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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 357210, 15 pages
Lattice Copies of in of a Vector Measure and Strongly Orthogonal Sequences
Instituto Universitario de Matemática Pura y Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain
Received 24 February 2012; Accepted 17 July 2012
Academic Editor: John R. Akeroyd
Copyright © 2012 E. Jiménez Fernández and E. A. Sánchez Pérez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let m be an -valued (countably additive) vector measure and consider the space (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 (m) and (m). Under certain requirements, our main result establishes that a normalized sequence in (m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in (), where is another -valued vector measure that satisfies (m) = (). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an -valued positive vector measure contains a lattice copy of .
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 space of integrable functions with respect to a suitable vector measure ([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 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 of -orthogonality are possible in the case of an -valued (countably additive) vector measure . A sequence in is said to be strongly -orthonormal if the integral of the product of two different functions is and the integral of each is , where is an orthonormal sequence in . In this paper, we center our attention in this strong version of -orthogonal sequence, giving a complete characterization of the spaces in which such sequences exist; actually, we will show that this fact is closely connected to the existence of lattice copies of in the corresponding space 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 —or —which are almost disjoint and have integrals that are almost orthogonal in : after an isomorphic change of vector measure, we obtain our results both for the existence of strongly -orthogonal sequences in and the existence of lattice copies of in .
The paper is organized as follows. After the preliminary Section 2, we analyze in Section 3 the existence of strongly orthonormal sequences in , 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 -valued measures with disjointly strictly singular integration maps. In the positive, we show in Theorem 3.7 that under reasonable requirements, given an -orthonormal sequence in , it is possible to construct another vector measure such that(1) is a strongly -orthonormal sequence in ,(2).
Combining with the Kadec-Pelczyński dichotomy, the requirement on the sequence of being -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 -valued vector measure and are not Hilbert spaces. For the case of positive measures, we show that the existence of strongly -orthonormal sequences is equivalent of the existence of lattice copies of in and lattice copies of in (Proposition 4.3 and Theorem 4.5). The translation of these results for the space gives the following result on its structure that can also be written in terms of the integration map (Theorem 4.7): the space 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 .
In this section, we introduce several definitions and comments regarding the spaces . We refer to  for definitions and basic results on vector measures. Let be a Banach space. We will denote by the unit ball of , that is . will be the topological dual of . Let be a -algebra on a nonempty set . Throughout the paper will be a countably additive vector measure. The semivariation of is the nonnegative function whose value on a set is given by . The variation of on a measurable set is given by for , where the supremum is computed over all finite measurable partitions of . The variation is a monotone countably additive function on —a positive scalar measure—, while the semivariation is a monotone subadditive function on , and for each we have that .
For each element , the formula , , defines a (countably additive) scalar measure. As usual, we say that a sequence of -measurable functions converges -almost everywhere if it converges pointwise in a set such that . A sequence converges -almost everywhere if it converges in a set that satisfies that the semivariation of in is .
Let be a positive scalar measure. The measure is absolutely continuous with respect to if ; in this case we write and we say that is a control measure for . Countably additive vector measures always have control measures. It is known that there exists always an element such that . We call such a scalar measure a Rybakov measure for (see [8, Ch.IX,2] ). If is a Rybakov measure for , a sequence of -measurable functions converges -almost everywhere if and only if it converges -almost everywhere.
A -measurable function is integrable with respect to if it is integrable with respect to each scalar measure , and for every there is an element such that for every . The set of all the (classes of —a.e. equal) —integrable functions defines an order continuous Banach function space with weak unit —in the sense of [9, p.28]—over any Rybakov measure for that is endowed with the norm The reader can find the definitions and fundamental results concerning the space in [2, 10–12].
The spaces are defined extending the definition above in a natural manner [1, 2, 13]. They are -convex order continuous Banach function spaces with weak unit over any Rybakov measure, with the norm It is also known that if , and , then the pointwise product belongs to (see for instance [2, Ch.3]). We will consider the integration operator associated to the vector measure , that is defined by , . The properties of the integration map have been largely studied in several recent papers (see [2, 14–17] and the references therein). If are indexes of a set , we write for the Kronecker delta as usual. A sequence in is called -orthogonal if for positive constants . If for all , it is called -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 -valued measures.
Definition 2.1. Let be a vector measure. We say that is a strongly -orthogonal sequence if for an orthonormal sequence in and for . If for every , we say that it is a strongly -orthonormal sequence.
