The geometry of L^p-spaces over atomless measure spaces and the Daugavet property
aa r X i v : . [ m a t h . F A ] J a n THE GEOMETRY OF L p -SPACES OVER ATOMLESSMEASURE SPACES AND THE DAUGAVET PROPERTY ENRIQUE A. S ´ANCHEZ P´EREZ AND DIRK WERNER
Abstract.
We show that L p -spaces over atomless measure spaces canbe characterized in terms of a p -concavity type geometric property thatis related with the Daugavet property. Introduction
A Banach space Y is said to have the Daugavet property if for every rankone operator T : Y → Y , the Daugavet equation k Id + T k = 1 + k T k is satisfied; in this case, it is known that the equation is satisfied for everyweakly compact operator ([9, Theorem 2.3]). Although for L ( µ ) spacesover an atomless measure µ this property is always fulfilled, it is known thatthis equation is only satisfied for a compact operator T on L p for 1 < p < ∞ when its norm is an eigenvalue of T ; this result can be extended to uniformlyconvex or uniformly smooth Banach spaces, and also to locally uniformlyconvex Banach spaces (see Corollary 2.4 and Theorem 2.7 in [1] or Section 4in [9]).After this negative result, some efforts have been made in order to find asimilar lower estimate for k Id + T k in terms of k T k in general Banach spacesor for the particular case of L p spaces. Based on the early ideas of Benyaminiand Lin in [4] several authors have been working in the direction of findingnice lower bounds for k Id + T k in terms of a function ψ : (0 , + ∞ ) → (1 , + ∞ )such that the inequality k Id + T k ≥ ψ ( k T k ) holds for all compact operators T : Y → Y (see for instance [6, 12, 13, 15, 17, 19] and [7] and the refer-ences therein). As in these cases, we are interested in this paper in findinga good alternative to the Daugavet equation for L p spaces, or in a moregeneral sense, for Banach function spaces satisfying certain p -convexity typerequirements. Our aim is to give a geometrical description of L p ( µ ) spacesdefined over measures µ without atoms in the same geometrical terms asfor spaces with the Daugavet property (slices and the geometry of the unitball). In order to do that, we use p -convexity and p -concavity propertiesof quasi-Banach function spaces for developing a sort of p -convexification Date : November 20, 2018.
File : LpDaugavet7.tex .2000
Mathematics Subject Classification.
Primary 46B04; secondary 46B25.
Key words and phrases.
Daugavet property, L p -space.The first author was partially supported by a grant from the Generalitat Valenciana(BEST/2009/108) and a grant from the Universidad Polit´ecnica de Valencia (PAID-00-09/2291). Support of the Ministerio de Educaci´on y Ciencia, under project ENRIQUE A. S ´ANCHEZ P´EREZ AND DIRK WERNER technique that allows us to obtain the desired geometrical description. Re-garding the Daugavet property for Banach function spaces the results thatare nowadays known are in a sense negative; for instance, in the class of Or-licz spaces over atomless finite measure spaces, the spaces that satisfy theDaugavet property with respect to the Luxemburg norm are isomorphic to L (see [3, Theorem 2.5]). However, it must be noted that there are Banachfunction spaces other than L ( µ ) and L ∞ ( µ ) over atomless measures µ thatsatisfy the Daugavet property (see for instance Section 5 in [5]; an explicitexample is c ( L [0 , c -sum of L [0 , ≤ p < ∞ . A p -convex and p -concave Banach lattice can beidentified isomorphically and in order with an L p -space; if the corresponding p -convexity and p -concavity constants are equal to 1, then this identificationis given by an isometry (see for instance [14, Theorem 2.7]). In this paperwe provide a Daugavet type geometric property which is more restrictivethan the p -concavity that is only satisfied for L p -spaces over measure spaceswithout atoms (see Theorem 2.8). In fact, it characterizes this class ofspaces.We remark that we deal with a different p -version of the Daugavet prop-erty in our paper [18].If Y is a Banach space, we denote as usual by B Y and S Y the (closed)unit ball and the unit sphere respectively. Y ∗ stands for its dual space. Theslice S ( y ∗ , ε ) of B Y defined by y ∗ ∈ B Y ∗ and ε > S ( y ∗ , ε ) = { y ∈ B Y : h y, y ∗ i ≥ − ε } . Notice that for the slice to be non-trivial it is enough to require that y ∗ ∈ S Y ∗ . Recall that Y has the Daugavet property if and only if the followinggeometric property is fulfilled: for every y ∈ S Y , every y ∗ ∈ S Y ∗ and every ε >
