About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
Volumeย 2012ย (2012), Article IDย 357210, 15 pages
http://dx.doi.org/10.1155/2012/357210
Research Article

Lattice Copies of โ„“๐Ÿ in ๐ฟ๐Ÿ of a Vector Measure and Strongly Orthogonal Sequences

Instituto Universitario de Matemรกtica Pura y Aplicada, Universidad Politรฉcnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain

Received 24 February 2012; Accepted 17 July 2012

Academic Editor: John R.ย Akeroyd

Copyright ยฉ 2012 E. Jimรฉnez Fernรกndez and E. A. Sรกnchez Pรฉrez. 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.

Abstract

Let m be an โ„“2-valued (countably additive) vector measure and consider the space ๐ฟ2(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-Pelczyล„ski dichotomy in the domain space and the Bessaga-Pelczyล„ski principle in the range space, we construct a two-sided disjointification method that allows to prove several structure theorems for the spaces ๐ฟ1(m) and ๐ฟ2(m). Under certain requirements, our main result establishes that a normalized sequence in ๐ฟ2(m) with a weakly null sequence of integrals has a subsequence that is strongly m-orthonormal in ๐ฟ2(๐ฆโˆ—), where ๐ฆโˆ— is another โ„“2-valued vector measure that satisfies ๐ฟ2(m)โ€‰=โ€‰๐ฟ2(๐ฆโˆ—). As an application of our technique, we give a complete characterization of when a space of integrable functions with respect to an โ„“2-valued positive vector measure contains a lattice copy of โ„“2.

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 space ๐ฟ2(๐ฆ) of integrable functions with respect to a suitable vector measure ๐ฆ ([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 ๐ฟ2(๐ฆ) that satisfy some orthogonality properties with respect to the vector valued integral become useful both for studying the geometry of the space [3โ€“5] and for applications, mainly in the context of the function approximation [3, 6, 7].

In contrast to the scalar case, several notions of ๐ฆ-orthogonality are possible in the case of an โ„“2-valued (countably additive) vector measure ๐ฆ. A sequence {๐‘“๐‘–}โˆž๐‘–=1 in ๐ฟ2(๐ฆ) is said to be strongly ๐ฆ-orthonormal if the integral of the product of two different functions is 0 and the integral of each ๐‘“2๐‘– is ๐‘’๐‘–, where {๐‘’๐‘–}โˆž๐‘–=1 is an orthonormal sequence in โ„“2. In this paper, we center our attention in this strong version of ๐ฆ-orthogonal sequence, giving a complete characterization of the spaces ๐ฟ2(๐ฆ) in which such sequences exist; actually, we will show that this fact is closely connected to the existence of lattice copies of โ„“2 in the corresponding space ๐ฟ1(๐ฆ) that is preserved by the integration map. In order to do this, we develop a sort of two-sided Kadec-Pelczyล„ski disjointification technique. Roughly speaking, this procedure allows to produce sequences of normalized functions in ๐ฟ2(๐ฆ)โ€”or ๐ฟ1(๐ฆ)โ€”which are almost disjoint and have integrals that are almost orthogonal in โ„“2: after an isomorphic change of vector measure, we obtain our results both for the existence of strongly ๐ฆ-orthogonal sequences in ๐ฟ2(๐ฆ) and the existence of lattice copies of โ„“2 in ๐ฟ1(๐ฆ).

The paper is organized as follows. After the preliminary Section 2, we analyze in Section 3 the existence of strongly orthonormal sequences in ๐ฟ2(๐ฆ), 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 โ„“2-valued measures with disjointly strictly singular integration maps. In the positive, we show in Theorem 3.7 that under reasonable requirements, given an ๐ฆ-orthonormal sequence {๐‘“๐‘–}โˆž๐‘–=1 in ๐ฟ2(๐ฆ), it is possible to construct another vector measure ๐ฆโˆ— such that(1){๐‘“๐‘–}โˆž๐‘–=1 is a strongly ๐ฆโˆ—-orthonormal sequence in ๐ฟ2(๐ฆ),(2)๐ฟ2(๐ฆ)=๐ฟ2(๐ฆโˆ—).

Combining with the Kadec-Pelczyล„ski dichotomy, the requirement on the sequence of being ๐ฆ-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 โ„“2-valued vector measure and are not Hilbert spaces. For the case of positive measures, we show that the existence of strongly ๐ฆ-orthonormal sequences is equivalent of the existence of lattice copies of โ„“4 in ๐ฟ2(๐ฆ) and lattice copies of โ„“2 in ๐ฟ1(๐ฆ) (Proposition 4.3 and Theorem 4.5). The translation of these results for the space ๐ฟ1(๐ฆ) gives the following result on its structure that can also be written in terms of the integration map (Theorem 4.7): the space ๐ฟ1(๐ฆ) 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 โ„“2.

2. Preliminaries

In this section, we introduce several definitions and comments regarding the spaces ๐ฟ2(๐ฆ). We refer to [8] for definitions and basic results on vector measures. Let ๐‘‹ be a Banach space. We will denote by ๐ต๐‘‹ the unit ball of ๐‘‹, that is ๐ต๐‘‹โˆถ={๐‘ฅโˆˆ๐‘‹โˆถโ€–๐‘ฅโ€–๐‘‹โ‰ค1}. ๐‘‹๎…ž will be the topological dual of ๐‘‹. Let ฮฃ be a ๐œŽ-algebra on a nonempty set ฮฉ. Throughout the paper ๐ฆโˆถฮฃโ†’๐‘‹ will be a countably additive vector measure. The semivariation of ๐ฆ is the nonnegative function โ€–๐ฆโ€– whose value on a set ๐ดโˆˆฮฃ is given by โ€–๐ฆโ€–(๐ด)โˆถ=sup{|โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ|(๐ด)โˆถ๐‘ฅ๎…žโˆˆ๐ต๐‘‹โ€ฒ}. The variation |๐ฆ| of ๐ฆ on a measurable set ๐ด is given by โˆ‘|๐ฆ|(๐ด)โˆถ=sup๐ตโˆˆฮ โ€–๐ฆ(๐ต)โ€– for ๐ดโˆˆฮฃ, where the supremum is computed over all finite measurable partitions ฮ  of ๐ด. The variation |๐ฆ| is a monotone countably additive function on ฮฃโ€”a positive scalar measureโ€”, while the semivariation โ€–๐ฆโ€– is a monotone subadditive function on ฮฃ, and for each ๐ดโˆˆฮฃ we have that โ€–๐ฆโ€–(๐ด)โ‰ค|๐ฆ|(๐ด).

For each element ๐‘ฅ๎…žโˆˆ๐‘‹๎…ž, the formula โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ(๐ด)โˆถ=โŸจ๐ฆ(๐ด),๐‘ฅ๎…žโŸฉ, ๐ดโˆˆฮฃ, defines a (countably additive) scalar measure. As usual, we say that a sequence of ฮฃ-measurable functions converges |โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ|-almost everywhere if it converges pointwise in a set ๐ดโˆˆฮฃ such that |โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ|(ฮฉโงต๐ด)=0. A sequence converges ๐ฆ-almost everywhere if it converges in a set ๐ด that satisfies that the semivariation of ๐ฆ in ฮฉโงต๐ด is 0.

Let ๐œ‡ be a positive scalar measure. The measure ๐ฆ is absolutely continuous with respect to ๐œ‡ if lim๐œ‡(๐ด)โ†’0๐ฆ(๐ด)=0; in this case we write ๐ฆโ‰ช๐œ‡ and we say that ๐œ‡ is a control measure for ๐ฆ. Countably additive vector measures always have control measures. It is known that there exists always an element ๐‘ฅ๎…žโˆˆ๐‘‹๎…ž such that ๐ฆโ‰ช|โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ|. We call such a scalar measure a Rybakov measure for ๐ฆ (see [8, Ch.IX,2] ). If |โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ| is a Rybakov measure for ๐ฆ, a sequence of ฮฃ-measurable functions converges ๐ฆ-almost everywhere if and only if it converges |โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ|-almost everywhere.

A ฮฃ-measurable function ๐‘“ is integrable with respect to ๐ฆ if it is integrable with respect to each scalar measure โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ, and for every ๐ดโˆˆฮฃ there is an element โˆซ๐ด๐‘“๐‘‘๐ฆโˆˆ๐‘‹ such that โŸจโˆซ๐ด๐‘“๐‘‘๐ฆ,๐‘ฅ๎…žโˆซโŸฉ=๐ด๐‘“๐‘‘โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ for every ๐‘ฅ๎…žโˆˆ๐‘‹๎…ž. The set of all the (classes of ๐ฆโ€”a.e. equal) ๐ฆโ€”integrable functions ๐ฟ1(๐ฆ) defines an order continuous Banach function space with weak unit ๐œ’ฮฉโ€”in the sense of [9, p.28]โ€”over any Rybakov measure for ๐ฆ that is endowed with the norm โ€–๐‘“โ€–๐ฟ1(๐ฆ)โˆถ=sup๐‘ฅโ€ฒโˆˆ๐ต๐‘‹โ€ฒ๎€œฮฉ||๐‘“||๐‘‘||๎ซ๐ฆ,๐‘ฅ๎…ž๎ฌ||,๐‘“โˆˆ๐ฟ1(๐ฆ).(2.1) The reader can find the definitions and fundamental results concerning the space ๐ฟ1(๐ฆ) in [2, 10โ€“12].

