Optimal domain of q-concave operators and vector measure representation of q-concave Banach lattices

arXiv:1511.02337v1 [math.FA] 7 Nov 2015 OPTIMAL DOMAIN OF q-CONCAVE OPERATORS AND VECTOR MEASURE REPRESENTATION OF q-CONCAVE BANACH LATTICES O. DELGADO AND E. A. S´ ANCHEZ P´ EREZ Abstract. Given a Banach space valued q-concave linear operator Tdefined on a σ-order continuous quasi-Banach function space, we provide a description of the optimal domain of Tpreserving q-concavity, that is, the largest σ-order continuous quasi-Banach function space to which Tcan be extended as a q-concave operator. We show in this way the existence of maximal extensions for q-concave operators. As an application, we show a representation theorem for q-concave Banach lattices through spaces of integrable functions with respect to a vector measure. This result culminates a series of representation theorems for Banach lattices using vector measures that have been obtained in the last twenty years. 1. Introduction Let X(µ) be a σ-order continuous quasi-Banach function space related to a positive measure µon a measurable space (Ω,Σ) such that there exists g∈X(µ) with g > 0µ-a.e. and let T:X(µ)→Ebe a continuous linear operator with values in a Banach space E. Considering the δ-ring ΣX(µ)of all sets A∈Σ satisfying that χA∈X(µ) and the vector measure mT: ΣX(µ)→Egiven by mT(A) = T(χA), it follows that the space L1(mT) of integrable functions with respect to mTis the optimal domain of Tpreserving continuity. That is, the largest σ-order continuous quasi-Banach function space to which Tcan be extended as a continuous operator still with values in E. Moreover, the extension of T to L1(mT) is given by the integration operator ImT. This fact was originally proved in [8, Corollary 3.3] for Banach function spaces X(µ) with µfinite and χΩ∈X(µ)⊂L1(µ), in which case ΣX(µ)coincides with the σ-algebra Σ. The extension for Banach function spaces (without extra assumptions) is deduced from [3, Proposition 4]. The jump to quasi- Banach function spaces appears in [26, Theorem 4.14] for the case when µis finite and χΩ∈X(µ) and in [16] for the general case. Some effort has been made in recent years to solve several versions of the following general problem: Suppose that the operator Thas a property P. Is there an optimal domain for Tpreserving P? that is, is there a function space Zsuch that Tcan be extended to Zas an operator with the property P in such a way that Zis the largest Date: July 20, 2018. 2010 Mathematics Subject Classification. Primary 47B38, 46G10. Secondary 46E30, 46B42. Key words and phrases. Banach lattices, q-concave operators, quasi-Banach function spaces, vector measures defined on a δ-ring. The first author gratefully acknowledges the support of the Ministerio de Econom´ıa y Competitividad (project #MTM2012-36732-C03-03) and the Junta de Andaluc´ıa (projects FQM-262 and FQM-7276), Spain. The second author acknowledges with thanks the support of the Ministerio de Econom´ıa y Competitividad (project #MTM2012-36740-C02-02), Spain. 