1 δ n < ε . We willconstruct ( n k ) k and ( ε k ) k by induction on N as follows. We set n = 1 and L = N and choose ε such that 2 q ε < δ . Suppose that n , . . . , n k and ε , . . . , ε k havebeen chosen for some k in N . Then Lemma 4.18 yields an L k ∈ [ L k − ] ∞ such thatfor every segment S there exist at most one n ∈ L k with | S ∗ ( x ) | ≥ ε k for some x ∈ F n . We then choose n k +1 ∈ L k with n k +1 > n k and ε k +1 < ε k such that(a) 2 q nk +1 ( k + 2) ε k +1 < δ k +1 (b) 2 q nm P k +1 i = m ( i + 1) ε i < δ m for every m ≤ k .It is easy to see that ( n k ) k and ( ε k ) k are as desired. (cid:3) Lemma 4.20.
Let ε > ε n ) n be a decreasing sequence of positive reals. Letalso ( F n ) n be a weakly null level block family with respect to ( Q [ p n ,q n ] ) n and assumethat sup n F n < ∞ and that the following hold.(i) For every n ∈ N and every initial segment S there exists at most one m > n such that | S ∗ ( x ) | ≥ ε n for some x ∈ F m .(ii) P ∞ n =1 q n P ∞ i = n ( i + 1) ε i < ε .Then for every n ∈ N and every choice of x , . . . , x n with x i ∈ F i and scalars a , . . . , a n we have that (cid:16) n X i =1 a i (cid:17) ≤ (cid:13)(cid:13)(cid:13) n X i =1 a i x i (cid:13)(cid:13)(cid:13) ≤ ( √ ε ) (cid:16) n X i =1 a i (cid:17) . Proof.
Let us first observe that if ( x n ) n is a sequence with each x n ∈ F n , then forevery n ∈ N and every segment S with | M ( x n − ) | < | min S | ≤ | M ( x n ) | , where for x ∈ c ( D ) we denote M ( x ) = max supp( x ), the following hold due to (i). (a) { i > n : | S ∗ ( x i ) | ≥ ε n } ≤ { i > n : ε k − > | S ∗ ( x i ) | ≥ ε k } ≤ k for every k > n .Now for each 1 ≤ i ≤ n , there exist pairwise disjoint segments S i , . . . , S im i suchthat S ij ⊂ Q [ p i ,q i ] and P m i j =1 ( S i ∗ j ( x i )) = k x i k and hence (cid:13)(cid:13)(cid:13) n X i =1 a i x i (cid:13)(cid:13)(cid:13) ≥ (cid:16) n X i =1 a i m i X j =1 (cid:0) S i ∗ j ( x i ) (cid:1) (cid:17) ≥ (cid:16) n X i =1 a i (cid:17) . Pick S , . . . , S m pairwise disjoint segments and b , . . . , b m reals with P mj =1 b j ≤ ≤ j ≤ m , we will denote by i j, the unique 1 ≤ i ≤ n such that | M ( x i j, − ) | < | min S j | ≤ | M ( x i j, ) | and also by i j, the unique, if there exists, i j, < i ≤ n such that | S ∗ j ( x i j, ) | ≥ ε i j, . We set S j,k = S j ∩ Q [ p ij,k ,q ij,k ] , for k = 1 ,
2, and S j, = S j \ ( S j, ∪ S j, ) and we also set J i = { j : i j, = i or i j, = i } ,for 1 ≤ i ≤ n . Note that, by (a), each j appears in J i for at most two i and so P ni =1 P j ∈ J i b j ≤ P mj =1 b j . We thus calculate: (cid:12)(cid:12)(cid:12) m X j =1 b j S ∗ j, (cid:16) n X i =1 a i x i (cid:17) + m X j =1 b j S ∗ j, (cid:16) n X i =1 a i x i (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n X i =1 a i X j ∈ J i b j S ∗ j ( x i ) (cid:12)(cid:12)(cid:12) ≤ (cid:16) n X i =1 a i (cid:17) (cid:16) n X i =1 (cid:0) X j ∈ J i b j S ∗ j ( x i ) (cid:1) (cid:17) ≤ (cid:16) n X i =1 a i (cid:17) (cid:16) n X i =1 X j ∈ J i b j (cid:17) ≤ √ (cid:16) n X i =1 a i (cid:17) . Finally, we set G i = { j : | M ( x i − ) | < | min S j | ≤ | M ( x i ) |} and we have that { , . . . , m } = ∪ ni =1 G i . Notice that G i ≤ q i and | S ∗ j, ( P nk =1 x k ) | < P ∞ k = i ( k + 1) ε k for any j ∈ G i . Then due to (b) and (ii), it follows that P mj =1 | S ∗ j, ( P ni =1 x i ) | < ε and hence we conclude that (cid:12)(cid:12)(cid:12) m X j =1 b j S ∗ j, (cid:16) n X i =1 a i x i (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) n X i =1 a i m X j =1 b j S ∗ j, ( x i ) (cid:12)(cid:12)(cid:12) < ε (cid:16) n X i =1 a i (cid:17) . (cid:3) Proof of Theorem 4.16.
Let ( x n ) n , . . . , ( x ln ) n in F ( JT ) be such that (( x in ) n ) li =1 generates a sequence ( e in ) li =1 ,n ∈ N as an l -joint spreading model and by a slidinghump argument we may assume that each sequence ( x in ) n is block. Hence we maychoose L ∈ [ N ] ∞ such that the family ( F n ) n ∈ L , with F n = { x n , . . . , x ln } , is a weaklynull level block family in JT which satisfies (i) and (ii) in Lemma 4.20. Then, since(( x in ) n ∈ L ) li =1 generates also generates ( e in ) li =1 ,n ∈ N as an l -joint spreading model, weconclude that ( e in ) li =1 ,n ∈ N is √ ℓ and thereforeany two l -joint spreading models, generated by sequences from F ( JT ), are 2-equivalent. (cid:3) Remark 4.21.
The notion of asymptotic models, which appeared in [HO], alsoconcerns the asymptotic behavior of countably many basic sequences. Althoughthe asymptotic models are different from joint spreading models, as B. Sari pointedout, a Banach space admits a uniformly unique asymptotic model with respect toa family F if and only if it admits a uniformly unique joint spreading model withrespect to F .5. uniform approximation of bounded operators We now pass to the study of the UALS property on certain classes of spaces. Firstwe consider spaces with very few operators, namely spaces with the scalar-plus-compact property. The second class includes spaces admitting a uniformly uniquejoint spreading model with respect to certain families of Schauder basic sequences.Here, the notion of joint spreading models and the UALS property come togetherin the sense that the first property yields the second one. The third subsection is
OINT SPREADING MODELS AND UNIFORM APPROXIMATION 23 devoted to the study of the UALS property under duality. A consequence of themain result, Theorem 5.23, is that the spaces C ( K ), with K countable compact,satisfy the UALS. In the fourth subsection we show that the spaces L p [0 , ≤ p ≤ ∞ and p = 2, and C ( K ) for uncountable compact metric spaces K fail theUALS property. We close with some final remarks and open problems. Definition 5.1.
We will say that a Banach space X satisfies the Uniform Approx-imation on Large Subspaces ( UALS ) property if there exists
C > W of L ( X ), every A ∈ L ( X )and ε > x ∈ B X , there is a B ∈ W such that k A ( x ) − B ( x ) k ≤ ε , then there exist a finite codimensional subspace Y of X and a B ∈ W such that k ( A − B ) | Y k L ( Y,X ) ≤ Cε . Definition 5.2.
A Banach space X will be called UALS-saturated if there exists
C > W of L ( X ), every A ∈ L ( X )and ε > x ∈ B X , there is a B ∈ W such that k ( A − B ) x k ≤ ε , it holds that for every subspace Y of X there exists a furthersubspace Z such that k ( A − B ) | Z k L ( Z,X ) ≤ Cε , for some B ∈ W .5.1. The UALS Property for Compact Operators.
The first class of spacessatisfying the UALS includes spaces with very few operators. We prove that Banachspaces with the scalar-plus-compact property satisfy the UALS and are in factUALS-saturated. Hence, the main result from [Ar et al. ] yields that a large classof spaces, that includes all superreflexive spaces, embed into spaces that satisfythe UALS. We start with the following variation of Mazur’s theorem [LT, Theorem1.a.5].
Lemma 5.3.
Let X be a Banach space, T ∈ L ( X ) and let ε > k T | Y k L ( Y,X ) > ε for every subspace Y of X of finite codimension. Then there existsa normalized sequence ( x n ) n in X such that ( T x n ) n is seminormalized Schauderbasic. Proof.
Let δ >
0. Pick x in the unit sphere of X with k T x k ≥ ε and assume that x , . . . , x n have been chosen for some n ∈ N . Let G be a finite subset of X ∗ such that,for every x ∈ span { T x , . . . , T x n } , we have that k x k ≤ (1 + δ ) max { g ( x ) : g ∈ G } and choose x n +1 in the unit sphere of ∩ g ∈ G ker T ∗ g with k T x n +1 k ≥ ε . It followsquite easily that ( T x n ) n is a Schauder basic sequence. (cid:3) Proposition 5.4.