The spaces ๐ฟ๐‘(๐ฆ) are defined extending the definition above in a natural manner [1, 2, 13]. They are ๐‘-convex order continuous Banach function spaces with weak unit ๐œ’ฮฉ over any Rybakov measure, with the norm โ€–๐‘“โ€–๐ฟ๐‘(๐ฆ)โ€–โ€–||๐‘“||โˆถ=๐‘โ€–โ€–๐ฟ1/๐‘1(๐ฆ),๐‘“โˆˆ๐ฟ๐‘(๐ฆ).(2.2) It is also known that if 1/๐‘+1/๐‘ž=1, ๐‘“1โˆˆ๐ฟ๐‘(๐ฆ) and ๐‘“2โˆˆ๐ฟ๐‘ž(๐ฆ), then the pointwise product ๐‘“1โ‹…๐‘“2 belongs to ๐ฟ1(๐ฆ) (see for instance [2, Ch.3]). We will consider the integration operator ๐ผ๐ฆโˆถ๐ฟ1(๐ฆ)โ†’๐‘‹ associated to the vector measure ๐ฆ, that is defined by ๐ผ๐ฆโˆซ(๐‘“)โˆถ=ฮฉ๐‘“๐‘‘๐ฆ, ๐‘“โˆˆ๐ฟ1(๐ฆ). The properties of the integration map have been largely studied in several recent papers (see [2, 14โ€“17] and the references therein). If ๐‘–,๐‘— are indexes of a set ๐ผ, we write ๐›ฟ๐‘–,๐‘— for the Kronecker delta as usual. A sequence {๐‘“๐‘–}โˆž๐‘–=1 in ๐ฟ2(๐ฆ) is called ๐ฆ-orthogonal if โ€–โˆซฮฉ๐‘“๐‘–๐‘“๐‘—๐‘‘๐ฆโ€–=๐›ฟ๐‘–,๐‘—๐‘˜๐‘– for positive constants ๐‘˜๐‘–. If โ€–๐‘“๐‘–โ€–๐ฟ2(๐ฆ)=1 for all ๐‘–โˆˆโ„•, it is called ๐ฆ-orthonormal. The properties of these sequences have been recently analyzed in a series of papers, and some applications have been already developed (see [3โ€“7, 18]). In this paper, we deal with the following more restrictive version of orthogonality for โ„“2-valued measures.

Definition 2.1. Let ๐ฆโˆถฮฃโ†’โ„“2 be a vector measure. We say that {๐‘“๐‘–}โˆž๐‘–=1โŠ‚๐ฟ2(๐ฆ) is a strongly ๐ฆ-orthogonal sequence if โˆซฮฉ๐‘“๐‘–๐‘“๐‘—๐‘‘๐ฆ=๐›ฟ๐‘–,๐‘—๐‘’๐‘–๐‘˜๐‘– for an orthonormal sequence {๐‘’๐‘–}โˆž๐‘–=1 in โ„“2 and for ๐‘˜๐‘–>0. If ๐‘˜๐‘–=1 for every ๐‘–โˆˆโ„•, we say that it is a strongly ๐ฆ-orthonormal sequence.

We need some elements on Banach-lattice-valued vector measures; in particular, on โ„“2-valued measures when the order in โ„“2 is considered. If ๐‘‹ is a Banach lattice, we say that a vector measure ๐ฆโˆถฮฃโ†’๐‘‹ is positive if ๐ฆ(๐ด)โ‰ฅ0 for all ๐ดโˆˆฮฃ. Note that if ๐ฆ is positive and ๐‘ฅ๎…ž is a positive element of the Banach lattice ๐‘‹๎…ž, then the measure โŸจ๐ฆ,๐‘ฅ๎…žโŸฉ 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 โ„“2-singular if this happens for subspaces isomorphic to โ„“2. 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 {๐‘ฅ๐‘›}โˆž๐‘›=1 in a Banach space ๐‘‹ is called a Schauder basis of ๐‘‹ (or simply a basis) if for every ๐‘ฅโˆˆ๐‘‹ there exists a unique sequence of scalars {๐›ผ๐‘›}โˆž๐‘›=1 such that ๐‘ฅ=lim๐‘›โ†’โˆžโˆ‘๐‘›๐‘˜=1๐›ผ๐‘˜๐‘ฅ๐‘˜. A sequence {๐‘ฅ๐‘›}โˆž๐‘›=1 which is a Schauder basis of its closed span is called a basic sequence. Let {๐‘ฅ๐‘›}โˆž๐‘›=1 and {๐‘ฆ๐‘›}โˆž๐‘›=1 be two basis for the Banach spaces ๐‘‹ and ๐‘Œ, respectively. Then {๐‘ฅ๐‘›}โˆž๐‘›=1 and {๐‘ฆ๐‘›}โˆž๐‘›=1 are equivalent if and only if there is an isomorphism between ๐‘‹ and ๐‘Œ that carries each ๐‘ฅ๐‘› to ๐‘ฆ๐‘›.

Let {๐‘ฅ๐‘–}โˆž๐‘–=1 be a basic sequence of a Banach space ๐‘‹ and take two sequences of positive integers {๐‘๐‘–}โˆž๐‘–=1 and {๐‘ž๐‘–}โˆž๐‘–=1 satisfying that ๐‘๐‘–<๐‘ž๐‘–<๐‘๐‘–+1 for every ๐‘–โˆˆโ„•. A block basic sequence {๐‘ฆ๐‘–}โˆž๐‘–=1 associated to {๐‘ฅ๐‘–}โˆž๐‘–=1 is a sequence of vectors of ๐‘‹ defined as finite linear combinations as ๐‘ฆ๐‘–=โˆ‘๐‘ž๐‘–๐‘˜=๐‘๐‘–๐›ผ๐‘–,๐‘˜๐‘ฅ๐‘˜, where ๐›ผ๐‘–,๐‘˜ 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 ๐ฟ2(๐ฆ)

This section is devoted to show how to construct strongly ๐ฆ-orthonormal sequences in ๐ฟ2(๐ฆ). Let us start with an example of the kind of sequences that we are interested in.

Example 3.1 3.1. Let ([0,โˆž),ฮฃ,๐œ‡) be Lebesgue measure space (Figure 1). Let ๐‘Ÿ๐‘˜(๐‘ฅ)โˆถ=sign{sin(2๐‘˜โˆ’1๐‘ฅ)} be the Rademacher function of period 2๐œ‹ defined at the interval ๐ธ๐‘˜=[2(๐‘˜โˆ’1)๐œ‹,2๐‘˜๐œ‹], ๐‘˜โˆˆโ„•. Consider the vector measure ๐ฆโˆถฮฃโ†’โ„“2 given by โˆ‘๐ฆ(๐ด)โˆถ=โˆž๐‘˜=1(โˆ’1/2๐‘˜โˆซ)(๐ดโˆฉ๐ธ๐‘˜๐‘Ÿ๐‘˜๐‘‘๐œ‡)๐‘’๐‘˜โˆˆโ„“2, ๐ดโˆˆฮฃ.
Note that if ๐‘“โˆˆ๐ฟ2(๐ฆ) then โˆซ[0,โˆž)๐‘“๐‘‘๐ฆ=((โˆ’1/2๐‘˜)โˆซ๐ธ๐‘˜๐‘“๐‘Ÿ๐‘˜๐‘‘๐œ‡)๐‘˜โˆˆโ„“2. Consider the sequence of functions ๐‘“1(๐‘ฅ)=sin(๐‘ฅ)โ‹…๐œ’[๐œ‹,2๐œ‹]๐‘“(๐‘ฅ)2๎€ท๐œ’(๐‘ฅ)=sin(2๐‘ฅ)โ‹…[0,2๐œ‹](๐‘ฅ)+๐œ’[(7/2)๐œ‹,4๐œ‹]๎€ธ๐‘“(๐‘ฅ)3(๎€ท๐œ’๐‘ฅ)=sin(4๐‘ฅ)โ‹…[0,4๐œ‹](๐‘ฅ)+๐œ’[(23/4)๐œ‹,6๐œ‹](๎€ธโ‹ฎ๐‘“๐‘ฅ)๐‘›๎€ท2(๐‘ฅ)=sin๐‘›โˆ’1๐‘ฅ๎€ธโ‹…๎€ท๐œ’[0,2(๐‘›โˆ’1)๐œ‹](๐‘ฅ)+๐œ’[(2๐‘›โˆ’2/2๐‘›)๐œ‹,2๐‘›๐œ‹]๎€ธ(๐‘ฅ),๐‘›โ‰ฅ2.(3.1)
This sequence can be used to define a strongly ๐ฆ-orthogonal sequence, since ๎ƒก๎€œ[0,โˆž)๐‘“2๐‘›๐‘‘๐ฆ,๐‘’๐‘›๎ƒข1=โˆ’2๐‘›๎€œ๐ธ๐‘›๐‘“2๐‘›๐‘Ÿ๐‘›๐œ‹๐‘‘๐œ‡=22๐‘›,๎ƒก๎€œ[0,โˆž)๐‘“2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜๎ƒข1=โˆ’2๐‘˜๎€œ๐ธ๐‘˜๐‘“2๐‘›๐‘Ÿ๐‘˜๎ƒก๎€œ๐‘‘๐œ‡=0,โˆ€๐‘˜โ‰ ๐‘›,[0,โˆž)๐‘“๐‘›๐‘“๐‘š๐‘‘๐ฆ,๐‘’๐‘˜๎ƒข1=โˆ’2๐‘˜๎€œ๐ธ๐‘˜๐‘“๐‘›๐‘“๐‘š๐‘Ÿ๐‘˜๐‘‘๐œ‡=0,for๐‘›โ‰ ๐‘šandโˆ€๐‘˜.(3.2) If we define the functions of the sequence {๐น๐‘›}โˆž๐‘›=1 by ๐น๐‘›(๐‘ฅ)โˆถ=(2๐‘›/โˆš๐œ‹)๐‘“๐‘›(๐‘ฅ), we get ๎€œ[0,โˆž)๐น2๐‘›๐‘‘๐ฆ=๐‘’๐‘›๎€œ,โˆ€๐‘›โˆˆโ„•[0,โˆž)๐น๐‘›๐น๐‘˜๐‘‘๐ฆ=0,โˆ€๐‘›,๐‘˜โˆˆโ„•,๐‘›โ‰ ๐‘˜.(3.3)

