# Lattice Copies ofℓ2

Authors: E. Jiménez Fernández; E. A. Sánchez Pérez
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

Letm be an ℓ2-valued (countably additive) vector measure and consider the space L2(m) of square integrable functions with respect to m. The integral with respect to m allows to define several notions of orthogonal sequence in these spaces. In this paper, we center our attention in the existence of strongly m-orthonormal sequences. Combining the use of the Kadec-Pelczyński dichotomy in the domain space and the Bessaga-Pelczyński principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces L1(m) and L2(m). Under certain requirements, our main result establishes that a normalized sequence in L2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in L2(m∗), where m∗ is another ℓ2-valued vector measure that satisfies L2(m) = L2(m∗). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an ℓ2-valued positive vector measure contains a lattice copy of ℓ2.


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## Body

## 1. Introduction

In recent years, vector measure integration has been shown to be a good framework for the analysis of the properties of Banach function spaces and the operators defined on them. In particular, it is a powerful tool for representing Banach function spaces providing an additional integration structure. For instance, every 2-convex order continuous Banach function space with weak unit can be written as a spaceL2(m) of integrable functions with respect to a suitable vector measure m ([1, Th. 2.4]; see also [2, Ch.3] for more information). As in the case of the Hilbert spaces of square integrable functions, sequences in L2(m) that satisfy some orthogonality properties with respect to the vector valued integral become useful both for studying the geometry of the space [3–5] and for applications, mainly in the context of the function approximation [3, 6, 7].In contrast to the scalar case, several notions ofm-orthogonality are possible in the case of an ℓ2-valued (countably additive) vector measure m. A sequence {fi}i=1∞ in L2(m) is said to be stronglym-orthonormal if the integral of the product of two different functions is 0 and the integral of each fi2 is ei, where {ei}i=1∞ is an orthonormal sequence in ℓ2. In this paper, we center our attention in this strong version of m-orthogonal sequence, giving a complete characterization of the spaces L2(m) in which such sequences exist; actually, we will show that this fact is closely connected to the existence of lattice copies of ℓ2 in the corresponding space L1(m) that is preserved by the integration map. In order to do this, we develop a sort of two-sided Kadec-Pelczyński disjointification technique. Roughly speaking, this procedure allows to produce sequences of normalized functions in L2(m)—or L1(m)—which are almost disjoint and have integrals that are almost orthogonal in ℓ2: after an isomorphic change of vector measure, we obtain our results both for the existence of strongly m-orthogonal sequences in L2(m) and the existence of lattice copies of ℓ2 in L1(m).The paper is organized as follows. After the preliminary Section2, we analyze in Section 3 the existence of strongly orthonormal sequences in L2(m), and we show that it is a genuine vector valued phenomenon, in the sense that they do not exist for scalar measures and in the case of their natural extensions, vector measures with compact integration maps. Actually, later on we prove that they do not exist for ℓ2-valued measures with disjointly strictly singular integration maps. In the positive, we show in Theorem 3.7 that under reasonable requirements, given an m-orthonormal sequence {fi}i=1∞ in L2(m), it is possible to construct another vector measure m* such that(1)
{fi}i=1∞ is a stronglym*-orthonormal sequence in L2(m),(2)
L2(m)=L2(m*).Combining with the Kadec-Pelczyński dichotomy, the requirement on the sequence of beingm-strongly orthogonal can be relaxed to being weakly null (Corollary 3.9), obtaining in this case a sequence of functions satisfying (1) and (2) that approximates a subsequence of the original one. Some examples and direct consequences of this result are also given. Finally, Section 4 is devoted to show some applications in the context of the structure theory of Banach function spaces, focusing our attention in Banach function lattices that are represented as spaces of square integrable functions with respect to an ℓ2-valued vector measure and are not Hilbert spaces. For the case of positive measures, we show that the existence of strongly m-orthonormal sequences is equivalent of the existence of lattice copies of ℓ4 in L2(m) and lattice copies of ℓ2 in L1(m) (Proposition 4.3 and Theorem 4.5). The translation of these results for the space L1(m) gives the following result on its structure that can also be written in terms of the integration map (Theorem 4.7): the space L1(m) contains a normalized weakly null sequence if and only if it has a reflexive sublattice if and only if it contains a lattice copy of ℓ2.


