Power concave operators and representation of p-convex and q-concave banach lattices

POWER CONCAVE OPERATORS AND REPRESENTATION OF p-CONVEX AND q-CONCAVE BANACH LATTICES OLVIDO DELGADO and ENRIQUE A. SÁNCHEZ PÉREZ Communicated by Florian-Horia Vasilescu ABSTRACT. As a consequence of the analysis of the class of (p,q)-power concave operators, we prove a general representation theorem for p-convex and q-concave Banach lattices as spaces of integrable functions with respect to vector measures. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years. KEYWORDS:Banach lattices, q-concave operators, quasi-Banach function spaces, vector measures, δ-ring. MSC (2010): Primary 47B38; Secondary 46G10, 46E30, 46B42. 1. INTRODUCTION Let 1 ⩽p⩽q<∞. In this paper we provide a new representation theorem for p-convex and q-concave Banach lattices in terms of integrable functions with respect to vector measures. This type of representation has its origin in Theorem 8 of [6], where it was proved that every order continuous Banach lattice Fwith a weak unit is order isometric to a space L1(ν)of a vector measure νon a σ-algebra. Later on, in Proposition 2.4 of [14] it was proved that if Fis also p-convex (with p-convexity constant equal to 1) then it is order isometric to Lp(m)for another vector measure m. Similar results work for Fwithout weak unit but in this case the vector measure used in the representation of Fis defined to act in a δ-ring, see Theorem 5 of [12] and Theorem 10 of [5]. Also, there are representation theorems for Freplacing σ-order continuity by the Fatou property, in this case through spaces of weakly integrable functions, see [7], [8], [12], [15]. Finally, some recent results on representation of quasi-Banach lattices using integration with respect to positive measures on quasi-Banach function spaces have been published in [30], showing that the vector measure representation technique can be extended to the quasi-Banach case successfully. As the reader will see, some of our results work — and are proved for — this case. 262 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ In Theorem 4.4 of this paper, we obtain that every q-concave and p-convex Banach lattice is order isomorphic to a space Lp(m)of an E-valued vector measure m defined on a δ-ring whose integration operator ImT:L1(mT)→E is (q/p)-concave; alternatively, and only for the case that the Banach lattice has a weak unit, this requirement can be substituted by ImT:Lp(mT)→E to be (q,p)-power concave. In a sense, this result closes the series of representation theorems for quasi-Banach lattices with specific geometric properties by adding the q-concavity requirement to the lattice to be represented. Other characterizations of p-convex and q-concave Banach function lattices are known. For example, Theorem 1.d.10 in [19] establishes that a Banach lattice is q-concave if and only if every positive operator from a C(K)-space in it, is psumming. Another one was provided by Reisner in [29]; in Theorem 1 of that paper, it is proved that a Banach function space X(µ)is p-convex and q-concave if and only if for every multiplication operator from Lq(µ)to Lp(µ), there is a factorization through X(µ)by means of multiplication operators in such a way that the product of their norms is uniformly bounded. However, although the general setting and the results are connected with our work, the arguments used for proving them are different. Our method is essentially based on the vector measure integration theory. Indeed, our main technical tool will be the specific class of operators from Banach lattices and having values on Banach spaces that are called (p,q)-power-concave operators, and was introduced in Chapter 6 of [24]. As will be shown in Section 3, the description of this class requires the spaces of vector measure p-integrable functions. We will prove a general optimal domain theorem for this class of operators (Theorem 3.4) — which contains the q-concave operators — that is also valid for the case of σ-finite measures. In general, we will extend the characterizations and study their main properties in the setting of non-finite measures, in order to cover the case of quasi-Banach function spaces over σ-finite measures, and in general, the case of vector measures acting in δ-rings instead of σ-algebras. Also, we will use the notion of (what we call) q-concave core of a Banach function space, that has been used in several papers related to factorization theorems for operators on Banach function spaces involving q-concavity and p-convexity properties. We will consider the construction provided for example Theorem 1 of [27] and Lemma 6 of [28], although the procedure given there uses abstract Banach lattice notions, while we prefer to give explicit descriptions in terms of measure theoretical tools. We develop our analysis using some elements of the circle of ideas of the classical Krivine’s theorem of factorization through Lp-space of