#### Abstract

We show a Dvoretzky-Rogers type theorem for the adapted version of the -summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

#### 1. Introduction

Summability in Banach spaces is one of the main topics in Functional Analysis, and results concerning the behavior of summable sequences are fundamental tool for its applications. Comparison between norm and weak absolutely summable series is at the origin of some classical problems in the theory of Banach spaces, and it was the starting point of the theory of -summing operators. In this paper we are interested in providing new elements for the analysis of summability in the case of Banach function spaces by using a vector valued duality that is provided by the vector measure integration theory on spaces of integrable functions with respect to a vector measure . These spaces represent, in fact, all order continuous -convex Banach lattices with weak unit. This theory supplies a distinguished element, the vector valued integral, for the study of summability in Banach spaces of measurable functions. It is well known that whenever and , . In this case, the integral determines a vector valued bilinear map that yields to duality: the vector valued duality between and (see [1, 2]).

This vector valued duality is the framework to study natural topologies on spaces of integrable functions with respect to a vector measure, as the topology generated by the seminorms and , when varying . This new vector valued point of view was first taken into consideration in the study of convergence of sequences: the relation between the convergence of sequences in spaces of vector measure integrable functions and the convergence of the corresponding vector valued integrals has been treated since the seventies (see, e.g., [3, 4], [5, Section ], [6], and the references therein). In this paper we are interested in the summability of sequences in spaces induced by the vector valued duality, that is, when the role played by the weak topology is assumed by the topology . It is worth mentioning that the -convexification () of the space of a vector measure was introduced as a tool for analyzing summability (see [1]), trying to bring together vector valued integration and the theory of -summing operators in Banach spaces (see also [7, 8]).

The classical Dvoretzky-Rogers theorem can be stated as follows: the identity map in a Banach space is absolutely -summing for some , if and only if is finite dimensional. This paper is devoted to proving an extension for Banach function spaces of this result. In our context, the usual scalar duality is replaced by the vector valued duality given by a vector measure and the role of the weak topology in the Banach space is assumed by the topology . In order to develop our study, we analyze some properties of the -summing operators that map summable sequences to norm summable sequences. Our main result shows the necessity of adding some topological requirements on local compactness to characterize finite dimensional spaces in terms of the -summability of the identity map. The last section shows an application to the study of subspaces of that are fixed by the integration operator. As a consequence of our Dvoretzky-Rogers type theorem, we prove that, under the local compactness hypotheses, only finite dimensional subspaces can be fixed by the integration map.

#### 2. Preliminaries

We use standard Banach space notation. Let . Then we write for the extended real number satisfying . We follow the definition of Banach function space over a finite measure given in [9, Def. 1.b.17, p. 28]. Throughout the paper will denote an infinite dimensional Banach function space over ; that is, is a Banach lattice of , a.e., equal classes of -integrable functions with a lattice norm and the a.e. order satisfying . We will also assume that is order continuous; that is, for each decreasing sequence in ,

Let be a real Banach space and let be a measurable space. If is a countably additive vector measure, we write for its range. The variation of is given by , where the supremum is computed over all finite measurable partitions of . is the semivariation of ; that is, , , where is the scalar measure given by . The Rybakov Theorem (see [10, Ch. IX]) establishes that there exists such that is absolutely continuous with respect to a so-called Rybakov measure that means that whenever . For , a (real) measurable function is said to be -integrable with respect to if is integrable with respect to all measures and for each there exists an element such that , .

The space , , is defined to be the Banach lattice of all (-equivalence classes of) measurable real functions defined on that are -integrable with respect to when the a.e. order and the norm are considered. It is an order continuous -convex Banach function space over any Rybakov measure for (see [1, Proposition ]; see also [11] and [6, Ch. ] for more information on these spaces). For the case , is defined as . A relevant fact is that, for each , (see [6, Prop. ] and [1, Sec. ]; see also [11]). Moreover, for each These relations allow defining the so-called vector measure duality by using the integration operator , which is given by We will use the symbol instead of throughout the paper. Relevant information on the properties of can be found in [12–14] and [6, Ch. 3] and the references therein. Since for all the inclusion always holds, the integration map can be defined also as an operator ; we use the same symbol in this case for this operator. It must be said that the spaces represent in fact the class of all order continuous -convex Banach lattices with a weak unit (see [11, Prop. 2.4] or [6, Prop. 3.30]) that means that our results can be applied to a broad class of Banach spaces.