## 2. Preliminaries

In this section, we introduce several definitions and comments regarding the spacesL2(m). We refer to [8] for definitions and basic results on vector measures. Let X be a Banach space. We will denote by BX the unit ball of X, that is BX:={x∈X:∥x∥X≤1}. X′ will be the topological dual of X. Let Σ be a σ-algebra on a nonempty set Ω. Throughout the paper m:Σ→X will be a countably additive vector measure. The semivariation of m is the nonnegative function ∥m∥ whose value on a set A∈Σ is given by ∥m∥(A):=sup{|〈m,x′〉|(A):x′∈BX′}. The variation|m| of m on a measurable set A is given by |m|(A):=sup∑B∈Π‍∥m(B)∥ for A∈Σ, where the supremum is computed over all finite measurable partitions Π of A. The variation |m| is a monotone countably additive function on Σ—a positive scalar measure—, while the semivariation ∥m∥ is a monotone subadditive function on Σ, and for each A∈Σ we have that ∥m∥(A)≤|m|(A).For each elementx′∈X′, the formula 〈m,x′〉(A):=〈m(A),x′〉, A∈Σ, defines a (countably additive) scalar measure. As usual, we say that a sequence of Σ-measurable functions converges |〈m,x′〉|-almost everywhere if it converges pointwise in a set A∈Σ such that |〈m,x′〉|(Ω∖A)=0. A sequence converges m-almost everywhere if it converges in a set A that satisfies that the semivariation of m in Ω∖A is 0.Letμ be a positive scalar measure. The measure m is absolutely continuous with respect to μ if limμ(A)→0m(A)=0; in this case we write m≪μ and we say that μ is a control measure for m. Countably additive vector measures always have control measures. It is known that there exists always an element x′∈X′ such that m≪|〈m,x′〉|. We call such a scalar measure a Rybakov measure for m (see [8, Ch.IX,2] ). If |〈m,x′〉| is a Rybakov measure for m, a sequence of Σ-measurable functions converges m-almost everywhere if and only if it converges |〈m,x′〉|-almost everywhere.AΣ-measurable function f is integrable with respect to m if it is integrable with respect to each scalar measure 〈m,x′〉, and for every A∈Σ there is an element ∫A‍fdm∈X such that 〈∫A‍fdm,x′〉=∫A‍fd〈m,x′〉 for every x′∈X′. The set of all the (classes of m—a.e. equal) m—integrable functions L1(m) defines an order continuous Banach function space with weak unit χΩ—in the sense of [9, p.28]—over any Rybakov measure for m that is endowed with the norm
(2.1)‖f‖L1(m):=sup⁡x′∈BX′∫Ω‍|f|d|〈m,x′〉|,f∈L1(m).
The reader can find the definitions and fundamental results concerning the space L1(m) in [2, 10–12].The spacesLp(m) are defined extending the definition above in a natural manner [1, 2, 13]. They are p-convex order continuous Banach function spaces with weak unit χΩ over any Rybakov measure, with the norm
(2.2)‖f‖Lp(m):=‖|f|p‖L1(m)1/p,f∈Lp(m).
It is also known that if 1/p+1/q=1, f1∈Lp(m) and f2∈Lq(m), then the pointwise product f1·f2 belongs to L1(m) (see for instance [2, Ch.3]). We will consider the integration operator Im:L1(m)→X associated to the vector measure m, that is defined by Im(f):=∫Ω‍fdm, f∈L1(m). The properties of the integration map have been largely studied in several recent papers (see [2, 14–17] and the references therein). If i,j are indexes of a set I, we write δi,j for the Kronecker delta as usual. A sequence {fi}i=1∞ in L2(m) is called m-orthogonal if ∥∫Ω‍fifjdm∥=δi,jki for positive constants ki. If ∥fi∥L2(m)=1 for all i∈ℕ, it is called m-orthonormal. The properties of these sequences have been recently analyzed in a series of papers, and some applications have been already developed (see [3–7, 18]). In this paper, we deal with the following more restrictive version of orthogonality for ℓ2-valued measures.Definition 2.1.
Letm:Σ→ℓ2 be a vector measure. We say that {fi}i=1∞⊂L2(m) is a stronglym-orthogonal sequence if ∫Ω‍fifjdm=δi,jeiki for an orthonormal sequence {ei}i=1∞ in ℓ2 and for ki>0. If ki=1 for every i∈ℕ, we say that it is a stronglym-orthonormal sequence.We need some elements on Banach-lattice-valued vector measures; in particular, onℓ2-valued measures when the order in ℓ2 is considered. If X is a Banach lattice, we say that a vector measure m:Σ→X is positive if m(A)≥0 for all A∈Σ. Note that if m is positive and x′ is a positive element of the Banach lattice X′, then the measure 〈m,x′〉 coincides with its variation. We refer to [2, 9, 19] for general questions concerning Banach lattices and Banach function spaces. An operator between Banach lattices is called strictly singular if no restriction to an infinite dimensional subspace give an isomorphism, and ℓ2-singular if this happens for subspaces isomorphic to ℓ2. It is called disjointly strictly singular if no restriction to the closed linear span of a disjoint sequence is an isomorphism.We use standard Banach spaces notation. A sequence{xn}n=1∞ in a Banach space X is called a Schauder basis of X (or simply a basis) if for every x∈X there exists a unique sequence of scalars {αn}n=1∞ such that x=limn→∞∑k=1n‍αkxk. A sequence {xn}n=1∞ which is a Schauder basis of its closed span is called a basic sequence. Let {xn}n=1∞ and {yn}n=1∞ be two basis for the Banach spaces X and Y, respectively. Then {xn}n=1∞ and {yn}n=1∞ are equivalent if and only if there is an isomorphism between X and Y that carries each xn to yn.Let{xi}i=1∞ be a basic sequence of a Banach space X and take two sequences of positive integers {pi}i=1∞ and {qi}i=1∞ satisfying that pi<qi<pi+1 for every i∈ℕ. A block basic sequence{yi}i=1∞ associated to {xi}i=1∞ is a sequence of vectors of X defined as finite linear combinations as yi=∑k=piqi‍αi,kxk, where αi,k are real numbers. We refer to [20, Ch.V] for the definition of block basic sequence and to [9, 20] for general questions concerning Schauder basis.


