aa r X i v : . [ m a t h . F A ] J a n A DUALITY OPERATORS/BANACH SPACES
MIKAEL DE LA SALLE
Abstract.
Given a set B of operators between subspaces of L p spaces, wecharacterize the operators between subspaces of L p spaces that remain boundedon the X -valued L p space for every Banach space on which elements of theoriginal class B are bounded.This is a form of the bipolar theorem for a duality between the class ofBanach spaces and the class of operators between subspaces of L p spaces,essentially introduced by Pisier. Introduction
All the Banach spaces appearing in this paper will be assumed to be separable,and will be over the field K of real or complex numbers.The local theory of Banach spaces studies infinite dimensional Banach spacesthrough their finite-dimensional subspaces. For example it cannot distinguish be-tween the (non linearly isomorphic if p = 2 [3, Theorem XII.3.8]) spaces L p ([0 , ℓ p ( N ), as they can both be written as the closure of an increasing sequenceof subspaces isometric to ℓ p ( { , . . . , n } ) : the subspace of L p ([0 , k n , k +12 n ], and the subspace of ℓ p ( N ) ofsequences that vanish oustide of { , . . . , n − } respectively.The relevant notions in the local theory of Banach spaces are the propertiesof a Banach space that depend only on the collection of his finite dimensionalsubspaces and not on the way they are organized. Said differently, the propertiesthat are inherited by finite representability. Such properties are called super-properties . The central question is to understand whether one super-propertyimplies another, see Section 2 for terminology, details and examples.The main result is Theorem 1.6, where a theoretical criterion is obtained forthe implication of two super-properties which are moreover stable under ℓ p -directsums, for some 1 ≤ p < ∞ which is fixed once and for all. A result by Hernandez[10] (Theorem 1.3 below) can be reformulated as: a superproperty P is stableunder ℓ p -direct sums if and only if it defined by p -homogeneous inequalities, i.e. if and only if there is an operator T between subspaces dom( T ) and ran( T ) of L p spaces L p (Ω , m ) and L p (Ω , m ) such that X satisfies P if and only if for every Date : 2021-01-20 01:41:47Z. n , every f , . . . , f n in the domain of T and every x , . . . , x n ∈ X , Z Ω k X i ( T f i )( ω ) x i k p dm ( ω ) ≤ Z Ω k X i f i ( ω ) x i k p dm ( ω ) . If one denotes by k T X k the (possibly infinite) norm of T ⊗ id X between thesubspaces dom( T ) ⊗ X and ran( T ) ⊗ X of L p (Ω i , m i ; X ), then this condition canbe shortly written as k T X k ≤
1. So our result characterizes, for two operators S and T between subspaces of L p spaces, when k T X k ≤ k S X k ≤ X ofcomplex separable Banach spaces up to isometry and the set T of norm ≤ L p spaces defined by the assignement ( T, X ) T X k . Indeed, adapting the standard terminology for locally convex topologicalvector spaces (see [4, II § Definition 1.1. If A ⊂ X is a class of Banach spaces, then its polar A ◦ is theclass of operators T ∈ T such that k T X k ≤ X in A . Definition 1.2. If B ⊂ T , then its polar ◦ B is the class of Banach spaces X ∈ X such that k T X k ≤ T in B .This duality is a variant of the one considered in [22], where Pisier restricts tooperators between L p spaces (and not subspaces of L p spaces). If one is interestedin the bipolar of a class of Banach spaces, the two dualities are very different.But a description of the bipolar for a class of operators for Pisier’s duality canbe obtained from our result, see Subsection 2.5 for details.In a locally convex topological space, the bipolar theorem ([4, II § C is equal to the closed convex hull of C ∪ { } . The inclusionof the closed convex hull of C ∪ { } in the bipolar of C is obvious; the content ofthe theorem is the other inclusion, which follows from the Hahn-Banach theorem.The aim of this paper is to state and prove a version of the bipolar theorem inthis setting, for the correct definition of “closed convex hull”. For the bipolar ofa class of Banach spaces, this is due to Hernandez. The methods we introduceallow us to give a new proof of it. Theorem 1.3. ( [10] ) The bipolar ◦ ( A ◦ ) of a class of Banach spaces A ⊂ X isthe class of Banach spaces finitely representable in the class of all finite ℓ p -directsums of elements in A . There is also an isomorphic version of the previous result.
Theorem 1.4. ( [10] ) Let A ⊂ X and X ∈ X . The following are equivalent: • k T X k < ∞ for every T ∈ A ◦ . • X is isomorphic to a space finitely representable in the class of finite ℓ p direct sums of spaces in A , i.e. to a space in ◦ A ◦ .In that case, the Banach-Mazur distance from X to a space in ◦ A ◦ is equal to sup T ∈ A ◦ k T X k . DUALITY OPERATORS/BANACH SPACES 3
Our main result is the bipolar theorem for sets of operators. To state it wehave to introduce some definition.