Let X be a Banach space and T ∈ L ( X ) be a compact operator.Then inf k T | Y k L ( Y,X ) = 0, where the infimum is taken over all subspaces Y of X offinite codimension. Proof.
Suppose that the conclusion is false, then the previous lemma yields that T [ B X ] contains a seminormalized Schauder basic sequence and this contradicts thefact that T is compact. (cid:3) Notation 5.5.
Let X be a Banach space, we will denote by K ( X ) the ideal of allcompact operators in the unital algebra L ( X ). Corollary 5.6.
Let X be a Banach space, W be a compact subset of K ( X ) and A ∈ L ( X ). Assume that there exists ε >
0, such that for every x ∈ B X , there is a B ∈ W such that k A ( x ) − B ( x ) k ≤ ε . Then, for every δ >
0, there exists a finitecodimensional subspace Y of X such that k ( A − B ) | Y k L ( Y,X ) ≤ ε + δ for all B ∈ W . Proof.
Let δ > { B i } ni =1 be a δ -net of W. Applying Proposition 5.4, we choosea finite codimensional subspace Y of X such that k B i | Y k < δ for every 1 ≤ i ≤ n .Note that k B | Y k ≤ δ for all B ∈ W . Then, for every x in the unit ball of Y , thereis a B in W such that k A ( x ) − B ( x ) k ≤ ε and hence k A ( x ) k < ε + 2 δ . That is, k A | Y k ≤ ε + 2 δ . Therefore k ( A − B ) | Y k L ( Y,X ) ≤ ε + 4 δ for every B ∈ W . (cid:3) Theorem 5.7.
Every Banach space with the scalar-plus-compact property satisfiesthe UALS.
Proof.
Let
W, A, ε be as in Definition 5.1, with A = λ A I + K A and K A compact.Let δ > { B i } ni =1 be a δ -net of W with B i = λ i I + K i and K i ∈ K ( X ), for i = 1 , . . . , n . Proposition 5.4 then yields a finite codimensional subspace Y of X such that k K A | Y k ≤ δ and k K i | Y k ≤ δ for all 1 ≤ i ≤ n . Pick an x ∈ Y with k x k = 1 and B ∈ W with B = λ B I + K B and k A ( x ) − B ( x ) k ≤ ε . Then, for1 ≤ i ≤ n such that k B − B i k ≤ δ , we have k A ( x ) − B i ( x ) k ≤ ε + δ and hence | λ A − λ i | = k λ A x − λ i x k ≤ k Ax − B i x k + k K A x + K i x k ≤ ε + 2 δ . Then, for every y in the unit ball of Y , we have that k A ( y ) − B i ( y ) k ≤ ε + 4 δ which proves thedesired result. (cid:3) Theorem 5.8.
Let X be a Banach space such that, for every A ∈ L ( X ), thereis a strictly singular operator S and λ ∈ R such that A = λI + S . Then X isUALS-saturated. Proof.
Let
W, A, ε be as in Definition 5.2, with A = λ A I + S A and S A a strictlysingular operator. Let δ > { B i } ni =1 be a δ -net of W with B i = λ i I + S i and S i strictly singular, 1 ≤ i ≤ n . Recall that for every infinite dimensional subspaceof X there exists a further subspace Y such that S A | Y and S i | Y , for 1 ≤ i ≤ n , arecompact operators. Applying the same arguments used in the previous proof weobtain the desired conclusion. (cid:3) Uniformly Unique Joint Spreading Models and the UALS Property.
In this subsection we study spaces that admit uniformly unique l -joint spreadingmodels with respect to families with sufficient stability properties, described in thefollowing definition, to deduce that in certain cases they satisfy the UALS property.Such spaces are, for example, all Asymptotic ℓ p spaces. This should be comparedto the examples of the following subsection that fail the UALS and the proof of thisfact is based on the existence of diverse plegma spreading sequences in these spaces.The families of sequences that we restrict our study to are very rich, in the sensethat any sequence has a subsequence the successive differences of which are in F and it is also closed under taking subsequences. Moreover, if a space has a uniformlyunique l -joint spreading model with respect to such a family then, as already shownin Proposition 4.9, it has to be at least unconditional and in most cases it has tobe some ℓ p or c . Definition 5.9.
Let X be a Banach space. A collection F of normalized andSchauder basic sequences in X will be called difference-including if(i) for every ( x n ) n in F any subsequence of ( x n ) n is in F and(ii) for every sequence ( x n ) n in X without a norm convergent subsequence thereexists an infinite subset L of N such that, for any further infinite subset M = { m k : k ∈ N } of L , if z k = k x m k − − x m k k − ( x m k − − x m k ), thenthe sequence ( z k ) k is in F . Remark 5.10.
A difference-including collection clearly satisfies (a) and (b) ofProposition 4.9. In fact, most naturally defined families of normalized Schauderbasic sequences in a Banach space X are difference-including. Such families are:(i) F ( X ), the collection of all normalized Schauder basic sequences in X .(ii) F (1+ ε ) ( X ), the collection of all normalized (1 + ε )-Schauder basic sequencesin X for some fixed ε > F , A for a countable subset A of the dual, where F , A = n ( x n ) n : ( x n ) n is normalized and lim n f ( x n ) = 0 for all f ∈ A o . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 25 (iv) ˜ F b ( X ) = F , ( e ∗ n ) n if X has a Schauder basis ( e n ) n , where ( e ∗ n ) n are thebiorthogonal functionals associated to the basis. Notice that a Banachspace X admits a uniformly unique l -joint spreading model with respect to F b ( X ) if and only if it does so with respect to ˜ F b ( X ).(v) F ( X ) if X does not contain ℓ .(vi) F su ( X ), the collection of all normalized Schauder basic sequences that gen-erate a 1-suppression unconditional spreading model.In certain cases, for X non-separable it is convenient to consider different collec-tions F Z for different separable subspaces Z of X . This is included in the statementof the following Theorem. Theorem 5.11.
Let X be a Banach space and assume that for every separable sub-space Z of X we have a difference-including collection F Z of normalized Schauderbasic sequences in Z . If there exists a uniform K ≥ Z admitsa K -uniformly unique l -joint spreading model with respect to F Z , then X satisfiesthe UALS property.We postpone the proof of Theorem 5.11 to first state and prove its corollaries.Note that if X is a Banach space and F is a collection of normalized Schauderbasic sequences in X with respect to which it admits a uniformly unique l -jointspreading model, then we may consider, for every separable subspace Z of X , thefamily F Z = { ( x n ) n in F : x i ∈ Z for all i ∈ N } . This is in fact sufficient to provemost cases stated bellow. Corollary 5.12.
In the following cases, a Banach space X and all of its subspacessatisfy the UALS property.(i) X has a Schauder basis and it admits a uniformly unique l -joint spreadingmodel with respect to F b ( X ).(ii) X is an arbitrary Banach space that admits a uniformly unique l -jointspreading model with respect to F ( X ).(iii) X does not contain ℓ and it admits a uniformly unique l -joint spreadingmodel with respect to F ( X ).(iv) X is an arbitrary Banach space and, for some ε >
0, it admits a uniformlyunique l -joint spreading model with respect to F (1+ ε ) ( X ). Proof.
All cases follow from Theorem 5.11. We describe some of the details. Thefirst case follows from the fact that such an X admits a uniformly unique l -jointspreading model with respect to ˜ F b ( X ) = F , ( e ∗ i ) i , which is difference-including.The second case follows directly from the fact that F ( X ) is difference-including.In case (iii), from Rosenthal’s ℓ theorem [Ro], we have that F ( X ) is difference-including. For case (iv), note that F (1+ ε ) is difference-including as well. (cid:3) Corollary 5.13.
The following Banach spaces and all of their subspaces satisfy theUALS property.(a) The space ℓ p (Γ), for 1 ≤ p < ∞ and any infinite set Γ.(b) The space c (Γ), for any infinite set Γ.(c) The James Tree space.(d) Every Asymptotic ℓ p space, for 1 ≤ p ≤ ∞ . Proof.
The case of ℓ p (Γ) follows from item (ii) of Corollary 5.12 for 1 < p < ∞ andfrom (i) for p = 1, while that of c (Γ) and the James Tree follows from item (iii).Moreover, for case (d), if 1 < p ≤ ∞ and X is an Asymptotic ℓ p space, then it doesnot contain ℓ and admits a uniformly unique l -joint spreading model with respectto F ( X ) so the result follows from case (iii) as well. Finally, if X is C -Asymptotic ℓ , use Theorem 4.11 to choose for every separable subspace Z of X a countablesubset A Z of Z ∗ such that Z admits a C -uniformly unique l -joint spreading modelwith respect to F Z = F , A Z . (cid:3) We break up the proof of Theorem 5.11 into several steps.
Lemma 5.14.