1 2 O. DELGADO AND E. A. S´ ANCHEZ P´ EREZ space for which this holds? And in this case, which is the relation among Zand mT? The answer to the first question is in general no. For example, in [25] it is proved that for compacness or weak compacness Thas an optimal domain only in the case when ImT is compact or weakly compact, respectively. In the same line it is shown in [5] that T has an optimal domain for AM-compacness if and only if ImTis AM-compact. However, other properties have got positive answers to our problem, see [5] for narrow operators, [3] for order-w continuous or Y(η)-extensible operators and [14] for positive order continuous operators. Also in [5] the problem is studied for Dunford-Pettis operators, but although some partial results are shown there, the question of the existence of a maximal extension is still open. In this paper we analyze this problem for the case of q-concave operators, obtaining a positive answer. Namely, if Tis q-concave we show how to compute explicitly the largest quasi-Banach function space to which Tcan be extended preserving q-concavity (Corollary 4.6). Even more, we prove that this optimal domain is in fact the q-concave core of the space L1(mT) and the maximal extension is given by the integration operator ImT. These results are obtained as a particular case of the more general Theorem 4.4 which gives the optimal domain for a class of operators (called (p, q)-power-concave) which contains the q-concave operators. As an application we obtain an improvement in some sense of the Maurey-Rosenthal factorization of q-concave operators acting in q-convex Banach function spaces (Corollary 4.7). The reader can find information about this nowadays classical topic for example in [11], [12] and the references therein. In the last section we provide a new representation theorem for q-concave Banach lattices in terms of a vector measure. This type of representation theorems has its origin in [7, Theorem 8], where it is proved that every order continuous Banach lattice Fwith a weak unit is order isometric to a space L1(ν) of a vector measure νdefined on a σ-algebra. Later in [17, Proposition 2.4] it is shown that if moreover Fis p-convex 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 measures used in the representations of Fare defined in a δ-ring, see [15, Theorem 5] and [6, Theorem 10]. Also there are representation theorems for Freplacing σ-order continuity by the Fatou property, in this case through spaces of weakly integrable functions, see [9], [10], [15] and [18]. For p, q ∈[1,∞), in Theorem 5.4 we obtain that every q-concave and p-convex Banach lattice is order isometric to a space Lp(m) of a vector measure mdefined on a δ-ring whose integration operator ImT is q p-concave. The converse is also true. In particular, every q-concave Banach lattice is order isometric to a space L1(m) of a vector measure mhaving a q-concave integration operator. 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. q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 3 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 µ are equivalent then L0(µ) = L0(λ) and [i] is the identity map i. 2.1. Quasi-Banach function spaces. Let Xbe a real vector space and k · kXaquasinorm 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. For 0 < r ≤1 being such that K= 21 r−1, it follows that    n X j=1 xj  X≤41 rn X j=1 kxjkr X1 r(2.1) for every finite subset (xj)n j=1 ⊂X, see [19, Lemma 1.1]. The quasi-norm k · kXinduces 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. Proposition 2.1. The following statements are equivalent: (a) Xis complete. (b) For every 0< r′≤r(ras in (2.1)) it follows that if (xn)⊂Xis such that Pkxnkr′ X<∞then Pxnconverges in X. (c) There exists r′>0satisfying that if (xn)⊂Xis such that Pkxnkr′ X<∞then Pxnconverges in X. Note that if a series Pxnconverges in Xthen   Xxn  X≤41 rKXkxnkr X1 r,(2.2) where ris as in (2.1). By using the map ||| · ||| given in [19, Theorem 1.2], it is routine to check that if xn→xin Xthen 4−1 rlim sup kxnkX≤ kxkX≤41 rlim inf kxnkX.