1. Introduction
The theory of Banach lattices, and in particular the theory of Banach function spaces, provides powerful specific tools in mathematical analysis. These tools can be added to those of the Banach space theory when the spaces considered have this supplementary structure. In this paper, we are interested in constructing a technique to transfer some of these techniques from the Banach lattices to the Banach space setting by means of some local identification of subspaces. Using Banach-space-type isomorphisms among substructures of the spaces involved (one of them a Banach function space, the other just a Banach space) we can identify lattice-type properties and constructions in Banach spaces.
The main notion that we develop is the
pth power of a Banach space. The concept of the
pth power of a Banach function space, sometimes called
p-concavification, is a useful construction both in the context of the study of the structure of the classical Banach spaces and that of the theory of operators on these spaces. It must be said that the notion of
p-concavification can be extended to abstract Banach lattices; the way of doing it is nowadays classical—see [
1,
2]. Some recent papers have extended it to some typical Banach space constructions (for example, tensor products, see [
3,
4,
5]), but always in a Banach lattice framework (Fremlin tensor products). As far as we know, the present paper is the first attempt of translating this notion to the Banach space setting.
This transfer makes it possible to find some new results of the factorization of operators between Banach spaces. Indeed, our main goal is to understand some factorization arguments that are performed for Banach function spaces—the so-called Maurey–Rosenthal theorems (see, for example, [
6,
7])—in the Banach space framework in order to show concrete representations of subspaces of Banach spaces as weighted
-spaces. The lattice geometric structure of the Banach function spaces regarding
p-convexity and
p-concavity inherited by the Banach spaces by means of our representation allows us to do this. The link between the factorization of operators and the concrete representation of Banach function spaces and operators is a classical tool in Banach lattice theory (see, for example, [
8,
9] and ([
10], Chapter III.H )). In our case, the arguments involve factorization of operators
(
E and
F Banach spaces) through inclusion maps
as
Note that for the cases , , this scheme can be understood as a strong version of the one that holds for maps that belong to some classical operator ideals (as, for instance, the ideal of p-integral operators). Finally, some applications in the representation of spaces of multiplication operators and spaces of polynomials are given.
From a technical point of view, we systematically use the identification provided by the integration with respect to vector measures. It is well known that order-continuous Banach function spaces are deeply related to spaces of integrable functions with respect to a vector measure. In fact, there is a representation theorem that allows the relation of both classes of spaces (see ([
11], Theorem 8), ([
12], Proposition 3.9). In particular, this provides a first representation result, using a linear isomorphism in this case: A Banach space is representable as an order-continuous Banach function space with a weak unit if and only if it is the range of an integration map of a (countably additive) vector measure which is an isomorphism. This technique has been widely used for the identification of the optimal domain of some relevant operators; the reader can find some examples and applications in [
12,
13,
14,
15,
16] and the references therein.
3. Order-Continuous Banach Function Subspaces and th Powers of Banach Spaces
Let us start by providing some technical tools that give the link between the structure of the Banach function spaces over which the Banach spaces E are represented and the vector measures associated to the identification map of and E. Recall that we are assuming that the measure is finite. First, we show a characterization of these representations in terms of vector measures over -algebras, which is in fact the main tool of the paper. For a vector measure we denote by the range of the measure .
Theorem 1. Let E be a Banach space. The following assertions are equivalent.
- (1)
There is a finite measure μ and an order-continuous Banach function space such that E is representable on .
- (2)
There is a measurable space and a vector measure such that
- (i)
is dense in E, and
- (ii)
for every sequence of simple functions, if is Cauchy in E, then is a Cauchy sequence in too.
Proof. (2) ⇒ (1) Assume that there is a vector measure m as in (2). Consider the space of integrable functions. The integration map is continuous, so it is enough to prove that it defines a bijection. Take . Then, by hypothesis, it is the limit of a sequence of elements of . For each n, can be written as a finite sum , and so it can be identified with the integral of the simple function , and so . Consequently, it is a Cauchy sequence of E and then, by hypothesis, is a Cauchy sequence in with limit . Clearly, , and this proves that the integration map is surjective.