Let X be a Banach space that admits a K -uniformly unique l -jointspreading model with respect to a difference-including collection F of normalizedSchauder basic sequences. Then for any D > K and any sequences ( z in ) n , ( y in ) n , i = 1 , . . . , l , in F , there exists an infinite subset L of N such that, for any scalars a , . . . , a l , θ , . . . , θ l and n < · · · < n l in L we have that(1) min ≤ i ≤ l | θ i | D (cid:13)(cid:13)(cid:13) l X i =1 a i z in i (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) l X i =1 a i θ i y in i (cid:13)(cid:13)(cid:13) ≤ max ≤ i ≤ l | θ i | D (cid:13)(cid:13)(cid:13) l X i =1 a i z in i (cid:13)(cid:13)(cid:13) . Proof.
Choose
C > K and pass to an infinite set L so that the l -tuples (( z in ) n ∈ L ) li =1 and (( y in ) n ∈ L ) li =1 generate some l -joint spreading models that are K -equivalentto one another. This means that we may perhaps pass to a further subset of L and have that, for any n < · · · < n l in L , the sequences ( z in i ) li =1 and ( y in i ) li =1 are C -equivalent and each of them is C -suppression unconditional and hence 2 C -unconditional. For any scalars a , . . . , a l , θ , . . . , θ l , we calculate: (cid:13)(cid:13)(cid:13) l X i =1 a i θ i y in i (cid:13)(cid:13)(cid:13) ≥ C min ≤ i ≤ l | θ i | (cid:13)(cid:13)(cid:13) l X i =1 a i y in i (cid:13)(cid:13)(cid:13) ≥ C min ≤ i ≤ l | θ i | (cid:13)(cid:13)(cid:13) l X i =1 a i z in i (cid:13)(cid:13)(cid:13) . The other inequality is obtained identically. Therefore, any
D > K satisfies theconclusion. (cid:3) The following is another variant of Mazur’s theorem [LT, Theorem 1.a.5].
Lemma 5.15.
Let X be a separable Banach space. Let also T ij ∈ L ( X ), for1 ≤ i ≤ n , 1 ≤ j ≤ m i and c > i = 1 , . . . , n and Y finitecodimensional subspace of X , there is an x i ∈ Y with k x i k = 1 and k T ij x i k > c for all 1 ≤ j ≤ m i . Then, for every ε >
0, there exist normalized sequences ( x ik ) k , i = 1 , . . . , n , in X such that if we set Z k = span {{ x ik } ni =1 ∪ { T ij x ik } n,m i i =1 ,j =1 } , for k ∈ N , and Z = span ∪ k Z k we have that:(i) ( Z k ) k forms an FDD for the space Z , with projection constant at most 1+ ε .(ii) k T ij ( x ik ) k > c for every i = 1 , . . . , n , j = 1 , . . . , m i , and k ∈ N . Proof.
Set A = { I } ∪ { T ij } n,m i i =1 ,j =1 and, for every 1 ≤ i ≤ n , choose a normalizedvector x i in X with k T ij x i k > c for all j = 1 , . . . , m i . Assume that we have chosen( x ik ) dk =1 up to some d ∈ N , for 1 ≤ i ≤ n , so that the spaces ( Z k ) dk =1 satisfy (i)for the space they generate and (ii) for 1 ≤ k ≤ d . Choose a finite subset G ofthe unit sphere of X ∗ such that, for all x in the linear span of ∪ dk =1 Z k , we have k x k ≤ (1 + ε ) max { g ( x ) : g ∈ G } and set F = ∪ T ∈A { T ∗ g : g ∈ G } . Finally, for i = 1 , . . . , n, choose x id +1 in the unit sphere of ∩ f ∈ F ker f such that k T ij x id +1 k > c for all 1 ≤ j ≤ m i . It follows then quite easily that the sequences are as desired. (cid:3) Lemma 5.16.
Let X be a Banach space and assume that for every separable sub-space Z of X we have a difference-including collection F Z of normalized Schauderbasic sequences in Z . Let T , . . . , T l be bounded linear operators on X and assumethat there is c > Y of X andevery i = 1 , . . . , l , we have that k T i | Y k L ( Y,X ) ≥ c . Assume moreover that, for some0 < δ < c and i = 1 , . . . , l , we have T i , . . . , T im i in L ( X ) with k T ij − T i k ≤ δ for j = 1 , . . . , m i . Then, if ˜ c = c − δ , there exist a separable subspace Z of X andnormalized sequences ( z ik ) k , i = 1 , . . . , l in F Z such that(i) for any n < · · · < n l , the sequence ( z in i ) li =1 is (9 / k T ij z ik k > ˜ c/ i = 1 , . . . , n , j = 1 , . . . , m i and k ∈ N , and(iii) if y ijk = k T ij z ik k − T ij z ik then ( y ijk ) k is in F Z for i = 1 , . . . , n , j = 1 , . . . , m i . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 27
Proof.
Note that, for any finite codimensional subspace Y of X , we may choose x i in the unit ball of Y such that k T i x i k > c − ( c − δ ) /
4, which means that, for j = 1 , . . . , m i , we have k T ij x i k > c − δ ) / c/
4. Apply Lemma 5.15 to findnormalized sequences ( x ik ) k , i = 1 , . . . , n , such that k T ij x ik k > c/ k ∈ N , i = 1 , . . . , n and j = 1 , . . . , m i , and the sequence ( Z k ) k defined in Lemma 5.15 isan FDD with constant 9 /
8. Let Z = span ∪ k Z k . Choose L such that, if we set z ik = k x im k − − x im k k − ( x im k − − x im k ), then ( z ik ) k as well as ( k T ij z ik k − T ij z ik ) k are in F Z for i = 1 , . . . , n , j = 1 , . . . , m i . By the fact that ( T ij x ik ) k is 9 / (cid:13)(cid:13) T ij z ik (cid:13)(cid:13) = 1 k x im k − − x im k k (cid:13)(cid:13) T ij x im k − − T ij x im k (cid:13)(cid:13) ≥
12 19 / (cid:13)(cid:13) T ij x im k − (cid:13)(cid:13) > c/ / . The fact that statement (i) is true follows from the fact ( Z k ) k is an FDD withconstant 9 / z in i ) li =1 is a block sequence. (cid:3) S. Kakutani [Ka] proved the finite dimensional analog of the following theorem,also known as Kakutani’s Fixed Point Theorem. We present the infinite dimensionalcase by H. F. Bohnenblust and S. Karlin [BK], which as mentioned already is akey ingredient in the proof of Theorem 5.11. Recall that a multivalued mapping φ : X ։ Y between topological spaces has closed graph if for every ( x n ) n ∈ X withlim x n = x and ( y n ) n ∈ Y with y n ∈ φ ( x n ) and lim y n = y , we have that y ∈ φ ( x ). Theorem 5.17.
Let X be a Banach space, K be a nonempty compact convexsubset of X and let the multivalued mapping φ : K ։ K have closed graph andnonempty convex values. Then φ has a fixed point, i.e. there exists x ∈ X suchthat x ∈ φ ( x ). Proof of Theorem 5.11.