0, there is an element x ∈ S ( y ∗ , ε ) such that k y + x k ≥ − ε (see[9, Lemma 2.1], [9, Lemma 2.2] or [8, Theorem 2.2]). The reader can findmore information on the geometric description of the Daugavet property in[8, 9, 10] and in [2, Ch. 11].We also use standard notation regarding quasi-Banach function spaces.A quasi-Banach space ( E, k · k E ) is a linear space that is complete withrespect to the topology induced by a quasi-norm k · k E . If E is also a linearlattice, we say that ( E, k · k E ) is a quasi-Banach lattice if k · k E is a latticequasi-norm in E , i.e., k x k E ≤ k y k E whenever x, y ∈ E and | x | ≤ | y | . Let(Ω , Σ , µ ) be a measure space. A quasi-Banach function space X ( µ ) over themeasure µ is an ideal of L ( µ ), the usual µ -a.e. order is considered, whichis a quasi-Banach space with a lattice quasi-norm k · k such that for every A ∈ Σ of finite measure, χ A ∈ X ( µ ) (see for instance [11, Chapter 1.b], [14,Chapter 2.6] and [16, Chapter 2] for definitions and main results regardingthese structures, but notice that the last property is not required in some ofthese references). In the case that k · k is a norm, we say that ( X ( µ ) , k · k ) isa Banach function space, see [11, Definition 1.b.17]. We shall write simply X for X ( µ ) if the measure is fixed in the context. HE GEOMETRY OF L p -SPACES 3 Let 0 < p < ∞ . A quasi-Banach lattice E is p -convex if there is a constant K such that for every finite sequence ( x i ) ni =1 in E , (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | x i | p (cid:17) /p (cid:13)(cid:13)(cid:13) E ≤ K (cid:16) n X i =1 k x i k pE (cid:17) /p . A quasi-Banach lattice E is p -concave if there is a constant k such that forevery finite sequence ( x i ) ni =1 in E , (cid:16) n X i =1 k x i k pE (cid:17) /p ≤ k (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | x i | p (cid:17) /p (cid:13)(cid:13)(cid:13) E . The best constants in the inequalities above are denoted by M ( p ) ( E ) and M ( p ) ( E ), respectively, and are called the p -convexity and the p -concavityconstants of E . Throughout the paper we will assume for a p -convex Banachfunction space that in fact M ( p ) ( E ) = 1, and when p -concavity is requiredthat M ( p ) ( E ) = 1. To indicate this, we will say that they are constant 1 p -convex or constant 1 p -concave, respectively.Let 0 < p < ∞ . Consider a quasi-Banach function space X ( µ ). Let X ( µ ) [ p ] := { h ∈ L ( µ ): | h | /p ∈ X ( µ ) } be its p -th power, which is a quasi-Banach function space when endowedwith the quasinorm k h k X [ p ] := k| h | /p k pX , h ∈ X [ p ] (see [16, Ch. 2]). For p ≥
1, if X is p -convex and M ( p ) ( X ) = 1, then X ( µ ) [ p ] is a Banach functionspace, since in this case k · k X [ p ] is a norm. If the Banach function space is p -convex, but the p -convexity constant is not 1, then k · k X [ p ] is not a norm,but it is equivalent to a norm (see for instance [16, Prop. 2.23]). It is alsowell known that every p -convex Banach lattice can be renormed in such away that the new norm is a lattice norm with p -convexity constant equalto 1 (see [11, Proposition 1.d.8]).2. Banach function spaces with p -th powers having theDaugavet property Let 0 < p < ∞ and let X ( µ ) be a constant 1 p -convex quasi-Banachfunction space. If f ∈ X ( µ ), we can always write it as f = sign { f }| f | . Thisallows us to define the (obviously non-linear) map i p : X ( µ ) → X ( µ ) [ p ] by i p ( f ) = sign { f }| f | p . Throughout the paper, we shall write f p := i p ( f )for the sake of simplicity, but notice that for even integers f p is in generalnot | f | p . The map i p is bijective and satisfies k i p ( f ) k X [ p ] = k sign { f }| f | p k X [ p ] = k f k pX , f ∈ X ( µ ) . (2.1)The inverse map i − p : X [ p ] → X coincides with i /p : Y → Y [1 /p ] , where Y = X [ p ] .In what follows we characterize the p -convex Banach function spaceswhose p -th powers satisfy the Daugavet property. The key idea to achieve ENRIQUE A. S ´ANCHEZ P´EREZ AND DIRK WERNER this is to introduce the notions of 1 /p -th power of a slice and p -convexificationof an operator T : X [ p ] → X [ p ] .If X ( µ ) is a constant 1 p -convex Banach function space, let S [ p ] ( x ∗ , ε ) bea slice in X ( µ ) [ p ] , where x ∗ ∈ B ( X ( µ ) [ p ] ) ∗ . Consider the set S /p [ p ] ( x ∗ , ε ) := { f ∈ X ( µ ): f p ∈ S [ p ] ( x ∗ , ε ) } . We call it the 1 /p -th power of the slice S [ p ] ( x ∗ , ε ).If T : X [ p ] → X [ p ] is an operator, we define its p -convexification ϕ T : X ( µ ) → X ( µ ) by ϕ T ( f ) := i − p ◦ T ◦ i p ( f ) = ( T ( f p )) /p , f ∈ X. We also define k ϕ T k := sup f ∈ B X k ϕ T ( f ) k . Notice that k ϕ T k = sup f ∈ B X k ϕ T ( f ) k X = sup f ∈ B X k ( T ( f p )) /p k X = sup h ∈ B X [ p ] k T ( h ) k /pX [ p ] = k T k /p . The following two lemmas provide a geometric description of the Dau-gavet property for a Banach function space in terms of slices of the p -convexification. Their proofs follow the lines of the ones of Lemmas 2.1,2.2 and 2.8 in [9]. However, we spell out the arguments that prove themain equivalences with some detail in order to show the role played by the p -convexity of the norm of X ( µ ). Lemma 2.1.