(2.3) 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 kT xkY≤MkxkXfor all x∈X, see [19, p. 8]. 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 4 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ g∈X(µ) and kgkX(µ)≤ kfkX(µ). If X(µ) is a Banach space we will refer 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→fin a quasi-B.f.s. X(µ), then there exists a subsequence fnj→f µ–a.e. Proof. Let rbe as in (2.1). We can take a strictly increasing sequence (nj)j≥1such that kf−fnjkX(µ)≤1 2j. For every m≥1, since X j≥m kf−fnjkr X(µ)≤X j≥m 1 2jr <∞, by Proposition 2.1 and (2.2), it follows that gm=Pj≥m|f−fnj|converges in X(µ) and kgmkX(µ)≤41 rK(Pj≥m 1 2jr )1 r. Fix N≥1 and let AN j={ω∈Ω : |f(ω)−fnj(ω)|>1 N}. Since χ∩m≥1∪j≥mAN j≤χ∪j≥mAN j≤X j≥m χAN j≤NX j≥m |f−fnj|=Ngm, then kχ∩m≥1∪j≥mAN jkX(µ)≤NkgmkX(µ)≤41 rNKX j≥m 1 2jr 1 r. Taking m→ ∞ we have that kχ∩m≥1∪j≥mAN jkX(µ)= 0 and so µ(∩m≥1∪j≥mAN j) = 0. Then µ(∪N≥1∩m≥1∪j≥mAN j) = 0, from which fnj→f µ-a.e.  A quasi-B.f.s. X(µ) is σ-order continuous if for every (fn)⊂X(µ) with fn↓0µ-a.e. it follows that kfnkX↓0. It has the σ-Fatou property if for every sequence (fn)⊂Xsuch that 0 ≤fn↑f µ-a.e. and supnkfnkX<∞we have that f∈Xand kfnkX↑ kfkX. A similar argument to that given in [21, p. 2] 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-norms (norms) kfkX(µ)∩Y(µ)= max kfkX(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 this 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 [16], see also [1, §3, Theorem 1.3] 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|pk 1 p X(µ). q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 5 The reader can find a complete explanation of the space Xp(µ) for instance in [26, §2.2] 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 X j=1 |fj|p1 p  X(µ)≤Cn X 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: (a) X(µ)pis σ-order continuous if and only if X(µ) is σ-order continuous. (b) If χΩ∈X(µ) and 0 < p ≤q < ∞then X(µ)q⊂X(µ)p. (c) If X(µ) is a B.f.s. then X(µ)pis p-convex. (d) 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. (e) 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 X j=1 kT(fj)kq E1 q≤C  n X 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. Note that if Tis q-concave then it is p-concave for all p > q. A proof of this fact can be found in [26, Proposition 2.54.(iv)] for the case when µis finite and χΩ∈X(µ). An adaptation of this 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 q < p, we only have to consider the case q≥1. Denote by Cthe q-concavity constant of X(µ) and consider (fn)⊂X(µ) such that fn↓0µ-a.e. For every strictly increasing subsequence (nk) we have that m X k=1 kfnk−fnk+1 kq X(µ)1 q≤C  m X k=1 |fnk−fnk+1 |q1 q  X(µ) ≤C   m X k=1 |fnk−fnk+1 |  X(µ) =Ckfn1−fnm+1 kX(µ) ≤Ckfn1kX(µ) 6 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ for all m≥1. Then, (fn) is a Cauchy sequence in X(µ), as in other case 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+1 kX(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→h µ–a.e. and so h= 0 µ-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)→Ewith values in a quasi-Banach space E. The operator Tis q-concave if and only if both T:X(µ)→Eand T:Y(µ)→Eare 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)→Eis q-concave. Suppose that T:X(µ)→Eand T:Y(µ)→Eare q-concave and denote by CXand CY their respective q-concavity constants. Write Kfor the constant satisfying the property (iii) of the quasi-norm k · kE. We will use the inequality: (a+b)t≤max{1,2t−1}(at+bt) (2.4) where 0 ≤a, b < ∞and 0 < t < ∞. Let (fj)n j=1 ⊂X(µ)+Y(µ). For h=Pn 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 X j=1 kT(fj)kq E≤Kq n X j=1 kT(fjχA)kE+kT(fjχΩ\A)kEq ≤Kqαq n X j=1 kT(fjχA)kq E+ n X 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 X j=1 kT(fjχA)kq E≤Cq X  n X 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 X j=1 kT(fjχΩ\A)kq E≤2qCq Ykh2kq Y(µ). Denoting C= max{CX, CY}and using again (2.4), it follows that n X j=1 kT(fj)kq E1 q≤2KCα 1 q qkh1kq X(µ)+kh2kq Y(µ)1 q ≤21+|1−1 q|KC kh1kX(µ)+kh2kY(µ). Taking infimum over all representations Pn j=1 |fj|q1 q=h1+h2µ-a.e. with h1∈X(µ) and h2∈Y(µ), we have that n X j=1 kT(fj)kq E1 q≤21+|1−1 q|KC   n X j=1 |fj|q1 q  X(µ)+Y(µ).  q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 7 Further information on Banach lattices and function spaces can be found for instance in [1, 19, 21, 22, 26] and [28]. 2.2. Integration with respect to a vector measure defined on a δ-ring. Let R be 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 Ω such 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 Pm(An) converges to m(∪An) in Efor each sequence (An)⊂ R of pairwise disjoint sets with ∪An∈ R. The variation of a real measure λ:R → Ris the measure |λ|:Rloc →[0,∞] given by |λ|(A) = sup nX|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=RΩ|f|d|λ|. The integral of an R–simple function ϕ=Pn j=1 ajχAjover A∈ Rloc is defined in the natural way by RAϕ dλ =Pn j=1 ajλ(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 RAf dλ = lim RAϕndλ for 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), 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 kmk is 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 [2, Theorem 3.2]). 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 RAf dm, such that x∗(xA) = ZA f dx∗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∗ZΩ |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 kfkm= sup    ZΩ fϕ dm   E :ϕ∈ S(R),|ϕ| ≤ 1.(2.5) 8 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ 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=Pn j=1 αjχAiit follows that RAϕ dm =Pn j=1 αim(Aj∩A) for all A∈ Rloc. The integration operator Im:L1(m)→Egiven by Im(f) = RΩf dm, is a continuous linear operator with kIm(f)kE≤ kfkm. If mis positive, that is m(A)≥0 for all A∈ R, then kfkm=kIm(|f|)kEfor 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 fg ∈L1(m), and in this case kfkL1(mg)=kfgkL1(m). For definitions and general results regarding integration with respect to a vector measure defined on a δ-ring we refer to [4, 13, 20, 23, 24]. 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). As noted in Remark 2.3, the space Lp(m) is a σ-order continuous quasi-B.f.s. with the quasi-norm kfkLp(m)=k |f|pk1/p L1(m). Moreover, if p≥1 then k · kLp(m)is a norm and so Lp(m) is a B.f.s. Direct proofs of these facts and some general results on the spaces Lp(m) can be found in [6]. 3. The q-concave core of a σ-order continuous quasi-B.f.s Let X(µ) be a σ-order continuous quasi-B.f.s. and q∈(0,∞). We define the space qX(µ) to be the set of functions f∈X(µ) such that kfkqX(µ)= sup n X j=1 kfjkq X(µ)1 q<∞, where the supremum is taken over all finite set (fj)n j=1 ⊂X(µ) satisfying |f|=Pn j=1 |fj|q1 q µ-a.e. Note that kfkX(µ)≤ kfkqX(µ). Proposition 3.1. The space qX(µ)is a quasi-B.f.s. with quasi-norm k · kqX(µ). Proof. First let us see that if f∈qX(µ) and g∈L0(µ) with |g| ≤ |f|µ–a.e. then g∈qX(µ) and kgkqX(µ)≤ kfkqX(µ). Note that g∈X(µ) as f∈X(µ). Let (gj)n j=1 ⊂X(µ) be