Let
D > K , i.e. a constant for which the conclusion ofLemma 5.14 can be applied to all families F Z . Set C = 7 D . Let W be a convexand compact subset of L ( X ), A ∈ L ( X ), and ε > x in the unitball of X , there is T ∈ W with k A ( x ) − T ( x ) k ≤ ε . We claim that there is a finitecodimensional subspace Y of X and T ∈ W such that k ( A − T ) | Y k L ( Y,X ) < Cε .Assume that the conclusion is false. Set c = Cε , δ = c/
2, and ˜ c = c − δ = C/ δ -separated subset ( T i ) li =1 of W , set η = ε/ (27 l ) and for i =1 , . . . , l choose a maximal η -separated subset ( T ij ) m i j =1 of B W ( T i , δ ) = { T ∈ W : k T i − T k ≤ δ } . Apply Lemma 5.16 to the operators A − T i and A − T ij , for i = 1 , . . . , l and j = 1 , . . . , m i , to find a separable subspace Z , normalized 9 / z ik ) k in F Z such that for i = 1 , . . . , l , j = 1 , . . . , m i if y ijk = k ( A − T ij ) z ik k − ( A − T ij ) z ik , for k ∈ N , then k ( A − T ij ) z ik k ≥ ˜ c/ y ijk ) k is in F Z . Iterate Lemma 5.14 to find an infinite subset L of N suchthat (1) is satisfied for ( z ik ) k ∈ L and ( y ij i k ) k ∈ L for all i = 1 , . . . , l and for any choiceof 1 ≤ j i ≤ m i .Fix k < · · · < k l in L and take a partition of unity f , . . . , f l of W subordinatedto T , . . . , T l . That is, f i : W → [0 ,
1] is continuous, P li =1 f i ( T ) = 1 for all T in W and f i ( T j ) = δ ij . We define a continuous mapping x : W → X given by x ( T ) = P li =1 f i ( T ) z in i (cid:13)(cid:13)(cid:13)P li =1 f i ( T ) z in i (cid:13)(cid:13)(cid:13) . Let T be an arbitrary element of W and if I T = { i = 1 , . . . , l : with k T − T i k ≤ δ } ,for i ∈ I T , choose 1 ≤ j i ≤ m i such that k T − T ij i k ≤ η . Recall that ( z in i ) i is(9 / (cid:13)(cid:13)(cid:13) l X i =1 f i ( T ) z in i (cid:13)(cid:13)(cid:13) ≥ l l X i =1 (cid:12)(cid:12) f i ( T ) (cid:12)(cid:12) = 49 l . We observe the following: k ( A − T ) x ( T ) k = 1 (cid:13)(cid:13)P i ∈ I T f i ( T ) z in i (cid:13)(cid:13) (cid:13)(cid:13)(cid:13) X i ∈ I T f i ( T ) ( A − T ) z in i (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)P i ∈ I T f i ( T ) ( A − T ij i ) z in i (cid:13)(cid:13)(cid:13)(cid:13)P i ∈ I T f i ( T ) z in i (cid:13)(cid:13) − P i ∈ I T | f i ( T ) | k T − T ij i k (cid:13)(cid:13)P i ∈ I T f i ( T ) z in i (cid:13)(cid:13) ≥ (cid:13)(cid:13)P i ∈ I T f i ( T ) ( A − T ij i ) z in i (cid:13)(cid:13)(cid:13)(cid:13)P i ∈ I T f i ( T ) x in i (cid:13)(cid:13) − η P i ∈ I T | f i ( T ) | / (9 l ) P i ∈ I T | f i ( T ) | (by (2))= (cid:13)(cid:13)P i ∈ I T f i ( T ) (cid:13)(cid:13) ( A − T ij i ) z in i (cid:13)(cid:13) y ijn i (cid:13)(cid:13)(cid:13)(cid:13)P i ∈ I T f i ( T ) x in i (cid:13)(cid:13) − lη ≥ min ≤ i ≤ l (cid:13)(cid:13) ( A − T ij i ) z in i (cid:13)(cid:13) D (cid:13)(cid:13)P i ∈ I T f i ( T ) x in i (cid:13)(cid:13)(cid:13)(cid:13)P i ∈ I T f i ( T ) x in i (cid:13)(cid:13) − lη ≥ ˜ c D − lη D D ε − ε = 1312 ε. (3)We now define a multivalued mapping φ : W ։ W with φ ( T ) = { S ∈ W : k ( A − S ) x ( T ) k ≤ ε } . By assumption, the values of φ are non-empty and they are also closed and convex.It also easily follows that φ has a closed graph. Hence, from Theorem 5.17, thereexists T ∈ W with T ∈ φ ( T ), i.e., k ( A − T ) x ( T ) k ≤ ε . This contradicts (3) whichcompletes the proof. (cid:3) The following lemma shows that if X is a Banach space with a shrinking FDDthat satisfies the UALS property, the finite codimensional subspaces of X on whichthe approximations happen, can be assumed to be tail subspaces. Lemma 5.18.
Let X be a Banach space with a shrinking FDD ( X n ) n and Y bea finite codimensional subspace of X . Then, for every ε >
0, there exists a tailsubspace Z of X such that B Z ⊂ B Y + εB X . Proof.
Let x , . . . , x n ∈ B X with X = Y ⊕ span { x , . . . , x n } and x ∗ , . . . , x ∗ n ∈ X ∗ be such that x ∗ i ( x j ) = δ ij for every 1 ≤ i, j ≤ n . Notice that Y = ∩ ni =1 ker x ∗ i . Since( X n ) n is a shrinking FDD, we may choose n ∈ N such that k x ∗ i − P ∗ n ( x ∗ i ) k < ε/l k x i k for all 1 ≤ i ≤ n , and set Z = span ∪ n>n X n . Pick a z in the unit ball of Z and set x = P li =1 x ∗ i ( z ) x i /ε . Then | x ∗ i ( z ) | < ε/l k x i k and x ∗ i ( x ) = x ∗ i ( z ) /ε for all 1 ≤ i ≤ n .Hence k x k < z − εx ∈ ∩ ni =1 ker x ∗ i , from which follows that z ∈ B Y +2 εB X . (cid:3) The next example demonstrates a shrinking FDD is necessary above to assumethat the uniform approximation happens on tail subspaces. Let us recall that thebasis of ℓ is not shrinking. Example 5.19.
Let ( e n ) n denote the unit vector basis of ℓ and consider theoperator A : ℓ → ℓ with A (cid:0) ( x n ) n (cid:1) = ∞ X n =1 x n − e + ∞ X n =1 x n e and for z ∈ ℓ , the operators B + z , B − z : ℓ → ℓ with B + z (cid:0) ( x n ) n (cid:1) = ∞ X n =1 x n z and B − z (cid:0) ( x n ) n (cid:1) = (cid:16) ∞ X n =1 x n − − ∞ X n =1 x n (cid:17) z. Set W = co (cid:8) B ± z : z ∈ span { e , e } and k z k ≤ (cid:9) .Let x ∈ ℓ with k x k ≤ A ( x ) = a e + a e , where a = P ∞ n =1 x n − and a = P ∞ n =1 x n . Suppose that A ( x ) = 0 and set a = max {| a + a | , | a − a |} . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 29
Notice that a = | a | + | a | . If a = | a + a | , setting z = a + a A ( x ), we have that k z k = 1 and B + z ( x ) = A ( x ). If a = | a − a | , then the same hold for z = a − a A ( x ).Hence, for every x ∈ ℓ with k x k ≤
1, there is a B ∈ W such that k ( A − B ) x k = 0.Pick any B ∈ W and n ∈ N . Then there exists a convex combination in W such that B = P ni =1 a i B + y i + P mi =1 b i B − z i and, for every k ∈ N , we have that B ( e k − ) = P ni =1 a i y i + P mi =1 b i z i and B ( e k ) = P ni =1 a i y i − P mi =1 b i z i and hence (cid:13)(cid:13)(cid:13)(cid:0) A − B (cid:1) e k − + e k (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) e + e − n X i =1 a i y i (cid:13)(cid:13)(cid:13) ≥ − n X i =1 a i . Similarly, k ( A − B ) e k − − e k k ≥ P ni =1 a i and thus, for any k ∈ N with n ≤ k − k ( A − B ) e k − + e k k ≥ / k ( A − B ) e k − − e k k ≥ /
2. Therefore, weconclude that k ( A − B ) | span { e n : n ≥ n } k ≥ / x , in the unit ball of ℓ there exists a B ∈ W such that k A ( x ) − B ( x ) k = 0.5.3. The UALS property and duality.
We make a connection between theUALS property of a space and its dual. In particular, for reflexive spaces with anFDD we show that the UALS for X is equivalent to the UALS for X ∗ . We also showthat if X has an FDD and X ∗ has a unique l -joint spreading model with respect toa difference-including family, then X must satisfy the UALS as well. This allowsus to show indirectly that certain spaces, such as L ∞ spaces with separable dual,satisfy the UALS. Proposition 5.20.
Let X be a Banach space, A ∈ L ( X ) and W be a convexand WOT-compact subset of L ( X ). If there is an ε > W ε -pointwiseapproximates A , then the set W ∗ = { T ∗ : T ∈ W } ε -pointwise approximates A ∗ . Proof.
If we assume that the conclusion is false, then there exists x ∗ in the unitsphere of X ∗ and δ > W ∗ x ∗ = { T ∗ x ∗ : T ∈ W } , then we have thatdist( A ∗ x ∗ , W ∗ x ∗ ) ≥ ε + δ . As W ∗ x ∗ is a convex and w ∗ -compact subset of X ∗ ,a separation theorem yields that there exists x in the unit sphere of X such that x ( A ∗ x ∗ ) + ( ε + δ/ ≤ inf T ∈ W x ( T ∗ x ∗ ) or k Ax − T x k ≥ x ∗ ( T x − Ax ) ≥ ε + δ/ T ∈ W . (cid:3) Remark 5.21.
The compactness of W is necessary in Proposition 5.20. To seethis, consider the case when X = ℓ , A is the identity operator, and W is the closedconvex hull of all natural projections onto finite subsets of N with respect to theunit vector basis.We state the main results and prove them afterwards. Theorem 5.22.
Let X be a reflexive Banach space with an FDD. Then X satisfiesthe UALS if and only if X ∗ does. Theorem 5.23.
Let X be a Banach space with an FDD. Assume that thereexist a uniform constant C > Z of X ∗ , adifference-including family F Z of normalized sequences in X ∗ such that Z admitsa C -uniformly unique l -joint spreading model with respect to F Z . Then X satisfiesthe UALS property.Recall that results from [H], [HS], [LS], and [S] yield that if X is an infinitedimensional L ∞ -space with separable dual then X ∗ is isomorphic to ℓ . Also thisis the case if and only if ℓ is not isomorphic to a subspace of X . As it was provenin [FOS], every Banach space with separable dual embeds in a L ∞ space withseparable dual. Corollary 5.24.
Every L ∞ -space with separable dual satisfies the UALS property.In particular:(i) every hereditarily indecomposable L ∞ -space satisfies the UALS,(ii) every Banach space with separable dual embeds in a space that satisfies theUALS, and(iii) for every countable compact metric space K , the space C ( K ) satisfies theUALS. Corollary 5.25. If X is Banach space such that X ∗ is an Asymptotic ℓ p space, forsome 1 ≤ p ≤ ∞ , then every quotient of X with an FDD satisfies the UALS. Lemma 5.26.