Suppose that it is not injective. Then, there is a non-null element such that . Take a sequence of simple functions converging to f in and consider the sequence defined for as and for all . Then, by the continuity of , is a Cauchy sequence, since it converges to 0. However, is not Cauchy in .
(1) ⇒ (2) Since
E is representable over
, there is a continuous bijection
. The Open Mapping Theorem gives that it is in fact an isomorphism. Define the vector measure
m by
,
, and note that the fact that
I is bijective implies that
is equivalent to
, that is, both have the same null sets. The Optimal Domain Theorem (see ([
12], Theorem 4.14)) asserts that
I can be extended to
with continuity, and so
. This gives that the isomorphism
I can be factored as
, where
is the inclusion map. Now, take a simple function
f. Taking into account that
, we have that
This and the order continuity of and give that is dense in E, since the order continuity implies that simple functions are dense. In addition, the representation implies that a sequence of functions is Cauchy in if and only if is so in E. This gives (1) ⇒ (2) and finishes the proof. □
Example 3. Let and let be a (Banach space valued) vector measure. Consider the space and the integration operator . The Banach function space , where is the conjugate exponent of p given by , can be isometrically identified with the space of operators . The identification (see ([12], Chapter 3)) is given by defined byfor each . Let us define the vector measure by . It can easily be seen that isometrically, and so (2) of Theorem 1 is satisfied by m. Let us now analyze when a particular subspace S of a Banach space E that satisfies that it is the range of an injective continuous linear map from a Banach function space can itself be identified with the Banach function space . In other words, let us see when the existence of a continuous inclusion from a Banach function space on a subspace S of Banach space E assures that is a (copy of a) Banach function subspace in E.
Proposition 1. Suppose that there is an order-continuous Banach function space such that there is a continuous injective map ι in a Banach space E, i.e., . Then, the following statements are equivalent.
- (1)
is a Banach subspace of E that can be represented over a Banach function space containing .
- (2)
There is a constant such that for every simple function ,
Proof. (2) ⇒ (1) By the same arguments that were used in the proof of (1) ⇒ (2) of Theorem 1, the inclusion map
can be extended to
, where
is the vector measure given by
,
. Since
gives another expression for the norm of
when
is a simple function (in this case we have that
), we have that the inequality in (2) gives
Since simple functions are dense in , these inequalities can be extended to the whole space , which gives the desired isomorphism among and . Note that the fact that is injective gives that is -determined, and so is equivalent to and to any Rybakov measure for . The identification of with gives the result.
(1) ⇒ (2) By hypothesis, we have that
can be extended to an isomorphism from
. This gives that there are constants
and
such that for every simple function
,
This gives the result for . □
Example 4. Let us consider again Example 3. If η is a Rybakov measure for m and , the space is a Banach function space over η included in . We have that is isometric to , which is also isometric to a subspace of . Clearly, for every ,and so (2) in Proposition 1 holds. Thus, . Note that the same argument also gives that is a Banach function subspace of , although it is not order continuous.
After the previous results, we are ready for defining structures that are naturally given for Banach function spaces, but which do not have a Banach space counterpart. In this paper, we will consider the construction of pth powers of Banach spaces. We start by showing that, as a direct consequence of Theorem 1, we can assure the existence and give a concrete description of the pth powers of representable Banach spaces. Recall that we are considering finite measures .
By definition of representability, there is a one-to-one identification that relates the elements of and the elements of E. For a Banach function space containing , the inclusion holds if , but the converse inclusion does not hold in general. This implies that we have to consider the cases and separately.
First consider the case
. We define the
pth power
of the Banach space
E as
A natural candidate for quasi-norm of the space is given by the expression
Note that this is well defined, since for each
,
that contains
. Moreover, for the elements
, we have
and so
makes sense and gives again an element of
E.
However, for , the pth power of a Banach space defined as above does not necessarily exist as a Banach subspace of E. For example, for , the 2nd power of is , which is not a subspace continuously included in : The inclusion is proper. Therefore, by definition, if we take , we have that , and so is the normed space . Actually, the pth power is not necessarily normed, but only quasi-normed: Consider, for example, instead of in the example.