operators factoring through p-convex and q-concave factors (see [17]), the so called Maurey–Rosenthal theorem (see for example [9], [10]) and other new results on factorization of operators acting in Banach function spaces (see Chapter 6 of [24]). On the other hand, the same type of requirements on p-convexity/q-concavity has been also used for finding extensions of regular operators from subspaces of a Banach lattice to the whole lattice (see [26]). POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 263 However, the key of our construction — the factorization of (p,q)-power concave operators from Banach function spaces over σ-finite measures — is essentially new. A good deal of the paper is devoted to show the properties of this new class, which are shown to have nice factorization/extension properties. This class of operators — that was introduced in Chapter 5 of [24] for the case of finite measures — is described by a concavity type inequality, and its main characterization is given — in the finite measure case — by factorization of an operator belonging to the class by a q-concave and a p-th factorable operator through a p-concave function lattice, and the natural adaptation in the case of non-finite measures. The reader can find all the required information on these concepts in Section 2 and Section 3 of the paper. 2. PRELIMINARIES In this section we establish the notation and present the basic results on quasi-Banach function spaces (including the proof of some of them for completeness) and on vector measure integration, which will be used through the whole paper. Let (Ω,Σ)be a fixed measurable space. For a measure µ:Σ→[0, ∞], we denote by L0(µ)the space of all Σ-measurable real valued functions on Ω, where functions which are equal µ-a.e. are identified. Given two set functions µ,λ:Σ→[0, ∞]we will write λµif µ(A) = 0 implies λ(A) = 0. If λµand µλwe will say that µand λare equivalent. If µ,λ:Σ→[0, ∞]are two measures with λµ, then the map [i]:L0(µ)→ L0(λ)which takes a µ-a.e. class in L0(µ)represented by finto the λ-a.e. class represented by the same f, is a well defined linear map. In order to simplify notation we will write [i]( f) = f. Note that if λand µare equivalent then L0(µ) = L0(λ)and [i]is the identity map i. 2.1. QUASI-BANACH FUNCTION SPACES. In this section we establish the setting of our work regarding quasi-Banach function spaces. The results are essentially known, but we prefer to include complete proofs since we did not find precise references for them in the literature. We refer to the general books that the reader can find in the bibliography and the paper [21] by Maligranda for more information about. Let Xbe a real vector space and k·kXaquasi-norm on X, that is a function k · kX:X→[0, ∞)satisfying the following conditions: (i) kxkX=0 if and only if x=0, (ii) kαxkX=|α|·kxkXfor all α∈Rand x∈X, and (iii) there is a constant K⩾1 such that kx+ykX⩽K(kxkX+kykX)for all x,y∈X. 264 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ For 0 <r⩽1 being such that K=2(1/r)−1, it follows that (2.1)    n ∑ j=1 xj  X ⩽41/rn ∑ j=1 kxjkr X1/r for every finite subset (xj)n j=1⊂X, see Lemma 1.1 of [16]. The quasi-norm k·kX induces a metrizable vector topology on Xwhere a base of neighborhoods of 0 is given by sets of the form {x∈X:kxkX⩽1/n}. So, a sequence (xn)converges to xin Xif and only if kx−xnkX→0. If such topology is complete then Xis said to be a quasi-Banach space (Banach space if K=1). Having in mind the inequality (2.1), standard arguments show the next result. Although the result is not the same, our ideas are similar to the ones that prove Theorem 1.1 in [21] (see also the rest of Section 1 in that paper). PROPOSITION 2.1. The following statements are equivalent: (i) X is complete. (ii) For every 0<r0⩽r (r as in (2.1)), if (xn)⊂X is such that ∑kxnkr0 X<∞, then ∑xnconverges in X. (iii) There exists r0>0satisfying that if (xn)⊂X is such that ∑kxnkr0 X<∞then ∑xnconverges in X. Note that if a series ∑xnconverges in Xthen (2.2)   ∑xn  X ⩽41/rK∑kxnkr X1/r, where ris as in (2.1). By using the map k| · |k given in Theorem 1.2 of [16], it is routine to check that if xn→xin Xthen (2.3) 4−1/rlim sup kxnkX⩽kxkX⩽41/rlim inf kxnkX. Also note that a linear map T:X→Ybetween quasi-Banach spaces is continuous if and only if there exists a constant M>0 such that kTxkY⩽MkxkX for all x∈X, see p. 8 of [16]. By a quasi-Banach function space (briefly, quasi-B.f.s.) we mean a quasi-Banach space X(µ)⊂L0(µ)satisfying that if f∈X(µ)and g∈L0(µ)with |g|⩽|f|µ- a.e. then g∈X(µ)and kgkX(µ)⩽kfkX(µ). If X(µ)is a Banach space we will refer to it as a Banach function space (briefly, B.f.s.). In particular, a quasi-B.f.s. is a quasi-Banach lattice for the µ-a.e. pointwise order, in which the convergence in quasi-norm of a sequence implies the convergence µ-a.e. for some subsequence. Let us prove this important fact. PROPOSITION 2.2. If fn→f in a quasi-B.f.s. X(µ), then there exists a subsequence