357210.fig.001
Figure 1: Functions ๐‘“1(๐‘ฅ), ๐‘“2(๐‘ฅ), and ๐‘“3(๐‘ฅ) in Example 3.1.

The starting point of our analysis is the Bessaga-Pelczyล„ski selection principle. It establishes that if {๐‘ฅ๐‘˜}โˆž๐‘˜=1 is a basis of the Banach space ๐‘‹ and {๐‘ฅโˆ—๐‘˜}โˆž๐‘˜=1 is the sequence of coefficient functionals, if we take a normalized sequence {๐‘ฆ๐‘›}โˆž๐‘›=1 such that lim๐‘›โŸจ๐‘ฆ๐‘›,๐‘ฅโˆ—๐‘˜โŸฉ=0, then {๐‘ฆ๐‘›}โˆž๐‘›=1 admits a basic subsequence that is equivalent to a block basic sequence of {๐‘ฅ๐‘›}โˆž๐‘›=1 (see for instance Theorem 3 in [20, 21], Ch.V). We adapt this result for sequences of square integrable functions {๐‘“๐‘›}โˆž๐‘›=1 in order to identify when the sequence of integrals {โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1โŠ‚โ„“2 is a basic sequence in โ„“2. 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 ๐ฆ-orthonormal sequences. The second condition constitutes the key of the problem.

Proposition 3.2. Let ๐ฆโˆถฮฃโ†’โ„“2 be a vector measure, and consider the canonical basis {๐‘’๐‘˜}โˆž๐‘˜=1 of โ„“2. Let {๐‘“๐‘›}โˆž๐‘›=1 be a sequence in ๐ฟ2(๐ฆ). If there is an ๐œ€>0 such that the sequence {โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1 satisfies(1)inf๐‘›โ€–โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆโ€–โ„“2=๐œ€>0, (2)lim๐‘›โŸจโˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0,โˆ€๐‘˜โˆˆโ„•, then {โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1 has a subsequence which is a basic sequence. Moreover, it is equivalent to a block basic sequence of {๐‘’๐‘˜}โˆž๐‘˜=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 (ฮฉ,ฮฃ,๐œ‡) be a probability measure space. Let us consider the following vector measure ๐ฆโˆถฮฃโ†’โ„“2, ๐ฆ(๐ด)โˆถ=โˆž๎“๐‘–=1๐œ‡๎€ท๐ดโˆฉ๐ด๐‘–๎€ธ๐‘’๐‘–โˆˆโ„“2,๐ดโˆˆฮฃ,(3.4) where {๐ด๐‘–}โˆž๐‘–=1 is a disjoint measurable partition of ฮฉ, with ๐œ‡(๐ด๐‘–)โ‰ 0 for all ๐‘–โˆˆโ„•. Notice that โˆซฮฉ๐‘“2โˆ‘๐‘‘๐ฆ=โˆž๐‘–=1(โˆซ๐ด๐‘–๐‘“2๐‘‘๐œ‡)๐‘’๐‘–โˆˆโ„“2 for all ๐‘“โˆˆ๐ฟ2(๐ฆ). Consider a sequence of norm one functions {๐‘“๐‘›}โˆž๐‘›=1 in ๐ฟ2(๐ฆ) such that ๐‘“๐‘›โˆถ=๐‘“๐‘›๐œ’๐ด๐‘› for all ๐‘›. For every ๐‘˜โˆˆโ„•, the following equalities hold: lim๐‘›๎ƒก๎€œฮฉ๐‘“2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜๎ƒข=lim๐‘›๎„”โˆž๎“๐‘–=1๎‚ต๎€œ๐ด๐‘–๐‘“2๐‘›๎‚ถ๐‘’๐‘‘๐œ‡๐‘–,๐‘’๐‘˜๎„•=lim๐‘›๎€œ๐ด๐‘˜๐‘“2๐‘›โ€–โ€–โ€–๎€œ๐‘‘๐œ‡=0,ฮฉ๐‘“2๐‘›โ€–โ€–โ€–๐‘‘๐ฆโ„“2=1โˆ€๐‘›โˆˆโ„•.(3.5)
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 ๐ฆ-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 ๐ฟ1(๐ฆ) is lattice isomorphic to the ๐ฟ1 space of the variation of ๐ฆ, that is a scalar measure (see [2, Prop.3.48]). We need the next formal requirement for the elements of the sequence {๐‘“๐‘›}โˆž๐‘›=1. We say that a function ๐‘“โˆˆ๐ฟ2(๐ฆ) is normed by the integral if โ€–๐‘“โ€–๐ฟ2(๐ฆ)โˆซ=โ€–ฮฉ๐‘“2๐‘‘๐ฆโ€–1/2. This happens for instance when the vector measure ๐ฆ is positive (see [22] or [2, Lemma 3.13]), since in this case the norm can be computed using the formula โ€–๐‘“โ€–๐ฟ1(๐ฆ)โˆซ=โ€–ฮฉ|๐‘“|๐‘‘๐ฆโ€– for all ๐‘“โˆˆ๐ฟ1(๐ฆ). 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 ๐ฆโˆถฮฃโ†’โ„“2 be a countably additive vector measure. If there exists a strongly ๐ฆ-orthonormal sequence in ๐ฟ2(๐ฆ) which elements are normed by the integrals, then the integration operator ๐ผ๐ฆโˆถ๐ฟ1(๐ฆ)โ†’โ„“2 is not compact. To see this, let {๐‘“๐‘–}โˆž๐‘–=1โŠ‚๐ฟ2(๐ฆ) be a strongly ๐ฆ-orthonormal sequence in ๐ฟ2(๐ฆ) and consider a orthonormal sequence {๐‘’๐‘–}โˆž๐‘–=1. Then โˆซฮฉ๐‘“๐‘–๐‘“๐‘—๐‘‘๐ฆ=๐›ฟ๐‘–,๐‘—๐‘’๐‘–, an thus ๎€œฮฉ๐‘“2๐‘–๐‘‘๐ฆ=๐‘’๐‘–=๐ผ๐ฆ๎€ท๐‘“2๐‘–๎€ธ.(3.6) Therefore {๐‘“2๐‘–}โˆž๐‘–=1โŠ‚๐ต๐ฟ1(๐ฆ), and so the sequence {๐ผ๐ฆ(๐‘“2๐‘–)}โˆž๐‘–=1 that satisfies that {๐ผ๐ฆ(๐‘“2๐‘–)}โˆž๐‘–=1โŠ‚๐ผ๐ฆ(๐ต๐ฟ1(๐ฆ))โŠ‚๐ผ๐ฆ(๐ต๐ฟ1(๐ฆ)) does not admit any convergent subsequence. It follows that ๐ผ๐ฆ(๐ต๐ฟ1(๐ฆ)) is not compact and so, ๐ผ๐ฆ(๐ต๐ฟ1(๐ฆ)) is not relatively compact. This allows to conclude that ๐ผ๐ฆ is not compact.

Theorem 3.7 below gives a necessary conditionโ€”and, in a sense, also a sufficient conditionโ€”for the existence of strongly orthonormal sequences in a space of functions ๐ฟ2(๐ฆ) starting from a given ๐ฆ-orthonormal sequence. The existence of such ๐ฆ-orthonormal sequences is always assured: just consider a sequence of normalized disjoint functions in ๐ฟ2(๐ฆ). The following result is an application of the Kadec-Pelczyล„ski disjointification procedure for order continuous Banach function spacesโ€”also called the Kadec-Pelczyล„ski dichotomy, see Theorem 4.1 in [23, 24]โ€”, in the following version, that can be found in [25] (see the comments after Proposition 1.1). Let ๐‘‹(๐œ‡) be an order continuous Banach function space over a finite measure ๐œ‡ with a weak unit (this implies ๐‘‹(๐œ‡)โ†ช๐ฟ1(๐œ‡)). Consider a normalized sequence {๐‘ฅ๐‘›}โˆž๐‘›=1 in ๐‘‹(๐œ‡). Then (1)either {โ€–๐‘ฅ๐‘›โ€–๐ฟ1(๐œ‡)}โˆž๐‘›=1 is bounded away from zero,(2)or there exists a subsequence {๐‘ฅ๐‘›๐‘˜}โˆž๐‘˜=1 and a disjoint sequence {๐‘ง๐‘˜}โˆž๐‘˜=1 in ๐‘‹(๐œ‡) such that โ€–๐‘ง๐‘˜โˆ’๐‘ฅ๐‘›๐‘˜โ€–โ†’๐‘˜0.

