Optimal domain of q -concave operators and vector measure representation of q -concave Banach lattices
aa r X i v : . [ m a t h . F A ] N ov OPTIMAL DOMAIN OF q -CONCAVE OPERATORS AND VECTORMEASURE 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 T defined on a σ -order continuous quasi-Banach function space, we provide a description of theoptimal domain of T preserving q -concavity, that is, the largest σ -order continuousquasi-Banach function space to which T can 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 latticesthrough spaces of integrable functions with respect to a vector measure. Thisresult culminates a series of representation theorems for Banach lattices using vectormeasures that have been obtained in the last twenty years. Introduction
Let X ( µ ) be a σ -order continuous quasi-Banach function space related to a positivemeasure µ on a measurable space (Ω , Σ) such that there exists g ∈ X ( µ ) with g > µ -a.e.and let T : X ( µ ) → E be a continuous linear operator with values in a Banach space E .Considering the δ -ring Σ X ( µ ) of all sets A ∈ Σ satisfying that χ A ∈ X ( µ ) and the vectormeasure m T : Σ X ( µ ) → E given by m T ( A ) = T ( χ A ), it follows that the space L ( m T ) ofintegrable functions with respect to m T is the optimal domain of T preserving continuity.That is, the largest σ -order continuous quasi-Banach function space to which T can beextended as a continuous operator still with values in E . Moreover, the extension of T to L ( m T ) is given by the integration operator I m T . This fact was originally proved in[8, Corollary 3.3] for Banach function spaces X ( µ ) with µ finite and χ Ω ∈ X ( µ ) ⊂ L ( µ ),in which case Σ X ( µ ) coincides with the σ -algebra Σ. The extension for Banach functionspaces (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 µ is finite and χ Ω ∈ X ( µ ) and in [16] for the general case.Some effort has been made in recent years to solve several versions of the followinggeneral problem: Suppose that the operator T has a property P. Is there an optimaldomain for T preserving P? that is, is there a function space Z such that T can beextended to Z as an operator with the property P in such a way that Z is 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, vectormeasures defined on a δ -ring.The first author gratefully acknowledges the support of the Ministerio de Econom´ıa y Competitividad(project space for which this holds? And in this case, which is the relation among Z and m T ?The answer to the first question is in general no. For example, in [25] it is proved thatfor compacness or weak compacness T has an optimal domain only in the case when I m T 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 I m T is 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 continuousoperators. Also in [5] the problem is studied for Dunford-Pettis operators, but althoughsome partial results are shown there, the question of the existence of a maximal extensionis still open.In this paper we analyze this problem for the case of q -concave operators, obtaining apositive answer. Namely, if T is q -concave we show how to compute explicitly the largestquasi-Banach function space to which T can be extended preserving q -concavity (Corollary4.6). Even more, we prove that this optimal domain is in fact the q -concave core of thespace L ( m T ) and the maximal extension is given by the integration operator I m T . Theseresults are obtained as a particular case of the more general Theorem 4.4 which gives theoptimal 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-Rosenthalfactorization of q -concave operators acting in q -convex Banach function spaces (Corollary4.7). 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 Banachlattices in terms of a vector measure. This type of representation theorems has its originin [7, Theorem 8], where it is proved that every order continuous Banach lattice F with aweak unit is order isometric to a space L ( ν ) of a vector measure ν defined on a σ -algebra.Later in [17, Proposition 2.4] it is shown that if moreover F is p -convex then it is orderisometric to L p ( m ) for another vector measure m . Similar results work for F without weakunit but in this case the vector measures used in the representations of F are defined in a δ -ring, see [15, Theorem 5] and [6, Theorem 10]. Also there are representation theoremsfor F replacing σ -order continuity by the Fatou property, in this case through spaces ofweakly integrable functions, see [9], [10], [15] and [18]. For p, q ∈ [1 , ∞ ), in Theorem5.4 we obtain that every q -concave and p -convex Banach lattice is order isometric to aspace L p ( m ) of a vector measure m defined on a δ -ring whose integration operator I m T is qp -concave. The converse is also true. In particular, every q -concave Banach lattice isorder isometric to a space L ( m ) of a vector measure m having a q -concave integrationoperator. 2. Preliminaries
In this section we establish the notation and present the basic results on quasi-Banachfunction spaces (including the proof of some of them for completeness) and on vectormeasure integration, which will be used through the whole paper. -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 L ( µ ) the space of all Σ–measurable real valued functions on Ω, where functions whichare equal µ –a.e. are identified.Given two set functions µ, λ : Σ → [0 , ∞ ] we will write λ ≪ µ if µ ( A ) = 0 implies λ ( A ) = 0. If λ ≪ µ and µ ≪ λ we will say that µ and λ are equivalent . If µ, λ : Σ → [0 , ∞ ]are two measures with λ ≪ µ , then the map [ i ] : L ( µ ) → L ( λ ) which takes a µ –a.e. classin L ( µ ) represented by f into the λ –a.e. class represented by the same f , is a well definedlinear map. In order to simplify notation we will write [ i ]( f ) = f . Note that if λ and µ are equivalent then L ( µ ) = L ( λ ) and [ i ] is the identity map i .2.1. Quasi-Banach function spaces.
Let X be a real vector space and k · k X a quasi-norm on X , that is a function k · k X : X → [0 , ∞ ) satisfying the following conditions:(i) k x k X = 0 if and only if x = 0,(ii) k αx k X = | α | · k x k X for all α ∈ R and x ∈ X , and(iii) there is a constant K ≥ k x + y k X ≤ K ( k x k X + k y k X ) for all x, y ∈ X .For 0 < r ≤ K = 2 r − , it follows that (cid:13)(cid:13)(cid:13) n X j =1 x j (cid:13)(cid:13)(cid:13) X ≤ r (cid:16) n X j =1 k x j k rX (cid:17) r (2.1)for every finite subset ( x j ) nj =1 ⊂ X , see [19, Lemma 1.1]. The quasi-norm k · k X inducesa metrizable vector topology on X where a base of neighborhoods of 0 is given by sets ofthe form { x ∈ X : k x k X ≤ n } . So, a sequence ( x n ) converges to x in X if and only if k x − x n k X →