Let X be a Banach space, let R : X → X be a finite rank operator,and let Q = I − R . If T is in L ( X ), then there exists a subspace Y of X of finitecodimension such that k T | Y k L ( Y,X ) ≤ k QT k . Proof.
Since RT is a finite rank operator the subspace Y = ker RT is of finitecodimension. Then, k T | Y k ≤ k RT | Y k + k QT | Y k ≤ k QT k . (cid:3) Proof of Theorem 5.22.
It is clearly enough to show one implication. Let us assumethat X ∗ satisfies the UALS with constant C > A ∈ L ( X ) and W be acompact and convex subset of L ( X ) that ε -approximates A . Then by Proposition5.20 we have that the set W ∗ = { T ∗ : T ∈ W } ε -approximates A ∗ and so thereexists a subspace Z of X ∗ of finite codimension such that k ( T ∗ − A ∗ ) | Z k ≤ Cε . ByLemma 5.18, and perhaps some additional error, we may assume that Z is a tailsubspace with an associated projection Q ∗ n and hence k Q n ( T − A ) k = k ( T ∗ − A ∗ ) Q ∗ n k ≤ C k Q n k ε. Applying Lemma 5.26, we may find a subspace Y of X of finite codimension suchthat k ( T − A ) | Y k ≤ k Q n ( T − A ) k and therefore k ( T − A ) | Y k ≤ C k Q n k ε . (cid:3) Lemma 5.27.
Let X be a Banach space with a bimonotone FDD and let ( Q n ) n denote the basis tail projections (i.e. Q n = I − P n , for all n ∈ N ). Assume thatfor every separable subspace Z of X ∗ we have a difference-including collection F Z of normalized Schauder basic sequences in Z . Let T , . . . , T l be bounded linearoperators on X and assume that there is c > n ∈ N andevery i = 1 , . . . , l , we have that k T ∗ i Q ∗ n k ≥ c . Assume moreover that, for some0 < δ < c and every i = 1 , . . . , l , we have T i , . . . , T im i in L ( X ) with k T ij − T i k ≤ δ for j = 1 , . . . , m i . Then, if ˜ c = c − δ , there exist a separable subspace Z of X ∗ andnormalized sequences ( z ik ) k , i = 1 , . . . , l in F Z such that(i) for any n < · · · < n l , the sequence ( z in i ) li =1 is (9 / k T ∗ ij z ik k > ˜ c/ i = 1 , . . . , n , j = 1 , . . . , m i and k ∈ N , and(iii) if y ijk = k T ∗ ij z ik k − T ∗ ij z ik , then ( y ijk ) k is in F Z for i = 1 , . . . , n , j = 1 , . . . , m i . Proof.
For every i = 1 , . . . , l , we choose a normalized sequence ( x i ∗ n ) n such that k T ∗ i Q ∗ n x i ∗ n k ≥ c − ( c − δ ) /
4. Since the FDD is bimonotone we may assume thatmin supp( x i ∗ n ) > n for all n ∈ N and 1 ≤ i ≤ l and that k T ∗ i x i ∗ n k ≥ c − ( c − δ ) / x i ∗ n ) n , ( T ∗ x i ∗ n ) n , for 1 ≤ i ≤ l , are w ∗ -null (by using w ∗ -continuity). We may now apply reasoning identical to that used in Lemmas 5.15and 5.16 to achieve the desired conclusion. (cid:3) Proof of Theorem 5.23.
We renorm the space X so that its FDD is bimonotone. Let D > K , i.e. a constant for which the conclusion of Lemma 5.14 can be applied toall families F Z for all separable subspaces Z of X ∗ . Set C = 14 D . We will show that X satisfies the UALS with constant C . Let A ∈ L ( X ) and W be a convex compactsubset of L of X that ε -approximates A . It is sufficient to find T ∈ W and n ∈ N such that k ( T ∗ − A ∗ ) Q ∗ n k < Cε . Indeed, then k Q n ( A − T ) k = k ( T ∗ − A ∗ ) Q ∗ n k < Cε and by Lemma 5.26 we will be done. If we assume that the conclusion is false, we OINT SPREADING MODELS AND UNIFORM APPROXIMATION 31 may follow the proof of Theorem 5.11 to the letter, only replacing Lemma 5.16 withLemma 5.27, to reach the desired conclusion. (cid:3)
Spaces Failing the UALS Property.
In this section we present an arche-typal example of a reflexive Banach space X that fails the UALS and admits aunique spreading model isometric to ℓ . The proof that X fails the property isbased on the the fact that it does not admit a uniformly unique joint spreadingmodel. This reasoning may then be modified and utilized to show that classicalspaces such as L p [0 , ≤ p ≤ ∞ and p = 2, and C ( K ), for uncountable compactmetric spaces K , fail the UALS. Definition 5.28.
For each n ∈ N , we set X n = ( P ni =1 ⊕ ℓ ) and Y n = ( P ni =1 ⊕ ℓ ) ∞ and let X = ( P ⊕ X n ⊕ Y n ) .For a vector x in X , we write x = P ∞ n =1 x n + y n to mean that x n ∈ X n and y n ∈ Y n and x n = P nj =1 x n ( j ) , y n = P nj =1 y n ( j ) to denote the coordinates of each x n and y n with respect to the natural decomposition of X n and Y n respectively.Under this notation we compute the norm of x as follows: k x k = ∞ X n =1 (cid:18) n X j =1 (cid:13)(cid:13) x n ( j ) (cid:13)(cid:13) (cid:19) + (cid:18) max ≤ j ≤ n (cid:13)(cid:13) y n ( j ) (cid:13)(cid:13) (cid:19) . By taking an orthonormal basis for each corresponding ℓ -component of the space X n as well as of the space Y n and taking a union over all n ∈ N and for j = 1 , . . . , n ,we obtain a 1-unconditional basis for the space X . Henceforth, when we say ( x k ) k is a block sequence in X , it will be understood that this is with respect to a fixedenumeration of the aforementioned basis. Proposition 5.29.
The space X fails the UALS property. Proof.
Assume that X satisfies the UALS with constant C > n ∈ N with C/n < . For G ⊂ { , . . . , n } , consider the bounded operator I G : X n → Y n with I G ( P ni =1 x i ) = P i ∈ G x i and set A n = I { ,..., n } and W n = co { I G : G = n } .Let x ∈ X n with x = P ni =1 x i and k x k = 1, that is P ni =1 k x i k = 1, and σ be apermutation of { , . . . , n } such that k x σ (1) k ≥ . . . ≥ k x σ (2 n ) k . Then notice that k x σ ( n +1) k ≤ n +1 and hence k A n ( x ) − I G ( x ) k ≤ n +1 , for G = { σ (1) , . . . , σ ( n ) } .The basis ( e n ) n of X is shrinking, since X is reflexive, and therefore Lemma 5.18yields a tail subspace Y = span { e n : n ≥ n } of X such that k ( A n − B ) | Y k < C/n for some B ∈ W . Then B is a convex combination B = P ki =1 λ i I G i and wehave that R P ki =1 λ i χ G i = , where the integral is with respect to the normalizedcounting measure on { , . . . , n } . Hence there exists a 1 ≤ j ≤ n such that P ki =1 λ i χ G i ( j ) ≤ . Pick any x ∈ X n ( j ) with k x k = 1 and supp( x ) ≥ n and noticethat k A n ( x ) − B ( x ) k ≥ − P ki =1 λ i χ G i ( j ). Thus k ( A n − B ) | Y k ≥ , which is acontradiction. (cid:3) The space X is a first example of a space failing the UALS property. As weshow next, it admits a uniformly unique spreading model while it fails to admit auniformly unique l -joint spreading model. We start with the following lemmas. Lemma 5.30.
Let ( x k ) k be a block sequence in X with x k = P n n = n x kn + y kn and assume that k x k n ( j ) k = k x k n ( j ) k and k y k n ( j ) k = k y k n ( j ) k for every k , k ∈ N , n ≤ n ≤ n and 1 ≤ j ≤ n . Set ε = k x k k , for k ∈ N . Then, for all m ∈ N and λ , . . . , λ m ∈ R , we have that k P mk =1 λ k x k k = ε ( P mk =1 λ k ) . Proof.
Let k ∈ N . For every k ∈ N and n ≤ n ≤ n , since ( x k ) k is block, we havethat (cid:13)(cid:13)(cid:13) m X k =1 λ k x kn ( j ) (cid:13)(cid:13)(cid:13) = m X k =1 λ k (cid:13)(cid:13) x kn ( j ) (cid:13)(cid:13) ! = (cid:13)(cid:13) x k n ( j ) (cid:13)(cid:13) m X k =1 λ k ! . We thus calculate(4) (cid:13)(cid:13)(cid:13) m X k =1 λ k x kn (cid:13)(cid:13)(cid:13) = n X j =1 (cid:13)(cid:13)(cid:13) m X k =1 λ k x kn ( j ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) x k n (cid:13)(cid:13) m X k =1 λ k ! and similarly(5) (cid:13)(cid:13)(cid:13) m X k =1 λ k y kn (cid:13)(cid:13)(cid:13) = max ≤ j ≤ n (cid:13)(cid:13)(cid:13) m X k =1 λ k y kn ( j ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) y k n (cid:13)(cid:13) m X k =1 λ k ! . Finally, using (4) and (5), we conclude that (cid:13)(cid:13)(cid:13) m X k =1 λ k x k (cid:13)(cid:13)(cid:13) = n X n = n (cid:18)(cid:13)(cid:13)(cid:13) m X k =1 λ k x kn (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) m X k =1 λ k y kn (cid:13)(cid:13)(cid:13) (cid:19) = m X k =1 λ k n X n = n (cid:16)(cid:13)(cid:13) x k n (cid:13)(cid:13) + (cid:13)(cid:13) y k n (cid:13)(cid:13) (cid:17) = m X k =1 λ k (cid:13)(cid:13) x k (cid:13)(cid:13) . (6) (cid:3) Lemma 5.31.