Recall that the space ๐ฟ2(๐ฆ) is an order continuous Banach function space over any Rybakov (finite) measure ๐œ‡=|โŸจ๐ฆ,๐‘ฅ๎…ž0โŸฉ| for ๐ฆ.

Proposition 3.5. Let {๐‘”๐‘›}โˆž๐‘›=1 be a normalized sequence in ๐ฟ2(๐ฆ). Suppose that there exists a Rybakov measure ๐œ‡=|โŸจ๐ฆ,๐‘ฅ๎…ž0โŸฉ| for ๐ฆ such that {โ€–๐‘”๐‘›โ€–๐ฟ1(๐œ‡)}โˆž๐‘›=1 is not bounded away from zero. Then there are a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1 and an ๐ฆ-orthonormal sequence {๐‘“๐‘˜}โˆž๐‘˜=1 such that โ€–๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜โ€–๐ฟ2(๐ฆ)โ†’๐‘˜0.

Proof. By the criterion given above, there is a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1 and a disjoint sequence {๐‘“โ€ฒ๐‘˜}โˆž๐‘˜=1 such that โ€–๐‘”๐‘›๐‘˜โˆ’๐‘“๎…ž๐‘˜โ€–๐ฟ2(๐ฆ)โ†’๐‘˜0. Consider the sequence given by the functions ๐‘“๐‘˜โˆถ=๐‘“๎…ž๐‘˜/โ€–๐‘“๎…ž๐‘˜โ€–. Then โ€–๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜โ€–๐ฟ2(๐ฆ)โ†’๐‘˜0. Since โˆซฮฉ๐‘“๐‘˜๐‘“๐‘—๐‘‘๐ฆ=0 for every ๐‘˜โ‰ ๐‘— due to the fact that they are disjoint, we obtain the result.

Although the existence of a strongly ๐ฆ-orthonormal subsequence of an ๐ฆ-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 ๐ฆโˆ— satisfying that ๐ฟ2(๐ฆ)=๐ฟ2(๐ฆโˆ—) and with respect to which there is a subsequence that is strongly ๐ฆโˆ—-orthonormal. We use the following lemma, which proof is elementary (see Lemma 3.27 in [2]).

Lemma 3.6. Let ๐ฆโˆถฮฃโ†’โ„“2 be a vector measure. Let ๐œ‘โˆถโ„“2โ†’๐ป be an isomorphism, where ๐ป is a separable Hilbert space, and consider the vector measure ๐ฆโˆ—=๐œ‘โˆ˜๐ฆ. Then the spaces ๐ฟ2(๐ฆ) and ๐ฟ2(๐ฆโˆ—) are isomorphic, and for every ๐‘“โˆˆ๐ฟ2(๐ฆ), โˆซฮฉ๐‘“2๐‘‘๐ฆโˆ—โˆซ=๐œ‘(ฮฉ๐‘“2๐‘‘๐ฆ).

Theorem 3.7. Let us consider a vector measure ๐ฆโˆถฮฃโ†’โ„“2 and an ๐ฆ-orthonormal sequence {๐‘“๐‘›}โˆž๐‘›=1 of functions in ๐ฟ2(๐ฆ) that are normed by the integrals. Let {๐‘’๐‘›}โˆž๐‘›=1 be the canonical basis of โ„“2. If lim๐‘›โŸจโˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0 for every ๐‘˜โˆˆโ„•, then there exists a subsequence {๐‘“๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘“๐‘›}โˆž๐‘›=1 and a vector measure ๐ฆโˆ—โˆถฮฃโ†’โ„“2 such that {๐‘“๐‘›๐‘˜}โˆž๐‘˜=1 is strongly ๐ฆโˆ—-orthonormal.
Moreover, ๐ฆโˆ— can be chosen to be as ๐ฆโˆ—=๐œ™โˆ˜๐ฆ for some Banach space isomorphism ๐œ™ from โ„“2 onto โ„“2, and so ๐ฟ2(๐ฆ)=๐ฟ2(๐ฆโˆ—).

Proof. Consider an ๐ฆ-orthonormal sequence {๐‘“๐‘›}โˆž๐‘›=1 in ๐ฟ2(๐ฆ) and the sequence of integrals {โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1. As an application of Proposition 3.2, we get a subsequence {โˆซฮฉ๐‘“2๐‘›๐‘˜๐‘‘๐ฆ}โˆž๐‘˜=1 that is equivalent to a block basic sequence {๐‘’๎…ž๐‘›๐‘˜}โˆž๐‘˜=1 of the canonical basis of โ„“2. Recall that according to the notation given in Section 2, ๐›ผ๐‘–,๐‘— are the constants that appear in the definition of the block basic sequence. Associated to this sequence, there is an isomorphism ๐œ‘๐ดโˆถ=๎€ท๐‘’span๎…ž๐‘›๐‘˜๎€ธโ„“2๐œ‘โˆ’โ†’๐ตโˆถ=๎‚ต๎€œspanฮฉ๐‘“2๐‘›๐‘˜๎‚ถ๐‘‘๐ฆโ„“2(3.7) such that ๐œ‘(๐‘’๎…ž๐‘›๐‘˜โˆซ)โˆถ=ฮฉ๐‘“2๐‘›๐‘˜๐‘‘๐ฆ, ๐‘˜โˆˆโ„•.
We can suppose without loss of generality that the elements of the sequence {๐‘’๎…ž๐‘›๐‘˜}โˆž๐‘˜=1 have norm one. To see this, it is enough to consider the following inequalities. First note that there are positive constants ๐‘„ and ๐พ such that for every ๐‘›โˆˆโ„•, โˆซ๐‘„=๐‘„โ€–ฮฉ๐‘“2๐‘›๐‘˜๐‘‘๐ฆโ€–โ‰คโ€–๐‘’๎…ž๐‘›๐‘˜โˆซโ€–โ‰ค๐พโ€–ฮฉ๐‘“2๐‘›๐‘˜๐‘‘๐ฆโ€–=๐พ as a consequence of the existence of the isomorphism ๐œ‘. Let {๐œ†๐‘–}โˆž๐‘–=1 be a sequence of real numbers. Then โ€–โ€–โ€–โ€–โˆž๎“๐‘–=1๐œ†๐‘–๐‘’๎…ž๐‘–โ€–โ€–๐‘’๎…ž๐‘–โ€–โ€–โ€–โ€–โ€–โ€–22=โˆž๎“๐‘–=1||๐œ†๐‘–||2๎‚€โˆ‘๐‘ž๐‘–๐‘—=๐‘๐‘–||๐›ผ๐‘–,๐‘—||2๎‚โ€–โ€–๐‘’๎…ž๐‘–โ€–โ€–2=โˆž๎“๐‘ž๐‘–=1๐‘–๎“๐‘—=๐‘๐‘–||๐œ†๐‘–||2||๐›ผ๐‘–,๐‘—||2โ€–โ€–๐‘’๎…ž๐‘–โ€–โ€–2.(3.8) The existence of an upper and a lower bound for the real numbers โ€–๐‘’๎…ž๐‘–โ€– given above provides the equivalence between this quantity and โ€–โˆ‘โˆž๐‘–=1๐œ†๐‘–๐‘’๎…ž๐‘–โ€–22 for each sequence of real numbers {๐œ†๐‘–}โˆž๐‘–=1.
Since each closed subspace of a Hilbert space is complemented, there is a subspace ๐ต๐‘ such that โ„“2=๐ตโŠ•2๐ต๐‘ isometrically, where this direct sum space is considered as a Hilbert space (with the adequate Hilbert space norm). We write ๐‘ƒ๐ต and ๐‘ƒ๐ต๐‘ for the corresponding projections. Let us consider the linear map ๐œ™โˆถ=๐œ‘โˆ’1โŠ•Idโˆถ๐ตโŠ•2๐ต๐‘๐œ™โˆ’โ†’๐ดโŠ•2๐ต๐‘, where Idโˆถ๐ต๐‘โ†’๐ต๐‘ is the identity map.
Note that ๐ปโˆถ=๐ดโŠ•2๐ต๐‘ is a Hilbert space with the scalar product ๎ซ๐‘ฅ+๐‘ฆ,๐‘ฅ๎…ž+๐‘ฆ๎…ž๎ฌ๐ป=๎ซ๐‘ฅ,๐‘ฅ๎…ž๎ฌ๐ป+๎ซ๐‘ฆ,๐‘ฆ๎…ž๎ฌ๐ป,๐‘ฅ+๐‘ฆ,๐‘ฅ๎…ž+๐‘ฆ๎…žโˆˆ๐ดโŠ•2๐ต๐‘,(3.9) that can be identified with โ„“2. Obviously, ๐œ™ is an isomorphism. Let us consider now the vector measure ๐ฆโˆ—โˆถ=๐œ™โˆ˜๐ฆโˆถฮฃ๐ฆโˆ’โ†’โ„“2๐œ™โˆ’โ†’๐ดโŠ•2๐ต๐‘. By Lemma 3.6, ๐ฟ2(๐ฆ)=๐ฟ2(๐œ™โˆ˜๐ฆ)=๐ฟ2(๐ฆโˆ—). Let us show that {๐‘“๐‘›๐‘˜}โˆž๐‘˜=1 is a strongly ๐ฆโˆ—-orthonormal sequence. We consider the orthonormal sequence {(๐‘’๎…ž๐‘›๐‘˜,0)}โˆž๐‘˜=1 in ๐ป. The first condition in the definition of strongly orthonormal sequence is fulfilled, since ๎€œฮฉ๐‘“2๐‘›๐‘˜๐‘‘๐ฆโˆ—=๎€œฮฉ๐‘“2๐‘›๐‘˜๎‚ต๐‘ƒ๐‘‘(๐œ™โˆ˜๐ฆ)=๐œ™๐ต๎‚ต๎€œฮฉ๐‘“2๐‘›๐‘˜๎‚ถ๐‘‘๐ฆ,๐‘ƒ๐ต๐‘๎‚ต๎€œฮฉ๐‘“2๐‘›๐‘˜=๎‚ต๐œ‘๐‘‘๐ฆ๎‚ถ๎‚ถโˆ’1๎‚ต๎€œฮฉ๐‘“2๐‘›๐‘˜๎‚ถ๎‚ถ=๎€ท๐‘’๐‘‘๐ฆ,0๎…ž๐‘›๐‘˜๎€ธ,,0(3.10) for every ๐‘˜โˆˆโ„•. The second one is given by the following calculations. For ๐‘˜โ‰ ๐‘™, โ€–โ€–โ€–๎€œฮฉ๐‘“๐‘›๐‘˜๐‘“๐‘›๐‘™๐‘‘๐ฆโˆ—โ€–โ€–โ€–=โ€–โ€–โ€–๎€œฮฉ๐‘“๐‘›๐‘˜๐‘“๐‘›๐‘™โ€–โ€–โ€–=โ€–โ€–โ€–๐œ™๎‚ต๎€œ๐‘‘(๐œ™โˆ˜๐ฆ)ฮฉ๐‘“๐‘›๐‘˜๐‘“๐‘›๐‘™๎‚ถโ€–โ€–โ€–=๐‘‘๐ฆโ€–๐œ™(0)โ€–=0,(3.11) since ๐œ™ is continuous and {๐‘“๐‘›๐‘˜}โˆž๐‘˜=1 is an ๐ฆ-orthonormal sequence. Thus we get โˆซฮฉ๐‘“๐‘›๐‘˜๐‘“๐‘›๐‘™๐‘‘๐ฆโˆ—=0. This proves the theorem.