0. If such topology is complete then X is 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) X is complete. (b) For every < r ′ ≤ r ( r as in (2.1) ) it follows that if ( x n ) ⊂ X is such that P k x n k r ′ X < ∞ then P x n converges in X . (c) There exists r ′ > satisfying that if ( x n ) ⊂ X is such that P k x n k r ′ X < ∞ then P x n converges in X . Note that if a series P x n converges in X then (cid:13)(cid:13)(cid:13) X x n (cid:13)(cid:13)(cid:13) X ≤ r K (cid:16) X k x n k rX (cid:17) r , (2.2)where r is as in (2.1). By using the map ||| · ||| given in [19, Theorem 1.2], it is routine tocheck that if x n → x in X then4 − r lim sup k x n k X ≤ k x k X ≤ r lim inf k x n k X . (2.3)Also note that a linear map T : X → Y between quasi-Banach spaces is continuous ifand only if there exists a constant M > k T x k Y ≤ M k x k X for 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 ) ⊂ L ( µ ) satisfying that if f ∈ X ( µ ) and g ∈ L ( µ ) with | g | ≤ | f | µ –a.e. then O. DELGADO AND E. A. S´ANCHEZ P´EREZ g ∈ X ( µ ) and k g k X ( µ ) ≤ k f k X ( µ ) . If X ( µ ) is a Banach space we will refer it as a Banachfunction space (briefly, B.f.s.). In particular, a quasi-B.f.s. is a quasi-Banach lattice forthe µ -a.e. pointwise order, in which the convergence in quasi-norm of a sequence impliesthe convergence µ -a.e. for some subsequence. Let us prove this important fact. Proposition 2.2. If f n → f in a quasi-B.f.s. X ( µ ) , then there exists a subsequence f n j → f µ –a.e.Proof. Let r be as in (2.1). We can take a strictly increasing sequence ( n j ) j ≥ such that k f − f n j k X ( µ ) ≤ j . For every m ≥
1, since X j ≥ m k f − f n j k rX ( µ ) ≤ X j ≥ m jr < ∞ , by Proposition 2.1 and (2.2), it follows that g m = P j ≥ m | f − f n j | converges in X ( µ ) and k g m k X ( µ ) ≤ r K ( P j ≥ m jr ) r . Fix N ≥ A Nj = { ω ∈ Ω : | f ( ω ) − f n j ( ω ) | > N } .Since χ ∩ m ≥ ∪ j ≥ m A Nj ≤ χ ∪ j ≥ m A Nj ≤ X j ≥ m χ A Nj ≤ N X j ≥ m | f − f n j | = Ng m , then k χ ∩ m ≥ ∪ j ≥ m A Nj k X ( µ ) ≤ N k g m k X ( µ ) ≤ r NK (cid:16) X j ≥ m jr (cid:17) r . Taking m → ∞ we have that k χ ∩ m ≥ ∪ j ≥ m A Nj k X ( µ ) = 0 and so µ ( ∩ m ≥ ∪ j ≥ m A Nj ) = 0.Then µ ( ∪ N ≥ ∩ m ≥ ∪ j ≥ m A Nj ) = 0, from which f n j → f µ -a.e. (cid:3) A quasi-B.f.s. X ( µ ) is σ -order continuous if for every ( f n ) ⊂ X ( µ ) with f n ↓ µ -a.e. itfollows that k f n k X ↓
0. It has the σ -Fatou property if for every sequence ( f n ) ⊂ X suchthat 0 ≤ f n ↑ f µ -a.e. and sup n k f n k X < ∞ we have that f ∈ X and k f n k X ↑ k f k X .A similar argument to that given in [21, p. 2] for Banach lattices shows that everypositive linear operator between quasi-Banach lattices is automatically continuous. Inparticular, 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) k f k X ( µ ) ∩ Y ( µ ) = max (cid:8) k f k X ( µ ) , k f k Y ( µ ) (cid:9) and k f k X ( µ )+ Y ( µ ) = inf (cid:0) k f k X ( µ ) + k f k Y ( µ ) (cid:1) , where the infimum is taken over all possible representations f = f + f µ -a.e. with f ∈ X ( µ ) and f ∈ 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 -ordercontinuous. Detailed proofs of these facts can be found in [16], see also [1, §
3, Theorem1.3] 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 = (cid:8) f ∈ L ( µ ) : | f | p ∈ X ( µ ) (cid:9) endowed with the quasi-norm k f k X ( µ ) p = k | f | p k p X ( µ ) . -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 5 The reader can find a complete explanation of the space X p ( µ ) for instance in [26, § µ is finite and χ Ω ∈ X ( µ ). The proofs given there, with the naturalmodifications, work in our general case. However, note that the notation is different: our p -powers here are the p -th powers there. This standard space can be found in differentsources, unfortunately, notation is not exactly the same in all of them.The following remark collects some results on the space X ( µ ) p which will be used in thenext sections. First, recall that a quasi-B.f.s. X ( µ ) is p -convex if there exists a constant C > (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | p (cid:17) p (cid:13)(cid:13)(cid:13) X ( µ ) ≤ C (cid:16) n X j =1 k f j k pX ( µ ) (cid:17) p for every finite subset ( f j ) nj =1 ⊂ X ( µ ). The smallest constant satisfying the previousinequality is called the p-convexity constant of X ( µ ) and is denoted by M ( p ) [ X ( µ )]. Remark . Let X ( µ ) be a quasi-B.f.s. The following statements hold:(a) X ( µ ) p is σ -order continuous if and only if X ( µ ) is σ -order continuous.(b) If χ Ω ∈ 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 ) p is p -convex.(d) If X ( µ ) is a B.f.s. and p ≥ k · k X ( µ ) p is a norm and so X ( µ ) p is a B.f.s.(e) If X ( µ ) is p -convex with M ( p ) [ X ( µ )] = 1 then k · k X ( µ ) p is a norm and so X ( µ ) p isa B.f.s.Let T : X ( µ ) → E be a linear operator defined on a quasi-B.f.s. X ( µ ) and with valuesin a quasi-Banach space E . For q ∈ (0 , ∞ ), the operator T is said to be q -concave if thereexists a constant C > (cid:16) n X j =1 k T ( f j ) k qE (cid:17) q ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) X ( µ ) for every finite subset ( f j ) nj =1 ⊂ X ( µ ). A quasi-B.f.s. X ( µ ) is q-concave if the identitymap i : X ( µ ) → X ( µ ) is q-concave.Note that if T is q -concave then it is p -concave for all p > q . A proof of this fact canbe found in [26, Proposition 2.54.(iv)] for the case when µ is finite and χ Ω ∈ X ( µ ). Anadaptation 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 thecase q ≥
1. Denote by C the q -concavity constant of X ( µ ) and consider ( f n ) ⊂ X ( µ ) suchthat f n ↓ µ -a.e. For every strictly increasing subsequence ( n k ) we have that (cid:16) m X k =1 k f n k − f n k +1 k qX ( µ ) (cid:17) q ≤ C (cid:13)(cid:13)(cid:13)(cid:16) m X k =1 | f n k − f n k +1 | q (cid:17) q (cid:13)(cid:13)(cid:13) X ( µ ) ≤ C (cid:13)(cid:13)(cid:13) m X k =1 | f n k − f n k +1 | (cid:13)(cid:13)(cid:13) X ( µ ) = C k f n − f n m +1 k X ( µ ) ≤ C k f n k X ( µ ) O. DELGADO AND E. A. S´ANCHEZ P´EREZ for all m ≥