Let ( n k ) k ≥ be an increasing sequence of naturals and ( x k ) k be ablock sequence in X such that(i) There exist c , c > c ≤ k x k k ≤ c for every k ∈ N .(ii) x k = P n n = n ( x kn + y kn ) + P n k +1 n = n k +1 ( x kn + y kn ) for every k ∈ N .(iii) k x k n ( j ) k = k x k n ( j ) k and k y k n ( j ) k = k y k n ( j ) k for every k , k ∈ N , n ≤ n ≤ n and 1 ≤ j ≤ n .Then, for all m ∈ N and λ , . . . , λ m ∈ R , we have that c m X k =1 λ k ! ≤ (cid:13)(cid:13)(cid:13) m X k =1 λ k x k (cid:13)(cid:13)(cid:13) ≤ c m X k =1 λ k ! . Proof.
Using (6), we have that (cid:13)(cid:13)(cid:13)(cid:13) m X k =1 λ k x k (cid:13)(cid:13)(cid:13)(cid:13) = n X n = n (cid:18)(cid:13)(cid:13)(cid:13) m X k =1 λ k x kn (cid:13)(cid:13)(cid:13) + (cid:13)(cid:13)(cid:13) m X k =1 λ k y kn (cid:13)(cid:13)(cid:13) (cid:19) + m X k =1 n k +1 X n = n k +1 (cid:18)(cid:13)(cid:13) λ k x kn (cid:13)(cid:13) + (cid:13)(cid:13) λ k y kn (cid:13)(cid:13) (cid:19) = m X k =1 λ k n X n = n (cid:16)(cid:13)(cid:13) x kn (cid:13)(cid:13) + k y kn (cid:13)(cid:13) (cid:17) + m X k =1 λ k n k +1 X n = n k +1 (cid:16)(cid:13)(cid:13) x kn (cid:13)(cid:13) + k y kn (cid:13)(cid:13) (cid:17) = m X k =1 λ k (cid:18) n X n = n (cid:16)(cid:13)(cid:13) x kn (cid:13)(cid:13) + k y kn (cid:13)(cid:13) (cid:17) + n k +1 X n = n k +1 (cid:16)(cid:13)(cid:13) x kn (cid:13)(cid:13) + k y kn (cid:13)(cid:13) (cid:17)(cid:19) = m X k =1 λ k (cid:13)(cid:13) x k (cid:13)(cid:13) which, due to (i), yields the desired result. (cid:3) Proposition 5.32.
Let ( x k ) k be a normalized block sequence in X . For every ε > x k ) k has a subsequence ( x k i ) i such that for every m ∈ N and λ , . . . , λ m ∈ R (1 − ε ) m X i =1 λ i ! ≤ (cid:13)(cid:13)(cid:13) m X i =1 λ i x k i (cid:13)(cid:13)(cid:13) ≤ (1 + ε ) m X i =1 λ i ! . OINT SPREADING MODELS AND UNIFORM APPROXIMATION 33
Proof.
We choose L ∈ [ N ] ∞ such that lim k ∈ L k x kn ( j ) k = a n,j and lim k ∈ L k y kn ( j ) k = b n,j for all n ∈ N and 1 ≤ j ≤ n . Set lim k ∈ L k x kn k = a n and lim k ∈ L k y kn k = b n .Since P ∞ n =1 k x kn k + k y kn k ≤ k ∈ N , then P ∞ n =1 a n + b n ≤ ε i ) i and ( δ i ) i be sequences of positive reals such that P ∞ i =1 ε i < ε and P ∞ i =1 δ i < ε . We then choose, by induction, ( n i ) i ⊂ N and ( k i ) i ⊂ L increasingsequences such that the following hold for every i ∈ N .(i) n i > max { n : x k i − n = 0 or y k i − n = 0 } when i > P n>n i a n + b n < ε i .(iii) P n i n =1 P nj =1 (cid:12)(cid:12) k x k i n ( j ) k − a n,j (cid:12)(cid:12) + (cid:12)(cid:12) k y k i n ( j ) k − b n,j (cid:12)(cid:12) < δ i .For each i ∈ N , due to (iii), we may assume that k x k i n ( j ) k = a n,j and k y k i n ( j ) k = b n,j for all 1 ≤ n ≤ n i and 1 ≤ j ≤ n , with an error δ i . Then Lemma 5.30 yields that (cid:13)(cid:13)(cid:13) m X i =2 m X j = i λ j n i X n = n i − +1 (cid:0) x k j n + y k j n (cid:1)(cid:13)(cid:13)(cid:13) ≤ m X i =2 ε i − m X j = i λ j ! + m X i =2 δ i m X j = i λ j ! . Hence, applying Lemma 5.31, we calculate (cid:13)(cid:13)(cid:13) m X i =1 λ i x k i (cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13) m X i =1 λ i (cid:16) n X n =1 (cid:0) x k i n + y k i n (cid:1) + n i +1 X n = n i +1 (cid:0) x k i n + y k i n (cid:1)(cid:17)(cid:13)(cid:13)(cid:13) − √ ε m X i =1 λ i ! ≥ √ − ε m X i =1 a i ! − √ ε m X i =1 a i ! and (cid:13)(cid:13)(cid:13) m X i =1 λ i x k i (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) m X i =1 λ i (cid:16) n X n =1 (cid:0) x k i n + y k i n (cid:1) + n i +1 X n = n i +1 (cid:0) x k i n + y k i n (cid:1)(cid:17)(cid:13)(cid:13)(cid:13) + 2 √ ε m X i =1 λ i ! ≤ √ ε m X i =1 λ i ! + 2 √ ε m X i =1 λ i ! . (cid:3) Corollary 5.33.
Every spreading model generated by a basic sequence in X isisometric to ℓ and hence X admits a uniformly unique spreading model with respectto F ( X ). Remark 5.34.
Using similar arguments we may show that every l -joint spreadingmodel generated by a basic sequence in X is isomorphic to ℓ , while this does nothappen with a uniform constant and as already shown X fails the UALS property.This shows a strong connection between the UALS and spaces with uniformly uniquejoint spreading models, which fails when the space only admits a uniformly uniquespreading model.As mentioned in Section 3, the space from [AM1] is another example of a spacethat admits a uniformly unique spreading model and fails to have a uniform constantfor which all of its l -joint spreading models, for every l ∈ N , are equivalent. Thisspace however satisfies the stronger property that every one of its subspaces doesnot admit a uniformly unique l -joint spreading model, contrary to the space X which contains ℓ .Motivated by the definition of space X , we modify the above arguments to showthat every L p [0 , ≤ p ≤ ∞ , as well as the space C ( K ) for an uncountablecompact metric space K fail the UALS. Proposition 5.35.
For every 1 < p < q < ∞ , the spaces ( P ⊕ ℓ p ) q and ( P ⊕ ℓ q ) p fail the UALS property. Proof.
For each n ∈ N , we set X n = ( P ni =1 ⊕ ℓ p ) p and Y n = ( P ni =1 ⊕ ℓ p ) q and X = ( P X n ⊕ Y n ) q . Assume that X satisfies the UALS property with constant C > n ∈ N with C/n r < , where r = ( q − p ) /pq .For every G ⊂ { , . . . , n } , consider the operator I G : X n → Y n such that I G ( P ni =1 a i x i ) = P i ∈ G a i x i and set A n = I { ,..., n } and W n = co { I G : G = n } .Let x ∈ X n with x = P ni =1 x i and P ni =1 k x i k p = 1 and let σ be a permutation of { , . . . , n } such that k x σ (1) k p ≥ . . . ≥ k x σ (2 n ) k p . Hence for G = { σ (1) , . . . , σ ( n ) } we have that k A n ( x ) − I G ( x ) k < /n r and using the same arguments as in the proofof Proposition 5.29, we derive a contradiction. The case of ( P ⊕ ℓ q ) p is similar. (cid:3) Remark 5.36.
It is immediate that if some infinite dimensional complementedsubspace of Banach space X fails the UALS property, then the same holds for X . Proposition 5.37.
The space L p [0 , < p < ∞ and p = 2, fails the UALSproperty. Proof.