Remark 3.8. In a certain sense, the converse of Theorem 3.7 also holds. Take as the vector measure ๐ฆโˆ— the measure ๐ฆ itself with values in โ„“2 and consider the canonical basis {๐‘’๐‘›}โˆž๐‘›=1. Clearly, every strongly ๐ฆ-orthonormal sequence is ๐ฆ-orthonormal and satisfies the condition lim๐‘›โŸจโˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆโˆ—,๐‘’๐‘˜โŸฉ=0, since ๎ƒก๎€œฮฉ๐‘“2๐‘›๐‘‘๐ฆโˆ—,๐‘’๐‘˜๎ƒข=โŸจ๐‘’๐‘›,๐‘’๐‘˜โŸฉ=0,๐‘˜โ‰ ๐‘›.(3.12)

Corollary 3.9. Let ๐ฆโˆถฮฃโ†’โ„“2 be a countably additive vector measure. Let {๐‘”๐‘›}โˆž๐‘›=1 be a normalized sequence of functions in ๐ฟ2(๐ฆ) that are normed by the integrals. Suppose that there exists a Rybakov measure ๐œ‡=|โŸจ๐ฆ,๐‘ฅ๎…ž0โŸฉ| for ๐ฆ such that {โ€–๐‘”๐‘›โ€–๐ฟ1(๐œ‡)}โˆž๐‘›=1 is not bounded away from zero. If lim๐‘›โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0 for every ๐‘˜โˆˆโ„•, then there is a (disjoint) sequence {๐‘“๐‘˜}โˆž๐‘˜=1 such that (1)lim๐‘˜โ€–๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜โ€–๐ฟ2(๐ฆ)=0 for a given subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1, and(2)it is strongly ๐ฆโˆ—-orthonormal for a certain Hilbert space valued vector measure ๐ฆโˆ— defined as in Theorem 3.7 that satisfies that ๐ฟ2(๐ฆ)=๐ฟ2(๐ฆโˆ—).

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 {๐‘”๐‘›}โˆž๐‘›=1 are normed by the integrals.

4. Applications: Copies of โ„“2 in ๐ฟ1(๐ฆ) That Are Preserved by the Integration Map

One of the consequences of the results of the previous section is that the existence of strongly ๐ฆ-orthonormal sequences in ๐ฟ2(๐ฆ) is closely related to the existence of lattice copies of โ„“2 in ๐ฟ1(๐ฆ). In this section, we show how to apply our arguments for finding some information on the structure of the spaces ๐ฟ1(๐ฆ) 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 โ„“2. 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-Pelczyล„ski 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 spacesโ€”๐ฟ๐‘-spaces, Lorentz spaces, Orlicz spaces, and general rearrangement invariant (r.i.) spacesโ€”these copies are related to subspaces generated by Rademacher-type sequences (see [27โ€“30] and the references therein). For instance, Corollary 2 in [27] states that for a r.i. Banach function space ๐ธ on [0,1], if the norms on ๐ธ and ๐ฟ1 are equivalent on some infinite dimensional subspace of ๐ธ, then the Rademacher functions span a copy of โ„“2 in ๐ธ. However, our construction generates copies of โ„“2 that are essentially different. Actually, they are defined by positive or even disjoint functions, and so the copies of โ„“2 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 โ„“2-singular.

On the other hand, it is well known that strongly orthonormal sequencesโ€”that are called ๐œ†-orthonormal systems in Definition 2 of [5]โ€”define isometric copies of โ„“4 in spaces ๐ฟ2(๐ฆ) of a positive vector measure ๐ฆ (see Propositions 8 and 11 in [5]). In particular, this makes clear that the existence of these sequences imply that ๐ฟ2(๐ฆ) is not a Hilbert space, and so ๐ฟ1(๐ฆ) is not an ๐ฟ1-space. However, there is a big class of Banach function spaces that can be represented as ๐ฟ1(๐ฆ) of an โ„“2-valued positive vector measure ๐ฆ (see for instance Example 10 in [5] or Example 8 in [4]). The ๐ฟ1(๐ฆ)-spaces associated to such vector measures are sometimes called โ„“-sums of ๐ฟ1-spaces. In Section 4 of [31], a first attempt to study 2-convex subspacesโ€”the natural extension of โ„“2-copies in this settingโ€”of โ„“-sums of ๐ฟ1-spaces was made. Also, a first analysis of the question of when ๐ฟ1(๐ฆ) is a Hilbert spaceโ€”based on the behavior of specific sequences tooโ€”was made in [11, Sectionโ€‰โ€‰4]. In what follows, we provide more information on the existence of copies of โ„“2 in spaces ๐ฟ1(๐ฆ) of a positive vector measure, and the closely related problem of the existence of โ„“4 in ๐ฟ2(๐ฆ). 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 ๐ผ๐ฆ to a particular class of operators and the structure properties of the space ๐ฟ1(๐ฆ). 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 mapโ€”mainly concavity and positive p-summing type propertiesโ€”, we refer to [33, 34] and [2, Ch.6].

For the aim of clarity, in this section we deal with positive vector measures, thatโ€”as we said in the previous sectionโ€”satisfy that all the elements of the spaces ๐ฟ2(๐ฆ) are normed by the integrals. In this case, it can be shown that there is an easy characterization of strongly ๐ฆ-orthonormal sequences, which simplifies the arguments.

Remark 4.1. Suppose that a vector measure satisfies that the set ๐‘ƒโˆถ={๐‘ฅโ€ฒโˆˆโ„“2โˆถโŸจ๐ฆ,๐‘ฅโ€ฒโŸฉโ‰ฅ0} separates the points of โ„“2 and assume that for a given sequence {๐‘“๐‘›}โˆž๐‘›=1, โŸจโˆซฮฉ๐‘“2๐‘›โˆซ๐‘‘๐ฆ,๐‘ฅโ€ฒโŸฉโ‹…โŸจฮฉ๐‘“2๐‘˜๐‘‘๐ฆ,๐‘ฅโ€ฒโŸฉ=0 for every ๐‘›,๐‘˜โˆˆโ„• such that ๐‘›โ‰ ๐‘˜ and ๐‘ฅโ€ฒโˆˆ๐‘ƒ. Then โˆซฮฉ๐‘“๐‘›๐‘“๐‘˜๐‘‘๐ฆ=0 for every ๐‘›โ‰ ๐‘˜. This is a direct consequence of Hรถlder's inequality and the integrability with respect to ๐ฆ of all the functions involved. For the particular case of positive vector measures, the standard basis {๐‘’๐‘›}โˆž๐‘›=1 of โ„“2 plays the role of ๐‘ƒ; this means that the requirement โˆซฮฉ๐‘“2๐‘›๐‘‘๐ฆ=๐‘’๐‘› for all ๐‘› automatically implies that {๐‘“๐‘›}โˆž๐‘›=1 is a strongly m-orthonormal sequence.