1. Then, ( f n ) is a Cauchy sequence in X ( µ ), as in other case we can find δ > n k ), ( m k ) such that n k < m k < n k +1 < m k +1 and δ < k f n k − f m k k X ( µ ) ≤ k f n k − f n k +1 k X ( µ ) for all k , which is a contradiction. Let h ∈ X ( µ )be such that f n → h in X ( µ ). From Proposition 2.2, there exists a subsequence f n j → hµ –a.e. and so h = 0 µ -a.e. Hence, k f n k X ( µ ) ↓ (cid:3) 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 -concaveif and only if both T : X ( µ ) → E and T : Y ( µ ) → E are q -concave.Proof. If T : X ( µ ) + Y ( µ ) → E is q -concave, since X ( µ ) ⊂ X ( µ ) + Y ( µ ) continuously, itfollows that T : X ( µ ) → E is q -concave. Similarly, T : Y ( µ ) → E is q -concave.Suppose that T : X ( µ ) → E and T : Y ( µ ) → E are q -concave and denote by C X and C Y their respective q -concavity constants. Write K for the constant satisfying the property(iii) of the quasi-norm k · k E . We will use the inequality:( a + b ) t ≤ max { , t − } ( a t + b t ) (2.4)where 0 ≤ a, b < ∞ and 0 < t < ∞ . Let ( f j ) nj =1 ⊂ X ( µ )+ Y ( µ ). For h = (cid:0) P nj =1 | f j | q (cid:1) q ∈ X ( µ ) + Y ( µ ), consider h ∈ X ( µ ) and h ∈ Y ( µ ) such that h = h + h µ -a.e. Taking theset A = (cid:8) ω ∈ Ω : h ( ω ) ≤ | h ( ω ) | (cid:9) , α q = max { , q − } and using (2.4), we have that n X j =1 k T ( f j ) k qE ≤ K q n X j =1 (cid:16) k T ( f j χ A ) k E + k T ( f j χ Ω \ A ) k E (cid:17) q ≤ K q α q n X j =1 k T ( f j χ A ) k qE + n X j =1 k T ( f j χ Ω \ A ) k qE ! . Note that ( f j χ A ) nj =1 ⊂ X ( µ ) as | f j | χ A ≤ hχ A ≤ | h | for all j . Then, n X j =1 k T ( f j χ A ) k qE ≤ C qX (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q χ A (cid:13)(cid:13)(cid:13) qX ( µ ) = C qX k hχ A k qX ( µ ) ≤ q C qX k h k qX ( µ ) . Similarly, ( f j χ Ω \ A ) nj =1 ⊂ Y ( µ ) as | f j | χ Ω \ A ≤ hχ Ω \ A ≤ | h | µ -a.e. for all j and so n X j =1 k T ( f j χ Ω \ A ) k qE ≤ q C qY k h k qY ( µ ) . Denoting C = max { C X , C Y } and using again (2.4), it follows that (cid:16) n X j =1 k T ( f j ) k qE (cid:17) q ≤ KCα q q (cid:0) k h k qX ( µ ) + k h k qY ( µ ) (cid:1) q ≤ | − q | KC (cid:0) k h k X ( µ ) + k h k Y ( µ ) (cid:1) . Taking infimum over all representations (cid:0) P nj =1 | f j | q (cid:1) q = h + h µ -a.e. with h ∈ X ( µ )and h ∈ Y ( µ ), we have that (cid:16) n X j =1 k T ( f j ) k qE (cid:17) q ≤ | − q | KC (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) X ( µ )+ Y ( µ ) . (cid:3) -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 7 Further information on Banach lattices and function spaces can be found for instancein [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 R loc be the σ –algebra of all subsets A of Ω such that A ∩ B ∈ R for all B ∈ R . Note that R loc = R whenever R is 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 → E is a vector measure ( real measure when E = R ) if P m ( A n ) converges to m ( ∪ A n ) in E for each sequence ( A n ) ⊂ R of pairwisedisjoint sets with ∪ A n ∈ R .The variation of a real measure λ : R → R is the measure | λ | : R loc → [0 , ∞ ] given by | λ | ( A ) = sup n X | λ ( A j ) | : ( A j ) finite disjoint sequence in R ∩ A o . The variation | λ | is finite on R . The space L ( λ ) of integrable functions with respect to λ is defined as the classical space L ( | λ | ) with the usual norm | f | λ = R Ω | f | d | λ | . Theintegral of an R –simple function ϕ = P nj =1 a j χ A j over A ∈ R loc is defined in the naturalway by R A ϕ dλ = P nj =1 a j λ ( A j ∩ A ). The space S ( R ) is dense in L ( λ ). This allows todefine the integral of a function f ∈ L ( λ ) over A ∈ R loc as R A f dλ = lim R A ϕ n dλ forany sequence ( ϕ n ) ⊂ S ( R ) converging to f in L ( λ ).The semivariation of a vector measure m : R → E is the function k m k : R loc → [0 , ∞ ]defined by k m k ( A ) = sup x ∗ ∈ B E ∗ | x ∗ m | ( A ) , where B E ∗ is the closed unit ball of the topological dual E ∗ of E and | x ∗ m | is the variationof the real measure x ∗ m given by the composition of m with x ∗ . The semivariation k m k is finite on R .A set A ∈ R loc is said to be m –null if m ( B ) = 0 for every B ∈ R ∩ A . This isequivalent to k m k ( A ) = 0. It is known that there exists a measure η : R loc → [0 , ∞ ]equivalent to k m k (see [2, Theorem 3.2]). Denote L ( m ) = L ( η ).The space L w ( m ) of weakly integrable functions with respect to m is defined as the spaceof ( m -a.e. equal) functions f ∈ L ( m ) such that f ∈ L ( x ∗ m ) for every x ∗ ∈ E ∗ . Thespace L ( m ) of integrable functions with respect to m consists in all functions f ∈ L w ( m )satisfying that for each A ∈ R loc there exists x A ∈ E , which is denoted by R A f dm , suchthat x ∗ ( x A ) = Z A f dx ∗ m, for all x ∗ ∈ E ∗ . The spaces L ( m ) and L w ( m ) are B.f.s.’ related to the measure space (Ω , R loc , η ), andthe expression k f k m = sup x ∗ ∈ B E ∗ Z Ω | f | d | x ∗ m | gives a norm for both spaces. The norm of f ∈ L ( m ) can also be computed by means ofthe formula k f k m = sup (cid:26)(cid:13)(cid:13)(cid:13)(cid:13)Z Ω fϕ dm (cid:13)(cid:13)(cid:13)(cid:13) E : ϕ ∈ S ( R ) , | ϕ | ≤ (cid:27) . (2.5) O. DELGADO AND E. A. S´ANCHEZ P´EREZ
Moreover, L ( m ) is σ -order continuous and contains S ( R ) as a dense subset and L w ( m )has the σ -Fatou property. For every R -simple function ϕ = P nj =1 α j χ A i it follows that R A ϕ dm = P nj =1 α i m ( A j ∩ A ) for all A ∈ R loc .The integration operator I m : L ( m ) → E given by I m ( f ) = R Ω f dm , is a continuouslinear operator with k I m ( f ) k E ≤ k f k m . If m is positive , that is m ( A ) ≥ A ∈ R ,then k f k m = k I m ( | f | ) k E for all f ∈ L ( m ).For every g ∈ L ( m ), the set function m g : R loc → E given by m g ( A ) = I m ( gχ A ) isa vector measure. Moreover, f ∈ L ( m g ) if and only if fg ∈ L ( m ), and in this case k f k L ( m g ) = k fg k L ( m ) .For definitions and general results regarding integration with respect to a vector mea-sure defined on a δ -ring we refer to [4, 13, 20, 23, 24].Let p ∈ (0 , ∞ ). We denote by L p ( m ) the p -power of L ( m ), that is, L p ( m ) = (cid:8) f ∈ L ( m ) : | f | p ∈ L ( m ) (cid:9) . As noted in Remark 2.3, the space L p ( m ) is a σ -order continuous quasi-B.f.s. with thequasi-norm k f k L p ( m ) = k | f | p k /pL ( m ) . Moreover, if p ≥ k · k L p ( m ) is a norm andso L p ( m ) is a B.f.s. Direct proofs of these facts and some general results on the spaces L p ( 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 k f k qX ( µ ) = sup (cid:16) n X j =1 k f j k qX ( µ ) (cid:17) q < ∞ , where the supremum is taken over all finite set ( f j ) nj =1 ⊂ X ( µ ) satisfying | f | = (cid:0) P nj =1 | f j | q (cid:1) q µ -a.e. Note that k f k X ( µ ) ≤ k f k qX ( µ ) . Proposition 3.1.