As we said in Introduction, duality and vector valued duality for the spaces are fundamental tools in this paper. Regarding duality, fix a Rybakov measure for . Due to the order continuity of , its dual space () allows an easy description; it coincides with its Köthe dual (or associate space) ; that is, , whereand the duality is given by Information about a precise description of can be found in [2, 7, 15–17]. It must be said here that and coincide only in very special situations, for instance, for being a scalar measure. We will write for the weak topology on .

Regarding vector valued duality relations between spaces, , the integration map defines the continuous bilinear map given by , , . Note that is both sides norming for and ; that is, for every , , and the same happens dually for the case of functions .

In this paper we will consider the topology of pointwise convergence of the integrals that is the locally convex topology defined by the seminorms , , . The topology of pointwise weak convergence of the integrals is defined by the seminorms , , , and . It is also a locally convex topology on . It is easy to see that the norm topology is finer than all the others, and and are finer than , although and are not comparable in general. An exhaustive analysis of the topology has been done recently and can be found in [18] (see also the references therein). The reader can find more information about it in [1, 6, 7, 11, 16, 19]. The following result establishes the basic relations between the quoted topologies.

Proposition 1 (See Proposition 1 in [18]). *Let . If is -compact then and coincide on bounded subsets of . Moreover, if and is -compact then the weak topology and coincide on bounded subsets of . Consequently, if , is -compact if and only if is reflexive and the weak topology and coincide on .*

In this paper we will make a local use of the duality defined by the integration bilinear map . For consider a subspace . We say that a subspace is an *-dual* for if is -norming for ; that is, the function gives an* equivalent* norm for . We write for such a space . In the same way, we say that a subspace of is -bidual of (with respect to -dual ) if and is -norming for . Notice that the inclusion is not necessary for to be -norming for . For instance, if is an order continuous Banach function space and is the vector measure given by , , then for the space generated by the function in is -norming for , and also the space generated by in is -norming for . However, is not included in . But note also that given , , and being norming, it can always be assumed that just by defining the new as the subspace of generated by We will use this example later.

We say that a triple of -dual spaces as above is an -dual system. We can define the topology over as the one induced by all the seminorms , , and the topology for given by the seminorms , . A quick look at the proof of Proposition in [18] shows that a local version of this result is also true, that is, a version of this result writing instead of and instead of , where is an -dual space.

Let us show some examples. A natural -dual space of is ; in this case, we write simply for the topology . However, an -dual space may be very small. For instance, if the integration map is isomorphism, then the subspace generated by is -dual for . Obviously, for every subspace , is -dual for .

Let us finish this section by defining a fundamental class of operators related to the summability of sequences with respect to the -topology. It generalizes the class considered in Lemma of [1] and in [7, Section ]. Theorem in [1] provides a Pietsch type domination/factorization theorem for this family of operators. The local version of this result becomes the main tool for the proof of our results.

*Definition 2. *Let , , be a subspace of and a Banach subspace of . Let be a Banach space. An operator is -summing if there is a constant such that, for any finite set of functions , Of course, the integration map is always -summing for all , . Indeed, if , then

#### 3. The Dvoretzky-Rogers Theorem for the -Summability

Throughout this section, , and are Banach spaces, is an -valued vector measure, is a subspace of , and is an -dual system. We will consider the following sequential properties associated with compactness with respect to the -topology.

*Definition 3. *An operator is -sequentially compact if every bounded sequence in has a subsequence such that is a Cauchy sequence for each .

*Definition 4. *An operator is -sequentially completely continuous if whenever is a bounded sequence such that for every .