This forces us to define the pth power of a Banach space for in a different way; in fact, it must be a “super" space such that .
Let
and suppose that
E is representable over a Banach function space
containing
. Define on
E the quasi-norm
by
It is a quasi-norm, since is so and is linear. Note that this formula only makes sense for elements of E, and an extension argument is needed for defining . We can define the pth power of E for as the completion of E with the quasi-norm , with the usual definitions. The map I can be extended to as a linear and continuous operator by continuity, since the following diagram commutes, where the vertical arrows are inclusions:
Consequently, the natural way of defining the quasi-norm for
is by means of the formula
which makes sense since
. The better way of understanding what this formula means is just to take into account that
I is a bijection, and so there is a dense set in
E for which the formula can be explicitly computed, since simple functions are dense both in
and in
and satisfy that
. Moreover, for the elements
e of
E that satisfy that
are simple functions, we have that
and
Consequently, we have that the extension by continuity of this formula for simple functions to the whole space works, since it is equivalent on a dense subset to the quasi-norm .
Summing up all the elements of this construction, we can formulate the following general definition for all cases .
Definition 3. Let . Let E be a Banach space that is representable over an order-continuous Banach function space that contains . We define the pth power of the Banach space E as the quasi-normed space given by the completion ofwith the quasi-norm defined by the formulafor the elements such that is a simple function, and by continuity for the rest of the elements of the space. Proposition 2. Let and let E be a Banach space that is representable over a p-convex order-continuous Banach function space . Then, the quasi-norm is equivalent to a norm, and so is normable.
Proof. By the arguments given just above, we know that
is a quasi-norm, so we only need to prove that it is equivalent to a norm. The candidate for comparing with
is given by the expression
It is clearly a semi-norm. Let us show that it is in fact a norm that is equivalent to
. Since by hypothesis, the space
is order continuous, we have that simple functions are dense, and so we can consider only elements of
in the computation and change
by
I. The result is a consequence of the following inequalities:
□
Corollary 1. Let and let E be a Banach space that is representable over an order-continuous Banach function space by means of the map . Then, is a Banach space when endowed with the quasi-norm , which is equivalent to a norm which is continuously included in E. Moreover, isomorphically and if, in addition, I is an isometry, then
- (i)
The function is in fact a norm, and
- (ii)
isometrically.
Proof. It is well known that the
pth power of
satisfies that
whenever
and
is finite; of course, there is a genuine inclusion, that is, the inclusion is injective since the equivalence classes are defined by the same measure. If
, we have that
, and so
. Therefore,
The same arguments (that is, the identification of
with
) allow us to prove that the space
is linear and that it is in fact a Banach space, since
is (see [
12], Chapter 2). Note that for
, every Banach function space
is
p-convex, and its
p-convexity constant
equals one. Clearly, by the definition of
, we get that
isomorphically. Finally, if
I is an isometry, then
. Therefore, the computations at the end of the proof of Proposition 2 show that
equals the norm appearing in this proof. Again, the quoted computations provide the isometry. □
Corollary 2. Let E be a Banach space that is representable on an order-continuous Banach function space containing and . Then, is a -convex Banach function subspace of E that is isomorphic to for a certain vector measure m that is equivalent to μ.
Proof. Since
, we have that
and, by using Corollary 1,
is isometric to
, with the isometry given by the restriction of
I to
. On the other hand, for the representing vector measure
given by
,
, we have that
, and so
. It is well known that this space is
-convex (see ([
12], Chapter 3)). □
For , a similar proof based on Proposition 2 gives the following result.
Corollary 3. Let . Let E be a Banach space that is representable over a p-convex Banach function space (with a p-convex constant equal to 1) by means of the map . Then, is a Banach space when endowed with the quasi-normwhich contains E continuously. Moreover, isomorphically as well as isometrically if I is an isomorphism. The last part of this section is devoted to showing some fundamental applications of the pth powers of Banach spaces. As the reader will see, this construction gives, for example, a systematic way for providing new canonical decompositions of a Banach space E as products of elements of some selected subspaces of E, as well as new interpolation formulae. To simplify, note that after Proposition 2, we can assume (and we do) without loss of generality that I is an isometry.