fnj→fµ-a.e. POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 265 Proof. Let rbe as in (2.1). We can take a strictly increasing sequence (nj)j⩾1 such that kf−fnjkX(µ)⩽1/2j. For every m⩾1, since ∑ j⩾m kf−fnjkr X(µ)⩽∑ j⩾m 1 2jr <∞, by Proposition 2.1 and (2.2), it follows that gm=∑ j⩾m |f−fnj|converges in X(µ) and kgmkX(µ)⩽41/rK∑ j⩾m (1/2jr)1/r. Fix N⩾1 and let AN j={ω∈Ω: |f(ω)−fnj(ω)|>1/N}. Since χTm⩾1Sj⩾mAN j ⩽χSj⩾mAN j ⩽∑ j⩾m χAN j ⩽N∑ j⩾m |f−fnj|=Ngm, then kχTm⩾1Sj⩾mAN jkX(µ)⩽NkgmkX(µ)⩽41/rNK∑ j⩾m 1 2jr 1/r. Taking m→∞we have that kχTm⩾1Sj⩾mAN jkX(µ)=0 and so µ(Tm⩾1Sj⩾mAN j) = 0. Then µ(SN⩾1Tm⩾1Sj⩾mAN j) = 0, from which fnj→fµ-a.e. A quasi-B.f.s. X(µ)is σ-order continuous if for every (fn)⊂X(µ)with fn↓0 µ-a.e. it follows that kfnkX↓0. It has the σ-Fatou property if for every sequence (fn)⊂Xsuch that 0 ⩽fn↑fµ-a.e. and sup n kfnkX<∞we have that f∈Xand kfnkX↑ kfkX. A similar argument to that given in p. 2 of [19] for Banach lattices shows that every positive linear operator between quasi-Banach lattices is automatically continuous. In particular, all inclusions between quasi-B.f.s. are continuous. The intersection X(µ)∩Y(µ)and the sum X(µ) + Y(µ)of two quasi-B.f.s.’ (B.f.s.’) X(µ)and Y(µ)are quasi-B.f.s.’ (B.f.s.’) endowed respectively with the quasi-norm (norms) kfkX(µ)∩Y(µ)=max{k fkX(µ),kfkY(µ)}and kfkX(µ)+Y(µ)=inf(kf1kX(µ)+kf2kY(µ)), where the infimum is taken over all possible representations f=f1+f2µ-a.e. with f1∈X(µ)and f2∈Y(µ). The σ-order continuity is also preserved by these operations: if X(µ)and Y(µ)are σ-order continuous then X(µ)∩Y(µ)and X(µ) + Y(µ)are σ-order continuous. Detailed proofs of these facts can be found in [13], see also Section 3, Theorem 1.3 of [1] for the standard parts. Let p∈(0, ∞). The p-power of a quasi-B.f.s. X(µ)is the quasi-B.f.s. X(µ)p={f∈L0(µ):|f|p∈X(µ)} endowed with the quasi-norm kfkX(µ)p=k |f|pk1/p X(µ). 266 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ The reader can find a complete explanation of the space Xp(µ)for instance in Section 2.2 of [24] for the case when µis finite and χΩ∈X(µ). The proofs given there, with the natural modifications, work in our general case. However, note that the notation is different: our p-powers here are the (1/p)-th powers there. This standard space can be found in different sources, unfortunately, notation is not exactly the same in all of them. The following remark collects some results on the space X(µ)pwhich will be used in the next sections. First, recall that a quasi-B.f.s. X(µ)is p-convex if there exists a constant C>0 such that   n ∑ j=1 |fj|p1/p  X(µ)⩽Cn ∑ j=1 kfjkp X(µ)1/p for every finite subset (fj)n j=1⊂X(µ). The smallest constant satisfying the previous inequality is called the p-convexity constant of X(µ)and is denoted by M(p)[X(µ)]. REMARK 2.3. Let X(µ)be a quasi-B.f.s. The following statements hold: (i) X(µ)pis σ-order continuous if and only if X(µ)is σ-order continuous. (ii) If χΩ∈X(µ)and 0 <p⩽q<∞then X(µ)q⊂X(µ)p. (iii) If X(µ)is a B.f.s. then X(µ)pis p-convex. (iv) If X(µ)is a B.f.s. and p⩾1 then k·kX(µ)pis a norm and so X(µ)pis a B.f.s. (v) If X(µ)is (1/p)-convex with M(1/p)[X(µ)] = 1 then k · kX(µ)pis a norm and so X(µ)pis a B.f.s. Let T:X(µ)→Ebe a linear operator defined on a quasi-B.f.s. X(µ)and with values in a quasi-Banach space E. For q∈(0, ∞), the operator Tis said to be q-concave if there exists a constant C>0 such that n ∑ j=1 kT(fj)kq E1/q ⩽C  n ∑ j=1 |fj|q1/q  X(µ) for every finite subset (fj)n j=1⊂X(µ). A quasi-B.f.s. X(µ)is q-concave if the identity map i:X(µ)→X(µ)is q-concave. The reader can find an excellent presentation of the main results regarding p-convexity and q-concavity in Banach function spaces in Section 4 of [21]. Note that if Tis q-concave then it is p-concave for all p>q. A proof of this fact can be found in Proposition 2.54(iv) of [24] for the case when µis finite and χΩ∈X(µ). An adaptation of that proof to our context works. PROPOSITION 2.4. If X(µ)is a q-concave quasi-B.f.s. then it is σ-order continuous. Proof. Since q-concavity implies p-concavity for every 0 <q<p, we only have to consider the case q⩾1. Denote by Cthe q-concavity constant of X(µ) POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 267 and consider (fn)⊂X(µ)such that fn↓0µ-a.e. For every strictly increasing subsequence (nk)we have that m ∑ k=1 kfnk−fnk+1kq X(µ)1/q ⩽C  m ∑ k=1 |fnk−fnk+1|q1/q  X(µ)⩽C   m ∑ k=1 |fnk−fnk+1|  X(µ) =Ckfn1−fnm+1kX(µ)⩽Ckfn1kX(µ) for all m⩾1. Then, (fn)is a Cauchy sequence in X(µ), as otherwise we can find δ>0 and two subsequences (nk),(mk)such that nk<mk<nk+1<mk+1 and δ<kfnk−fmkkX(µ)⩽kfnk−fnk+1kX(µ)for all k, which is a contradiction. Let h∈X(µ)be such that fn→hin X(µ). From Proposition 2.2, there exists a subsequence fnj→hµ-a.e. and so h=0µ-a.e. Hence, kfnkX(µ)↓0. LEMMA 2.5. Let X(µ)and Y(µ)be two quasi-B.f.s.’ and consider a linear operator T :X(µ) + Y(µ)→E with values in a quasi-Banach space E. The operator T is q-concave if and only if both T :X(µ)→E