Let X ( µ ) be a quasi-Banach function space and let < p < ∞ .The following assertions are equivalent: (0) The space satisfies the following. (i) ( X [ p ] ) ∗ = { } . (ii) For every finite family of rank-one continuous operators T i : X [ p ] → X [ p ] , sup f ∈ B X (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 ϕ T i ( f ) p (cid:17) /p (cid:13)(cid:13)(cid:13) ≤ (cid:16) n X i =1 k ϕ T i k p (cid:17) /p . (iii) For every rank-one operator T : X [ p ] → X [ p ] , sup f ∈ B X k| f p + ϕ T ( f ) p | /p k X = (1 + k ϕ T k p ) /p . (1) X is constant p -convex and for every rank-one operator T : X [ p ] → X [ p ] , sup f ∈ B X k| f p + ϕ T ( f ) p | /p k X = (1 + k ϕ T k p ) /p . (2.2)(2) X is constant p -convex and for every f ∈ S X , every x ∗ ∈ S ( X [ p ] ) ∗ and every ε > there is an element g ∈ S /p [ p ] ( x ∗ , ε ) such that k| f p + g p | /p k pX ≥ − ε. (3) X ( µ ) [ p ] if a Banach function space over µ with the Daugavet prop-erty. HE GEOMETRY OF L p -SPACES 5 (4) X is constant p -convex and for every f ∈ S X and every slice S ( x ∗ , ε ) of B X [ p ] there is another non-trivial slice S [ p ] ( x ∗ , ε ) ⊂ S [ p ] ( x ∗ , ε ) such that for every g ∈ S /p [ p ] ( x ∗ , ε ) the inequality k ( f p + g p ) /p k pX ≥ − ε holds.Proof. Let us start with (0) ⇒ (1). Take a finite set f , . . . , f n ∈ X . Sincethe dual space ( X [ p ] ) ∗ contains a non-trivial element x ∗ , we can assume that x ∗ ∈ S ( X [ p ] ) ∗ , and we can consider the operators T i : X [ p ] → X [ p ] given by T i ( h ) := h h, x ∗ i| f i | p . They are obviously continuous and k T i k = k f pi k X [ p ] = k f i k pX . Thus, by (ii), we have (cid:16) n X i =1 k f i k pX (cid:17) /p = (cid:16) n X i =1 k ϕ T i k p (cid:17) /p ≥ sup f ∈ B X (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 ϕ pT i ( f ) (cid:17) /p (cid:13)(cid:13)(cid:13) = sup f ∈ B X (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 T i ( f p ) (cid:17) /p (cid:13)(cid:13)(cid:13) = sup f ∈ B X (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 h f p , x ∗ i| f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) = sup h ∈ B X [ p ] ( h h, x ∗ i ) /p (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) X . Consequently, X is p -convex and M ( p ) ( X ) = 1, and so (1) is obtained.For the converse, since X is p -convex and has p -convexity constant equalto 1, X [ p ] is a Banach function space (see for instance [16, Proposition 2.23]),and so its dual space is non-trivial. It only remains to prove (ii). Take anyfinite set of rank-one operators T i : X [ p ] → X [ p ] , i = 1 , . . . , n . Each of themcan be written as T i = x ∗ i ⊗ f pi , where k x ∗ i k = 1 and f i ∈ X . Then for every f ∈ B X , (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 ( ϕ T i ( f )) p (cid:17) /p (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 |h f, x ∗ i i| p | f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) ≤ (cid:16) n X i =1 k f i k p (cid:17) /p = (cid:16) n X i =1 k ϕ T i k p (cid:17) /p . This gives (0).Let us now prove the equivalence of (1) and (2). First notice that (1) isequivalent to the fact that for every rank-one operator T : X [ p ] → X [ p ] ,sup g ∈ B X k| g p + T ( g p ) | /p k pX = 1 + k T k . ENRIQUE A. S ´ANCHEZ P´EREZ AND DIRK WERNER
For (1) ⇒ (2), take f ∈ S X ( µ ) , x ∗ ∈ S ( X ( µ ) [ p ] ) ∗ and ε >