(a) Decomposition Theorem (Product Theorem for Banach Spaces)
Let us introduce the notion of a pointwise product of Banach spaces. Recall that given a pair of Banach function spaces
and
over the same measure
, the product
is defined as follows. Consider the space of all the functions in
for which the function
is finite. Under some requirements, this is a normed space of classes of
-a.e. equal measurable functions, and its completion is what is called the product space
. It is a Banach function space over the same measure
. If the spaces
and
satisfy adequate
p-convexity requirements, the norm can be computed just by using single product decompositions instead of sums of such products. The reader can find all the information that is needed in ([
28], §2) and [
26] (see also ([
29], §2) for a slightly different definition and main properties, and in [
30] for a general setting for the pointwise-type products of Banach spaces).
Let us show that our construction allows us to define the (pointwise) product of Banach spaces. Let
E be a Banach space that is representable by the Banach function space
with a representation given by
. Consider
and
two Banach function subspaces of
E (with different norms that of
E) such that
. We define the product
as
with the norm
given by:
We can say that an element e of E is a pointwise product of two elements and if .
Using the product decomposition of the elements of a Banach function space and Corollary 2, we directly obtain the following canonical decomposition of Banach spaces that can be represented as Banach function spaces.
Corollary 4. Let E be a Banach space that is representable isometrically as an order-continuous Banach function space over a finite measure μ and . Then,isometrically. Consequently, each element of E can be decomposed as a pointwise product of an element of and an element of . Proof. Take
and
. It can be easily seen that
isometrically just by using Hölder’s inequality for Banach lattices (see [
2], Proposition 1.d.2). If
, we have that
, and then
can be written as
Since and , and are Banach function spaces with the corresponding pth power and th power quasi-norms being norms, and I is an isometry, we get the result. □
A similar result should be obtained by considering the construction for different spaces
F and
H as being representable as Banach function spaces
and
by using the transference of the lattice structure on these spaces provided by isometry
I; the results in ([
28], §2), [
26,
29,
31,
32] may be used for this aim.
(b) Lozanovskii Theorem for Banach Spaces
The representation technique presented in this section can also be applied for obtaining a Banach space version of the well-known Lozanovskii Decomposition Theorem, which establishes that for a finite measure can be written as the pointwise product of every order-continuous Banach function space and its Köthe dual.
Corollary 5. Let E be a Banach space that is representable as for a certain finite measure μ and with I being an isometry. Suppose that is the range by I of an order-continuous Banach function subspace of with the Fatou property. Then, is also isometric to a Banach function subspace of , andisometrically. Proof. Recall that
is assumed to have the Fatou property. Since we have that
is an order-continuous Banach function space included in
, we have that
isometrically, and also
. Moreover, since
I is an isomorphism, we have that
is so, and
Thus,
is isomorphic to
. Define
. Then, the fact that
I is an isometry provides that
is isometric to
, and by using the Lozanovskii Theorem (see ([
32], Theorem 6)), we get that this is equal to
, which is isometric to
E. This gives the result. □
(c) Interpolation Theorem
A particular case of product space combined with
and
th powers of the involved spaces
and
provides the so-called Calderón–Lozanovskii interpolation of Banach function spaces. Indeed, it is well known that for an interpolation couple of Banach function spaces
and
, the Calderón–Lozanovskii interpolation space
of index
is defined as
with the norm
It is well known that, under some requirements, this space coincides with the complex interpolation space of
and
with index
(see [
33]).
Using the main interpolation theorem for operators applied to the present context, we obtain the following results.
Corollary 6. Let . Suppose that the Banach space E is representable as the Banach function space by means of an isometry I, and and are Banach function subspaces of E that are represented by the Banach function subspaces and of , respectively by means of I, which also defines isometries from into , . Then, the complex interpolation space can be represented as the Calderón–Lozanovskii space by means of I.
Proof. Just consider the isometric equalities
□
Corollary 7. Let . Suppose that the Banach space E is representable as the Banach function space . Then, the complex interpolation space can be described as Proof. This is a consequence of the isometries
□