If we assume that (we can always make big enough to have it), then is -sequentially continuous. In the classical summing operators theory it is well known that any summing operator is weakly compact. However, not every -summing operator is -sequentially compact. For instance, given a Banach function space , define , . Then is an isomorphism which is -summing but it is not -sequentially compact in general as in this case the norm topology and the topology coincide. Then is not -compact unless is finite dimensional. Let us see that, under some compactness assumptions, the -summing operators behave similarly as absolutely summing operators. We need first an easy lemma.

Lemma 5. *Let and let be a subspace of and an -norming subspace for . Consider a Banach space valued -summing operator . Then is -summing for each .*

* Proof. *Let be such that . Take a finite set of functions . Then where is the constant associated with the -summability of .

Theorem 6. *Let . Let be a -summing operator. The following statements hold. *(i)*If is -compact then is -sequentially completely continuous.*(ii)*If is -compact and is -compact then is completely continuous.*(iii)*Finally, if is -compact and is reflexive, then is also weakly compact.*

* Proof. *(i) We have that satisfies that, for every finite set ,Taking into account that is an -dual system, it can be shown as in the case of Pietsch’s Domination Theorem for -summing operators (see Lemma in [1] and make the obvious modifications) that there is measure on the compact space such thatThis easily gives that factorizes through the following scheme (see Theorem in [1]):

where is the subspace of given by the functions , is the isomorphism given by the identification of a function with the corresponding vector valued function in , is the closure of the image of by the natural inclusion/quotient map where is a Radon probability measure on , and is the map that closes the diagram. Using this scheme, an argument based on the Dominated Convergence Theorem gives the result. Let be a bounded sequence in such that the sequence of integrals is null for every . It is enough to prove that the sequence of functions satisfies . For each , the function belongs to the space of scalar continuous functions defined on the compact set . Since there is a constant such that for all and , we can apply the Lebesgue Dominated Convergence Theorem to obtain that Therefore, using the factorization we obtain that and so is -sequentially completely continuous.

(ii) Let be a weakly null sequence in . Since is compact, using the factorization given in (i) and taking into account that each continuous operator is weak-to-weak continuous we get that for each element , , , we have that Due to an easy adaptation of Proposition 1, since we are assuming that is compact, the topologies , generated by the seminorms when varying and , and coincide on .

Consequently for each , Using the domination in (i), we obtain the result on the complete continuity.

(iii) Finally, by Lemma 5 if is -summing it is -summing for , and so the reflexivity of implies the reflexivity of . Thus, the factorization of through a subspace of gives that is weakly compact.

The following result is a direct consequence of statements (ii) and (iii) of Theorem 6.

Corollary 7. *Suppose that is an -valued vector measure and is reflexive. Let be a -summing operator, and suppose that is -compact and is -compact. Then is compact.*

In particular, if is isomorphism in Corollary 7, we obtain that has to be finite dimensional.

*Example 8. *Let us show an example of a proper infinite dimensional subspace of a space with an -dual system in which , , and coincide. However, the identity map is not -summing for any . Take an infinite nontrivial measurable partition of the Lebesgue space , and define the vector measure given by , where is the canonical basis of and (see Example in [20]). Consider the (infinite dimensional closed) subspace of generated by the functions , . A direct calculation shows that, for each ,and so is isometric to (see Proposition in [20]). We can define the -dual space and the -bidual space as . It is clear that norms and norms . However, the identity map is not -summing for any . In order to see this, consider the sequence of functions . Then, if , for each we get but This gives a contradiction and shows that the identity map cannot be -summing for any . Note that the range of is relatively compact, since it can be included in the convex hull of a null sequence of . Corollary in [18] establishes that for a reflexive and separable space (our space satisfies both requirements) relative compactness of the range of implies compactness of . is -closed, since by Proposition 1, is finer than the weak topology on . This gives compactness of , since the topology is weaker than the topology on , and so compactness of . The topological requirements of Corollary 7 are then satisfied and is reflexive, but obviously the identity map is not compact. Since is not a Schur space, the identity map is not completely continuous. This shows that the summability condition in Theorem 6(ii) and in Corollary 7 cannot be dropped.