We need some elements on Banach-lattice-valued vector measures; in particular, on -valued measures when the order in is considered. If is a Banach lattice, we say that a vector measure is positive if for all . Note that if is positive and is a positive element of the Banach lattice , then the measure 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 -singular if this happens for subspaces isomorphic to . 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 in a Banach space is called a Schauder basis of (or simply a basis) if for every there exists a unique sequence of scalars such that . A sequence which is a Schauder basis of its closed span is called a basic sequence. Let and be two basis for the Banach spaces and , respectively. Then and are equivalent if and only if there is an isomorphism between and that carries each to .
Let be a basic sequence of a Banach space and take two sequences of positive integers and satisfying that for every . A block basic sequence associated to is a sequence of vectors of defined as finite linear combinations as , where 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 in
This section is devoted to show how to construct strongly -orthonormal sequences in . Let us start with an example of the kind of sequences that we are interested in.
Example 3.1 3.1. Let be Lebesgue measure space (Figure 1). Let be the Rademacher function of period defined at the interval , . Consider the vector measure given by , .
Note that if then . Consider the sequence of functions
This sequence can be used to define a strongly -orthogonal sequence, since If we define the functions of the sequence by , we get
The starting point of our analysis is the Bessaga-Pelczyński selection principle. It establishes that if is a basis of the Banach space and is the sequence of coefficient functionals, if we take a normalized sequence such that , then admits a basic subsequence that is equivalent to a block basic sequence of (see for instance Theorem 3 in [20, 21], Ch.V). We adapt this result for sequences of square integrable functions in order to identify when the sequence of integrals is a basic sequence in . 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 -orthonormal sequences. The second condition constitutes the key of the problem.
Proposition 3.2. Let be a vector measure, and consider the canonical basis of . Let be a sequence in . If there is an such that the sequence satisfies(1), (2), then has a subsequence which is a basic sequence. Moreover, it is equivalent to a block basic sequence of .
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 be a probability measure space. Let us consider the following vector measure ,
where is a disjoint measurable partition of , with for all . Notice that for all . Consider a sequence of norm one functions in such that for all . For every , the following equalities hold:
Therefore, condition (2.2) of Proposition 3.2 is fulfilled in this example: the role of disjointness is clear.
In what follows, we show that if the integration operator is compact then there are no strongly -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 is lattice isomorphic to the space of the variation of , that is a scalar measure (see [2, Prop.3.48]). We need the next formal requirement for the elements of the sequence . We say that a function is normed by the integral if . This happens for instance when the vector measure is positive (see  or [2, Lemma 3.13]), since in this case the norm can be computed using the formula for all . 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. Let be a countably additive vector measure. If there exists a strongly -orthonormal sequence in which elements are normed by the integrals, then the integration operator is not compact. To see this, let be a strongly -orthonormal sequence in and consider a orthonormal sequence . Then , an thus Therefore , and so the sequence that satisfies that does not admit any convergent subsequence. It follows that is not compact and so, is not relatively compact. This allows to conclude that is not compact.
Theorem 3.7 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 starting from a given -orthonormal sequence. The existence of such -orthonormal sequences is always assured: just consider a sequence of normalized disjoint functions in . 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  (see the comments after Proposition 1.1). Let be an order continuous Banach function space over a finite measure with a weak unit (this implies ). Consider a normalized sequence in . Then (1)either is bounded away from zero,(2)or there exists a subsequence and a disjoint sequence in such that .
Recall that the space is an order continuous Banach function space over any Rybakov (finite) measure for .
Proposition 3.5. Let be a normalized sequence in . Suppose that there exists a Rybakov measure for such that is not bounded away from zero. Then there are a subsequence of and an -orthonormal sequence such that .
Proof. By the criterion given above, there is a subsequence of and a disjoint sequence such that . Consider the sequence given by the functions . Then . Since for every due to the fact that they are disjoint, we obtain the result.
Although the existence of a strongly -orthonormal subsequence of an -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 satisfying that and with respect to which there is a subsequence that is strongly -orthonormal. We use the following lemma, which proof is elementary (see Lemma 3.27 in ).
Lemma 3.6. Let be a vector measure. Let be an isomorphism, where is a separable Hilbert space, and consider the vector measure . Then the spaces and are isomorphic, and for every , .
Theorem 3.7. Let us consider a vector measure and an -orthonormal sequence of functions in that are normed by the integrals. Let be the canonical basis of . If for every , then there exists a subsequence of and a vector measure such that is strongly -orthonormal.
Moreover, can be chosen to be as for some Banach space isomorphism from onto , and so .
Proof. Consider an -orthonormal sequence in and the sequence of integrals . As an application of Proposition 3.2, we get a subsequence that is equivalent to a block basic sequence of the canonical basis of . Recall that according to the notation given in Section 2, are the constants that appear in the definition of the block basic sequence. Associated to this sequence, there is an isomorphism
such that , .