and T :Y(µ)→E are q-concave. Proof. If T:X(µ) + Y(µ)→Eis q-concave, since X(µ)⊂X(µ) + Y(µ) continuously, it follows that T:X(µ)→Eis q-concave. Similarly, T:Y(µ)→E is q-concave. Suppose that T:X(µ)→Eand T:Y(µ)→Eare q-concave and denote by CXand CYtheir respective q-concavity constants. Write Kfor the constant satisfying the property (iii) of the quasi-norm k · kE. We will use the inequality: (2.4) (a+b)t⩽max{1, 2t−1}(at+bt) where 0 ⩽a,b<∞and 0 <t<∞. Let (fj)n j=1⊂X(µ) + Y(µ). For h= n ∑ j=1|fj|q1/q∈X(µ) + Y(µ), consider h1∈X(µ)and h2∈Y(µ)such that h= h1+h2µ-a.e. Taking the set A={ω∈Ω:h(ω)⩽2|h1(ω)|},αq=max{1,2q−1} and using (2.4), we have that n ∑ j=1 kT(fj)kq E⩽Kqn ∑ j=1 (kT(fjχA)kE+kT(fjχΩ\A)kE)q ⩽Kqαqn ∑ j=1 kT(fjχA)kq E+ n ∑ j=1 kT(fjχΩ\A)kq E. Note that (fjχA)n j=1⊂X(µ)as |fj|χA⩽hχA⩽2|h1|for all j. Then, n ∑ j=1 kT(fjχA)kq E⩽Cq X  n ∑ j=1 |fj|q1/qχA   q X(µ)=Cq XkhχAkq X(µ)⩽2qCq Xkh1kq X(µ). Similarly, (fjχΩ\A)n j=1⊂Y(µ)as |fj|χΩ\A⩽hχΩ\A⩽2|h2|µ-a.e. for all jand so n ∑ j=1 kT(fjχΩ\A)kq E⩽2qCq Ykh2kq Y(µ). 268 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ Denoting C=max{CX,CY}and using again (2.4), it follows that n ∑ j=1 kT(fj)kq E1/q ⩽2KCα1/q q(kh1kq X(µ)+kh2kq Y(µ))1/q ⩽21+|1−(1/q)|KC (kh1kX(µ)+kh2kY(µ)). Taking infimum over all representations n ∑ j=1|fj|q1/q=h1+h2µ-a.e. with h1∈ X(µ)and h2∈Y(µ), we have that n ∑ j=1 kT(fj)kq E1/q ⩽21+|1−(1/q)|KC   n ∑ j=1 |fj|q1/q  X(µ)+Y(µ). Further information on Banach lattices and function spaces can be found for instance in [1], [16], [19], [20], [21], [24], [31]. 2.2. INTEGRATION WITH RESPECT TO A VECTOR MEASURE DEFINED ON A δ-RING. Let Rbe a δ-ring of subsets of Ω(i.e. a ring closed under countable intersections) and let Rloc be the σ-algebra of all subsets Aof Ωsuch that A∩B∈ R for all B∈ R. Note that Rloc =Rwhenever Ris a σ-algebra. Write S(R)for the space of all R-simple functions (i.e. simple functions supported in R). A Banach space valued set function m:R → Eis a vector measure (real measure when E=R) if ∑m(An)converges to m(SAn)in Efor each sequence (An)⊂ R of pairwise disjoint sets with SAn∈ R. The variation of a real measure λ:R → Ris the measure |λ|:Rloc →[0, ∞] given by |λ|(A) = sup n∑|λ(Aj)|:(Aj)finite disjoint sequence in R ∩ 2Ao. The variation |λ|is finite on R. The space L1(λ)of integrable functions with respect to λis defined as the classical space L1(|λ|)with the usual norm |f|λ=R Ω |f|d|λ|. The integral of an R-simple function ϕ=n ∑ j=1ajχAjover A∈ Rloc is defined in the natural way by R A ϕdλ=n ∑ j=1ajλ(Aj∩A). The space S(R)is dense in L1(λ). This allows to define the integral of a function f∈L1(λ)over A∈ Rloc as R A fdλ=lim R A ϕndλfor any sequence (ϕn)⊂ S(R)converging to fin L1(λ). The semivariation of a vector measure m:R → Eis the function kmk:Rloc →[0, ∞] defined by kmk(A) = sup x∗∈BE∗ |x∗m|(A), POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 269 where BE∗is the closed unit ball of the topological dual E∗of Eand |x∗m|is the variation of the real measure x∗mgiven by the composition of mwith x∗. The semivariation kmkis finite on R. A set A∈ Rloc is said to be m-null if m(B) = 0 for every B∈ R ∩ 2A. This is equivalent to kmk(A) = 0. It is known that there exists a measure η:Rloc → [0, ∞]equivalent to kmk(see Theorem 3.2 of [2]). Denote L0(m) = L0(η). The space L1 w(m)of weakly integrable functions with respect to mis defined as the space of (m-a.e. equal) functions f∈L0(m)such that f∈L1(x∗m)for every x∗∈E∗. The space L1(m)of integrable functions with respect to mconsists in all functions f∈L1 w(m)satisfying that for each A∈ Rloc there exists xA∈E, which is denoted by R A fdm, such that x∗(xA) = Z A fdx∗m, for all x∗∈E∗. The spaces L1(m)and L1 w(m)are B.f.s.’ related to the measure space (Ω,Rloc,η), and the expression kfkm=sup x∗∈BE∗Z Ω |f|d|x∗m| gives a norm for both spaces. The norm of f∈L1(m)can also be computed by means of the formula (2.5) kfkm=sup n  Z Ω fϕdm  E:ϕ∈ S(R),|ϕ|⩽1o. Moreover, L1(m)is σ-order continuous and contains S(R)as a dense subset and L1 w(m)has the σ-Fatou property. For every R-simple function ϕ=n ∑ j=1αjχAiit follows that R A ϕdm=n ∑ j=1αim(Aj∩A)for all A∈ Rloc. The integration operator Im:L1(m)→Egiven by Im(f) = R Ω fdm, is a continuous linear operator with kIm(f)kE⩽kfkm. If mis positive, that is, it is defined on a Banach lattice and m(A)⩾0 for all A∈ R, then kfkm=kIm(|f|)kE for all f∈L1(m). For every g∈L1(m), the set function mg:Rloc →Egiven by mg(A) = Im(gχA)is a vector measure. Moreover, f∈L1(mg)if and only if f g ∈L1(m), and in this case kfkL1(mg)=kf gkL1(m). For definitions and general results regarding integration with respect to a vector measure defined on a δ-ring we refer to [4], [11], [18], [22], [23]. Let p∈(0, ∞). We denote by Lp(m)the p-power of L1(m), that is, Lp(m) = {f∈L0(m):|f|p∈L1(m)}. 