The space qX ( µ ) is a quasi-B.f.s. with quasi-norm k · k qX ( µ ) .Proof. First let us see that if f ∈ qX ( µ ) and g ∈ L ( µ ) with | g | ≤ | f | µ –a.e. then g ∈ qX ( µ )and k g k qX ( µ ) ≤ k f k qX ( µ ) . Note that g ∈ X ( µ ) as f ∈ X ( µ ). Let ( g j ) nj =1 ⊂ X ( µ )be such that | g | = (cid:0) P nj =1 | g j | q (cid:1) q µ -a.e. and take h = (cid:12)(cid:12) | f | q − | g | q (cid:12)(cid:12) q ∈ X ( µ ). Since | f | = (cid:0) P nj =1 | g j | q + | h | q (cid:1) q µ -a.e., we have that (cid:16) n X j =1 k g j k qX ( µ ) (cid:17) q ≤ (cid:16) n X j =1 k g j k qX ( µ ) + k h k qX ( µ ) (cid:17) q ≤ k f k qX ( µ ) . Taking supremum over all ( g j ) nj =1 ⊂ X with | g | = (cid:0) P nj =1 | g j | q (cid:1) q µ -a.e., we have that g ∈ qX ( µ ) with k g k qX ( µ ) ≤ k f k qX ( µ ) .It is direct to check that k · k qX ( µ ) satisfies the properties (i) and (ii) of a quasi-norm.Let K be the constant satisfying the property (iii) of a quasi-norm for k · k X ( µ ) . Given f, g ∈ qX ( µ ) and ( h j ) nj =1 ⊂ X such that | f + g | = (cid:0) P nj =1 | h j | q (cid:1) q µ -a.e., by taking -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 9 A = (cid:8) ω ∈ Ω : | f ( ω ) + g ( ω ) | ≤ | f ( ω ) | (cid:9) , α q = max { , q − } and using (2.4), we have that n X j =1 k h j k qX ( µ ) ≤ K q n X j =1 (cid:0) k h j χ A k X ( µ ) + k h j χ Ω \ A k X ( µ ) (cid:1) q ≤ K q α q (cid:16) n X j =1 k h j χ A k qX ( µ ) + n X j =1 k h j χ Ω \ A k qX ( µ ) (cid:17) . Note that | f + g | χ A , | f + g | χ Ω \ A ∈ qX ( µ ) as | f + g | χ A ≤ | f | and | f + g | χ Ω \ A ≤ | g | .Then, n X j =1 k h j k qX ( µ ) ≤ K q α q (cid:16)(cid:13)(cid:13) | f + g | χ A (cid:13)(cid:13) qqX ( µ ) + (cid:13)(cid:13) | f + g | χ Ω \ A (cid:13)(cid:13) qqX ( µ ) (cid:17) ≤ q K q α q (cid:0) k f k qqX ( µ ) + k g k qqX ( µ ) (cid:1) . By using again (2.4), we have that (cid:16) n X j =1 k h j k qX ( µ ) (cid:17) q ≤ Kα q q (cid:0) k f k qqX ( µ ) + k g k qqX ( µ ) (cid:1) q ≤ | − q | K (cid:0) k f k qX ( µ ) + k g k qX ( µ ) (cid:1) . Taking supremum over all ( h j ) nj =1 ⊂ X with | f + g | = (cid:0) P nj =1 | h j | q (cid:1) q µ -a.e., we have that k f + g k qX ( µ ) ≤ | − q | K (cid:0) k f k qX ( µ ) + k g k qX ( µ ) (cid:1) . (3.1)Finally, let us prove that qX ( µ ) is complete. Denote by r and r ′ the constants satisfying(2.1) for X ( µ ) and qX ( µ ) respectively. Note that r ′ < r as 2 | − q | K > K . Let ( f n ) ⊂ qX ( µ ) be such that P k f n k r ′ qX ( µ ) < ∞ . Since k · k X ( µ ) ≤ k · k qX ( µ ) , from Proposition2.1, we have that P kj =1 f j → g and P kj =1 | f j | → ˜ g in X ( µ ). From Proposition 2.2, itfollows that P kj =1 f j → g and P kj =1 | f j | → ˜ g pointwise except on a µ -null set Z . Fixany γ > A k = (cid:8) ω ∈ Ω : | g ( ω ) | ≤ γ P kj =1 | f j ( ω ) | (cid:9) . Note thatΩ \ ∪ A k ⊂ Z and so it is µ -null. Indeed, if ω Z and | g ( ω ) | > γ P kj =1 | f j ( ω ) | for all k (in particular P | f n ( ω ) | 6 = 0), then γ P | f n ( ω ) | ≤ | g ( ω ) | ≤ P | f n ( ω ) | < ∞ , which is acontradiction. Also note that gχ A k ∈ qX ( µ ) as | g | χ A k ≤ γ P kj =1 | f j | . Given ( h j ) nj =1 ⊂ X with | g | = (cid:0) P nj =1 | h j | q (cid:1) q µ -a.e., we have that (cid:16) n X j =1 k h j χ A k k qX ( µ ) (cid:17) q ≤ k gχ A k k qX ( µ ) ≤ γ (cid:13)(cid:13)(cid:13) k X j =1 | f j | (cid:13)(cid:13)(cid:13) qX ( µ ) ≤ r ′ γ (cid:16) k X j =1 k f j k r ′ qX ( µ ) (cid:17) r ′ ≤ r ′ γ (cid:16) X k f n k r ′ qX ( µ ) (cid:17) r ′ . On other hand, since X ( µ ) is σ -order continuous and | h j | χ A k ↑ | h j | µ -a.e. as k → ∞ , wehave that h j χ A k → h j in X ( µ ) as k → ∞ . Taking limit as k → ∞ in the above inequalityand applying (2.3), we obtain that (cid:16) n X j =1 k h j k qX ( µ ) (cid:17) q ≤ r + r ′ γ (cid:16) X k f n k r ′ qX ( µ ) (cid:17) r ′ . Now, taking supremum over all ( h j ) nj =1 ⊂ X with | g | = (cid:0) P nj =1 | h j | q (cid:1) q µ -a.e., it followsthat g ∈ qX ( µ ) with k g k qX ( µ ) ≤ r + r ′ γ (cid:0) P k f n k r ′ qX ( µ ) (cid:1) r ′ . Even more, since γ is arbitrary,taking γ → (cid:13)(cid:13)(cid:13) X f n (cid:13)(cid:13)(cid:13) qX ( µ ) ≤ r + r ′ (cid:16) X k f n k r ′ qX ( µ ) (cid:17) r ′ . Of course P nj =1 f j → g in qX ( µ ) as (cid:13)(cid:13)(cid:13) g − n X j =1 f j (cid:13)(cid:13)(cid:13) qX ( µ ) = (cid:13)(cid:13)(cid:13) X j>n f j (cid:13)(cid:13)(cid:13) qX ( µ ) ≤ r + r ′ (cid:16) X j>n k f j k r ′ qX ( µ ) (cid:17) r ′ → . Therefore, from Proposition 2.1 it follows that qX ( µ ) is complete. (cid:3) Proposition 3.2.
The space qX ( µ ) is q -concave. In consequence, it is also σ -ordercontinuous.Proof. Let ( f j ) nj =1 ⊂ qX ( µ ) and consider ( h jk ) m j k =1 ⊂ X ( µ ) with | f j | = (cid:0) P m j k =1 | h jk | q (cid:1) q µ -a.e. for each j . Since (cid:0) P nj =1 | f j | q (cid:1) q = (cid:0) P nj =1 P m j k =1 | h jk | q (cid:1) q µ -a.e., it follows that n X j =1 m j X k =1 k h jk k qX ( µ ) ≤ (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) qqX ( µ ) . Taking supremum for each j = 1 , ..., n over all ( h jk ) m j k =1 ⊂ X ( µ ) with | f j | = (cid:0) P m j k =1 | h jk | q (cid:1) q µ -a.e., we have that n X j =1 k f j k qqX ( µ ) ≤ (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) qqX ( µ ) and so qX ( µ ) is q -concave. The σ -order continuity is given by Proposition 2.4. (cid:3) Even more, the following proposition shows that qX ( µ ) is in fact the q -concave core of X ( µ ), that is, the largest q -concave quasi-B.f.s. related to µ contained in X ( µ ). Inparticular, qX ( µ ) = X ( µ ) whenever X ( µ ) is q -concave. Proposition 3.3.