We can suppose without loss of generality that the elements of the sequence have norm one. To see this, it is enough to consider the following inequalities. First note that there are positive constants and such that for every , as a consequence of the existence of the isomorphism . Let be a sequence of real numbers. Then The existence of an upper and a lower bound for the real numbers given above provides the equivalence between this quantity and for each sequence of real numbers .
Since each closed subspace of a Hilbert space is complemented, there is a subspace such that isometrically, where this direct sum space is considered as a Hilbert space (with the adequate Hilbert space norm). We write and for the corresponding projections. Let us consider the linear map , where is the identity map.
Note that is a Hilbert space with the scalar product that can be identified with . Obviously, is an isomorphism. Let us consider now the vector measure . By Lemma 3.6, . Let us show that is a strongly -orthonormal sequence. We consider the orthonormal sequence in . The first condition in the definition of strongly orthonormal sequence is fulfilled, since for every . The second one is given by the following calculations. For , since is continuous and is an -orthonormal sequence. Thus we get . This proves the theorem.
Remark 3.8. In a certain sense, the converse of Theorem 3.7 also holds. Take as the vector measure the measure itself with values in and consider the canonical basis . Clearly, every strongly -orthonormal sequence is -orthonormal and satisfies the condition , since
Corollary 3.9. Let be a countably additive vector measure. Let be a normalized sequence of functions in that are normed by the integrals. Suppose that there exists a Rybakov measure for such that is not bounded away from zero. If for every , then there is a (disjoint) sequence such that (1) for a given subsequence of , and(2)it is strongly -orthonormal for a certain Hilbert space valued vector measure defined as in Theorem 3.7 that satisfies that .
This is a direct consequence of Proposition 3.5 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 are normed by the integrals.
4. Applications: Copies of in That Are Preserved by the Integration Map
One of the consequences of the results of the previous section is that the existence of strongly -orthonormal sequences in is closely related to the existence of lattice copies of in . In this section, we show how to apply our arguments for finding some information on the structure of the spaces 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 to . 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—-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  states that for a r.i. Banach function space on , if the norms on and are equivalent on some infinite dimensional subspace of , then the Rademacher functions span a copy of in . However, our construction generates copies of that are essentially different. Actually, they are defined by positive or even disjoint functions, and so the copies of 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 -singular.
On the other hand, it is well known that strongly orthonormal sequences—that are called -orthonormal systems in Definition 2 of —define isometric copies of in spaces of a positive vector measure (see Propositions 8 and 11 in ). In particular, this makes clear that the existence of these sequences imply that is not a Hilbert space, and so is not an -space. However, there is a big class of Banach function spaces that can be represented as of an -valued positive vector measure (see for instance Example 10 in  or Example 8 in ). The -spaces associated to such vector measures are sometimes called -sums of -spaces. In Section 4 of , a first attempt to study -convex subspaces—the natural extension of -copies in this setting—of -sums of -spaces was made. Also, a first analysis of the question of when 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 in spaces of a positive vector measure, and the closely related problem of the existence of in . 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 to a particular class of operators and the structure properties of the space . 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 spaces are normed by the integrals. In this case, it can be shown that there is an easy characterization of strongly -orthonormal sequences, which simplifies the arguments.
Remark 4.1. Suppose that a vector measure satisfies that the set separates the points of and assume that for a given sequence , for every such that and . Then for every . This is a direct consequence of Hölder's inequality and the integrability with respect to of all the functions involved. For the particular case of positive vector measures, the standard basis of plays the role of ; this means that the requirement for all automatically implies that is a strongly m-orthonormal sequence.
Lemma 4.2. Let be a positive vector measure, and suppose that the bounded sequence in satisfies that for all . Then there is a Rybakov measure for such that .
Proof. Take for instance the sequence . Since is positive, the measure is positive and defines a Rybakov measure for . Since for all and the requirement on imply that it is weakly null, we obtain by Hölder's inequality that .
Proposition 4.3. Let be a positive (countably additive) vector measure. Let be a normalized sequence in such that for every , . Then contains a lattice copy of . In particular, there is a subsequence of that is equivalent to the unit vector basis of .
Proof. By Lemma 4.2, we can use Corollary 3.9 to produce a disjoint sequence in that approximates a subsequence of and is strongly -orthogonal. The same computations that can be found in the proof of Proposition 8 in  show that for finite sums , the norm in is equivalent to the norm of in . Consequently, the closure of these finite sums in provides a copy of . The disjointness of implies that in fact it is a lattice copy. Note also that is equivalent to and so to the unit vector basis of .