The following is our main result and gives a vector measure version of the Dvoretzky-Rogers theorem.

Theorem 9. *Let be a Banach space, a subspace of , and isomorphism. The following statements are equivalent. *(i)*There is an -dual system such that is -sequentially compact, is -compact, and is -summing for some and then, for all.*(ii)* has finite dimension.*

* Proof. *(i) (ii) Assume that is -summing for fixed . Let us show that the composition is compact. As a consequence of Theorem 6(i), we know that is -sequentially completely continuous. Since is -sequentially compact, is -sequentially compact. Then the identity map is compact, and so has finite dimension.

(ii) (i) Since is finite dimensional, we have that is compact. The norm topology is finer than , and so the unit sphere is compact too. For each element , take a norm one function that satisfies that . Consider the -open covering of given by the sets There is finite subcovering given by a finite set of such functions . Then we define to be the subspace generated by Note that for each there is an index such that and so Consequently, for each ,Therefore, the space is -norming for and is -sequentially compact since the norm topology and coincide in the finite dimensional space .

Note that we can also define a finite dimensional subspace containing that is -norming for following the same procedure in the definition of . The finite dimension of proves also that is -compact.

Finally, let us see that is -summing for all . By Lemma 5 it suffices to prove that is -summing. Write now for the (usual topological) dual of . Since is finite dimensional, we have that the identity map is -summing, and so for each finite family where is the -summing norm of the identity map and the constant 4 comes from (19). Therefore, is -summing and so -summing for every .

When is a scalar measure then the spaces and , , are reflexive and hence their closed unit balls are weakly compact or, equivalently, -compact. Besides, in this case -summability coincides with the usual absolute -summability for operators. Therefore Theorem 9 can be considered an extension of the classical Dvoretzky-Rogers Theorem to spaces of integrable functions with respect to a vector measure.

Let us present some examples that show that all the requirements in (i) are needed for the result to be true. Recall that is an order continuous Banach function space over a finite measure space .

*Remark 10. * *(1)**-Sequential Compactness of ** Is a Necessary Requirement.* Consider the vector measure given by , . In this case, and the integration map is isomorphism. Take that is not finite dimensional by assumption. The subspace of generated by is -norming for . Consider the -bidual space for defined as . Obviously, is -compact. Since the seminorm on defined by coincides with the norm, we have that is -norming for but clearly is not -sequentially compact. Note that any other -dual space for containing a function for some satisfies the same property: is not compact for the topology . Observe also that the identity is -summing for each , since for each finite set ,Note that the identity is sequentially completely continuous trivially. This example shows clearly the difference between -summing and -summing operators. In the first case, Alaoglu’s Theorem assures that the unit ball of the dual space is -compact, and this is enough to prove the Dvoretzky-Rogers theorem via Pietsch’s Factorization Theorem. In the second case, the topological properties for the unit balls of the spaces involved must be given as additional requirements. This means that the corresponding summability property for the isomorphism does not assure our Dvoretzky-Rogers type theorem to hold.*(2) Not All the **-Dual Systems for a Finite Dimensional Space ** Satisfy the Requirements of Theorem 9.* Consider again the vector measure given in the example given in (1). Take as the (finite dimensional) subspace of generated by . First, take the -dual system , with the understanding that and are subspaces of and is a subspace of . In this case, is -sequentially compact, is -compact, and the identity map on that coincides with the integration operator is -summable for each , providing all the requirements in (i) of Theorem 9.