Let Z ( ξ ) be a quasi-B.f.s. with µ ≪ ξ . The following statements areequivalent: (a) [ i ] : Z ( ξ ) → X ( µ ) is well defined and q -concave. (b) [ i ] : Z ( ξ ) → qX ( µ ) is well defined.In particular, qX ( µ ) is the q -concave core of X ( µ ) .Proof. (a) ⇒ (b) Denote by C the q -concavity constant of the operator [ i ] : Z ( ξ ) → X ( µ ).Let f ∈ Z ( ξ ) (so f ∈ X ( µ )) and ( f j ) nj =1 ⊂ X ( µ ) with | f | = (cid:0) P nj =1 | f j | q (cid:1) q except on a µ -null set N . Since | f j | χ Ω \ N ≤ | f | pointwise (so ξ -a.e.), then f j χ Ω \ N ∈ Z ( ξ ). Noting that f j = f j χ Ω \ N µ -a.e., it follows that (cid:16) n X j =1 k f j k qX ( µ ) (cid:17) q = (cid:16) n X j =1 k f j χ Ω \ N k qX ( µ ) (cid:17) q ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q χ Ω \ N (cid:13)(cid:13)(cid:13) Z ( ξ ) ≤ C k f k Z ( ξ ) . Hence f ∈ qX ( µ ) with k f k qX ( µ ) ≤ C k f k Z ( ξ ) . -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 11 (b) ⇒ (a) Clearly [ i ] : Z ( ξ ) → X ( µ ) is well defined as qX ( µ ) ⊂ X ( µ ). Denote by M the continuity constant of [ i ] : Z ( ξ ) → qX ( µ ) (recall that every positive operator betweenquasi-B.f.s.’ is continuous). For every ( f j ) nj =1 ⊂ Z ( ξ ) we have that (cid:0) P nj =1 | f j | q (cid:1) q is in qX ( µ ) as it is in Z ( ξ ), and so (cid:16) n X j =1 k f j k qX ( µ ) (cid:17) q ≤ (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) qX ( µ ) ≤ M (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) Z ( ξ ) . Hence, [ i ] : Z ( ξ ) → X ( µ ) is q -concave.In particular, if Z ( µ ) is a q -concave quasi-B.f.s. such that Z ( µ ) ⊂ X ( µ ), we have that i : Z ( µ ) → X ( µ ) is well defined, continuous and so q -concave. Then, from (a) ⇒ (b) wehave that Z ( µ ) ⊂ qX ( µ ). (cid:3) For p ∈ (0 , ∞ ), the p -power of qX ( µ ) can be described in terms of the p -power of X ( µ ). Proposition 3.4.
The equality (cid:0) qX ( µ ) (cid:1) p = qpX ( µ ) p holds with equal norms.Proof. Let f ∈ (cid:0) qX ( µ ) (cid:1) p . Since | f | p ∈ qX ( µ ), in particular | f | p ∈ X ( µ ) and so f ∈ X ( µ ) p .Consider ( f j ) nj =1 ⊂ X ( µ ) p satisfying that | f | = (cid:0) P nj =1 | f j | qp (cid:1) qp µ -a.e. Noting that( | f j | p ) nj =1 ⊂ X ( µ ) and | f | p = (cid:0) P nj =1 ( | f j | p ) q (cid:1) q µ -a.e., we have that (cid:16) n X j =1 k f j k qpX ( µ ) p (cid:17) qp = (cid:16) n X j =1 k | f j | p k qX ( µ ) (cid:17) qp ≤ k | f | p k p qX ( µ ) = k f k ( qX ( µ )) p . Then, f ∈ qpX ( µ ) p and k f k qpX ( µ ) p ≤ k f k ( qX ( µ )) p .Let now f ∈ qpX ( µ ) p . In particular f ∈ X ( µ ) p and so | f | p ∈ X ( µ ). Consider ( f j ) nj =1 ⊂ X ( µ ) satisfying that | f | p = (cid:0) P nj =1 | f j | q (cid:1) q µ -a.e. Noting that ( | f j | p ) nj =1 ⊂ X ( µ ) p and | f | = (cid:0) P nj =1 ( | f j | p ) qp (cid:1) qp µ -a.e., we have that (cid:16) n X j =1 k f j k qX ( µ ) (cid:17) q = (cid:16) n X j =1 k | f j | p k qpX ( µ ) p (cid:17) q ≤ k f k pqpX ( µ ) p . Then, | f | p ∈ qX ( µ ) and k | f | p k qX ( µ ) ≤ k f k pqpX ( µ ) p . Hence, f ∈ (cid:0) qX ( µ ) (cid:1) p and k f k ( qX ( µ )) p = k | f | p k p qX ( µ ) ≤ k f k qpX ( µ ) p . (cid:3) Optimal domain for ( p, q ) -power-concave operators Let X ( µ ) be a σ -order continuous quasi-B.f.s. satisfying what we call the σ -property :Ω = ∪ Ω n with χ Ω n ∈ X ( µ ) for all n, and let T : X ( µ ) → E be a continuous linear operator with values in a Banach space E .We consider the δ -ring Σ X ( µ ) = (cid:8) A ∈ Σ : χ A ∈ X ( µ ) (cid:9) and the vector measure m T : Σ X ( µ ) → E given by m T ( A ) = T ( χ A ). Note that the σ -property of X ( µ ) guarantees that Σ locX ( µ ) = Σ and since k m T k ≪ µ we have that[ i ] : L ( µ ) → L ( m T ) is well defined. Also note that a quasi-B.f.s. has the σ -propertyif and only if it contains a function g > µ -a.e. As an extension of [3, §
3] to quasi-B.f.s.’, in [16] it is proved that [ i ] : X ( µ ) → L ( m T )is well defined and T = I m T ◦ [ i ]. Even more, L ( m T ) is the largest σ -order continuousquasi-B.f.s. with this property. That means, if Z ( ξ ) is a σ -order continuous quasi-B.f.s.with ξ ≪ µ and T factors as X ( µ ) T / / [ i ] " " EZ ( ξ ) S > > with S being a continuous linear operator, then [ i ] : Z ( ξ ) → L ( m T ) is well defined and S = I m T ◦ [ i ]. In other words, L ( m T ) is the optimal domain to which T can be extendedpreserving continuity.In this section we present the main results of the paper, including a description ofthe optimal domain for T (when T is q -concave) preserving q -concavity. First, we haveto provide a natural non-finite measure version of the so called p -th power factorableoperators, which were developed for the first time in [26, § p ∈ (0 , ∞ ), we say that T is a p -th power factorable operator with acontinuous extension if there is a continuous linear extension of T to X ( µ ) p + X ( µ ), i.e. T factors as X ( µ ) T / / i & & EX ( µ ) p + X ( µ ) S for a continuous linear operator S .Regarding this definition and having in mind Remark 2.3.(b), two standard cases mustbe considered whenever χ Ω ∈ X ( µ ). If 1 < p we have that X ( µ ) p + X ( µ ) = X ( µ ) p ,and then the definition of p -th power factorable operator with a continuous extensioncoincides with the one given in [26, Definition 5.1]. However, if p ≤ X ( µ ) p + X ( µ ) = X ( µ ) and so p -th power factorable operators with continuous extensionsare just continuous operators.The following result, which is proved in [16] in order to find the optimal domain for p -th power factorable operators, will be the starting point of our work in this section. Theproof is an adaptation to our setting of the proof given in [26, Theorem 5.7] for the casewhen µ is finite, χ Ω ∈ X ( µ ) and p ≥ Theorem 4.1.