0. Consider T = x ∗ ⊗ f p . The equality above can be written assup g p ∈ B X [ p ] k g p + T ( g p ) k X [ p ] = 1 + k T k . (2.3)In particular, this implies that we can assume by Lemma 11.4 in [2] (or [20,p. 78]) that T and hence x ∗ and f are of norm one. Take h ∈ S X such that k| h p + T ( h p ) | /p k pX ≥ − ε. We can also assume that h h p , x ∗ i ≥ h by − h ). Noticefirst that since X ( µ ) is p -convex with constant 1,1 + h h p , x ∗ i = k| h p | /p k pX + k| T ( h p ) | /p k pX ≥ − ε, which implies that h h p , x ∗ i ≥ − ε . Consequently, h ∈ S /p [ p ] ( g ∗ , ε ). On theother hand, using again the constant 1 p -convexity of X ( µ ),2 − ε ≤ k| h p + T ( h p ) | /p k pX ≤ k| h p + f p | /p k pX + k| T ( h p ) − f p | /p k pX = k| h p + f p | /p k pX + k| ( h h p , x ∗ i − f p | /p k pX ≤ k| h p + f p | /p k pX + (1 − h h p , x ∗ i ) ≤ k| h p + f p | /p k pX + ε. This gives (2).For the converse, we can suppose again that T is defined as T = x ∗ ⊗ f p for two norm one elements x ∗ and f . Let ε > h ∈ S /p [ p ] ( x ∗ , ε ) suchthat k| f p + h p | /p k pX ≥ − ε. Then, by the constant 1 p -convexity of X ( µ ),2 − ε ≤ k| f p + h p | /p k pX = k| f p − T ( h p ) + T ( h p ) + h p | /p k pX ≤ k| f p − T ( h p ) | /p k pX + k| T ( h p ) + h p | /p k pX ≤ (1 − h h p , x ∗ i ) + k| T ( h p ) + h p | /p k pX ≤ ε + k| T ( h p ) + h p | /p k pX . Since this holds for every ε > i p , the definitionof the norm k · k X [ p ] and (2.3), the equivalence of (1) and (3) becomes obvi-ous. Notice that the fact that X ( µ ) [ p ] is a Banach function space over µ isequivalent to the fact that X ( µ ) = ( X ( µ ) [ p ] ) [1 /p ] is constant 1 p -convex (seefor instance [16, Proposition 2.23(ii)]).Similar arguments prove (3) ⇒ (4); a direct proof can be given usingLemma 2.1(a) in [9], the definition of the norm in X [ p ] and the fact thatevery element h ∈ X [ p ] can be written as f p for some f ∈ X . (4) ⇒ (2) isobvious. (cid:3) HE GEOMETRY OF L p -SPACES 7 Remark . L p ( µ ) spaces over a non-atomic measure µ satisfy the equivalentstatements of Lemma 2.1; this is a direct consequence of ( L p ( µ )) [ p ] = L ( µ )and the well-known fact that L ( µ ) satisfies the Daugavet property (see forinstance [1, Theorem 3.2], or the example after Theorem 2.3 in [9] for asimple proof). However, we can easily construct Banach function spaceswhich are not L p spaces but their p -th powers have the Daugavet property.For instance, consider a σ -finite atomless measure space (Ω , Σ , µ ) and aninfinite measurable partition { A i } of Ω and take a Banach space F with a1-unconditional normalized Schauder basis endowed with its natural Banachfunction space structure given by the pointwise order. Consider the Banachspace X defined as the F -sum of the spaces L ( µ | A i ), where µ | A i denotes therestriction of µ to A i , i ∈ N , that is, X is the space of sequences ( f i ) such that f i ∈ L ( µ | A i ) and ( k f i k ) ∈ F . If F has the positive Daugavet property (i.e.,every positive rank one operator on F satisfies the Daugavet equation), thenTheorem 5.1 in [5] ensures that the F -sum has the Daugavet property. Thespaces ℓ and ℓ ∞ satisfy the positive Daugavet property, but the reader canfind other examples in [5, Section 5]. It is easy to see that the 1 /p -th powerof X is also a Banach function space and it can be identified isometricallywith the F [1 /p ] -sum of the spaces L p ( µ | A i ). Since ( X [1 /p ] ) [ p ] = X has theDaugavet property, X [1 /p ] satisfies the assertions of Lemma 2.1.Other examples can be constructed using the fact that spaces of Bochnerintegrable functions over atomless measures satisfy the Daugavet property(see again the example after Theorem 2.3 in [9]). Let Y be a Banach lattice,let µ be a measure without atoms and consider the Bochner space L ( µ, Y ).It is a Banach lattice when the natural order is considered; assume that itis also an order