## 3. Strongly m-Orthogonal Sequences inL2(m)

This section is devoted to show how to construct stronglym-orthonormal sequences in L2(m). Let us start with an example of the kind of sequences that we are interested in.Example 3.13.1.
Let([0,∞),Σ,μ) be Lebesgue measure space (Figure 1). Let rk(x):=sign{sin(2k-1x)} be the Rademacher function of period 2π defined at the interval Ek=[2(k-1)π,2kπ], k∈ℕ. Consider the vector measure m:Σ→ℓ2 given by m(A):=∑k=1∞‍(-1/2k)(∫A∩Ek‍rkdμ)ek∈ℓ2, A∈Σ.
Note that iff∈L2(m) then ∫[0,∞)‍fdm=((-1/2k)∫Ek‍frkdμ)k∈ℓ2. Consider the sequence of functions
(3.1)f1(x)=sin⁡(x)⋅χ[π,2π](x)f2(x)=sin⁡(2x)⋅(χ[0,2π](x)+χ[(7/2)π,4π](x))f3(x)=sin⁡(4x)⋅(χ[0,4π](x)+χ[(23/4)π,6π](x))⋮fn(x)=sin⁡(2n-1x)⋅(χ[0,2(n-1)π](x)+χ[(2n-2/2n)π,2nπ](x)),n≥2.
This sequence can be used to define a stronglym-orthogonal sequence, since
(3.2)〈∫[0,∞)fn2dm,en〉=-12n∫En‍fn2rndμ=π22n,〈∫[0,∞)‍fn2dm,ek〉=-12k∫Ek‍fn2rkdμ=0,∀k≠n,〈∫[0,∞)‍fnfmdm,ek〉=-12k∫Ek‍fnfmrkdμ=0,forn≠mand∀k.
If we define the functions of the sequence {Fn}n=1∞ by Fn(x):=(2n/π)fn(x), we get
(3.3)∫[0,∞)‍Fn2dm=en,∀n∈N∫[0,∞)‍FnFkdm=0,∀n,k∈N,n≠k.Figure 1
Functionsf1(x), f2(x), and f3(x) in Example 3.1.The starting point of our analysis is the Bessaga-Pelczyński selection principle. It establishes that if{xk}k=1∞ is a basis of the Banach space X and {xk*}k=1∞ is the sequence of coefficient functionals, if we take a normalized sequence {yn}n=1∞ such that limn〈yn,xk*〉=0, then {yn}n=1∞ admits a basic subsequence that is equivalent to a block basic sequence of {xn}n=1∞ (see for instance Theorem 3 in [20, 21], Ch.V). We adapt this result for sequences of square integrable functions {fn}n=1∞ in order to identify when the sequence of integrals {∫Ω‍fn2dm}n=1∞⊂ℓ2 is a basic sequence in ℓ2. The following result is a direct consequence of the principle mentioned above. Notice that the first requirement in Proposition 3.2 is obviously satisfied in the case of m-orthonormal sequences. The second condition constitutes the key of the problem.Proposition 3.2.
Letm:Σ→ℓ2 be a vector measure, and consider the canonical basis {ek}k=1∞ of ℓ2. Let {fn}n=1∞ be a sequence in L2(m). If there is an ɛ>0 such that the sequence {∫Ω‍fn2dm}n=1∞ satisfies(1)
infn∥∫Ω‍fn2dm∥ℓ2=ɛ>0,(2)
limn〈∫Ω‍fn2dm,ek〉=0,∀k∈ℕ, then {∫Ω‍fn2dm}n=1∞ has a subsequence which is a basic sequence. Moreover, it is equivalent to a block basic sequence of {ek}k=1∞.Let us highlight with an example the geometrical meaning of the requirements above. This is, in a sense, the canonical situation involving disjointness.Example 3.3 ([5, Ex.10]).
Let(Ω,Σ,μ) be a probability measure space. Let us consider the following vector measure m:Σ→ℓ2,
(3.4)m(A):=∑i=1∞‍μ(A∩Ai)ei∈l2,A∈Σ,
where {Ai}i=1∞ is a disjoint measurable partition of Ω, with μ(Ai)≠0 for all i∈ℕ. Notice that ∫Ω‍f2dm=∑i=1∞‍(∫Ai‍f2dμ)ei∈ℓ2 for all f∈L2(m). Consider a sequence of norm one functions {fn}n=1∞ in L2(m) such that fn:=fnχAn for all n. For every k∈ℕ, the following equalities hold:
(3.5)lim⁡n〈∫Ω‍fn2dm,ek〉=lim⁡n〈∑i=1∞‍(∫Ai‍fn2dμ)ei,ek〉=lim⁡n∫Ak‍fn2dμ=0,‖∫Ω‍fn2dm‖l2=1∀n∈N.