The following statements are equivalent. (a) T is p -th power factorable with a continuous extension. (b) [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is well defined. (c) [ i ] : X ( µ ) → L p ( m T ) ∩ L ( m T ) is well defined. (d) There exists
M > such that k T f k E ≤ M k f k X ( µ ) p + X ( µ ) for all f ∈ X ( µ ) .Moreover, if (a)-(d) holds, the extension of T to X ( µ ) p + X ( µ ) coincides with the inte-gration operator I m T ◦ [ i ] . -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 13 In a brief overview, (a) implies (b) and the fact that the extension of T to X ( µ ) p + X ( µ )is just I m T ◦ [ i ] follow from the optimality of L ( m T ). Note that X ( µ ) p + X ( µ ) is σ -ordercontinuous as X ( µ ) is so. The equivalence between (b) and (c) is a direct check. Statement(b) implies (d) since [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is continuous (as it is positive) and T = I m T ◦ [ i ]. Finally, (d) implies (a) is based on a standard argument which use theapproximation of a measurable function through functions in X ( µ ) (possible by the σ -property) to construct an extension of T to X ( µ ) p + X ( µ ). For a detailed proof ofTheorem 4.1 see [16], where moreover it is proved that if T is p -th power factorable witha continuous extension then L p ( m T ) ∩ L ( m T ) is the optimal domain to which T can beextended preserving this property.Now, let us go to the new results on optimal domains. We consider the followingproperty stronger than p -th power factorable and look for its optimal domain.For p, q ∈ (0 , ∞ ), we say that T is ( p, q ) -power-concave if there exists a constant C > (cid:16) n X j =1 k T ( f j ) k qp E (cid:17) pq ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) for every finite subset ( f j ) nj =1 ⊂ X ( µ ). If χ Ω ∈ X ( µ ) and p ≥ X ( µ ) p + X ( µ ) = X ( µ ) p , and then our definition of ( p, q )-power-concave operator coincides withthe one given in [26, Definition 6.1]. Remark . The following statements hold:(i) A (1 , q )-power-concave operator is just a q -concave operator.(ii) If T is ( p, q )-power-concave then T is qp -concave, as X ( µ ) ⊂ X ( µ ) p + X ( µ ) contin-uously.(iii) If χ Ω ∈ X ( µ ) and p <
1, since X ( µ ) p + X ( µ ) = X ( µ ), we have that ( p, q )-power-concavity coincides with qp -concavity.(iv) If T is ( p, q )-power-concave then T is p -th power factorable with a continuous ex-tension. Indeed, the ( p, q )-power-concave inequality applied to an unique functionis just the item (d) of Theorem 4.1As we will see in the next result, ( p, q )-power-concavity is close related to the followingproperty. We say that T is p -th power factorable with a q -concave extension if there existsa q -concave linear extension of T to X ( µ ) p + X ( µ ), i.e. T factors as X ( µ ) T / / i & & EX ( µ ) p + X ( µ ) S with S being a q -concave linear operator. In this case, it is direct to check that T is q -concave. Theorem 4.3.
The following statements are equivalent: (a) T is ( p, q ) -power-concave. (b) T is p -th power factorable with a qp -concave extension. (c) [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is well defined and qp -concave. (d) [ i ] : X ( µ ) → L ( m T ) is well defined and qp -concave, and [ i ] : X ( µ ) → L p ( m T ) is welldefined and q -concave. (e) [ i ] : X ( µ ) → qp L ( m T ) ∩ qL p ( m T ) is well defined.Moreover, if (a)-(e) holds, the extension of T to X ( µ ) p + X ( µ ) coincides with the inte-gration operator I m T ◦ [ i ] .Proof. First note that qp L ( m T ) ∩ qL p ( m T ) is σ -order continuous as a consequence ofProposition 2.4.(a) ⇒ (b) From Remark 4.2.(iv) we have that T is p -th power factorable with a con-tinuous extension. Let S : X ( µ ) p + X ( µ ) → E be a continuous linear operator extending T . We are going to see that S is qp -concave. Since T is ( p, q )-power-concave and S = T on X ( µ ), there exists C > (cid:16) n X j =1 k S ( f j ) k qp E (cid:17) pq ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) for all finite subset ( f j ) nj =1 ⊂ X ( µ ). Consider ( f j ) nj =1 ⊂ X ( µ ) p + X ( µ ) with f j ≥ µ -a.e. for all j . The σ -property of X ( µ ) allows to find for each j = 1 , ..., n a sequence( h jk ) ⊂ X ( µ ) such that 0 ≤ h jk ↑ f j µ -a.e. as k → ∞ (see [16] for the details). For every k ,we have that (cid:16) n X j =1 k S ( h jk ) k qp E (cid:17) pq ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | h jk | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) . On other hand, since X ( µ ) p + X ( µ ) is σ -order continuous, it follows that h jk → f j in X ( µ ) p + X ( µ ) as k → ∞ , and so S ( h jk ) → S ( f j ) in E as k → ∞ . Hence, taking limit as k → ∞ in the above inequality, it follows that (cid:16) n X j =1 k S ( f j ) k qp E (cid:17) pq ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) . For a general ( f j ) nj =1 ⊂ X ( µ ) p + X ( µ ), write f j = f + j − f − j where f + j and f − j are thepositive and negative parts respectively of each f j . By using inequality (2.4) and denoting α p,q = max { , − pq } , we have that (cid:16) n X j =1 k S ( f j ) k qp E (cid:17) pq ≤ (cid:16) n X j =1 (cid:0) k S ( f + j ) k E + k S ( f − j ) k E (cid:1) qp (cid:17) pq ≤ α p,q (cid:16) n X j =1 k S ( f + j ) k qp E + n X j =1 k S ( f − j ) k qp E (cid:17) pq ≤ α p,q C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f + j | qp + n X j =1 | f − j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) = α p,q C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 15 (for the last equality note that | f j | qp = | f + j | qp + | f − j | qp as f + j and f − j have disjoint support).(b) ⇒ (c) Since T is p -th power factorable with a qp -concave (and so continuous)extension, from Theorem 4.1, the map [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is well defined.Let S : X ( µ ) p + X ( µ ) → E be a qp -concave linear operator extending T . Note that S = I m T ◦ [ i ] (Theorem 4.1). Denote by C the qp -concavity constant of S . Consider( f j ) nj =1 ⊂ X ( µ ) p + X ( µ ) and fix ε >
0. For each j , by (2.5), we can take ϕ j ∈ S (cid:0) Σ X ( µ ) (cid:1) such that | ϕ j | ≤ k f j k L ( m T ) ≤ (cid:16) ε j (cid:17) pq + k I m T ( f j ϕ j ) k E . Since f j ϕ j ∈ X ( µ ) p + X ( µ ) as | f j ϕ j | ≤ | f j | , then I m T ( f j ϕ j ) = S ( f j ϕ j ). So, by usinginequality (2.4) and the qp -concavity of S , we have that n X j =1 k f j k qp L ( m T ) ≤ n X j =1 (cid:18)(cid:16) ε j (cid:17) pq + k S ( f j ϕ j ) k E (cid:19) qp ≤ max { , qp − } n X j =1 ε j + n X j =1 k S ( f j ϕ j ) k qp E ! ≤ max { , qp − } ε + C qp (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j ϕ j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) qp X ( µ ) p + X ( µ ) ! ≤ max { , qp − } ε + C qp (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) qp X ( µ ) p + X ( µ ) ! . Taking limit as ε →
0, we obtain n X j =1 k f j k qp L ( m T ) ≤ C qp max { , qp − } (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) qp X ( µ ) p + X ( µ ) and so (cid:16) n X j =1 k f j k qp L ( m T ) (cid:17) pq ≤ C max { , − pq } (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) . Hence, [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is qp -concave.(c) ⇔ (d) From Theorem 4.1, we have that [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is welldefined if and only if [ i ] : X ( µ ) → L p ( m T ) ∩ L ( m T ) is well defined, which is equivalentto [ i ] : X ( µ ) → L ( m T ) and [ i ] : X ( µ ) → L p ( m T ) well defined. By Lemma 2.5 we havethat [ i ] : X ( µ ) p + X ( µ ) → L ( m T ) is qp -concave if and only if [ i ] : X ( µ ) p → L ( m T ) and[ i ] : X ( µ ) → L ( m T ) are qp -concave. On other hand, it is straightforward to verify that[ i ] : X ( µ ) p → L ( m T ) is qp -concave if and only if [ i ] : X ( µ ) → L p ( m T ) is q -concave.(d) ⇔ (e) follows from Proposition 3.3.(c) ⇒ (a) Denote by C the qp -concavity constant of [ i ] : X ( µ ) p + X ( µ ) → L ( m T ).Consider ( f j ) nj =1 ⊂ X ( µ ) and note that f j ∈ L ( m T ) with I m T ( f j ) = T ( f j ) for all j . Then, (cid:16) n X j =1 k T ( f j ) k qp E (cid:17) pq = (cid:16) n X j =1 k I m T ( f j ) k qp E (cid:17) pq ≤ (cid:16) n X j =1 k f j k qp L ( m T ) (cid:17) pq ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | qp (cid:17) pq (cid:13)(cid:13)(cid:13) X ( µ ) p + X ( µ ) . (cid:3) Note that qL p ( m T ) = (cid:0) qp L ( m T ) (cid:1) p (see Proposition 3.4). In particular, in the casewhen T is ( p, q )-power-concave and χ Ω ∈ X ( µ ) (so χ Ω ∈ qp L ( m T )), from Remark 2.3.(b) itfollows that qp L ( m T ) ∩ qL p ( m T ) = qL p ( m T ) if p ≥ qp L ( m T ) ∩ qL p ( m T ) = qp L ( m T )if p < Theorem 4.4.