continuous Banach lattice with a weak unit. Then it canbe represented as a Banach function space Z (see for instance [11, Theo-rem 1.b.14]). Since the Daugavet property is preserved under isometries, weobtain that Z [1 /p ] satisfies the statements of Lemma 2.1. Remark . Note that although the assertions in Lemma 2.1 have beenstated in terms of rank one operators, the equivalences also hold when otherclasses of operators satisfying the Daugavet equation in X [ p ] are considered.Therefore, it includes for instance the weakly compact operators and furtherclasses, see for example [9] and [10].The following “sign independent” inequality is crucial for the computa-tions regarding the p -convexification of the Daugavet property.Given 1 ≤ p < ∞ , we denote by p ′ the conjugate exponent defined by1 /p + 1 /p ′ = 1. Also, we let k ( p ) = 1 if p ≥ p ′ and k ( p ) = 2 ( p ′ /p ) − if p < p ′ .It follows ( a p/p ′ + b p/p ′ ) p ′ /p ≤ k ( p )( a + b )for real numbers a, b ≥ Lemma 2.4.
Let ≤ p < ∞ and consider two elements f and g in the unitball of the constant p -convex Banach function space X . Then k| f p − g p | /p k p ≤ k f − g k p + p (2 k ( p )) p/p ′ ) k f − g k . Consequently, the map i p : X → X [ p ] mapping f to f p is continuous. ENRIQUE A. S ´ANCHEZ P´EREZ AND DIRK WERNER
Proof.
Let 1 ≤ p < ∞ and consider c p := sign { c }| c | p for every c ∈ R . Let a, b ∈ R . Then we have to take into account two cases:1) sign { a } 6 = sign { b } . Suppose without loss of generality that a ≥ b ≤
0. Then | a p − b p | /p = | a p + | b | p | /p ≤ | a + | b || = | a − b | .
2) sign { a } = sign { b } . Then it is known that | a p − b p | /p ≤ (cid:0) p | a p − + b p − | · | a − b | (cid:1) /p (see for instance [16, Section 2.2]).Take now two functions f, g ∈ B X and put A = { ω : sign { f ( ω ) } 6 =sign { g ( ω ) }} and B = { ω : sign { f ( ω ) } = sign { g ( ω ) }} . Then by case 1) k| f p − g p | /p χ A k p ≤ k| f − g | χ A k p . Since p − p/p ′ , by the H¨older inequality for the Banach lattice X (seefor instance Proposition 1.d.2 in [11]), we obtain (see also [16, Section 2.2]for the pointwise inequalities involved) k| f p − g p | /p χ B k p ≤ p k (cid:0) | f p − + g p − | · | f − g | (cid:1) /p χ B k p ≤ p k ( | f p/p ′ + g p/p ′ | p ′ /p ) /p ′ · | f − g | /p χ B k p ≤ p k ( | f p/p ′ + g p/p ′ | p ′ /p ) χ B k p/p ′ · k| f − g | χ B k≤ pk ( p ) p/p ′ k| f + g | χ B k p/p ′ · k| f − g | χ B k≤ p (2 k ( p )) p/p ′ k| f − g | χ B k . Therefore, since by the constant 1 p -convexity of X the inequality k| f p − g p | /p k p ≤ k| f p − g p | /p χ A k p + k| f p − g p | /p χ B k p is satisfied, we obtain the result. (cid:3) Remark . We can relate our property for rank one operators with thegeneral ψ -Daugavet property for Banach spaces that has been quoted in theIntroduction (see [6, 15, 19]). For example in Theorem 2.1 of [15] inequalitieslike k Id + T k ≥ (1 + c p k T k p ) /p for a compact operator T : X → X areconsidered, where c p is a non-negative constant. In our case we obtainthe following similar estimate in terms of the p -convexification ϕ T . Forinstance, if X [ p ] is a Banach function space with the Daugavet property and T : X [ p ] → X [ p ] is weakly compact, we obtain(1 + k ϕ T k p ) /p ≤ sup f ∈ B X k| f p + ϕ T ( f ) p | /p k X or equivalently (1 + k T k ) /p ≤ sup f ∈ B X k| f p + T ( f p ) | /p k X . Clearly in the case of positive operators, and using the estimate given in theproof of Lemma 2.4 for the case of different signs, this inequality gives also(1 + k ϕ T k p ) /p ≤ sup f ∈ B X k f + ϕ T ( f ) k X . The following lemma is similar to Lemma 2.8 in [9].