Lemma 4.2. Let ๐ฆโˆถฮฃโ†’โ„“2 be a positive vector measure, and suppose that the bounded sequence {๐‘”๐‘›}โˆž๐‘›=1 in ๐ฟ2(๐ฆ) satisfies that lim๐‘›โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0 for all ๐‘˜โˆˆโ„•. Then there is a Rybakov measure ๐œ‡ for ๐ฆ such that lim๐‘›โ€–๐‘”๐‘›โ€–๐ฟ1(๐œ‡)=0.

Proof. Take for instance the sequence ๐‘ฅ๎…ž0={(1/2)๐‘›/2}โˆž๐‘›=1โˆˆโ„“2. Since ๐ฆ is positive, the measure ๐œ‡โˆถ=โŸจ๐ฆ,๐‘ฅ๎…ž0โŸฉ is positive and defines a Rybakov measure for ๐ฆ. Since โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘ฅ๎…ž0โŸฉ=โ€–๐‘”๐‘›โ€–2๐ฟ2(๐œ‡) for all ๐‘›โˆˆโ„• and the requirement on {โˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1 imply that it is weakly null, we obtain by Hรถlder's inequality that lim๐‘›โ€–๐‘”๐‘›โ€–๐ฟ1(๐œ‡)โ‰คlim๐‘›โ€–๐‘”๐‘›โ€–๐ฟ2(๐œ‡)โ€–๐ฆโ€–1/2=0.

Proposition 4.3. Let ๐ฆโˆถฮฃโ†’โ„“2 be a positive (countably additive) vector measure. Let {๐‘”๐‘›}โˆž๐‘›=1 be a normalized sequence in ๐ฟ2(๐ฆ) such that for every ๐‘˜โˆˆโ„•, lim๐‘›โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0. Then ๐ฟ2(๐ฆ) contains a lattice copy of โ„“4. In particular, there is a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1 that is equivalent to the unit vector basis of โ„“4.

Proof. By Lemma 4.2, we can use Corollary 3.9 to produce a disjoint sequence {๐‘“๐‘˜}โˆž๐‘˜=1 in ๐ฟ2(๐ฆ) that approximates a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1 and is strongly ๐ฆโˆ—-orthogonal. The same computations that can be found in the proof of Proposition 8 in [5] show that for finite sums โˆ‘๐‘๐‘˜=1๐›ผ๐‘˜๐‘“๐‘˜, the norm in ๐ฟ2(๐ฆ) is equivalent to the norm of {๐›ผ๐‘˜}๐‘๐‘˜=1 in โ„“4. Consequently, the closure of these finite sums in ๐ฟ2(๐ฆ) provides a copy of โ„“4. The disjointness of {๐‘“๐‘˜}โˆž๐‘˜=1 implies that in fact it is a lattice copy. Note also that {๐‘“๐‘˜}โˆž๐‘˜=1 is equivalent to {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 and so to the unit vector basis of โ„“4.

As a direct consequence, we obtain that for a positive vector measure ๐ฆ, the existence of a normalized sequence of functions such that the sequence of square integrals is weakly null implies that ๐ฟ2(๐ฆ) cannot be a Hilbert space. On the other hand, if the integration map is compact, then ๐ฟ1(๐ฆ)=๐ฟ1(|๐ฆ|) isomorphically (see Proposition 3.48 in [2]), and thus ๐ฟ2(๐ฆ) is (isomorphic to) a Hilbert space. Notice first the following obvious consequence of this fact: ๐ฟ1(๐ฆ)is isomorphic to an ๐ฟ1(๐œ‡)-space of a finite measure ๐œ‡ if and only if there is a positive โ„“2-valued vector vector measure ๐ฆ0โ€‰โ€‰such that ๐ฟ1(๐ฆ)=๐ฟ1(๐ฆ0)โ€‰โ€‰such that the integration map is compact; the converse statement is proved by considering the vector measure ๐‘›(๐ด)โˆถ=๐œ‡(๐ด)๐‘’1โˆˆโ„“2, ๐ดโˆˆฮฃ. However, as the next example shows, there are spaces ๐ฟ2(๐ฆ) for positive โ„“2-valued vector measures with noncompact integration map that are Hilbert spaces. We will find in Corollary 4.6 that this conclusionโ€”๐ฟ1(๐ฆ) not being an ๐ฟ1-space, and so ๐ฟ2(๐ฆ) not to be a Hilbert spaceโ€”can be extended to the case of strictly singular integration maps.

Example 4.4. (1) An โ„“2-valued measure such that ๐ฟ1(๐ฆ) is a Hilbert space and the integration map is not compact. Consider the Hilbert space ๐ฟ2[0,1] and a orthonormal basis ๐‘† for it. Consider the associated isomorphism ๐œ™๐‘†โˆถ๐ฟ2[0,1]โ†’โ„“2 that carries each function to the 2-summable sequence of its fourier coefficients. Take the vector measure ๐‘š๐‘†โˆถฮฃโ†’โ„“2 given by ๐‘š๐‘†(๐ด)โˆถ=๐œ™๐‘†(๐œ’๐ด) for each Lebesgue measurable set ๐ดโˆˆฮฃ. Then ๐ฟ1(๐‘š๐‘†)=๐ฟ2[0,1], although ๐ผ๐‘š๐‘† is in fact an isomorphism.
(2) A positive โ„“2-valued measure with noncompact integration map such that ๐ฟ1(๐ฆ) is a Hilbert space. Consider a vector measure ๐ฆ as in Example 3.3 and define the positive measure ๐‘›โˆถฮฃโ†’โ„“2 by ๐‘›(๐ด)โˆถ=๐œ‡(๐ด)๐‘’1+๐ฆ(๐ด), ๐ดโˆˆฮฃ. A direct computation shows that the norm in ๐ฟ1(๐‘›) is equivalent to the one in ๐ฟ1(๐œ‡). Then ๐ฟ2(๐‘›)=๐ฟ2(๐œ‡) isomorphically, and ๐ผ๐‘›โˆถฮฃโ†’โ„“2 is clearly noncompact.

Next result shows the consequences on the structure of ๐ฟ1(๐ฆ) of our arguments about the existence of strongly orthonormal sequences in ๐ฟ2(๐ฆ).

Theorem 4.5. Let ๐ฆโˆถฮฃโ†’โ„“2 be a positive (countably additive) vector measure. Let {๐‘”๐‘›}โˆž๐‘›=1 be a normalized sequence in ๐ฟ2(๐ฆ) such that for every ๐‘˜โˆˆโ„•, lim๐‘›โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0 for all ๐‘˜โˆˆโ„•. Then there is a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 such that {๐‘”2๐‘›๐‘˜}โˆž๐‘˜=1 generates an isomorphic copy of โ„“2 in ๐ฟ1(๐ฆ) that is preserved by the integration map. Moreover, there is a normalized disjoint sequence {๐‘“๐‘˜}โˆž๐‘˜=1 that is equivalent to the previous one and {๐‘“2๐‘˜}โˆž๐‘˜=1 gives a lattice copy of โ„“2 in ๐ฟ1(๐ฆ) that is preserved by ๐ผ๐ฆโˆ—.