such that |g|=Pn j=1 |gj|q1 qµ-a.e. and take h=|f|q− |g|q 1 q∈X(µ). Since |f|=Pn j=1 |gj|q+|h|q1 qµ-a.e., we have that n X j=1 kgjkq X(µ)1 q≤n X j=1 kgjkq X(µ)+khkq X(µ)1 q≤ kfkqX(µ). Taking supremum over all (gj)n j=1 ⊂Xwith |g|=Pn j=1 |gj|q1 qµ-a.e., we have that g∈qX(µ) with kgkqX(µ)≤ kfkqX(µ). It is direct to check that k · kqX(µ)satisfies the properties (i) and (ii) of a quasi-norm. Let Kbe the constant satisfying the property (iii) of a quasi-norm for k · kX(µ). Given f, g ∈qX(µ) and (hj)n j=1 ⊂Xsuch that |f+g|=Pn j=1 |hj|q1 qµ-a.e., by taking q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 9 A=ω∈Ω : |f(ω) + g(ω)| ≤ 2|f(ω)|,αq= max{1,2q−1}and using (2.4), we have that n X j=1 khjkq X(µ)≤Kq n X j=1 khjχAkX(µ)+khjχΩ\AkX(µ)q ≤Kqαqn X j=1 khjχAkq X(µ)+ n X j=1 khjχΩ\Akq X(µ). Note that |f+g|χA,|f+g|χΩ\A∈qX(µ) as |f+g|χA≤2|f|and |f+g|χΩ\A≤2|g|. Then, n X j=1 khjkq X(µ)≤Kqαq |f+g|χA  q qX(µ)+ |f+g|χΩ\A  q qX(µ) ≤2qKqαqkfkq qX(µ)+kgkq qX(µ). By using again (2.4), we have that n X j=1 khjkq X(µ)1 q≤2Kα 1 q qkfkq qX(µ)+kgkq qX(µ)1 q ≤21+|1−1 q|KkfkqX(µ)+kgkqX(µ). Taking supremum over all (hj)n j=1 ⊂Xwith |f+g|=Pn j=1 |hj|q1 qµ-a.e., we have that kf+gkqX(µ)≤21+|1−1 q|KkfkqX(µ)+kgkqX(µ).(3.1) Finally, let us prove that qX(µ) is complete. Denote by rand r′the constants satisfying (2.1) for X(µ) and qX(µ) respectively. Note that r′< r as 21+|1−1 q|K > K. Let (fn)⊂ qX(µ) be such that Pkfnkr′ qX(µ)<∞. Since k · kX(µ)≤ k · kqX(µ), from Proposition 2.1, we have that Pk j=1 fj→gand Pk j=1 |fj| → ˜gin X(µ). From Proposition 2.2, it follows that Pk j=1 fj→gand Pk j=1 |fj| → ˜gpointwise except on a µ-null set Z. Fix any γ > 1 and consider the sets Ak=ω∈Ω : |g(ω)| ≤ γPk j=1 |fj(ω)|. Note that Ω\ ∪ Ak⊂Zand so it is µ-null. Indeed, if ω6∈ Zand |g(ω)|> γ Pk j=1 |fj(ω)|for all k(in particular P|fn(ω)| 6= 0), then γP|fn(ω)| ≤ |g(ω)| ≤ P|fn(ω)|<∞, which is a contradiction. Also note that gχAk∈qX(µ) as |g|χAk≤γPk j=1 |fj|. Given (hj)n j=1 ⊂X with |g|=Pn j=1 |hj|q1 qµ-a.e., we have that n X j=1 khjχAkkq X(µ)1 q≤ kgχAkkqX(µ)≤γ   k X j=1 |fj|  qX(µ) ≤41 r′γk X j=1 kfjkr′ qX(µ)1 r′ ≤41 r′γXkfnkr′ qX(µ)1 r′. On other hand, since X(µ) is σ-order continuous and |hj|χAk↑ |hj|µ-a.e. as k→ ∞, we have that hjχAk→hjin X(µ) as k→ ∞. Taking limit as k→ ∞ in the above inequality and applying (2.3), we obtain that n X j=1 khjkq X(µ)1 q≤41 r+1 r′γXkfnkr′ qX(µ)1 r′. 16 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ Then, n X j=1 kT(fj)k q p Ep q=n X j=1 kImT(fj)k q p Ep q ≤n X j=1 kfjk q p L1(mT)p q ≤C  n X j=1 |fj|q pp q  X(µ) 1 p+X(µ).  Note that qLp(mT) = q pL1(mT)p(see Proposition 3.4). In particular, in the case when Tis (p, q)-power-concave and χΩ∈X(µ) (so χΩ∈q pL1(mT)), from Remark 2.3.(b) it follows that q pL1(mT)∩qLp(mT) = qLp(mT) if p≥1 and q pL1(mT)∩qLp(mT) = q pL1(mT) if p < 1. Theorem 4.4. Suppose that Tis (p, q)-power-concave. Then, Tfactors as X(µ)T// [i] '' E q pL1(mT)∩qLp(mT) ImT 88 (4.1) with ImTbeing (p, q)-power-concave. Moreover, the factorization is optimal in the 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 >> (4.2) with Sbeing a (p, q)-power-concave linear operator                                =⇒[i]: Z(ξ)→q pL1(mT)∩qLp(mT) is well defined and S=ImT◦[i]. Proof. The factorization (4.1) follows from Theorem 4.3. The space q pL1(mT)∩qLp(mT) is σ-order continuous as noted before and satisfies the σ-property as X(µ) does. Since ImT:q pL1(mT)∩qLp(mT)→Eis continuous (as ImT:L1(mT)→Eis so), we can apply Theorem 4.3 to see that it is (p, q)-power-concave. Note that ΣX(µ)⊂Σq pL1(mT)∩qLp(mT) and