HE GEOMETRY OF L p -SPACES 9 Lemma 2.6.
Suppose that X [ p ] is a Banach space with the Daugavet prop-erty. Then for every finite dimensional subspace X of X , every ε > andevery x ∗ ∈ ( X [ p ] ) ∗ there is an element g ∈ S /p [ p ] ( x ∗ , ε ) such that for every f ∈ X and t ∈ R k (( tg ) p + f p ) /p k p ≥ (1 − ε )( | t | p + k f k p ) . Proof.
Take δ > δ p + p (2 k ( p )) p/p ′ δ ≤ ε/
2, where k ( p ) is defined asabove, a finite dimensional subspace X of X and a finite δ -net { f , . . . , f n } in S X . Applying Lemma 2.1(4) we find a sequence of slices S [ p ] ( x ∗ n , ε n ) ⊂ . . . ⊂ S [ p ] ( x ∗ , ε ) ⊂ S [ p ] ( x ∗ , ε ) such that k ( f pk + g p ) /p k pX ≥ − δ p for all g ∈ S /p [ p ] ( x ∗ k , ε k ), k = 1 , . . . , n . If we consider elements g in S /p [ p ] ( x ∗ n , ε n ),these inequalities are true for all k = 1 , . . . , n . Consequently, by Lemma 2.4and the constant 1 p -convexity of X , for every g ∈ S /p [ p ] ( x ∗ n , ε n ) and f ∈ S X there is an index k ∈ { , . . . , n } such that k ( f p + g p ) /p k pX ≥ k ( f pk + g p ) /p k pX − k| f pk − f p | /p k pX ≥ − δ p − ε/ ≥ − ε. Now, if 0 ≤ s ≤ t are real numbers such that t p + s p = 1, then for all such g and f , k (( tg ) p + ( sf ) p ) /p k p = k ( t p ( g p + f p ) − | s p − t p | f p ) /p k p ≥ t p k ( g p + f p ) /p k p − | s p − t p |k f k p ≥ t p (2 − ε ) + s p − t p = 1 − ε. Since the same calculations can be done for t ≤ s , we obtain the followinginequality for every t ≥ f ∈ X : k (( tg ) p + f p ) /p k p = (cid:13)(cid:13)(cid:13)(cid:16) ( tg ) p + (cid:16) k f k f k f k (cid:17) p (cid:17) /p (cid:13)(cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13)(cid:16) t p g p t p + k f k p + k f k p ( f / k f k ) p t p + k f k p (cid:17) /p (cid:13)(cid:13)(cid:13) p · ( t p + k f k p ) ≥ (1 − ε )( t p + k f k p ) . The symmetry of the norm allows to obtain the same inequality for every t ∈ R , replacing t by | t | . (cid:3) The following lemma makes it clear that the Daugavet type equation (2.2)fails in the presence of atoms.
Lemma 2.7.
Let (Ω , Σ , µ ) be a measure space. Let < p < ∞ , let X ( µ ) bea constant p -convex quasi-Banach function space over µ and suppose that µ has an atom. Then there is a rank one operator T : X [ p ] → X [ p ] such that sup f ∈ B X k ( f p + T ( f p )) /p k X < (1 + k T k ) /p . Proof.
Recall that by the constant 1 p -convexity of X , X [ p ] is a Banachfunction space. Let { a } be an atom for µ . Then 0 < µ ( { a } ) < ∞ andthe characteristic function χ { a } belongs to X [ p ] , and defines a (continuous) functional of ( X [ p ] ) ∗ by h h, χ { a } i = R χ { a } h dµ = h ( a ) µ ( { a } ), h ∈ X [ p ] . Let T be the non-null rank one operator T = − χ { a } µ ( { a } ) ⊗ χ { a } . Thensup f ∈ B X k ( f p + T ( f p )) /p k = sup f ∈ B X k ( f p ( a ) χ { a } + f p χ Ω \{ a } − f p ( a ) χ { a } ) /p k = sup f ∈ B X k f χ Ω \{ a } k ≤ < (1 + k T k ) /p , as claimed. (cid:3) The following result provides the desired geometric characterization of L p spaces over atomless measure spaces. Recall that an abstract L p space is aBanach lattice E for which for every couple of disjoint elements x, y ∈ E , theequality k x + y k p = k x k p + k y k p holds (see for instance [11, Definition 1.b.1]). Theorem 2.8.