Therefore, condition (2.2) of Proposition 3.2 is fulfilled in this example: the role of disjointness is clear.In what follows, we show thatif the integration operator is compact then there are no stronglym-orthonormal sequences. In particular, this shows that the existence of such sequences is a pure vector measure phenomenon, since the integration map is obviously compact when the measure is scalar. Compactness of the integration map is nowadays well characterized (see [2, Ch.3] and the references therein); it is a strong property, in the sense that it implies that the space L1(m) is lattice isomorphic to the L1 space of the variation of m, that is a scalar measure (see [2, Prop.3.48]). We need the next formal requirement for the elements of the sequence {fn}n=1∞. We say that a function f∈L2(m) is normed by the integral if ∥f∥L2(m)=∥∫Ω‍f2dm∥1/2. This happens for instance when the vector measure m is positive (see [22] or [2, Lemma 3.13]), since in this case the norm can be computed using the formula ∥f∥L1(m)=∥∫Ω‍|f|dm∥ for all f∈L1(m). We impose this requirement for the aim of clarity; some of the results could be adapted using a convenient renorming process in order to avoid it.Remark 3.4.
Letm:Σ→ℓ2 be a countably additive vector measure. If there exists a strongly m-orthonormal sequence in L2(m) which elements are normed by the integrals, then the integration operator Im:L1(m)→ℓ2 is not compact. To see this, let {fi}i=1∞⊂L2(m) be a strongly m-orthonormal sequence in L2(m) and consider a orthonormal sequence {ei}i=1∞. Then ∫Ω‍fifjdm=δi,jei, an thus
(3.6)∫Ω‍fi2dm=ei=Im(fi2).
Therefore {fi2}i=1∞⊂BL1(m), and so the sequence {Im(fi2)}i=1∞ that satisfies that {Im(fi2)}i=1∞⊂Im(BL1(m))⊂Im(BL1(m))¯ does not admit any convergent subsequence. It follows that Im(BL1(m))¯ is not compact and so, Im(BL1(m)) is not relatively compact. This allows to conclude that Im is not compact.Theorem3.7 below gives a necessary condition—and, in a sense, also a sufficient condition—for the existence of strongly orthonormal sequences in a space of functions L2(m) starting from a given m-orthonormal sequence. The existence of such m-orthonormal sequences is always assured: just consider a sequence of normalized disjoint functions in L2(m). The following result is an application of the Kadec-Pelczyński disjointification procedure for order continuous Banach function spaces—also called the Kadec-Pelczyński dichotomy, see Theorem 4.1 in [23, 24]—, in the following version, that can be found in [25] (see the comments after Proposition 1.1). LetX(μ)be an order continuous Banach function space over a finite measureμwith a weak unit(this impliesX(μ)↪L1(μ)). Consider a normalized sequence{xn}n=1∞inX(μ). Then(1)
either{∥xn∥L1(μ)}n=1∞is bounded away from zero,(2)
or there exists a subsequence{xnk}k=1∞and a disjoint sequence{zk}k=1∞inX(μ)such that∥zk-xnk∥→k0.Recall that the spaceL2(m) is an order continuous Banach function space over any Rybakov (finite) measure μ=|〈m,x0′〉| for m.Proposition 3.5.
Let{gn}n=1∞ be a normalized sequence in L2(m). Suppose that there exists a Rybakov measure μ=|〈m,x0′〉| for m such that {∥gn∥L1(μ)}n=1∞ is not bounded away from zero. Then there are a subsequence {gnk}k=1∞ of {gn}n=1∞ and an m-orthonormal sequence {fk}k=1∞ such that ∥gnk-fk∥L2(m)→k0.Proof.
By the criterion given above, there is a subsequence{gnk}k=1∞ of {gn}n=1∞ and a disjoint sequence {f′k}k=1∞ such that ∥gnk-fk′∥L2(m)→k0. Consider the sequence given by the functions fk:=fk′/∥fk′∥. Then ∥gnk-fk∥L2(m)→k0. Since ∫Ω‍fkfjdm=0 for every k≠j due to the fact that they are disjoint, we obtain the result.Although the existence of a stronglym-orthonormal subsequence of an m-orthogonal sequence cannot be assured in general, we show in what follows that under the adequate requirements it is possible to find a vector measure m* satisfying that L2(m)=L2(m*) and with respect to which there is a subsequence that is strongly m*-orthonormal. We use the following lemma, which proof is elementary (see Lemma 3.27 in [2]).Lemma 3.6.
Letm:Σ→ℓ2 be a vector measure. Let φ:ℓ2→H be an isomorphism, where H is a separable Hilbert space, and consider the vector measure m*=φ∘m. Then the spaces L2(m) and L2(m*) are isomorphic, and for every f∈L2(m), ∫Ω‍f2dm*=φ(∫Ω‍f2dm).Theorem 3.7.
Let us consider a vector measurem:Σ→ℓ2 and an m-orthonormal sequence {fn}n=1∞ of functions in L2(m) that are normed by the integrals. Let {en}n=1∞ be the canonical basis of ℓ2. If limn〈∫Ω‍fn2dm,ek〉=0 for every k∈ℕ, then there exists a subsequence {fnk}k=1∞ of {fn}n=1∞ and a vector measure m*:Σ→ℓ2 such that {fnk}k=1∞ is strongly m*-orthonormal.