mImT(A) = ImT(χA) = T(χA) = mT(A) for all A∈ΣX(µ). That is, mTis the restriction of mImT: Σq pL1(mT)∩qLp(mT)→Eto ΣX(µ). From [3, Lemma 3], it follows that L1(mImT) = L1(mT). Then, [i]: q pL1(mT)∩qLp(mT)→q pL1(mImT)∩qLp(mImT) is well defined as q pL1(mImT)∩qLp(mImT) = q pL1(mT)∩qLp(mT). Let Z(ξ) satisfy (4.2). In particular, Z(ξ) has the σ-property. From Theorem 4.3 applied to the operator S:Z(ξ)→E, we have that [i]: Z(ξ)→q pL1(mS)∩qLp(mS) is well defined and S=ImS◦[i]. Since ΣX(µ)⊂ΣZ(ξ)and mS(A) = S(χA) = T(χA) = mT(A) q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 17 for all A∈ΣX(µ)(i.e. mTis the restriction of mS: ΣZ(ξ)→Eto ΣX(µ)), from [3, Lemma 3], it follows that L1(mS) = L1(mT) and ImS=ImT. Therefore, [i]: Z(ξ)→q pL1(mS)∩qLp(mS) = q pL1(mT)∩qLp(mT) is well defined and S=ImS◦[i] = ImT◦[i].  We can rewrite Theorem 4.4 in terms of optimal domains. Corollary 4.5. Suppose that Tis (p, q)-power-concave. Then q pL1(mT)∩qLp(mT) is the largest σ-order continuous quasi-B.f.s. to which Tcan be extended as a (p, q)- power-concave operator still with values in E. Moreover, the extension of Tto the space q pL1(mT)∩qLp(mT)is given by the integration operator ImT. Recalling that the (1, q)-power-concave operators coincide with the q-concave operators, we obtain our main result. Corollary 4.6. Suppose that Tis q-concave. Then qL1(mT)is the largest σ-order continuous quasi-B.f.s. to which Tcan be extended as a q-concave operator still with values in E. Moreover, the extension of Tto qL1(mT)is given by the integration operator ImT. Let us give now 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, by Corollary 4.6, the integration operator ImTextends Tto the space qL1(mT). 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 [11, Corollary 5], 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 4.7. Let 1≤q < ∞. Assume that µis σ-finite and that X(µ)is q-convex and has the σ-Fatou property. If Tis 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. 5. Vector measure representation of q-concave Banach lattices In this last section we look for a 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 can be order isometrically represented as spaces Lpof a vector measure defined on a δring (see [6, Theorem 10]). 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. First let us show two results concerning concavity for the integration operator of a vector measure which will be needed later. 18 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ 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 5.1. The integration operator Im:L1(m)→Eis q-concave if and only if L1(m)is q-concave. Proof. Suppose that Im:L1(m)→Eis q-concave and denote by Cits q-concavity 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 X j=1 kIm(fjϕj)kq E1 q≤C  n X j=1 |fjϕj|q1 q  L1(m)≤C  n X j=1 |fj|q1 q  L1(m). Taking supremum for each j= 1, .., n over all ϕj∈ S(R) with |ϕj| ≤ 1, from (2.5), if follows that n X j=1 kfjkq L1(m)1 q≤C  n X 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 qconcavity for q≥1 always implies (q, 1)-concavity (see the definition for instance in [27, p. 61]). Thus, by [27, Proposition 7.9], 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 5.2. Suppose that χΩ∈L1(m)and p≥1. The integration operator Im:Lp(m)→Eis (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 σ-property (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 4.3, we have that [i]: Lp(m)→q pL1(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 pL1(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 pconcave. Since Lp(m)⊂L1(m), the integration operator Im:L1(m)→Eis continuous and Lp(m)1 p+Lp(m) = L1(m), it follows that Im:Lp(m)→Esatisfies 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.’