Suppose that T is ( p, q ) -power-concave. Then, T factors as X ( µ ) T / / [ i ] ' ' E qp L ( m T ) ∩ qL p ( m T ) I mT (4.1) with I m T being ( 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 ≪ µ and X ( µ ) T / / [ i ] " " EZ ( ξ ) S > > (4.2) with S being a ( p, q ) -power-concave linearoperator = ⇒ [ i ] : Z ( ξ ) → qp L ( m T ) ∩ qL p ( m T ) is well defined and S = I m T ◦ [ i ] .Proof. The factorization (4.1) follows from Theorem 4.3. The space qp L ( m T ) ∩ qL p ( m T )is σ -order continuous as noted before and satisfies the σ -property as X ( µ ) does. Since I m T : qp L ( m T ) ∩ qL p ( m T ) → E is continuous (as I m T : L ( m T ) → E is so), we can applyTheorem 4.3 to see that it is ( p, q )-power-concave. Note that Σ X ( µ ) ⊂ Σ qp L ( m T ) ∩ qL p ( m T ) and m I mT ( A ) = I m T ( χ A ) = T ( χ A ) = m T ( A ) for all A ∈ Σ X ( µ ) . That is, m T is therestriction of m I mT : Σ qp L ( m T ) ∩ qL p ( m T ) → E to Σ X ( µ ) . From [3, Lemma 3], it followsthat L ( m I mT ) = L ( m T ). Then,[ i ] : qp L ( m T ) ∩ qL p ( m T ) → qp L ( m I mT ) ∩ qL p ( m I mT )is well defined as qp L ( m I mT ) ∩ qL p ( m I mT ) = qp L ( m T ) ∩ qL p ( m T ).Let Z ( ξ ) satisfy (4.2). In particular, Z ( ξ ) has the σ -property. From Theorem 4.3applied to the operator S : Z ( ξ ) → E , we have that [ i ] : Z ( ξ ) → qp L ( m S ) ∩ qL p ( m S ) is welldefined and S = I m S ◦ [ i ]. Since Σ X ( µ ) ⊂ Σ Z ( ξ ) and m S ( A ) = S ( χ A ) = T ( χ A ) = m T ( A ) -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 17 for all A ∈ Σ X ( µ ) (i.e. m T is the restriction of m S : Σ Z ( ξ ) → E to Σ X ( µ ) ), from [3, Lemma3], it follows that L ( m S ) = L ( m T ) and I m S = I m T . Therefore,[ i ] : Z ( ξ ) → qp L ( m S ) ∩ qL p ( m S ) = qp L ( m T ) ∩ qL p ( m T )is well defined and S = I m S ◦ [ i ] = I m T ◦ [ i ]. (cid:3) We can rewrite Theorem 4.4 in terms of optimal domains.
Corollary 4.5.
Suppose that T is ( p, q ) -power-concave. Then qp L ( m T ) ∩ qL p ( m T ) 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 qp L ( m T ) ∩ qL p ( m T ) is given by the integration operator I m T . Recalling that the (1 , q )-power-concave operators coincide with the q -concave operators,we obtain our main result. Corollary 4.6.
Suppose that T is q -concave. Then qL ( m T ) is the largest σ -order con-tinuous quasi-B.f.s. to which T can be extended as a q -concave operator still with valuesin E . Moreover, the extension of T to qL ( m T ) is given by the integration operator I m T . 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 T is q -concave, byCorollary 4.6, the integration operator I m T extends T to the space qL ( m T ). Note thatthe map [ i ] : X ( µ ) → qL ( m T ) is q -concave as it is continuous and qL ( m T ) is q -concave.From a variant of the Maurey-Rosenthal theorem proved in [11, Corollary 5], under someextra conditions, if X ( µ ) is q -convex then [ i ] : X ( µ ) → qL ( m T ) factors through the space L q ( µ ). So, we obtain the following improvement of the usual factorization of q -concaveoperators on q -convex quasi-B.f.s.’. Corollary 4.7.
Let ≤ q < ∞ . Assume that µ is σ -finite and that X ( µ ) is q -convex andhas the σ -Fatou property. If T is q -concave then it can be factored as X ( µ ) T / / M g (cid:15) (cid:15) EL q ( µ ) M g − / / qL ( m T ) I mT O O for positive multiplication operators M g and M g − . The converse is also true. Vector measure representation of q -concave Banach lattices In this last section we look for a characterization of the class of Banach lattices whichare p -convex and q -concave in terms of spaces of integrable functions with respect to avector measure. For 1 < p , it is known that order continuous p -convex Banach latticescan be order isometrically represented as spaces L p of a vector measure defined on a δ -ring (see [6, Theorem 10]). We will see that the addition of the q -concavity propertyto the represented Banach lattice translates to adding some concavity property to thecorresponding integration map.First let us show two results concerning concavity for the integration operator of avector measure which will be needed later. Let m : R → E be a vector measure defined on a δ –ring R of subsets of Ω and withvalues in a Banach space E . Proposition 5.1.
The integration operator I m : L ( m ) → E is q -concave if and only if L ( m ) is q -concave.Proof. Suppose that I m : L ( m ) → E is q -concave and denote by C its q -concavity con-stant. Take ( f j ) nj =1 ⊂ L ( m ) and ( ϕ j ) nj =1 ⊂ S ( R ) with | ϕ j | ≤ j . Since( f j ϕ j ) nj =1 ⊂ L ( m ), as | f j ϕ j | ≤ | f j | for all j , we have that (cid:16) n X j =1 k I m ( f j ϕ j ) k qE (cid:17) q ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j ϕ j | q (cid:17) q (cid:13)(cid:13)(cid:13) L ( m ) ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) L ( m ) . Taking supremum for each j = 1 , .., n over all ϕ j ∈ S ( R ) with | ϕ j | ≤
1, from (2.5), iffollows that (cid:16) n X j =1 k f j k qL ( m ) (cid:17) q ≤ C (cid:13)(cid:13)(cid:13)(cid:16) n X j =1 | f j | q (cid:17) q (cid:13)(cid:13)(cid:13) L ( m ) . The converse is obvious as I m is continuous. (cid:3) Direct useful consequences can be deduced of the fact that the integration map I m : L ( m ) → E is q -concave. Assume that m is defined on a σ -algebra and note that q -concavity for q ≥ q, I m is q -concave for q ≥ weaklycompletely continuous (i.e. it maps weak Cauchy sequences into weakly convergent se-quences). Moreover, this implies that L ( m ) coincides with the space L w ( m ) and so ithas the σ -Fatou property.In the case when χ Ω ∈ L ( m ) (for instance if m is defined on a σ -algebra), we obtaina further result regarding ( p, q )-power-concave operators. Proposition 5.2.