Moreover,m* can be chosen to be as m*=ϕ∘m for some Banach space isomorphism ϕ from ℓ2 onto ℓ2, and so L2(m)=L2(m*).Proof.
Consider anm-orthonormal sequence {fn}n=1∞ in L2(m) and the sequence of integrals {∫Ω‍fn2dm}n=1∞. As an application of Proposition 3.2, we get a subsequence {∫Ω‍fnk2dm}k=1∞ that is equivalent to a block basic sequence {enk′}k=1∞ of the canonical basis of ℓ2. Recall that according to the notation given in Section 2, αi,j are the constants that appear in the definition of the block basic sequence. Associated to this sequence, there is an isomorphism φ(3.7)A:=span⁡(enk′)¯l2⟶φB:=span⁡(∫Ω‍fnk2dm)¯l2
such that φ(enk′):=∫Ω‍fnk2dm, k∈ℕ.
We can suppose without loss of generality that the elements of the sequence{enk′}k=1∞ have norm one. To see this, it is enough to consider the following inequalities. First note that there are positive constants Q and K such that for every n∈ℕ, Q=Q∥∫Ω‍fnk2dm∥≤∥enk′∥≤K∥∫Ω‍fnk2dm∥=K as a consequence of the existence of the isomorphism φ. Let {λi}i=1∞ be a sequence of real numbers. Then
(3.8)‖∑i=1∞λiei′‖ei′‖‖22=∑i=1∞|λi|2(∑j=piqi|αi,j|2)‖ei′‖2=∑i=1∞∑j=piqi|λi|2|αi,j|2‖ei′‖2.
The existence of an upper and a lower bound for the real numbers ∥ei′∥ given above provides the equivalence between this quantity and ∥∑i=1∞‍λiei′∥22 for each sequence of real numbers {λi}i=1∞.
Since each closed subspace of a Hilbert space is complemented, there is a subspaceBc such that ℓ2=B⊕2Bc isometrically, where this direct sum space is considered as a Hilbert space (with the adequate Hilbert space norm). We write PB and PBc for the corresponding projections. Let us consider the linear map ϕ:=φ-1⊕Id:B⊕2Bc→ϕA⊕2Bc, where Id:Bc→Bc is the identity map.
Note thatH:=A⊕2Bc is a Hilbert space with the scalar product
(3.9)〈x+y,x′+y′〉H=〈x,x′〉H+〈y,y′〉H,x+y,x′+y′∈A⊕2Bc,
that can be identified with ℓ2. Obviously, ϕ is an isomorphism. Let us consider now the vector measure m*:=ϕ∘m:Σ→mℓ2→ϕA⊕2Bc. By Lemma 3.6, L2(m)=L2(ϕ∘m)=L2(m*). Let us show that {fnk}k=1∞ is a strongly m*-orthonormal sequence. We consider the orthonormal sequence {(enk′,0)}k=1∞ in H. The first condition in the definition of strongly orthonormal sequence is fulfilled, since
(3.10)∫Ω‍fnk2dm*=∫Ω‍fnk2d(ϕ∘m)=ϕ(PB(∫Ω‍fnk2dm),PBc(∫Ω‍fnk2dm))=(φ-1(∫Ω‍fnk2dm),0)=(enk′,0),
for every k∈ℕ. The second one is given by the following calculations. For k≠l,
(3.11)‖∫Ω‍fnkfnldm*‖=‖∫Ω‍fnkfnld(ϕ∘m)‖=‖ϕ(∫Ω‍fnkfnldm)‖=‖ϕ(0)‖=0,
since ϕ is continuous and {fnk}k=1∞ is an m-orthonormal sequence. Thus we get ∫Ω‍fnkfnldm*=0. This proves the theorem.Remark 3.8.
In a certain sense, the converse of Theorem3.7 also holds. Take as the vector measure m* the measure m itself with values in ℓ2 and consider the canonical basis {en}n=1∞. Clearly, every strongly m-orthonormal sequence is m-orthonormal and satisfies the condition limn〈∫Ω‍fn2dm*,ek〉=0, since
(3.12)〈∫Ω‍fn2dm*,ek〉=〈en,ek〉=0,k≠n.Corollary 3.9.
Letm:Σ→ℓ2 be a countably additive vector measure. Let {gn}n=1∞ be a normalized sequence of functions in L2(m) that are normed by the integrals. Suppose that there exists a Rybakov measure μ=|〈m,x0′〉| for m such that {∥gn∥L1(μ)}n=1∞ is not bounded away from zero. If limn〈∫Ω‍gn2dm,ek〉=0 for every k∈ℕ, then there is a (disjoint) sequence {fk}k=1∞ such that  (1)
limk∥gnk-fk∥L2(m)=0 for a given subsequence {gnk}k=1∞ of {gn}n=1∞, and(2)
it is stronglym*-orthonormal for a certain Hilbert space valued vector measure m* defined as in Theorem 3.7 that satisfies that L2(m)=L2(m*).This is a direct consequence of Proposition3.5 and Theorem 3.7. For the proof, just take into account the continuity of the integration map and the fact that the elements of the sequence {gn}n=1∞ are normed by the integrals.