As a direct consequence, we obtain that for a positive vector measure , the existence of a normalized sequence of functions such that the sequence of square integrals is weakly null implies that cannot be a Hilbert space. On the other hand, if the integration map is compact, then isomorphically (see Proposition 3.48 in ), and thus is (isomorphic to) a Hilbert space. Notice first the following obvious consequence of this fact: is isomorphic to an -space of a finite measure if and only if there is a positive -valued vector vector measure such that such that the integration map is compact; the converse statement is proved by considering the vector measure , . However, as the next example shows, there are spaces for positive -valued vector measures with noncompact integration map that are Hilbert spaces. We will find in Corollary 4.6 that this conclusion— not being an -space, and so not to be a Hilbert space—can be extended to the case of strictly singular integration maps.
Example 4.4. (1) An -valued measure such that is a Hilbert space and the integration map is not compact. Consider the Hilbert space and a orthonormal basis for it. Consider the associated isomorphism that carries each function to the -summable sequence of its fourier coefficients. Take the vector measure given by for each Lebesgue measurable set . Then , although is in fact an isomorphism.
(2) A positive -valued measure with noncompact integration map such that is a Hilbert space. Consider a vector measure as in Example 3.3 and define the positive measure by , . A direct computation shows that the norm in is equivalent to the one in . Then isomorphically, and is clearly noncompact.
Next result shows the consequences on the structure of of our arguments about the existence of strongly orthonormal sequences in .
Theorem 4.5. Let be a positive (countably additive) vector measure. Let be a normalized sequence in such that for every , for all . Then there is a subsequence such that generates an isomorphic copy of in that is preserved by the integration map. Moreover, there is a normalized disjoint sequence that is equivalent to the previous one and gives a lattice copy of in that is preserved by .
Proof. By Corollary 3.9 and Lemma 4.2, there is a (normalized) disjoint sequence in that is equivalent to a subsequence of . Let us prove directly that generates an isomorphic copy of in . Let the vector measure given in Corollary 3.9 and let be the norm of . Since for every there is a subsequence of the one above (that we denote as the previous one) that satisfies that Fix an . We have that, by Hölder inequality, This means that Similar computations give the converse inequality. The construction of and the disjointness of the functions of the sequence give the last statement.
Corollary 4.6. Let be a positive (countably additive) vector measure. The following assertions are equivalent. (1)There is a normalized sequence in satisfying that for all the elements of the canonical basis of . (2)There is an -valued vector measure an isomorphism—such that and there is a disjoint sequence in that is strongly -orthonormal.(3)There is a subspace that is fixed by the integration map which satisfies that there are positive functions such that is an orthonormal basis for .(4)There is an -valued vector measure defined as an isomorphism—such that and a subspace of such that the restriction of to is a lattice isomorphism in .
Proof. is a direct consequence of Theorem 4.5. For , just notice that the strong -orthogonality of a disjoint sequence implies that gives a lattice copy of preserved by the integration map . Since , we obtain that is a basis for .
. There is a bounded sequence in such that , where is a orthonormal basis of closure of the subspace , and an isomorphism from to such that . By composing with the integration map, the copy of that is fixed by can be considered in such a way that . Consequently, for all , and so for all . It is enough to take .
. Take the normalized sequence of positive functions in such that is equivalent to the standard basis of , and define . The weak to weak continuity of gives the result.
We have shown that the existence of lattice copies of in is directly connected with the existence of lattice copies of in . Thus, and summarizing the results in this section, we finish the paper with a complete characterization of this property for of a positive -valued vector measure .
Theorem 4.7. The following assertions for a positive vector measure are equivalent. (1) contains a lattice copy of . (2) has a reflexive infinite dimensional sublattice. (3) has a relatively weakly compact normalized sequence of disjoint functions. (4) contains a weakly null normalized sequence. (5)There is a vector measure defined by such that integration map fixes a copy of . (6)There is a vector measure defined as that is not disjointly strictly singular.
Proof. are obvious. For , 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 ).
. Take a weakly null normalized sequence in . By (the arguments used in) Lemma 4.2 we can find a Rybakov measure for such that . 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 of and a disjoint sequence in the unit sphere of such that . Notice that also converges weakly to , so by taking a subsequence and after restricting the supports of the functions and renorming if necessary, we obtain a normalized weakly null positive disjoint sequence . This gives the copy of that is fixed by the integration map associated to a vector measure satisfying by for the canonical basis of . Finally, and are evident.
The support of the Ministerio de Economía y Competitividad, under Project no.MTM2009-14483-C02-02 (Spain) is gratefully acknowledged.
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