276 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ In a brief overview, (i) implies (ii) and the fact that the extension of Tto X(µ)1/p+X(µ)is just ImT◦[i]follow from the optimality of L1(mT). Indeed, by the results in [13] quoted above, L1(mT)is the largest σ-order continuous quasi- B.f.s to which Tcan be extended, and so it must contain X(µ)1/p+X(µ); note that X(µ)1/p+X(µ)is σ-order continuous as X(µ)is so. The equivalence between (ii) and (iii) is a direct check. Statement (ii) implies (iv) since [i]:X(µ)1/p+X(µ)→ L1(mT)is continuous (as it is positive) and T=ImT◦[i]. Finally, (iv) implies (i) is based on a standard argument which uses the approximation of a measurable function through functions in X(µ)(possible by the σ-property) to construct an extension of Tto X(µ)1/p+X(µ). For a detailed proof of Theorem 3.1, see [13], where moreover it is proved that if Tis p-th power factorable with a continuous extension then Lp(mT)∩L1(mT)is the optimal domain to which Tcan be extended preserving this property. Now, let us go to the new results on optimal domains. We consider the following property stronger than p-th power factorable and look for its optimal domain. For p,q∈(0, ∞), we say that Tis (p,q)-power-concave if there exists a constant C>0 such that n ∑ j=1 kT(fj)kq/p Ep/q ⩽C  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ) for every finite subset (fj)n j=1⊂X(µ). If χΩ∈X(µ)and p⩾1 we have that X(µ)1/p+X(µ) = X(µ)1/p, and then our definition of (p,q)-power-concave operator coincides with the one given in Definition 6.1 of [24]. REMARK 3.2. The following statements hold: (i) A (1, q)-power-concave operator is just a q-concave operator. (ii) If Tis (p,q)-power-concave then Tis (q/p)-concave, as X(µ)⊂X(µ)1/p+ X(µ)continuously. (iii) If χΩ∈X(µ)and 0 <p<1, since X(µ)1/p+X(µ) = X(µ), we have that (p,q)-power-concavity coincides with (q/p)-concavity. (iv) If Tis (p,q)-power-concave then Tis p-th power factorable with a continuous extension. Indeed, the (p,q)-power-concave inequality applied to a unique function is just the item (iv) of Theorem 3.1 As we will see in the next result, (p,q)-power-concavity is closely related to the following property. We say that Tis p-th power factorable with a q-concave extension if there exists a q-concave linear extension of Tto X(µ)1/p+X(µ), i.e. T factors as X(µ)T// i&& E X(µ)1/p+X(µ) S 99 POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 277 with Sbeing a q-concave linear operator. In this case, one checks directly that T is q-concave. THEOREM 3.3. The following statements are equivalent: (i) T is (p,q)-power-concave. (ii) T is p-th power factorable with a (q/p)-concave extension. (iii) [i]:X(µ)1/p+X(µ)→L1(mT)is well defined and (q/p)-concave. (iv) [i]:X(µ)→L1(mT)is well defined and (q/p)-concave, and [i]:X(µ)→ Lp(mT)is well defined and q-concave. (v) [i]:X(µ)→(q/p)L1(mT)∩qLp(mT)is well defined. Moreover, if (i)–(v) holds, the extension of T to X(µ)1/p+X(µ)coincides with the integration operator ImT◦[i]. Proof. First note that (q/p)L1(mT)∩qLp(mT)is σ-order continuous as a consequence of Proposition 2.4. (i) ⇒(ii) From Remark 3.2(iv) we have that Tis p-th power factorable with a continuous extension. Let S:X(µ)1/p+X(µ)→Ebe a continuous linear operator extending T. We are going to see that Sis (q/p)-concave. Since Tis (p,q)-power-concave and S=Ton X(µ), there exists C>0 such that n ∑ j=1 kS(fj)kq/p Ep/q ⩽C  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ) for any finite subset (fj)n j=1⊂X(µ). Consider (fj)n j=1⊂X(µ)1/p+X(µ)with fj⩾0µ-a.e. for all j. The σ-property of X(µ)allows to find for each j=1, . . . , n a sequence (hj k)⊂X(µ)such that 0 ⩽hj k↑fjµ-a.e. as k→∞(see [13] for the details). For every k, we have that n ∑ j=1 kS(hj k)kq/p Ep/q ⩽C  n ∑ j=1 |hj k|q/pp/q  X(µ)1/p+X(µ) ⩽C  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ). On the other hand, since X(µ)1/p+X(µ)is σ-order continuous, it follows that hj k→fjin X(µ)1/p+X(µ)as k→∞, and so S(hj k)→S(fj)in Eas k→∞. Hence, taking limit as k→∞in the above inequality, it follows that n ∑ j=1 kS(fj)kq/p Ep/q ⩽C  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ). For a general (fj)n j=1⊂X(µ)1/p+X(µ), write fj=f+ j−f− jwhere f+ jand f− jare the positive and negative parts respectively of each fj. By using inequality (2.4) 278 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ and denoting αp,q=max{1,21−(p/q)}, we have that n ∑ j=1 kS(fj)kq/p Ep/q ⩽n ∑ j=1 (kS(f+ j)kE+kS(f− j)kE)q/pp/q ⩽αp,qn ∑ j=1 kS(f+ j)kq/p E+ n ∑ j=1 kS(f− j)kq/p Ep/q ⩽αp,qC  n ∑ j=1 |f+ j|q/p+ n ∑ j=1 |f− j|q/pp/q  X(µ)1/p+X(µ) =αp,qC  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ) (for the last equality note that |fj|q/p=|f+ j|q/p+|f− j|q/pas f+ jand f− jhave disjoint support). (ii) ⇒(iii) Since Tis p-th power factorable with a (q/p)-concave (and so continuous) extension, from Theorem 3.1, the map [i]:X(µ)1/p+X(µ)→L1(mT)is well defined. Let S:X(µ)1/p+X(µ)→Ebe a (q/p)-concave linear operator extending T. Note that S=ImT◦[i](Theorem 3.1). Denote by Cthe (q/p)-concavity constant of S. Consider (fj)n j=1⊂X(µ)1/p+X(µ)and fix ε>0. For each j, by (2.5), we can