Suppose that χ Ω ∈ L ( m ) and p ≥ . The integration operator I m : L p ( m ) → E is ( p, q ) -power-concave if and only if L p ( m ) is q -concave.Proof. First note that under the hypothesis it follows that L p ( m ) has the σ -property(in fact χ Ω ∈ L p ( m )) and L p ( m ) ⊂ L ( m ). So, I m : L p ( m ) → E is well defined andcontinuous.Suppose that I m : L p ( m ) → E is ( p, q )-power-concave. From Theorem 4.3, we havethat [ i ] : L p ( m ) → qp L ( m I m ) ∩ qL p ( m I m ) is well defined. Note that ( R loc ) L p ( m ) = R loc and so m I m coincides with m χ Ω (see Preliminaries). Then, L ( m I m ) = L ( m ) and so L p ( m ) ⊂ qp L ( m ) ∩ qL p ( m ) ⊂ qL p ( m ). Hence, L p ( m ) is q -concave as L p ( m ) = qL p ( m ).Suppose now that L p ( m ) is q -concave. Then, it is direct to check that L ( m ) is qp -concave. Since L p ( m ) ⊂ L ( m ), the integration operator I m : L ( m ) → E is continuousand (cid:0) L p ( m ) (cid:1) p + L p ( m ) = L ( m ), it follows that I m : L p ( m ) → E satisfies the inequalityof the definition of ( p, q )-power-concave operator. (cid:3) Let us go now to the representation of q -concave Banach lattices as spaces of integrablefunctions. 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 p -convex in the case when p > . Then, Z ( ξ ) coincides with the space L p ( m ) of a Banach -CONCAVE OPERATORS AND q -CONCAVE BANACH LATTICES 19 space valued vector measure m : R → E defined on a δ -ring whose integration operator I m : L ( m ) → E is qp -concave. Moreover, if χ Ω ∈ Z ( ξ ) , the vector measure m is definedon a σ -algebra.Proof. Note that if p ≤ Z ( ξ ) p is 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 constantof Z ( ξ ) is equal to 1 (see [21, Proposition 1.d.8]), and so Z ( ξ ) p is a B.f.s. (see Remark2.3.(e)). Consider the δ -ring Σ Z ( ξ ) = (cid:8) A ∈ Σ : χ A ∈ Z ( ξ ) (cid:9) and the finitely additive setfunction m : Σ Z ( ξ ) → Z ( ξ ) p given by m ( A ) = χ A . Since Z ( ξ ) p is σ -order continuous,as Z ( ξ ) is so by Proposition 2.4, it follows that m is a vector measure. Let us see that L ( m ) = Z ( ξ ) p with equal norms and so we will have that Z ( ξ ) coincides with L p ( m ).For ϕ ∈ S (Σ Z ( ξ ) ) we have that ϕ ∈ Z ( ξ ) p and I m ( ϕ ) = ϕ . Moreover, since m is positive, k ϕ k L ( m ) = k I m ( | ϕ | ) k Z ( ξ ) p = k ϕ k Z ( ξ ) p . (5.1)In particular, by taking ϕ = χ A , we obtain that k m k is equivalent to ξ . Given f ∈ L ( m ),since S (Σ Z ( ξ ) ) is dense in L ( m ), we can take ( ϕ n ) ⊂ S (Σ Z ( ξ ) ) such that ϕ n → f in L ( m ) and m -a.e. From (5.1), we have that ( ϕ n ) is a Cauchy sequence in Z ( ξ ) p and sothere is h ∈ Z ( ξ ) p such that ϕ n → h in Z ( ξ ) p . Taking a subsequence ϕ n j → h ξ -a.e. wesee that f = h ∈ Z ( ξ ) p and k f k Z ( ξ ) p = lim k ϕ n k Z ( ξ ) p = lim k ϕ n k L ( m ) = k f k L ( m ) . Let now f ∈ Z ( ξ ) p and take ( ϕ n ) ⊂ S (Σ) such that 0 ≤ ϕ n ↑ | f | . For any n , writing ϕ n = P mj =1 α j χ A j with ( A j ) mj =1 being pairwise disjoint and α j > j , we seethat χ A j ≤ α − /pj | f | /p and so ϕ n ∈ S (Σ Z ( ξ ) ). On other hand, since Z ( ξ ) p is σ -ordercontinuous, we have that ϕ n → f in Z ( ξ ) p . From (5.1), we have that ( ϕ n ) is a Cauchysequence in L ( m ) and so there is h ∈ L ( m ) such that ϕ n → h in L ( m ). Taking asubsequence ϕ n j → h m -a.e. we see that f = h ∈ L ( m ).Hence, L ( m ) = Z ( ξ ) p with equal norms and, since Z ( ξ ) is q -concave, it follows that L ( m ) is qp -concave. From Proposition 5.1, the integration operator I m : L ( m ) → E is qp -concave.Note that if χ Ω ∈ Z ( ξ ), then Σ Z ( ξ ) = Σ and so m is defined on a σ -algebra. (cid:3) 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 forB.f.s.’. A Banach lattice F is said to be order continuous if for every downwards directedsystem ( x τ ) ⊂ F with x τ ↓ k x τ k F ↓ σ -complete ifevery order bounded sequence in F has a supremum. A Banach lattice which is σ -ordercontinuous and σ -complete at the same time is order continuous, see [21, Proposition 1.a.8].A weak unit of a Banach lattice F is an element 0 ≤ e ∈ F such that inf { x, e } = 0 implies x = 0. An operator T : F → F between Banach lattices is said to be an order isometry ifit is linear, one to one, onto, k T x k F = k x k F for all x ∈ F and T (inf { x, y } ) = inf { T x, T y } for all x, y ∈ F . 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 -convexityand q -concavity whenever p, q ≥ Theorem 5.4.
Let p, q ∈ [1 , ∞ ) and let F be a Banach lattice. The following statementsare equivalent: (a) F is q -concave and p -convex. (b) F is order isometric to a space L p ( m ) of a Banach space valued vector measure m : R → E defined on a δ -ring whose integration operator I m : L ( m ) → E is qp -concave.Moreover, (a) holds with F having a weak unit if and only if (b) holds with m defined ona σ -algebra. In this last case I m : L p ( m ) → E is ( p, q ) -power-concave.Proof. (a) ⇒ (b) Since F is q -concave, it satisfies a lower q -estimate (see [21, Definition1.f.4]) and then it is σ -complete and σ -order continuous (see the proof of [21, Proposition1.f.5]). So, F is order continuous. From [15, Theorem 5] we have that F is order isometricto a space L ( ν ) of a Banach space valued vector measure ν defined on a δ -ring. Then, L ( ν ) is a B.f.s. satisfying the conditions of Proposition 5.3 and so L ( ν ) = L p ( m ) with m : R → E being a vector measure defined on a δ -ring R and with values in a Banachspace E , whose integration operator I m : L ( m ) → E is qp -concave.(b) ⇒ (a) Since L p ( m ) is p -convex (Remark ?? .(c)) and q -concave (as L ( m ) is qp -concave by Proposition 5.1), F also is.Now suppose that (a) holds with F having a weak unit. From [7, Theorem 8] we havethat F is order isometric to a space L ( ν ) of a Banach space valued vector measure ν defined on a σ -algebra. Since χ Ω ∈ L ( ν ), from Proposition 5.3 we have that (b) holdswith m defined on a σ -algebra.Conversely, if (b) holds with m defined on a σ -algebra then χ Ω ∈ L p ( m ) (as χ Ω ∈ L ( m )). So, the image of χ Ω by the order isometry is a weak unit in F . Moreover, fromProposition 5.2 it follows that I m : L p ( m ) → E is ( p, q )-power-concave. (cid:3) In particular, from Theorem 5.4 we obtain that a Banach lattice is q -concave (with q ≥
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Departamento de Matem´atica Aplicada I, E. T. S. de Ingenier´ıa de Edificaci´on, Universidad deSevilla, Avenida de Reina Mercedes, 4 A, Sevilla 41012, Spain
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