Let ≤ p < ∞ and let X ( µ ) be a quasi-Banach functionspace over µ . The following statements are equivalent: (i) X is an abstract L p space such that X [ p ] has the Daugavet property. (ii) X is equal to L p ( h dµ ) for some < h ∈ L ( µ ) and the measure µ does not have atoms. (iii) ( X [ p ] ) ∗ = { } , X is constant p -concave, and for every finite set ofoperators T i : X [ p ] → X [ p ] , i ∈ { , . . . , n } we have that sup f ∈ B X (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 ϕ T i ( f ) p (cid:17) /p (cid:13)(cid:13)(cid:13) ≤ (cid:16) n X i =1 k ϕ T i k p (cid:17) /p and sup f ∈ B X k| f p + ϕ T i ( f ) p | /p k X = (1 + k ϕ T i k p ) /p . (iv) X is constant p -convex, constant p -concave and for every rankone operator T : X [ p ] → X [ p ] , sup f ∈ B X k| f p + ϕ T ( f ) p | /p k X = (1 + k ϕ T k p ) /p . (v) X is constant p -convex and for every slice S [ p ] ( x ∗ , δ ) , every ε > and every finite dimensional subspace X of X there is an element g ∈ S /p [ p ] ( x ∗ , δ ) such that for every f , . . . , f n ∈ X and α i ≥ satisfying P ni =1 α pi = 1 , (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | ( α i g ) p + f pi | (cid:17) /p (cid:13)(cid:13)(cid:13) p ≥ (1 − ε ) (cid:16) k g k p + n X i =1 k f i k p (cid:17) . (vi) X is constant p -convex and for every slice S [ p ] ( x ∗ , δ ) , every ε > and every finite dimensional subspace X of X there is an element g ∈ S /p [ p ] ( x ∗ , δ ) and an element x ∗ ∈ B ( X [ p ] ) ∗ such that (cid:16) | f p + g p |k f k p + k g k p (cid:17) /p ∈ S /p [ p ] ( x ∗ , ε ) for every f ∈ X . HE GEOMETRY OF L p -SPACES 11 Proof. (i) ⇒ (ii). Since X is an abstract L p space and 1 ≤ p < ∞ , X isin particular a σ -order continuous Banach function space. Using a Maurey-Rosenthal type factorization argument (see for example Corollary 6.17 in [16,Ch. 6]), we find that there is a function 0 < g such that the identity mapId: X ( µ ) → X ( µ ) factorizes through L p ( µ ) by means of the multiplicationoperators M g : X → L p ( µ ) and M /g : L p ( µ ) → X ; in fact, M g definesan isometry (notice that for applying the Corollary 6.17 quoted above itis necessary to take into account that the operator M g always has denserange). Therefore, the space X ( µ ) can be identified isometrically and inorder with L p ( h dµ ), h = g p , and its elements are the same functions. So, X [ p ] = L ( h dµ ) has the Daugavet property, and therefore by Lemma 2.7with p = 1 the measure h dµ does not have atoms. Consequently, µ doesnot have atoms either.For (ii) ⇒ (i), just recall that an L ( ν )-space over an atomless measurespace has the Daugavet property (see for instance [1, Theorem 3.2] or theexample after Theorem 2.3 in [9]).By Lemma 2.1, (ii) implies the equivalent statements (iii) and (iv), takinginto account that X can be written as a Banach function space over themeasure h dµ . Clearly, (iv) implies (i).Let us now show (i) ⇒ (v). Assume that X is an L p -space and X [ p ] has theDaugavet property. Then Lemma 2.6 provides for every finite dimensionalsubspace X of X , every ε > x ∗ ∈ ( X [ p ] ) ∗ an element g ∈ S /p [ p ] ( x ∗ , ε ) such that for every f ∈ X and t ∈ R k (( tg ) p + f p ) /p k pX ≥ (1 − ε )( | t | p + k f k pX ) . Thus, taking into account that X is an L p -space (and then also constant 1 p -concave), for every finite set of elements f , . . . , f n ∈ X and positive realnumbers α i such that P ni =1 α pi = 1, we obtain (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | f pi + ( α i g ) p | (cid:17) /p (cid:13)(cid:13)(cid:13) pX = n X i =1 k| f pi + ( α i g ) p | /p k pX ≥ (1 − ε ) (cid:16) k g k pX + n X i =1 k f i k pX (cid:17) . (2.4)For (v) ⇒ (vi) we apply the following separation argument. Considerthe convex set B ( X [ p ] ) ∗ , which is a compact Hausdorff space when endowedwith the weak* topology, and the family