. Proposition 5.3. Let p, q ∈(0,∞)and let Z(ξ)be a q-concave B.f.s. which is also pconvex in the case when p > 1. Then, Z(ξ)coincides with the space Lp(m)of a Banach q-CONCAVE OPERATORS AND q-CONCAVE BANACH LATTICES 19 space valued vector measure m:R → Edefined on a δ-ring whose integration operator Im:L1(m)→Eis q p-concave. Moreover, if χΩ∈Z(ξ), the vector measure mis defined on a σ-algebra. Proof. Note that if p≤1 then Z(ξ)1 pis a B.f.s. (see Remark 2.3.(d)). 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 [21, Proposition 1.d.8]), and so Z(ξ)1 pis a B.f.s. (see Remark 2.3.(e)). 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, kϕkL1(m)=kIm(|ϕ|)kZ(ξ) 1 p=kϕkZ(ξ) 1 p.(5.1) 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 (5.1), 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→h ξ-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=Pm j=1 αjχAjwith (Aj)m j=1 being pairwise disjoint and αj>0 for all j, we see that χAj≤α−1/p j|f|1/p and so ϕn∈ S(ΣZ(ξ)). On other hand, since Z(ξ)1 pis σ-order continuous, we have that ϕn→fin Z(ξ)1 p. From (5.1), 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 5.1, the integration operator Im:L1(m)→Eis q p-concave. Note that if χΩ∈Z(ξ), then ΣZ(ξ)= Σ and so mis defined on a σ-algebra.  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 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 [21, Proposition 1.a.8]. Aweak 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, kT xkF2=kxkF1for all x∈F1and T(inf{x, y}) = inf{T x, T y} for all x, y ∈F1. In particular, an order isometry is a positive operator. So, by using [21, Proposition 1.d.9], it is direct to check that every order isometry preserves p-convexity and q-concavity whenever p, q ≥1. 20 O. DELGADO AND E. A. S ´ ANCHEZ P´ EREZ Theorem 5.4. Let p, q ∈[1,∞)and let Fbe a Banach lattice. The following statements are equivalent: (a) Fis q-concave and p-convex. (b) Fis order isometric to a space Lp(m)of a Banach space valued vector measure m:R → Edefined on a δ-ring whose integration operator Im:L1(m)→Eis q pconcave. Moreover, (a) holds with Fhaving a weak unit if and only if (b) holds with mdefined on aσ-algebra. In this last case Im:Lp(m)→Eis (p, q)-power-concave. Proof. (a) ⇒(b) Since Fis q-concave, it satisfies a lower q-estimate (see [21, Definition 1.f.4]) and then it is σ-complete and σ-order continuous (see the proof of [21, Proposition 1.f.5]). So, Fis order continuous. From [15, Theorem 5] 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 5.3 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. (b) ⇒(a) Since Lp(m) is p-convex (Remark ??.(c)) and q-concave (as L1(m) is q pconcave by Proposition 5.1), Falso is. Now suppose that (a) holds with Fhaving a weak unit. From [7, Theorem 8] we have that Fis order isometric to a space L1(ν) of a Banach space valued vector measure ν defined on a σ-algebra. Since χΩ∈L1(ν), from Proposition 5.3 we have that (b) holds with mdefined on a σ-algebra. 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S. de Ingenier´ ıa de Edificaci´ on, Universidad de Sevilla, Avenida de Reina Mercedes, 4 A, Sevilla 41012, Spain E-mail address:[email protected] Instituto Universitario de Matem´ atica Pura y Aplicada, Universitat Polit` ecnica de Val` encia, Camino de Vera s/n, 46022 Valencia, Spain. E-mail address:[email protected]