## 4. Applications: Copies ofℓ2 in L1(m) That Are Preserved by the Integration Map

One of the consequences of the results of the previous section is that the existence of stronglym-orthonormal sequences in L2(m) is closely related to the existence of lattice copies of ℓ2 in L1(m). In this section, we show how to apply our arguments for finding some information on the structure of the spaces L1(m) and the properties of the associated integration map.Our motivation has its roots in the general problem of finding subspaces of Banach function spaces that are isomorphic toℓ2. It is well known that in general these copies are related to weakly null normalized sequences; the arguments that prove this relation go back to the Kadec-Pelczyński dichotomy and have been applied largely in the study of strictly singular embeddings between Banach function spaces [25, 26]. In some relevant classes of Banach function spaces—Lp-spaces, Lorentz spaces, Orlicz spaces, and general rearrangement invariant (r.i.) spaces—these copies are related to subspaces generated by Rademacher-type sequences (see [27–30] and the references therein). For instance, Corollary 2 in [27] states that for a r.i. Banach function space E on [0,1], if the norms on E and L1 are equivalent on some infinite dimensional subspace of E, then the Rademacher functions span a copy of ℓ2 in E. However, our construction generates copies of ℓ2 that are essentially different. Actually, they are defined by positive or even disjoint functions, and so the copies of ℓ2 that our results produce allow to conclude that if there is a normalized sequence of positive functions with a weakly null sequence of integrals, the integration map is neither disjointly strictly singular nor ℓ2-singular.On the other hand, it is well known that strongly orthonormal sequences—that are calledλ-orthonormal systems in Definition 2 of [5]—define isometric copies of ℓ4 in spaces L2(m) of a positive vector measure m (see Propositions 8 and 11 in [5]). In particular, this makes clear that the existence of these sequences imply that L2(m) is not a Hilbert space, and so L1(m) is not an L1-space. However, there is a big class of Banach function spaces that can be represented as L1(m) of an ℓ2-valued positive vector measure m (see for instance Example 10 in [5] or Example 8 in [4]). The L1(m)-spaces associated to such vector measures are sometimes called ℓ-sums of L1-spaces. In Section 4 of [31], a first attempt to study 2-convex subspaces—the natural extension of ℓ2-copies in this setting—of ℓ-sums of L1-spaces was made. Also, a first analysis of the question of when L1(m) is a Hilbert space—based on the behavior of specific sequences too—was made in [11, Section  4]. In what follows, we provide more information on the existence of copies of ℓ2 in spaces L1(m) of a positive vector measure, and the closely related problem of the existence of ℓ4 in L2(m). After that, some contributions to the analysis to the study of strictly singular integration maps are given. Recently, a new considerable effort has been made in order to find the links between the belonging of the integration map Im to a particular class of operators and the structure properties of the space L1(m). For integration maps belonging to relevant operator ideals, this has been done in [16, 17, 32] (see also [2, Ch.5] and the references therein). For geometric and order properties of the integration map—mainly concavity and positive p-summing type properties—, we refer to [33, 34] and [2, Ch.6].For the aim of clarity, in this section we deal with positive vector measures, that—as we said in the previous section—satisfy that all the elements of the spacesL2(m) are normed by the integrals. In this case, it can be shown that there is an easy characterization of strongly m-orthonormal sequences, which simplifies the arguments.Remark 4.1.
Suppose that a vector measure satisfies that the setP:={x′∈ℓ2:〈m,x′〉≥0} separates the points of ℓ2 and assume that for a given sequence {fn}n=1∞, 〈∫Ω‍fn2dm,x′〉·〈∫Ω‍fk2dm,x′〉=0 for every n,k∈ℕ such that n≠k and x′∈P. Then ∫Ω‍fnfkdm=0 for every n≠k. This is a direct consequence of Hölder's inequality and the integrability with respect to m of all the functions involved. For the particular case of positive vector measures, the standard basis {en}n=1∞ of ℓ2 plays the role of P; this means that the requirement ∫Ω‍fn2dm=en for all n automatically implies that {fn}n=1∞ is a strongly m-orthonormal sequence.Lemma 4.2.
Letm:Σ→ℓ2 be a positive vector measure, and suppose that the bounded sequence {gn}n=1∞ in L2(m) satisfies that limn〈∫Ω‍gn2dm,ek〉=0 for all k∈ℕ. Then there is a Rybakov measure μ for m such that limn∥gn∥L1(μ)=0.Proof.
Take for instance the sequencex0′={(1/2)n/2}n=1∞∈ℓ2. Since m is positive, the measure μ:=〈m,x0′〉 is positive and defines a Rybakov measure for m. Since 〈∫Ω‍gn2dm,x0′〉=∥gn∥L2(μ)2 for all n∈ℕ and the requirement on {∫Ω‍gn2dm}n=1∞ imply that it is weakly null, we obtain by Hölder's inequality that limn∥gn∥L1(μ)≤limn∥gn∥L2(μ)∥m∥1/2=0.Proposition 4.3.
Letm:Σ→ℓ2 be a positive (countably additive) vector measure. Let {gn}n=1∞ be a normalized sequence in L2(m) such that for every k∈ℕ, limn〈∫Ω‍gn2dm,ek〉=0. Then L2(m) contains a lattice copy of ℓ4. In particular, there is a subsequence {gnk}k=1∞ of {gn}n=1∞ that is equivalent to the unit vector basis of ℓ4.Proof.