of all functions Φ f ,...,f n ; α ,...,α n : B ( X [ p ] ) ∗ → R , n ∈ N , f , . . . , f n ∈ X , α , . . . , α n ∈ R , P ni =1 α pi = 1, definedbyΦ f ,...,f n ; α ,...,α n ( x ∗ ) := (1 − ε ) (cid:16) k g k p + n X i =1 k f i k p (cid:17) − D n X i =1 | f pi + α pi g p | , x ∗ E . Each function defined in this way is clearly convex, and the family ofall such functions is concave, since each convex combination of two suchfunctions can be written again as a function of the same family; indeed for0 ≤ β ≤ β Φ f ,...,f n ; α ,...,α n + (1 − β )Φ f ,...,f m ; α ,...,α m = Φ β /p f ,...,β /p f n , (1 − β ) /p f ,..., (1 − β ) /p f m ; β /p α ,...,β /p α n , (1 − β ) /p α ,..., (1 − β ) /p α m . These functions are continuous with respect to the weak* topology, andby (2.4) and the Hahn-Banach Theorem for each of them there is an x ∗ ∈ B ( X [ p ] ) ∗ such that Φ f ,...,f n ; α ,...,α n ( x ∗ ) ≤
0, so an application of Ky Fan’sLemma (see for instance [16, Lemma 6.12]) gives an element x ∗ such thatΦ f ,...,f n ; α ,...,α n ( x ∗ ) ≤ ⇒ (ii). Take a finite set of elements f , . . . , f n ∈ X and consider the finite dimensional subspace X generated by them. Takeany slice S [ p ] ( x ∗ , ε ) generated by a norm one element x ∗ and an ε >
0. Thenan application of (vi) gives a g ∈ S /p [ p ] ( x ∗ , ε ) and an element x ∗ ∈ B ( X [ p ] ) ∗ such that n X i =1 h| f pi + g p | , x ∗ i ≥ (1 − ε ) (cid:16) n X i =1 k f i k pX + n k g k pX (cid:17) . Thus, (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) pX + n k g k pX ≥ (1 − ε ) (cid:16) n X i =1 k f i k pX + n k g k pX (cid:17) and therefore, (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) pX + εn k g k pX ≥ (1 − ε ) (cid:16) n X i =1 k f i k pX (cid:17) . Since this construction can be done for every ε >
0, we obtain that (cid:13)(cid:13)(cid:13)(cid:16) n X i =1 | f i | p (cid:17) /p (cid:13)(cid:13)(cid:13) = (cid:16) n X i =1 k f i k pX (cid:17) /p . Consequently, X is an abstract L p space. Also, for ε >
0, taking a singlefunction f ∈ S X and the subspace X generated by it and an x ∗ ∈ S ( X [ p ] ) ∗ we obtain by (vi) an x ∗ ∈ B ( X [ p ] ) ∗ and a function g ∈ S /p [ p ] ( x ∗ , ε ) such that k| f p + g p | /p k pX ≥ h| f p + g p | , x ∗ i ≥ − ε ) . Thus, Lemma 2.1 gives that X [ p ] has the Daugavet property. (cid:3) Corollary 2.9.
Let ≤ p < ∞ . Every separable quasi-Banach functionspace satisfying the equivalent statements of Theorem 2.8 is order isomorphicand isometric to L p ([0 , . This is a direct consequence of Theorem 2.8 and the characterization ofatomless separable L p -spaces (see [14, Theorem 2.7.3]). Remark . Note that using Kakutani’s representation theorem (see forinstance [11, Theorem 1.b.2] or [14, Theorem 2.7.1]) Theorem 2.8 can beapplied in a more abstract setting, without the requirement for X to be aquasi-Banach function space. If X is just a Banach lattice that is also anabstract L p space, then X is order isometric to an L p ( µ ) space over somemeasure space (Ω , Σ , µ ), so in this case the condition of L p ( µ ) [ p ] = L ( µ ) HE GEOMETRY OF L p -SPACES 13 having the Daugavet property, i.e., µ having no atoms, is characterized bythe equivalent statemens of the theorem. Therefore, Corollary 2.9 can alsobe stated for Banach lattices via the atomic properties of the representingmeasure that Kakutani’s theorem gives. References [1]
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Instituto Universitario de Matem´atica Pura y Aplicada, UniversidadPolit´ecnica de Valencia, Camino de Vera s/n, 46071 Valencia, Spain.
Current address : Department of Mathematics, Freie Universit¨at Berlin, Arnimallee 6,D-14 195 Berlin, Germany.
E-mail address : [email protected] Department of Mathematics, Freie Universit¨at Berlin, Arnimallee 6,D-14 195 Berlin, Germany
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