Recall that, as follows from Khintchine’s inequality, ℓ embeds isomorphi-cally as a complemented subspace into L p [0 , < p < ∞ . If p >
2, foreach n ∈ N set X n = ( P ni =1 ⊕ ℓ ) and Y n = ( P ni =1 ⊕ ℓ ) p and if p < X n = ( P ni =1 ⊕ ℓ ) p and Y n = ( P ni =1 ⊕ ℓ ) . Then, following the proof of Proposition5.29, we may show that the space X = ( P ⊕ X n ⊕ Y n ) p fails the UALS and since it iscomplemented into L p [0 ,
1] = ( P ⊕ L p [0 , p , the latter also fails the property. (cid:3) Proposition 5.38.
The space L [0 ,
1] fails the UALS property.
Proof.
Assume that L [0 ,
1] satisfies the UALS with constant
C > n ∈ N with C/n < . Set X n = ( P ni =1 ⊕ ℓ ) and Y n = ( P ni =1 ⊕ ℓ ) . Since ℓ is isometricto a complemented subspace of L [0 , P ⊕ X n ) . More-over, Khintchine’s inequality yields that ℓ embeds isomorphically into L [0 ,
1] andhence so does ( P ⊕ Y n ) .For every G ⊂ { , . . . , n } , consider the operator I G : X n → Y n such that I G ( P ni =1 a i x i ) = P i ∈ G a i x i and set A n = I { ,..., n } and W n = co { I G : G = n } .As above, for all x ∈ X n with k x k ≤
1, we may find G ⊂ { , . . . , n } such that k A n ( x ) − I G ( x ) k < /n . Let B = P ki =1 λ i I G i in W n and Y be a finite codimensionalsubspace of L [0 ,
1] such that k ( A n − B ) | Y k < C/n and choose, as in the proof ofProposition 5.29, 1 ≤ j ≤ n with P ki =1 λ i χ G i ( j ) ≤ . Let x ∗ , . . . , x ∗ l ∈ L ∞ [0 , Y = ∩ li =1 ker x ∗ i . Denote by ( e m ) m the basis of X n ( j ) and choose M ∈ [ N ] ∞ such that ( x ∗ i ( e m )) m ∈ M converges for all 1 ≤ i ≤ l . Using Lemma 2.1 we choose m , m ∈ M such that d ( x, Y ) < , for x = ( e m − e m ) /
2. Then k A n ( x ) − B ( x ) k ≥ and hence k ( A n − B ) | Y k ≥ , which is a contradiction. (cid:3) Proposition 5.39.
The space L ∞ [0 ,
1] fails the UALS property.
Proof.
Fix n ∈ N . The σ -algebra B [0 ,
1] of all Borel sets of [0 ,
1] is homeomorphic tothat of [0 , n and hence L ∞ [0 ,
1] is isometric to L ∞ [0 , n . For 1 ≤ i ≤ n , denoteby B i the σ -algebra generated by { B ∈ Q ni =1 B [0 ,
1] : B j = [0 ,
1] for j > i } and for f ∈ L ∞ [0 , n set E i ( f ) = E [ f |B i ] and consider the operator ∆ i : L ∞ [0 , n → L ([0 , i , ⊗ j ≤ i λ ) with ∆ i ( f ) = E i ( f ) − E i − ( f ), where E ( f ) = 0 and λ denotesthe Lebesgue measure on [0 , G ⊂ { , . . . , n } , let ∆ G : L ∞ [0 , n → ( P ni =1 ⊕ L ([0 , i , ⊗ j ≤ i λ )) ∞ with ∆ G = P i ∈ G ∆ i and set A n = ∆ { ,..., n } and W n = co { ∆ G : G = n } .Observe that ( P ni =1 ⊕ L ([0 , i , ⊗ j ≤ i λ )) ∞ embeds isometrically into L ∞ [0 , n andhence we have that ∆ G : L ∞ [0 , → L ∞ [0 , f ∈ L ∞ [0 , n and notice that( E i ( f )) ni =1 is a martingale, since B i is a subalgebra of B j for every 1 ≤ i < j ≤ n .Then for the martingale differences (∆ i ( f )) ni =1 the Burkholder inequality [B] yields OINT SPREADING MODELS AND UNIFORM APPROXIMATION 35 a c > Z
10 2 n X i =1 ∆ i ( f ) ! ≤ c k f k . Claim 1 : For every ε >
0, there exists n ∈ N such that, for every n ≥ n and f ∈ L ∞ [0 , n , there is a B ∈ W n such that k ( A n − B ) f k ≤ ε k f k . Proof of Claim 1.
Pick n ∈ N such that c / √ n < ε . Let n ≥ n and f in L ∞ [0 , n with k f k = 1. Then, as a direct consequence of (7), we have that { i : k ∆ i ( f ) k > c / √ n + 1 } ≤ n . Let σ be permutation of { , . . . , n } such that k ∆ σ (1) ( f ) k ≥ . . . ≥ k ∆ σ (2 n ) ( f ) k . Hence for G = { σ (1) , . . . , σ ( n ) } , we concludethat k ( A n − ∆ G ) f k < c / √ n and this yields the desired result. Claim 2 : For every n ∈ N , every finite codimensional subspace Y of L ∞ [0 , n and B ∈ W n , we have that k ( A − B ) | Y k ≥ / Proof of Claim 2.
There exist x ∗ , . . . , x ∗ l ∈ ( L ∞ [0 , n ) ∗ with Y = ∩ li =1 ker x ∗ i andalso B is a convex combination B = P ki =1 λ i ∆ G i in W n . Then, as in Proposi-tion 5.29, choose 1 ≤ j ≤ n with P ki =1 λ i χ G i ( j ) ≤ . Denote by ( R m ) m theRademacher system and consider a natural extension ( ˜ R m ) m into L ∞ [0 , n suchthat ˜ R m ( t , . . . , t n ) = R m ( t j ). We choose M ∈ [ N ] ∞ such that ( x ∗ i ( ˜ R m )) m ∈ M converges for all 1 ≤ i ≤ l , and applying Lemma 2.1 we may find m , m ∈ M suchthat d ( f, Y ) < /
8, for f = ( ˜ R m − ˜ R m ) /
2. We recall that ( ˜ R m ) m is isomorphicto the unit vector basis of ℓ in the L ∞ -norm and hence k f k ∞ = 1. Notice that,for every m ∈ M , we have that ∆ i ( ˜ R m ) = δ ij ˜ R m and k ˜ R m k = 1 and since ( R m ) m are orthogonal, k ˜ R m − ˜ R m k = ( k ˜ R m k + k ˜ R m k ) . Hence k ( A n − B ) f k ≥ / k ( A n − B ) | Y k ≥ /
9, since d ( f, Y ) < / L ∞ [0 ,
1] satisfies the UALS with constant
C > ε >
Cε < /
9. The first claim yields an n ∈ N such that, for every f ∈ L ∞ [0 , n with k f k ≤
1, there exists B ∈ W n with k ( A n − B ) f k < ε . Hence thereexist a subspace Y of L ∞ [0 , n of finite codimension and a B ∈ W n such that k ( A − B ) | Y k < Cε and this contradicts our second claim, since Cε < / (cid:3) Proposition 5.40.
Let K be an uncountable compact metrizable space. Then thespace C ( K ) fails the UALS property. Proof.
We set Ω = {− , } N and Milutin’s Theorem [M] yields that the space C ( K )is isomorphic to C (Ω) for every K uncountable compact metrizable. We now fix n ∈ N , consider a partition of N into disjoint infinite sets N , . . . , N n and setΩ i = {− , } N i , for 1 ≤ i ≤ n . Clearly C (Ω) is isometric to C ( Q ni =1 Ω i ).In a similar manner as in the previous proposition, for every 1 ≤ i ≤ n , wedefine E i , ∆ i : C ( Q ni =1 Ω i ) → L ( Q j ≤ i Ω j , ⊗ j ≤ i µ j ), where by µ j we denote theHaar probability measure on Ω j . Moreover, for every G ⊂ { , . . . , n } , we define theoperator ∆ G : C ( Q ni =1 Ω i ) → ( P ni =1 L ( Q j ≤ i Ω j , ⊗ j ≤ i µ j )) ∞ with ∆ G = P i ∈ G ∆ i .Observe that ( P ni =1 L ( Q j ≤ i Ω j , ⊗ j ≤ i µ j )) ∞ is isometric to a subspace of C (Ω) andhence ∆ G : C (Ω) → C (Ω). Also set A n = ∆ { ,..., n } and W n = co { ∆ G : G = n } .The family ( π n ) n of the projections of Ω onto its coordinates corresponds tothe Rademacher system in L ∞ [0 , C (Ω) satisfies theUALS property, we arrive at a contradiction applying the corresponding argumentsof Proposition 5.39. (cid:3) Final Remarks.
This last subsection contains some final remarks and openproblems concerning the UALS property. We start with the following examplesuggested by W. B. Johnson which shows that in the definition of the UALS, wecannot expect the uniform approximation to happen on the whole space.
Example 5.41.