However, take now and . Assume that the vector measure does not have relatively compact range. This happens, for example, when , (see Example in [6]). Then is -sequentially compact but is not -compact, since the topology induced on by coincides with the topology of on this space. To see this, just consider the seminormThus if is -compact, this would imply compactness of with respect to the topology of , and so it would imply that the range of the vector measure is relatively compact, since it is included in .* (3) The Topological Requirements for the **-Dual System Are Not Enough: The Assumption on the **-Summability of the Isomorphism Is Also Needed. *Consider the vector measure defined as Lebesgue measure on . Take any and consider . Then we have that is -dual for , and so the topology gives the weak topology for the reflexive space (see Proposition 1). If we define the -bidual , we have that the topology for is given by the weak topology for . So both topological requirements in (i) of Theorem 9 are satisfied. Of course, no isomorphism from is -summing for any , and so no isomorphism is -summing, since in this case both definitions of summability coincide.

*Example 11. ** The Vector Measure Associated with the Volterra Operator.* Let and let be the Volterra measure, that is, the vector measure associated with the Volterra operator. This measure is defined as (see the explanation in [6, p. 113]; all the information about this measure can be found in different sections of [6]). It is known that the range of is relatively compact. This is a consequence of the compactness of the Volterra operator (see the comments after [6, Proposition ]).

Let , , and consider a subspace of . Assume that there is an -dual space for such that for certain (e.g., a subspace generated by a finite set of functions in with -norm greater than ). Take as . Then is -compact as a consequence of Theorem in [18]. In this case, we have a simplified version of our Dvoretzky-Rogers type theorem for the subspace : is finite dimensional if there is such that the identity map is -summing and is -sequentially compact.

#### 4. An Application: Subspaces of That Are Fixed by the Integration Map

In what follows we use our results in order to obtain information about subspaces of spaces that are fixed by the integration map . This topic has been studied since the very beginning of the investigations on the structure of the spaces of integrable functions with respect to a vector measure, and several papers on this topic have been published recently (mainly regarding subspaces that are isomorphic to and , see [21] and the references therein). Let us show an easy example.

*Example 12. *Consider as in Remark 10 for the vector measure given by , . Consider the subspace generated by the Rademacher sequence in . By the Khintchine inequalities, is a subspace in that is isomorphic to . Recall that and the integration map is isomorphism. Obviously the restriction of the integral operator to is in fact the identity map. For we have that , and again by the Khintchine inequalities is a subspace of that is fixed by the integration map .

As we noted after the definition of -summing operator, the integration map from for any is always -summing for every ; in fact it is in a sense the canonical example of this kind of operators. Thus, our Dvoretzky-Rogers type result can be directly applied to obtain negative results on the existence of infinite dimensional subspaces of that are fixed by . We say that a subspace of is* fixed by the integration map* if is isomorphism.

The following result shows that, under some compactness requirements, any subspace of that is fixed by has to be finite dimensional. For the case in which the -dual system that is considered is and , conditions under which the balls of these spaces are compact are given in Corollary of [18].

Corollary 13. *Let , and let be a subspace of that is fixed by the integration map. If there is an -dual system for such that is -sequentially compact and is -compact, then is finite dimensional.*

* Proof. *It is a consequence of Theorem 9 and the fact that the integration map is -summing for every .

In particular, the subspace generated by the Rademacher functions that has been shown in Example 12 does not have an -dual system satisfying the compactness requirements in Corollary 13.

*Remark 14. *By [11, Theorem ], if the vector measure has relatively compact range and , then the restriction of the integration map to is compact. Thus, if is a subspace of that is fixed by the integration map, it has always finite dimension.

To finish, let us remark that as a consequence of the following result the ideas that prove Corollary 13 can be applied to maps acting in a subspace that is fixed by the integration map, other than the inclusion map.

Proposition 15. *Let . Let be a subspace of that is fixed by the integration map and let be an -dual space of containing . Then every operator with values on a Banach space is -summable for every .*

* Proof. *Let be an operator with values on a Banach space , and let . Then This gives the result.

#### Competing Interests

The authors declare that they have no competing interests.

#### Acknowledgments

This work was supported by the Ministerio de Economía y Competitividad (Spain) under Grants MTM2015-66823-C2-2-P (P. Rueda) and MTM2012-36740-C02-02 (E. A. Sánchez Pérez).