By Lemma4.2, we can use Corollary 3.9 to produce a disjoint sequence {fk}k=1∞ in L2(m) that approximates a subsequence {gnk}k=1∞ of {gn}n=1∞ and is strongly m*-orthogonal. The same computations that can be found in the proof of Proposition 8 in [5] show that for finite sums ∑k=1N‍αkfk, the norm in L2(m) is equivalent to the norm of {αk}k=1N in ℓ4. Consequently, the closure of these finite sums in L2(m) provides a copy of ℓ4. The disjointness of {fk}k=1∞ implies that in fact it is a lattice copy. Note also that {fk}k=1∞ is equivalent to {gnk}k=1∞ and so to the unit vector basis of ℓ4.As a direct consequence, we obtain that for a positive vector measurem, the existence of a normalized sequence of functions such that the sequence of square integrals is weakly null implies that L2(m) cannot be a Hilbert space. On the other hand, if the integration map is compact, then L1(m)=L1(|m|) isomorphically (see Proposition 3.48 in [2]), and thus L2(m) is (isomorphic to) a Hilbert space. Notice first the following obvious consequence of this fact: L1(m)is isomorphic to anL1(μ)-space of a finite measureμif and only if there is a positiveℓ2-valued vector vector measurem0  such thatL1(m)=L1(m0)  such that the integration map is compact; the converse statement is proved by considering the vector measure n(A):=μ(A)e1∈ℓ2, A∈Σ. However, as the next example shows, there are spaces L2(m) for positive ℓ2-valued vector measures with noncompact integration map that are Hilbert spaces. We will find in Corollary 4.6 that this conclusion—L1(m) not being an L1-space, and so L2(m) not to be a Hilbert space—can be extended to the case of strictly singular integration maps.Example 4.4.
(1)Anℓ2-valued measure such thatL1(m)is a Hilbert space and the integration map is not compact. Consider the Hilbert space L2[0,1] and a orthonormal basis S for it. Consider the associated isomorphism ϕS:L2[0,1]→ℓ2 that carries each function to the 2-summable sequence of its fourier coefficients. Take the vector measure mS:Σ→ℓ2 given by mS(A):=ϕS(χA) for each Lebesgue measurable set A∈Σ. Then L1(mS)=L2[0,1], although ImS is in fact an isomorphism.
(2)A positiveℓ2-valued measure with noncompact integration map such thatL1(m)is a Hilbert space. Consider a vector measure m as in Example 3.3 and define the positive measure n:Σ→ℓ2 by n(A):=μ(A)e1+m(A), A∈Σ. A direct computation shows that the norm in L1(n) is equivalent to the one in L1(μ). Then L2(n)=L2(μ) isomorphically, and In:Σ→ℓ2 is clearly noncompact.Next result shows the consequences on the structure ofL1(m) of our arguments about the existence of strongly orthonormal sequences in L2(m).Theorem 4.5.
Letm:Σ→ℓ2 be a positive (countably additive) vector measure. Let {gn}n=1∞ be a normalized sequence in L2(m) such that for every k∈ℕ, limn〈∫Ω‍gn2dm,ek〉=0 for all k∈ℕ. Then there is a subsequence {gnk}k=1∞ such that {gnk2}k=1∞ generates an isomorphic copy of ℓ2 in L1(m) that is preserved by the integration map. Moreover, there is a normalized disjoint sequence {fk}k=1∞ that is equivalent to the previous one and {fk2}k=1∞ gives a lattice copy of ℓ2 in L1(m) that is preserved by Im*.Proof.
By Corollary3.9 and Lemma 4.2, there is a (normalized) disjoint sequence {fk}k=1∞ in L2(m) that is equivalent to a subsequence {gnk}k=1∞ of {gn}n=1∞. Let us prove directly that {gnk2}k=1∞ generates an isomorphic copy of ℓ2 in L1(m). Let m*=ϕ∘m the vector measure given in Corollary 3.9 and let K be the norm of ϕ-1. Since limk∥gnk-fk∥L2(m)=0 for every ɛ>0 there is a subsequence of the one above (that we denote as the previous one) that satisfies that
(4.1)(∑k=1n‍‖∫Ω‍|gnk-fk|2dm‖)1/2<ɛ2.
Fix an ɛ>0. We have that, by Hölder inequality,
(4.2)(∑k=1n‍‖∫Ω‍|gnk2-fk2|dm‖l22)1/2=(∑k=1n‍‖∫Ω‍|gnk-fk|⋅|gnk+fk|dm‖l22)1/2≤(∑k=1n‍‖∫Ω‍|gnk-fk|2dm‖⋅‖∫Ω‍|gnk+fk|2dm‖)1/2≤2(∑k=1n‍‖∫Ω‍|gnk-fk|2dm‖)1/2<ɛ.
This means that
(4.3)‖∫Ω‍|∑k=1n‍αkgnk2|dm‖l2≤‖∫Ω‍|∑k=1n‍αk(gnk2-fk2)|dm‖l2+‖∫Ω‍∑k=1n‍|αk|fk2dm‖l2≤(∑k=1n‍αk2)1/2ɛ+K‖∫Ω‍∑k=1n‍|αk|fk2dm*‖l2≤(ɛ+K)(∑k=1n‍αk2)1/2.