Let k · k be a norm on R and for x, x ∗ ∈ R define the operator x ∗ ⊗ x : R → R with x ∗ ⊗ x ( y ) = x ∗ ( y ) x and set W = co { x ∗ ⊗ x : x, x ∗ ∈ R and k x k , k x ∗ k ≤ } . Let y ∈ R with k y k ≤ x ∗ ∈ R with k x ∗ k = 1 such that x ∗ ( y ) = k y k . Thenfor x = y/ k y k , we have that x ∗ ⊗ x ∈ W and k x ∗ ⊗ x ( y ) − I ( y ) k = 0, where I denotes the identity operator.For any B ∈ W , there exists a convex combination P i =1 a i B i in W such that B = P i =1 a i B i . Then a i ≥ / ≤ i ≤
5, and for x ∈ ker B i with k x k = 1 we have that k x − P i =1 a i B i ( x ) k ≥ − P i =1 a i . Hence k I − B k ≥ / B ∈ W .This example is extended to every Banach space with dimension greater thantwo, by the following easy modification. Proposition 5.42.
Let X be a Banach space with dim X ≥
2. There exist
C > W of L ( X ) with the property that, for every x ∈ B X ,there exists a B ∈ W such that k x − B ( x ) k = 0 whereas k I − B k ≥ C for all B ∈ W ,where I : X → X denotes the identity operator. Proof.
Let e , e be linearly independent vectors in X , denote by Y their linearspan and let Z be a subspace of X such that X = Y ⊕ Z . Set W = co { x ∗ ⊗ x | Y + I | Z : x, x ∗ ∈ Y and k x k , k x ∗ k ≤ } and notice that, using similar arguments to those in the previous example, we obtainthe desired result. (cid:3) Remark 5.43.
I. Gasparis pointed out that in the case of c , the UALS can beproved without the use of Kakutani’s theorem. This is a consequence of the followingfact. Let T ∈ L ( c ) and ( x in ) n , 1 ≤ i ≤ l , be normalized block sequences such thatfor some ε >
0, we have that k T ( x n ) k ≥ ε for all n ∈ N . Then, for every δ >
0, thereexists a choice n < . . . < n l such that k T ( P li =1 x in i ) k > ε − δ . Assume now that T , . . . , T l ∈ L ( c ) and ε >
0, such that for every x in the unit ball of c , there exists1 ≤ i ≤ l such that k T i ( x ) k ≤ ε . Then, for every ε ′ > ε , there exist 1 ≤ i ≤ l and n ∈ N such that k T i | span { e n : n ≥ n } k ≤ ε ′ . If not, we may choose for each i = 1 , . . . , l a normalized block sequence ( x in ) n such that k T i ( x in ) k ≥ ε ′ for all n ∈ N . Thenapplying simultaneously the above observation, for the operators T , . . . , T l , we mayselect n < . . . < n l such that k T i ( P li =1 x in i ) k > ε for all i = 1 , . . . , l , and this yieldsa contradiction. Remark 5.44.
There exist Banach spaces which satisfy the UALS while this is nottrue for all of their subspaces. As already shown, every L p [0 ,
1] for 1 < p < ∞ and p = 2, fails the UALS whereas item (ii) of Corollary 5.24 yields that it embeds in aspace satisfying the property.Another open problem in a similar context as the above remark is the following.Notice that all spaces in the previous subsection failing the UALS contain a subspacewhich satisfies the property. Problem 2.
Does there exist a Banach space that none of its subspaces satisfy theUALS property?
OINT SPREADING MODELS AND UNIFORM APPROXIMATION 37
References [AI] I. Ameniya and T. Ito,
Weakly null sequences in James spaces on trees , Kodai Math. J. 4(1981), 418–425.[Ar et al. ] S. A. Argyros, D. Freeman, R. Haydon, E. Odell, Th. Raikoftsalis, Th. Schlumprechtand D. Zisimopoulou,
Embedding uniformly convex spaces into spaces with very few opera-tors , Journal of Functional Analysis 262 (2012), 825–849.[AH] S. A. Argyros and R. G. Haydon,
A hereditarily indecomposable L ∞ space that solves thescalar-plus-compact-problem , Acta Math. 206 (1) (2011), 1–54.[AKT] S. A. Argyros, V. Kanellopoulos and K. Tyros, Finite order spreading models , Advancesin Mathematics 234 (2013), 574–617.[AM2] S. A. Argyros and P. Motakis,
A dual method of constructing hereditarily indecomposableBanach spaces , Positivity 20 (2016),625–662.[AM1] S. A. Argyros and P. Motakis,
On the complete separation of asymptotic structures inBanach spaces , arXiv:1902.10092 (preprint).[BL] B. Beauzamy and J.T. Laprest´e,
Mod`eles ´etal´es des espaces de Banach , Travaux en Cours,Hermann, Paris, 1984, iv+210 pp.[Be] G. Berg,
On James spaces , Ph.D. Thesis, The University of Texas, Austin, Texas, 1996.[BK] H. F. Bohnenblust and S. Karlin,
On a theorem of Ville , Contributions to the Theory ofGames, Annals of Mathematics Studies, no. 24, Princeton University Press, Princeton, N.J., 1950, pp. 155–160.[BS] A. Brunel and L. Sucheston,
On B-convex Banach spaces , Math. Systems Theory 7 (1974),no. 4, 294–299.[B] D. L. Burkholder,
Distribution Function Inequalities for Martingales , Annals of Probability1 (1973), 19–42.[CN] W. W. Comfort and S. Negrepontis,
The theory of ultrafilters , Springer (1974).[FJ] T. Figiel and W. B. Johnson,
A uniformly convex Banach space which contains no ℓ p ,Compositio Math. 29 (1974), 179–190.[FOS] D. Freeman, E. Odell and Th. Schlumprecht, The universality of ℓ as a dual space , Math.Ann. (2011), no. 1, 149-186.[GM] W. T. Gowers and B. Maurey, Banach Spaces with Small Spaces of Operators , Mathema-tische Annalen 307 (1997), no. 4, 543–568.[H] J. Hagler,
Some more Banach spaces which contain L , Studia Math., (1973), 35-42.[HS] J. Hagler and C. Stegall, Banach spaces whose duals contain complemented subspaces iso-morphic to ( C [0 , ∗ , J. Funct. Anal., (1973), 233-251.[HO] L. Halbeisen and E. Odell, On asymptotic models in Banach spaces , Israel J. Math. 139,(2004), 253–291.[HB] F. Helga and B. Gamboa de Buen,
Spreading sequences in JT , Studia Mathematica 125.1(1997), 57–66.[J1] R. C. James,
Bases and reflexivity of Banach spaces , Ann. of Math. 52 (1950), 518–527.[J2] R. C. James,
A separable somewhat reflexive Banach space with nonseparable dual , Bull.Amer. Math. Soc. 80 (1974), 738–743.[Ka] S. Kakutani,
A generalization of Brouwers fixed point theorem , Duke Math. J. 8 (1941), no.3, 457–459.[Kr] J. L. Krivine,
Sous espaces de dimension finie des espaces de Banach r´eticul´es , Ann. ofMath. 104 (1976), 1-29[L] H. Lemberg,
Nouvelle d´emonstration d’un th´eor`eme de J. L. Krivine sur la finierepr´esentation de ℓ p dans un espace de Banach , Israel J. Math. 39 (1981), 341–348.[LS] D. Lewis and C. Stegall, Banach spaces whose duals are isomorphic to ℓ (Γ), J. FunctionalAnalysis (1973), 177-187.[LR] J. Lindenstrauss and H. P. Rosenthal, The L p spaces , Israel J. Math. 7 (1969), 325–249.[LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: sequence spaces , Springer Verlag(1977).[MMT] B. Maurey, V. D. Milman, and N. Tomczak-Jaegermann,
Asymptotic infinite-dimensionaltheory of Banach spaces , Geometric aspects of functional analysis (Israel, 1992-1994), 149-175, Oper. Theory Adv. Appl., , Birkh¨auser, Basel, 1995.[MR] B. Maurey and H. P. Rosenthal, Normalized weakly null sequence with no unconditionalsubsequence , Studia Math. 61 (1977), 77–98.[MT] V. D. Milman and N. Tomczak-Jaegermann,
Asymptotic ℓ p spaces and bounded distortion ,Banach spaces (M´erida, 1992) Contemp. Math., vol. 144, Amer. Math. Soc., Providence, RI,1993, pp. 173-195.[M] A. A. Milutin, Isomorphisms of spaces of continuous functions on compacts of power con-tinuum , Tieoria Func. (Kharov), 2 (1966), 150–156 (Russian).[Ra] F. P. Ramsey,
On a problem of formal logic , Proc. London Math. Soc. 30, (1929), 264–286. [Ro] H. P. Rosenthal,
A characterization of Banach spaces containing ℓ , Proc. Nat. Acad. Sci.U.S.A. 71 (1974), 2411–2413.[S] C. Stegall, Banach spaces whose duals contain ℓ (Γ) with applications to the study of dual L ( µ ) spaces , Trans. Amer. Math. Soc. (1973), 463-477.[T] B. S. Tsirelson, Not every Banach space contains ℓ p or c , Funct. Anal. Appl. 8 (1974),138–141. National Technical University of Athens, Faculty of Applied Sciences, Departmentof Mathematics, Zografou Campus, 157 80, Athens, Greece
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