take ϕj∈ S(ΣX(µ))such that |ϕj|⩽1 and kfjkL1(mT)⩽ε 2jp/q+kImT(fjϕj)kE. Since fjϕj∈X(µ)1/p+X(µ)as |fjϕj|⩽|fj|, then ImT(fjϕj) = S(fjϕj). So, by using inequality (2.4) and the (q/p)-concavity of S, we have that n ∑ j=1 kfjkq/p L1(mT)⩽ n ∑ j=1ε 2jp/q+kS(fjϕj)kEq/p ⩽max{1, 2(q/p)−1}n ∑ j=1 ε 2j+ n ∑ j=1 kS(fjϕj)kq/p E ⩽max{1, 2(q/p)−1}ε+Cq/p  n ∑ j=1 |fjϕj|q/pp/q   q/p X(µ)1/p+X(µ) ⩽max{1, 2(q/p)−1}ε+Cq/p  n ∑ j=1 |fj|q/pp/q   q/p X(µ)1/p+X(µ). Taking limit as ε→0, we obtain n ∑ j=1 kfjkq/p L1(mT)⩽Cq/pmax{1, 2(q/p)−1}  n ∑ j=1 |fj|q/pp/q   q/p X(µ)1/p+X(µ) POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 279 and so n ∑ j=1 kfjkq/p L1(mT)p/q ⩽Cmax{1, 21−(p/q)}  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ). Hence, [i]:X(µ)1/p+X(µ)→L1(mT)is (q/p)-concave. (iii) ⇔(iv) From Theorem 3.1, we have that [i]:X(µ)1/p+X(µ)→L1(mT) is well defined if and only if [i]:X(µ)→Lp(mT)∩L1(mT)is well defined, which is equivalent to [i]:X(µ)→L1(mT)and [i]:X(µ)→Lp(mT)well defined. By Lemma 2.5 we have that [i]:X(µ)1/p+X(µ)→L1(mT)is (q/p)-concave if and only if [i]:X(µ)1/p→L1(mT)and [i]:X(µ)→L1(mT)are (q/p)-concave. On the other hand, it is straightforward to verify that [i]:X(µ)1/p→L1(mT)is (q/p)-concave if and only if [i]:X(µ)→Lp(mT)is q-concave. (iv) ⇔(v) follows from Proposition 2.8. (iii) ⇒(i) Denote by Cthe (q/p)-concavity constant of [i]:X(µ)1/p+ X(µ)→L1(mT). Consider (fj)n j=1⊂X(µ)and note that fj∈L1(mT)with ImT(fj) = T(fj)for all j. Then, n ∑ j=1 kT(fj)kq/p Ep/q=n ∑ j=1 kImT(fj)kq/p Ep/q ⩽n ∑ j=1 kfjkq/p L1(mT)p/q ⩽C  n ∑ j=1 |fj|q/pp/q  X(µ)1/p+X(µ). Note that qLp(mT) = ((q/p)L1(mT))p(see Proposition 2.9). In particular, in the case when Tis (p,q)-power-concave and χΩ∈X(µ)(so χΩ∈(q/p)L1(mT)), from Remark 2.3(ii) it follows that (q/p)L1(mT)∩qLp(mT) = qLp(mT)if p⩾1 and (q/p)L1(mT)∩qLp(mT) = (q/p)L1(mT)if 0 <p<1. THEOREM 3.4. Suppose that T is (p,q)-power-concave. Then, T factors as X(µ)T// [i](( E (q/p)L1(mT)∩qLp(mT) ImT 77 (3.1) with ImTbeing (p,q)-power-concave. Moreover, the factorization is optimal in the following sense: if Z(ξ)is a σ-order continuous quasi-B.f.s. such that ξµand X(µ)T// [i]"" E Z(ξ) S >> (3.2) with S being a (p,q)-power-concave linear operator then we have that [i]:Z(ξ)→ (q/p)L1(mT)∩qLp(mT)is well defined and S =ImT◦[i]. 280 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ Proof. The factorization (3.1) follows from Theorem 3.3. The space (q/p)L1(mT)∩qLp(mT) is σ-order continuous as noted before and satisfies the σ-property as X(µ)does. Since ImT:(q/p)L1(mT)∩qLp(mT)→Eis continuous (as ImT:L1(mT)→Eis so), we can apply Theorem 3.3 to see that it is (p,q)-power-concave. Note that ΣX(µ)⊂Σ(q/p)L1(mT)∩qLp(mT)and mImT(A) = ImT(χA) = T(χA) = mT(A)for all A∈ΣX(µ). That is, mTis the restriction of mImT:Σ(q/p)L1(mT)∩qLp(mT)→Eto ΣX(µ). From Lemma 3 of [3], it follows that L1(mImT) = L1(mT). Then, [i]:(q/p)L1(mT)∩qLp(mT)→(q/p)L1(mImT)∩qLp(mImT) is well defined as (q/p)L1(mImT)∩qLp(mImT) = (q/p)L1(mT)∩qLp(mT). Let Z(ξ)satisfy (3.2). In particular, Z(ξ)has the σ-property. From Theorem 3.3 applied to the operator S:Z(ξ)→E, we have that [i]:Z(ξ)→ (q/p)L1(mS)∩qLp(mS)is well defined and S=ImS◦[i]. Since ΣX(µ)⊂ΣZ(ξ)and mS(A)=S(χA)=T(χA)=mT(A)for all A∈ΣX(µ)(i.e. mTis the restriction of mS:ΣZ(ξ)→E to ΣX(µ)), from Lemma 3 of [3], it follows that L1(mS) = L1(mT)and ImS=ImT. Therefore, [i]:Z(ξ)→(q/p)L1(mS)∩qLp(mS) = (q/p)L1(mT)∩qLp(mT) is well defined and S=ImS◦[i] = ImT◦[i]. We can rewrite Theorem 3.4 in terms of optimal domains. COROLLARY 3.5. Suppose that T is (p,q)-power-concave. Then (q/p)L1(mT)∩ qLp(mT)is the largest σ-order continuous quasi-B.f.s. to which T can be extended as a (p,q)-power-concave operator still with values in E. Moreover, the extension of T to the space (q/p)L1(mT)∩qLp(mT)is given by the integration operator ImT. Recalling that (1, q)-power-concave operators coincide with q-concave operators, we obtain the following result, that is already well-known, as a particular case of our construction. The reader can find arguments that implicitly prove our result in the paper by Reisner [28] (Proposition 5, Lemma 6). Indeed, the space Mappearing in the factorization of Proposition 5 in that paper is q-concave. A look at how it is defined makes clear that it is also optimal, since any other qconcave factorization space is included in it. We notice also that the result itself can be directly deduced from Theorem 1 and Proposition 2 in [27] for the class of Banach lattices. The space qL1(mT)provides an explicit representation of the factorization space appearing in these results as a space of integrable functions. COROLLARY 3.6. Suppose that T is q-concave. Then qL1(mT)is the largest σorder continuous quasi-B.f.s. to which T can be extended as a q-concave operator still POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 281 with values in E. Moreover, the extension of T to qL1(mT)is given by the integration operator ImT. To end this section we give a direct application related to the Maurey– Rosenthal factorization of q-concave operators defined on a q-convex quasi-B.f.s. In the case when Tis q-concave, the integration operator ImTextends Tto the space qL1(mT)by Corollary 3.6. Note that the map [i]:X(µ)→qL1(mT)is q-concave as it is continuous and qL1(mT)is q-concave. From a variant of the Maurey–Rosenthal theorem proved in Corollary 5 of [9], under some extra conditions, if X(µ)is q-convex then [i]:X(µ)→qL1(mT)factors through the space Lq(µ). So, we obtain the following improvement of the usual factorization of q-concave operators on q-convex quasi-B.f.s.’