Similar computations give the converse inequality. The construction of m* and the disjointness of the functions of the sequence {fk2}k=1∞ give the last statement.Corollary 4.6.
Letm:Σ→ℓ2 be a positive (countably additive) vector measure. The following assertions are equivalent.  (1)
There is a normalized sequence inL2(m) satisfying that limn〈∫Ω‍gn2dm,ek〉=0 for all the elements of the canonical basis {ek}k=1∞ of ℓ2.(2)
There is anℓ2-valued vector measure m*=ϕ∘m—ϕ an isomorphism—such that L2(m)=L2(m*) and there is a disjoint sequence in L2(m) that is strongly m*-orthonormal.(3)
There is a subspaceS⊆L1(m) that is fixed by the integration map Im which satisfies that there are positive functions hn∈S such that {∫Ω‍hndm}n=1∞ is an orthonormal basis for Im(S).(4)
There is anℓ2-valued vector measure m* defined as m*=ϕ∘m—ϕ an isomorphism—such that L1(m)=L1(m*) and a subspace S of L1(m) such that the restriction of Im* to S is a lattice isomorphism in ℓ2.Proof.
(
1
)
⇒
(
2
) is a direct consequence of Theorem 4.5. For (2)⇒(3), just notice that the strong m*-orthogonality of a disjoint sequence {gn}n=1∞ implies that {gn2}n=1∞ gives a lattice copy of ℓ2 preserved by the integration map Im*. Since ϕ-1∘m*=m, we obtain that {ϕ-1(∫Ω‍gn2dm*)}n=1∞={∫Ω‍gn2dm}n=1∞ is a basis for ℓ2.
(
3
)
⇒
(
1
). There is a bounded sequence {hn}=1∞ in L1(m) such that ∫Ω‍hndm=an, where an is a orthonormal basis of closure of the subspace A:=span{∫Ω‍hndm:n∈ℕ}, and an isomorphism ϕ from A to ℓ2 such that ϕ(∫Ω‍hndm)=en. By composing with ϕ the integration map, the copy of ℓ2 that is fixed by Im can be considered in such a way that ϕ(∫Ω‍gn2dm)=en. Consequently, limn〈ϕ(∫Ω‍hndm),ek〉=0 for all k∈ℕ, and so limn〈ek,∫Ω‍hndm〉=0 for all k∈ℕ. It is enough to take gn=hn1/2.
(
3
)
⇒
(
4
) is obvious.
(
4
)
⇒
(
1
). Take the normalized sequence of positive functions {hn}n=1∞ in S such that {∫Ω‍hndϕ∘m}n=1∞ is equivalent to the standard basis of ℓ2, and define gn:=hn1/2. The weak to weak continuity of ϕ gives the result.We have shown that the existence of lattice copies ofℓ4 in L2(m) is directly connected with the existence of lattice copies of ℓ2 in L1(m). Thus, and summarizing the results in this section, we finish the paper with a complete characterization of this property for L1(m) of a positive ℓ2-valued vector measure m.Theorem 4.7.
The following assertions for a positive vector measurem:Σ→ℓ2 are equivalent.  (1)
L1(m) contains a lattice copy of ℓ2.(2)
L1(m) has a reflexive infinite dimensional sublattice.(3)
L1(m) has a relatively weakly compact normalized sequence of disjoint functions.(4)
L1(m) contains a weakly null normalized sequence.(5)
There is a vector measurem* defined by m*=ϕ∘m such that integration map Im* fixes a copy of ℓ2.(6)
There is a vector measurem* defined as m*=ϕ∘m that is not disjointly strictly singular.Proof.
(
1
)
⇒
(
2
)
⇒
(
3
) are obvious. For (3)⇒(4), just take into account that disjoint normalized sequences in weakly compact sets of Banach lattices are weakly null (see for instance the proof of Proposition 3.6.7 in [19]).
(
4
)
⇒
(
5
). Take a weakly null normalized sequence {gj}j=1n in L1(m). By (the arguments used in) Lemma 4.2 we can find a Rybakov measure μ for m such that ∥gj∥L1(μ)→0. Now we use the same arguments that lead to Theorem 3.7 and Corollary 3.9; by the Kadec-Pelczyński dichotomy, there exists a subsequence {gjl}l=1∞ of {gj}j=1∞ and a disjoint sequence {zl}l=1∞ in the unit sphere of L1(m) such that liml∥gjl-zl∥=0. Notice that {zl}l=1∞ also converges weakly to 0, so by taking a subsequence and after restricting the supports of the functions zl and renorming if necessary, we obtain a normalized weakly null positive disjoint sequence {vk}k=1∞. This gives the copy of ℓ2 that is fixed by the integration map associated to a vector measure m*=ϕ∘m satisfying L1(m)=L1(m*) by Im(vk)=ek for the canonical basis {ek}k=1∞ of ℓ2. Finally, (5)⇒(6) and (6)⇒(1) are evident.



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*Source: 357210-2012-09-02.xml*