Proof. By Corollary 3.9 and Lemma 4.2, there is a (normalized) disjoint sequence {๐‘“๐‘˜}โˆž๐‘˜=1 in ๐ฟ2(๐ฆ) that is equivalent to a subsequence {๐‘”๐‘›๐‘˜}โˆž๐‘˜=1 of {๐‘”๐‘›}โˆž๐‘›=1. Let us prove directly that {๐‘”2๐‘›๐‘˜}โˆž๐‘˜=1 generates an isomorphic copy of โ„“2 in ๐ฟ1(๐ฆ). Let ๐ฆโˆ—=๐œ™โˆ˜๐ฆ the vector measure given in Corollary 3.9 and let ๐พ be the norm of ๐œ™โˆ’1. Since lim๐‘˜โ€–๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜โ€–๐ฟ2(๐ฆ)=0 for every ๐œ€>0 there is a subsequence of the one above (that we denote as the previous one) that satisfies that ๎ƒฉ๐‘›๎“๐‘˜=1โ€–โ€–โ€–๎€œฮฉ|๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜|2โ€–โ€–โ€–๎ƒช๐‘‘๐ฆ1/2<๐œ€2.(4.1) Fix an ๐œ€>0. We have that, by Hรถlder inequality, ๎ƒฉ๐‘›๎“๐‘˜=1โ€–โ€–โ€–๎€œฮฉ||๐‘”2๐‘›๐‘˜โˆ’๐‘“2๐‘˜||โ€–โ€–โ€–๐‘‘๐ฆ2โ„“2๎ƒช1/2=๎ƒฉ๐‘›๎“๐‘˜=1โ€–โ€–โ€–๎€œฮฉ||๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜||โ‹…||๐‘”๐‘›๐‘˜+๐‘“๐‘˜||โ€–โ€–โ€–๐‘‘๐ฆ2โ„“2๎ƒช1/2โ‰ค๎ƒฉ๐‘›๎“๐‘˜=1โ€–โ€–โ€–๎€œฮฉ||๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜||2โ€–โ€–โ€–โ‹…โ€–โ€–โ€–๎€œ๐‘‘๐ฆฮฉ||๐‘”๐‘›๐‘˜+๐‘“๐‘˜||2โ€–โ€–โ€–๎ƒช๐‘‘๐ฆ1/2๎ƒฉโ‰ค2๐‘›๎“๐‘˜=1โ€–โ€–โ€–๎€œฮฉ||๐‘”๐‘›๐‘˜โˆ’๐‘“๐‘˜||2โ€–โ€–โ€–๎ƒช๐‘‘๐ฆ1/2<๐œ€.(4.2) This means that โ€–โ€–โ€–โ€–๎€œฮฉ|||||๐‘›๎“๐‘˜=1๐›ผ๐‘˜๐‘”2๐‘›๐‘˜|||||โ€–โ€–โ€–โ€–๐‘‘๐ฆโ„“2โ‰คโ€–โ€–โ€–โ€–๎€œฮฉ|||||๐‘›๎“๐‘˜=1๐›ผ๐‘˜๎€ท๐‘”2๐‘›๐‘˜โˆ’๐‘“2๐‘˜๎€ธ|||||โ€–โ€–โ€–โ€–๐‘‘๐ฆโ„“2+โ€–โ€–โ€–โ€–๎€œฮฉ๐‘›๎“๐‘˜=1||๐›ผ๐‘˜||๐‘“2๐‘˜โ€–โ€–โ€–โ€–๐‘‘๐ฆโ„“2โ‰ค๎ƒฉ๐‘›๎“๐‘˜=1๐›ผ2๐‘˜๎ƒช1/2โ€–โ€–โ€–โ€–๎€œ๐œ€+๐พฮฉ๐‘›๎“๐‘˜=1||๐›ผ๐‘˜||๐‘“2๐‘˜๐‘‘๐ฆโˆ—โ€–โ€–โ€–โ€–โ„“2๎ƒฉโ‰ค(๐œ€+๐พ)๐‘›๎“๐‘˜=1๐›ผ2๐‘˜๎ƒช1/2.(4.3) Similar computations give the converse inequality. The construction of ๐ฆโˆ— and the disjointness of the functions of the sequence {๐‘“2๐‘˜}โˆž๐‘˜=1 give the last statement.

Corollary 4.6. Let ๐ฆโˆถฮฃโ†’โ„“2 be a positive (countably additive) vector measure. The following assertions are equivalent. (1)There is a normalized sequence in ๐ฟ2(๐ฆ) satisfying that lim๐‘›โŸจโˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆ,๐‘’๐‘˜โŸฉ=0 for all the elements of the canonical basis {๐‘’๐‘˜}โˆž๐‘˜=1 of โ„“2. (2)There is an โ„“2-valued vector measure ๐ฆโˆ—=๐œ™โˆ˜๐ฆ---๐œ™ an isomorphismโ€”such that ๐ฟ2(๐ฆ)=๐ฟ2(๐ฆโˆ—) and there is a disjoint sequence in ๐ฟ2(๐ฆ) that is strongly ๐ฆโˆ—-orthonormal.(3)There is a subspace ๐‘†โŠ†๐ฟ1(๐ฆ) that is fixed by the integration map ๐ผ๐ฆ which satisfies that there are positive functions โ„Ž๐‘›โˆˆ๐‘† such that {โˆซฮฉโ„Ž๐‘›๐‘‘๐ฆ}โˆž๐‘›=1 is an orthonormal basis for ๐ผ๐ฆ(๐‘†).(4)There is an โ„“2-valued vector measure ๐ฆโˆ— defined as ๐ฆโˆ—=๐œ™โˆ˜๐ฆ---๐œ™ an isomorphismโ€”such that ๐ฟ1(๐ฆ)=๐ฟ1(๐ฆโˆ—) and a subspace ๐‘† of ๐ฟ1(๐ฆ) such that the restriction of ๐ผ๐ฆโˆ— to ๐‘† is a lattice isomorphism in โ„“2.

Proof. (1)โ‡’(2) is a direct consequence of Theorem 4.5. For (2)โ‡’(3), just notice that the strong ๐ฆโˆ—-orthogonality of a disjoint sequence {๐‘”๐‘›}โˆž๐‘›=1 implies that {๐‘”2๐‘›}โˆž๐‘›=1 gives a lattice copy of โ„“2 preserved by the integration map ๐ผ๐ฆโˆ—. Since ๐œ™โˆ’1โˆ˜๐ฆโˆ—=๐ฆ, we obtain that {๐œ™โˆ’1(โˆซฮฉ๐‘”2๐‘›๐‘‘๐ฆโˆ—)}โˆž๐‘›=1โˆซ={ฮฉ๐‘”2๐‘›๐‘‘๐ฆ}โˆž๐‘›=1 is a basis for โ„“2.
(3)โ‡’(1). There is a bounded sequence {โ„Ž๐‘›}โˆž=1 in ๐ฟ1(๐ฆ) such that โˆซฮฉโ„Ž๐‘›๐‘‘๐ฆ=๐‘Ž๐‘›, where ๐‘Ž๐‘› is a orthonormal basis of closure of the subspace โˆซ๐ดโˆถ=span{ฮฉโ„Ž๐‘›๐‘‘๐ฆโˆถ๐‘›โˆˆโ„•}, and an isomorphism ๐œ™ from ๐ด to โ„“2 such that โˆซ๐œ™(ฮฉโ„Ž๐‘›๐‘‘๐ฆ)=๐‘’๐‘›. By composing with ๐œ™ the integration map, the copy of โ„“2 that is fixed by ๐ผ๐ฆ can be considered in such a way that โˆซ๐œ™(ฮฉ๐‘”2๐‘›๐‘‘๐ฆ)=๐‘’๐‘›. Consequently, lim๐‘›โˆซโŸจ๐œ™(ฮฉโ„Ž๐‘›๐‘‘๐ฆ),๐‘’๐‘˜โŸฉ=0 for all ๐‘˜โˆˆโ„•, and so lim๐‘›โŸจ๐‘’๐‘˜,โˆซฮฉโ„Ž๐‘›๐‘‘๐ฆโŸฉ=0 for all ๐‘˜โˆˆโ„•. It is enough to take ๐‘”๐‘›=โ„Ž๐‘›1/2.
(3)โ‡’(4) is obvious.
(4)โ‡’(1). Take the normalized sequence of positive functions {โ„Ž๐‘›}โˆž๐‘›=1 in ๐‘† such that {โˆซฮฉโ„Ž๐‘›๐‘‘๐œ™โˆ˜๐ฆ}โˆž๐‘›=1 is equivalent to the standard basis of โ„“2, and define ๐‘”๐‘›โˆถ=โ„Ž๐‘›1/2. The weak to weak continuity of ๐œ™ gives the result.

We have shown that the existence of lattice copies of โ„“4 in ๐ฟ2(๐ฆ) is directly connected with the existence of lattice copies of โ„“2 in ๐ฟ1(๐ฆ). Thus, and summarizing the results in this section, we finish the paper with a complete characterization of this property for ๐ฟ1(๐ฆ) of a positive โ„“2-valued vector measure ๐ฆ.

Theorem 4.7. The following assertions for a positive vector measure ๐ฆโˆถฮฃโ†’โ„“2 are equivalent. (1)๐ฟ1(๐ฆ) contains a lattice copy of โ„“2. (2)๐ฟ1(๐ฆ) has a reflexive infinite dimensional sublattice. (3)๐ฟ1(๐ฆ) has a relatively weakly compact normalized sequence of disjoint functions. (4)๐ฟ1(๐ฆ) contains a weakly null normalized sequence. (5)There is a vector measure ๐ฆโˆ— defined by ๐ฆโˆ—=๐œ™โˆ˜๐ฆ such that integration map ๐ผ๐ฆโˆ— fixes a copy of โ„“2. (6)There is a vector measure ๐ฆโˆ— defined as ๐ฆโˆ—=๐œ™โˆ˜๐ฆ that is not disjointly strictly singular.

Proof. (1)โ‡’(2)โ‡’(3) are obvious. For (3)โ‡’(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)โ‡’(5). Take a weakly null normalized sequence {๐‘”๐‘—}๐‘›๐‘—=1 in ๐ฟ1(๐ฆ). By (the arguments used in) Lemma 4.2 we can find a Rybakov measure ๐œ‡ for ๐ฆ such that โ€–๐‘”๐‘—โ€–๐ฟ1(๐œ‡)โ†’0. Now we use the same arguments that lead to Theorem 3.7 and Corollary 3.9; by the Kadec-Pelczyล„ski dichotomy, there exists a subsequence {๐‘”๐‘—๐‘™}โˆž๐‘™=1 of {๐‘”๐‘—}โˆž๐‘—=1 and a disjoint sequence {๐‘ง๐‘™}โˆž๐‘™=1 in the unit sphere of ๐ฟ1(๐ฆ) such that lim๐‘™โ€–๐‘”๐‘—๐‘™โˆ’๐‘ง๐‘™โ€–=0. Notice that {๐‘ง๐‘™}โˆž๐‘™=1 also converges weakly to 0, so by taking a subsequence and after restricting the supports of the functions ๐‘ง๐‘™ and renorming if necessary, we obtain a normalized weakly null positive disjoint sequence {๐‘ฃ๐‘˜}โˆž๐‘˜=1. This gives the copy of โ„“2 that is fixed by the integration map associated to a vector measure ๐ฆโˆ—=๐œ™โˆ˜๐ฆ satisfying ๐ฟ1(๐ฆ)=๐ฟ1(๐ฆโˆ—) by ๐ผ๐ฆ(๐‘ฃ๐‘˜)=๐‘’๐‘˜ for the canonical basis {๐‘’๐‘˜}โˆž๐‘˜=1 of โ„“2. Finally, (5)โ‡’(6) and (6)โ‡’(1) are evident.