. COROLLARY 3.7. Let 1⩽q<∞. Assume that µis σ-finite and that X(µ)is q-convex and has the σ-Fatou property. If T is q-concave then it can be factored as X(µ)T// Mg  E Lq(µ) Mg−1//qL1(mT) ImT OO for positive multiplication operators Mgand Mg−1. The converse is also true. 4. VECTOR MEASURE REPRESENTATION OF q-CONCAVE BANACH LATTICES In this last section we show our characterization of the class of Banach lattices which are p-convex and q-concave in terms of spaces of integrable functions with respect to a vector measure. For 1 <p, it is known that order continuous p-convex Banach lattices with p-convexity constant equal to 1 can be order isometrically represented as spaces Lpof a vector measure defined on a δ-ring (see Theorem 10 of [5]). We will see that the addition of the q-concavity property to the represented Banach lattice translates to adding some concavity property to the corresponding integration map. For the aim of clarity, we will assume in all this section that the p-convexity constants of the p-convex lattices appearing are 1; this is not a real restriction to our results, since it is well-known that each p-convex Banach lattice can be renormed with a Banach lattice norm for which the space has p-convexity constant equal to 1 (see Proposition 1.d.8 of [19]). First let us show two results concerning concavity for the integration operator of a vector measure which will be needed later. Let m:R → Ebe a vector measure defined on a δ-ring Rof subsets of Ω and with values in a Banach space E. PROPOSITION 4.1. The integration operator Im:L1(m)→E is q-concave if and only if L1(m)is q-concave. 282 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ Proof. Suppose that Im:L1(m)→Eis q-concave and denote by Cits qconcavity constant. Take (fj)n j=1⊂L1(m)and (ϕj)n j=1⊂ S(R)with |ϕj|⩽1 for all j. Since (fjϕj)n j=1⊂L1(m), as |fjϕj|⩽|fj|for all j, we have that n ∑ j=1 kIm(fjϕj)kq E1/q ⩽C  n ∑ j=1 |fjϕj|q1/q  L1(m)⩽C  n ∑ j=1 |fj|q1/q  L1(m). Taking supremum for each j=1, . . . , nover all ϕj∈ S(R)with |ϕj|⩽1, from (2.5), if follows that n ∑ j=1 kfjkq L1(m)1/q ⩽C  n ∑ j=1 |fj|q1/q  L1(m). The converse is obvious as Imis continuous. Direct useful consequences can be deduced of the fact that the integration map Im:L1(m)→Eis q-concave. Assume that mis defined on a σ-algebra and note that q-concavity for q⩾1 always implies (q,1)-concavity (see the definition for instance in p. 61 of [25]). Thus, by Proposition 7.9 of [25], if Imis q-concave for q⩾1 then it is weakly completely continuous (i.e. it maps weak Cauchy sequences into weakly convergent sequences). Moreover, this implies that L1(m)coincides with the space L1 w(m)and so it has the σ-Fatou property. In the case when χΩ∈L1(m)(for instance if mis defined on a σ-algebra), we obtain a further result regarding (p,q)-power-concave operators. PROPOSITION 4.2. Suppose that χΩ∈L1(m)and p ⩾1. The integration operator Im:Lp(m)→E is (p,q)-power-concave if and only if Lp(m)is q-concave. Proof. First note that under the hypothesis it follows that Lp(m)has the σproperty (in fact χΩ∈Lp(m)) and Lp(m)⊂L1(m). So, Im:Lp(m)→Eis well defined and continuous. Suppose that Im:Lp(m)→Eis (p,q)-power-concave. From Theorem 3.3, we have that [i]:Lp(m)→(q/p)L1(mIm)∩qLp(mIm)is well defined. Note that (Rloc)Lp(m)=Rloc and so mImcoincides with mχΩ(see Preliminaries). Then, L1(mIm) = L1(m)and so Lp(m)⊂(q/p)L1(m)∩qLp(m)⊂qLp(m). Hence, Lp(m)is q-concave as Lp(m) = qLp(m). Suppose now that Lp(m)is q-concave. Then, it is direct to check that L1(m) is (q/p)-concave. Since Lp(m)⊂L1(m), the integration operator Im:L1(m)→E is continuous and (Lp(m))1/p+Lp(m) = L1(m), it follows that Im:Lp(m)→E satisfies the inequality of the definition of (p,q)-power-concave operator. Let us go now to the representation of q-concave Banach lattices as spaces of integrable functions. We begin by considering B.f.s.’. POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 283 PROPOSITION 4.3. Let p,q∈(0, ∞)and let Z(ξ)be a q-concave B.f.s. which is also p-convex in the case when p >1. Then, Z(ξ)coincides with the space Lp(m)of a Banach space valued vector measure m :R → E defined on a δ-ring whose integration operator Im:L1(m)→E is (q/p)-concave. Moreover, if χΩ∈Z(ξ), the vector measure m is defined on a σ-algebra. Proof. Note that if p⩽1 then Z(ξ)1/pis a B.f.s. (see Remark 2.3(iv)). In the case when p>1, renorming Z(ξ)if it is necessary, we can assume that the p-convexity constant of Z(ξ)is equal to 1 (see Proposition 1.d.8 of [19]), and so Z(ξ)1/pis a B.f.s. (see Remark 2.3(v)). Consider the δ-ring ΣZ(ξ)={A∈Σ: χA∈Z(ξ)}and the finitely additive set function m:ΣZ(ξ)→Z(ξ)1/pgiven by m(A) = χA. Since Z(ξ)1/pis σ-order continuous, as Z(ξ)is so by Proposition 2.4, it follows that mis a vector measure. Let us see that L1(m) = Z(ξ)1/pwith equal norms and so we will have that Z(ξ)coincides with Lp(m). For ϕ∈ S(ΣZ(ξ))we have that ϕ∈Z(ξ)1/pand Im(ϕ) = ϕ. Moreover, since mis positive, (4.1) kϕkL1(m)=kIm(|ϕ|)kZ(ξ)1/p=kϕkZ(ξ)1/p. In particular, by taking ϕ=χA, we obtain that kmkis equivalent to ξ. Given f∈L1(m), since S(ΣZ(ξ))is dense in L1(m), we can take (ϕn)⊂ S(ΣZ(ξ))such that ϕn→fin L1(m)and m-a.e. From (4.1), we have that (ϕn)is a Cauchy sequence in Z(ξ)1/pand so there is h∈Z(ξ)1/psuch that ϕn→hin Z(ξ)1/p. Taking a subsequence ϕnj→hξ-a.e. we see that f=h∈Z(ξ)1/pand kfkZ(ξ)1/p=lim kϕnkZ(ξ)1/p=lim kϕnkL1(m)=kfkL1(m). Let now f∈Z(ξ)1/pand take (ϕn)⊂ S(Σ)such that 0 ⩽ϕn↑ | f|. For any n, writing ϕn=m ∑ j=1αjχAjwith (Aj)m j=1being pairwise disjoint and αj>0 for all j, we see that χAj⩽α−1/p j|f|1/pand so ϕn∈ S(ΣZ(ξ)). On the other hand, since Z(ξ)1/pis σ-order continuous, we have that ϕn→fin Z(ξ)1/p. From (4.1), we have that (ϕn)is a Cauchy sequence in L1(m)and so there is h∈L1(m) such that ϕn→hin L1(m). Taking a subsequence ϕnj→h m-a.e. we see that f=h∈L1(m). Hence, L1(m) = Z(ξ)1/pwith equal norms and, since Z(ξ)is q-concave, it follows that L1(m)is (q/p)-concave. From Proposition 4.1, the integration operator Im:L1(m)→Eis (q/p)-concave. Note that if χΩ∈Z(ξ), then ΣZ(ξ)=Σand so mis defined on a σalgebra. For the final result we need some concepts related to Banach lattices. The definitions of p-convexity, q-concavity and σ-order continuity for Banach lattices are the same that for B.f.s.’. A Banach lattice Fis said to be order continuous if for every downwards directed system (xτ)⊂Fwith xτ↓0 it follows that kxτkF↓0 284 OLVIDO DELGADO AND ENRIQUE A. SÁNCHEZ PÉREZ and is said to be σ-complete if every order bounded sequence in Fhas a supremum. A Banach lattice which is σ-order continuous and σ-complete at the same time is order continuous, see Proposition 1.a.8 of [19]. A weak unit of a Banach lattice Fis an element 0 ⩽e∈Fsuch that inf{x,e}=0 implies x=0. An operator T:F1→F2between Banach lattices is said to be an order isometry if it is linear, one to one, onto, kTxkF2=kxkF1for all x∈F1and T(inf{x,y}) = inf{Tx,Ty}for all x,y∈F1. In particular, an order isometry is a positive operator. So, by using Proposition 1.d.9 of [19], it is direct to check that every order isometry preserves p-convexity and q-concavity whenever p,q⩾1. THEOREM 4.4. Let p,q∈[1, ∞)and let F be a Banach lattice. The following statements are equivalent: (i) F is q-concave and p-convex. (ii) F is order isometric to a space Lp(m)of a Banach space valued vector measure m:R → E defined on a δ-ring whose integration operator Im:L1(m)→E is (q/p)- concave. Moreover, (i) holds with F having a weak unit if and only if (ii) holds with m defined on aσ-algebra. In this last case Im:Lp(m)→E is (p,q)-power-concave. Proof. (i) ⇒(ii) Since Fis q-concave, it satisfies a lower q-estimate (see Definition 1.f.4 of [19]) and then it is σ-complete and σ-order continuous (see the proof of Proposition 1.f.5 of [19]). So, Fis order continuous. From Theorem 5 of [12] we have that Fis order isometric to a space L1(ν)of a Banach space valued vector measure νdefined on a δ-ring. Then, L1(ν)is a B.f.s. satisfying the conditions of Proposition 4.3 and so L1(ν) = Lp(m)with m:R → Ebeing a vector measure defined on a δ-ring Rand with values in a Banach space E, whose integration operator Im:L1(m)→Eis (q/p)-concave. (ii) ⇒(i) Since Lp(m)is p-convex (Remark 2.3(iii)) and q-concave (as L1(m) is (q/p)-concave by Proposition 4.1), Falso is. Now suppose that (i) holds with Fhaving a weak unit. From Theorem 8 of [6] we have that Fis order isometric to a space L1(ν)of a Banach space valued vector measure νdefined on a σ-algebra. Since χΩ∈L1(ν), from Proposition 4.3 we have that (ii) holds with mdefined on a σ-algebra. Conversely, if (ii) holds with mdefined on a σ-algebra then χΩ∈Lp(m) (as χΩ∈L1(m)). So, the image of χΩby the order isometry is a weak unit in F. Moreover, from Proposition 4.2 it follows that Im:Lp(m)→Eis (p,q)-power- concave. In particular, from Theorem 4.4 we obtain that a Banach lattice is q-concave (with q⩾1) if and only if it is order isometric to a space L1(m)of a vector measure mwith a q-concave integration operator. Acknowledgements. The first author gratefully acknowledges the support of the Ministerio de Economía y Competitividad (project #MTM2012-36732-C03-03) and the Junta POWER CONCAVE OPERATORS AND REPRESENTATION BANACH LATTICES 285 de Andalucía (projects FQM-262 and FQM-7276), Spain. 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