Acknowledgment

The support of the Ministerio de Economรญa y Competitividad, under Project no.MTM2009-14483-C02-02 (Spain) is gratefully acknowledged.

References

  1. A. Fernández, F. Mayoral, F. Naranjo, C. Sáez, and E. A. Sánchez-Pérez, โ€œSpaces of p-integrable functions with respect to a vector measure,โ€ Positivity, vol. 10, no. 1, pp. 1โ€“16, 2006. View at Publisher ยท View at Google Scholar
  2. S. Okada, W. J. Ricker, and E. A. Sánchez Pérez, Optimal Domain and Integral Extension of Operators, vol. 180, Birkhäuser, Basel, Switzerland, 2008.
  3. L. M. García-Raffi, D. Ginestar, and E. A. Sánchez-Pérez, โ€œIntegration with respect to a vector measure and function approximation,โ€ Abstract and Applied Analysis, vol. 5, no. 4, pp. 207โ€“226, 2000. View at Publisher ยท View at Google Scholar
  4. S. Oltra, E. A. Sánchez Pérez, and O. Valero, โ€œSpaces L2(λ) of a positive vector measure λ and generalized Fourier coefficients,โ€ The Rocky Mountain Journal of Mathematics, vol. 35, no. 1, pp. 211โ€“225, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  5. E. A. Sánchez Pérez, โ€œVector measure orthonormal functions and best approximation for the 4-norm,โ€ Archiv der Mathematik, vol. 80, no. 2, pp. 177โ€“190, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  6. L. M. García-Raffi, D. Ginestar, and E. A. Sánchez Pérez, โ€œVector measure orthonormal systems and self-weighted functions approximation,โ€ Kyoto University, vol. 41, no. 3, pp. 551โ€“563, 2005. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  7. L. M. García Raffi, E. A. Sánchez Pérez, and J. V. Sánchez Pérez, โ€œCommutative sequences of integrable functions and best approximation with respect to the weighted vector measure distance,โ€ Integral Equations and Operator Theory, vol. 54, no. 4, pp. 495โ€“510, 2006. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  8. J. Diestel and J. J. Uhl, Vector Measures, vol. 15, American Mathematical Society, Providence, RI, USA, 1977.
  9. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I and II, Springer, Berlin, Germany, 1996.
  10. R. G. Bartle, N. Dunford, and J. T. Schwartz, โ€œWeak compactness and vector measures,โ€ Canadian Journal of Mathematics, vol. 7, pp. 289โ€“305, 1955. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  11. G. P. Curbera, โ€œBanach space properties of L1 of a vector measure,โ€ Proceedings of the American Mathematical Society, vol. 123, no. 12, pp. 3797โ€“3806, 1995. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  12. D. R. Lewis, โ€œIntegration with respect to vector measures,โ€ Pacific Journal of Mathematics, vol. 33, pp. 157โ€“165, 1970. View at Zentralblatt MATH
  13. E. A. Sánchez Pérez, โ€œCompactness arguments for spaces of p-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces,โ€ Illinois Journal of Mathematics, vol. 45, no. 3, pp. 907โ€“923, 2001.
  14. R. del Campo, A. Fernández, I. Ferrando, F. Mayoral, and F. Naranjo, โ€œMultiplication operators on spaces on integrable functions with respect to a vector measure,โ€ Journal of Mathematical Analysis and Applications, vol. 343, no. 1, pp. 514โ€“524, 2008. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  15. R. del Campo, A. Fernandez, I. Ferrando, F. Mayoral, and F. Naranjo, โ€œCompactness of multiplication operators on spaces of integrable functions with respect to a vector measure,โ€ in Vector Measures, Integration and Related Topics, vol. 201 of Oper. Theory Adv. Appl., pp. 109โ€“113, Birkhäuser Verlag, Basel, 2010.
  16. S. Okada and W. J. Ricker, โ€œThe range of the integration map of a vector measure,โ€ Archiv der Mathematik, vol. 64, no. 6, pp. 512โ€“522, 1995. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  17. S. Okada, W. J. Ricker, and L. Rodríguez-Piazza, โ€œCompactness of the integration operator associated with a vector measure,โ€ Studia Mathematica, vol. 150, no. 2, pp. 133โ€“149, 2002. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  18. E. Jiménez Fernández and E. A. Sánchez Pérez, โ€œWeak orthogonal sequences in L2 of a vector measure and the Menchoff-Rademacher Theorem,โ€ Bulletin of the Belgian Mathematical Society Simon Stevin, vol. 19, no. 1, pp. 63โ€“80, 2012.
  19. P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, Germany, 1991. View at Publisher ยท View at Google Scholar
  20. J. Diestel, Sequences and Series in Banach Spaces, Springer, New York, NY, USA, 1984. View at Publisher ยท View at Google Scholar
  21. C. Bessaga and A. Pełczyński, โ€œOn bases and unconditional convergence of series in Banach spaces,โ€ Polska Akademia Nauk, vol. 17, pp. 151โ€“164, 1958. View at Zentralblatt MATH
  22. R. del Campo and E. A. Sánchez-Pérez, โ€œPositive representations of L1 of a vector measure,โ€ Positivity, vol. 11, no. 3, pp. 449โ€“459, 2007. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  23. T. Figiel, W. B. Johnson, and L. Tzafriri, โ€œOn Banach lattices and spaces having local unconditional structure, with applications to Lorentz function spaces,โ€ Journal of Approximation Theory, vol. 13, pp. 395โ€“412, 1975. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  24. M. I. Kadec and A. Pełczyński, โ€œBases, lacunary sequences and complemented subspaces in the spaces Lp,โ€ Studia Mathematica, vol. 21, pp. 161โ€“176, 1962.
  25. J. Flores, F. L. Hernández, N. J. Kalton, and P. Tradacete, โ€œCharacterizations of strictly singular operators on Banach lattices,โ€ Journal of the London Mathematical Society, vol. 79, no. 3, pp. 612โ€“630, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  26. W. B. Johnson and G. Schechtman, โ€œMultiplication operators on L(Lp) and p-strictly singular operators,โ€ Journal of the European Mathematical Society, vol. 10, no. 4, pp. 1105โ€“1119, 2008. View at Publisher ยท View at Google Scholar
  27. F. L. Hernandez, S. Ya. Novikov, and E. M. Semenov, โ€œStrictly singular embeddings between rearrangement invariant spaces,โ€ Positivity, vol. 7, no. 1-2, pp. 119โ€“124, 2003. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  28. S. J. Montgomery-Smith and E. M. Semenov, โ€œEmbeddings of rearrangement invariant spaces that are not strictly singular,โ€ Positivity, vol. 4, no. 4, pp. 397โ€“402, 2000. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  29. S. Ya. Novikov, E. M. Semenov, and F. L. Hernández, โ€œStrictly singular embeddings,โ€ Functional Analysis and Its Applications, vol. 36, no. 1, pp. 71โ€“73, 2002.
  30. S. J. Dilworth, โ€œSpecial Banach lattices and their applications,โ€ in Handbook of the Geometry of Banach Spaces I, chapter 12, Elsevier, Amsterdam, The Netherlands, 2001.
  31. A. Fernández, F. Mayoral, F. Naranjo, C. Sáez, and E. A. Sánchez-Pérez, โ€œVector measure Maurey-Rosenthal-type factorizations and l-sums of L1-spaces,โ€ Journal of Functional Analysis, vol. 220, no. 2, pp. 460โ€“485, 2005. View at Publisher ยท View at Google Scholar
  32. S. Okada, W. J. Ricker, and L. Rodríguez-Piazza, โ€œOperator ideal properties of vector measures with finite variation,โ€ Studia Mathematica, vol. 250, no. 3, pp. 215โ€“249, 2011.
  33. J. M. Calabuig, J. Rodríguez, and E. A. Sánchez-Pérez, โ€œOn the structure of L1 of a vector measure via its integration operator,โ€ Integral Equations and Operator Theory, vol. 64, no. 1, pp. 21โ€“33, 2009. View at Publisher ยท View at Google Scholar ยท View at Zentralblatt MATH
  34. J. M. Calabuig, J. Rodr guez, and E. A. Sánchez-Pérez, โ€œInterpolation subspaces of L1 of a vector measure and norm inequalities for the integration operator,โ€ Contemporary Mathematics, vol. 561, pp. 155โ€“163, 2012.