Definition 1.5. A spatial isometry between finite dimensional subspaces of L p spaces is a composition of isometries of the form: • (Change of density) Restriction to a subspace of L p (Ω , m ) of the multi-plication by a nonvanishing measurable function h : Ω → K ∗ , i.e. f ∈ L p (Ω , m ) hf ∈ L p (Ω , | h | − p m ). • (Equimeasurability outside of 0) Maps of the form T : dom( T ) ⊂ L p (Ω , m ) → L p (Ω ′ , m ′ ) such that for every finite family f , . . . , f n ∈ dom( T ) andevery Borel subset E ⊂ K n \ { } , m ( { x, ( f ( x ) , . . . , f n ( x )) ∈ E } ) = m ′ ( { x, ( T f ( x ) , . . . , T f n ( x )) ∈ E } ).It is not hard to prove (see Lemma 4.10 and Remark 4.11) that every spatialisometry is of the form C EC for C , C changes of phase and measure and E equimeasurable outside of 0.It is important that we require 0 / ∈ E , as we want for example that f ∈ L p ([0 , f χ [0 , ∈ L p ([0 , p is not an even integer, it is known that every isometry between (sep-arable) subspaces of L p spaces is a spatial isometry ([9]). The idea developpedin this article allows to recover this result, and to generalize it to arbitrary p :a linear map T is a spatial isometry if and only if it is a regular isometry, i.e. k T X k = k T − X k = 1 for all X (see Remark A.2 and Corollary A.3).We can now state the version of the bipolar theorem for sets of operators. Theorem 1.6.
Let B ⊂ T and T : dom( T ) ⊂ L p (Ω , m ) → L p (Ω , m ) be alinear map, and f , f , . . . , be a sequence generating a dense subspace of dom( T ) .The following are equivalent : • For every X ∈ Banach , sup S ∈ B k S X k ≤ ⇒ k T X k ≤ . • For every n and ε > , there exist – an operator S = S ⊕ S ⊕ · · · ⊕ S k with S of regular norm and S . . . , S k ∈ B , – spatial isometries U : dom( U ) ⊂ L p (Ω × [0 , ⊕ p L p ([0 , → dom( S ) ,V : dom( V ) = S (ran U ) → L p (Ω × [0 , ⊕ p L p ([0 , , – for every i = 1 . . . , n there are g i ∈ L p (Ω × [0 , , g ′ i ∈ L p (Ω × [0 , and h i ∈ L p ([0 , MIKAEL DE LA SALLE such that ( g i , h i ) ∈ dom( U ) , V ◦ S ◦ U ( g i , h i ) = ( g ′ i , h i ) and (cid:18)Z Ω × [0 , | f i ( ω ) − g i ( ω, s ) | p dm ( ω ) ds (cid:19) p ≤ ε (cid:18)Z Ω × [0 , | ( T f i )( ω ) − g ′ i ( ω, s ) | p dm ( ω ) ds (cid:19) p ≤ ε. It is instructing to work out explicitly a very simple case of this theorem,namely for the obvious implication max( k S X k , k T X k ) ≤ ⇒ k ( T ◦ S ) X k ≤ Corollary 1.7.
The bipolar ( ◦ B ) ◦ of a class B ⊂ T is the smallest class B ′ ⊂ T containing B and satisfying the following properties :(i) B ′ contains { T ∈ T , sup X ∈X k T X k ≤ } .(ii) B ′ is stable under finite ℓ p -direct sums.(iii) If T ∈ B ′ and U, V are spatial isometries then U ◦ T ◦ V ∈ B ′ .(iv) Let T ∈ B ′ such that T : dom( T ) ⊂ L p (Ω , m ) ⊕ L p (Ω , m ) → L p (Ω , m ) ⊕ L p (Ω , m ) is of the form ( f, g ) ( Sf, g ) for some S ∈ T with domain equalto the image of dom( T ) by the first coordinate projection. Then S ∈ B ′ .(v) If T ∈ T is an operator between subspaces of L p (Ω , m ) and L p (Ω ′ , m ′ ) andif, for every finite family f , . . . , f n in the domain of T and every ε > ,there is S ∈ B ′ with domain contained in L p (Ω , m ) and range contained in L p (Ω ′ , m ′ ) and elements g , . . . , g n ∈ dom( S ) such that k f i − g i k ≤ ε and k T f i − Sg i k ≤ ε , then T ∈ B ′ . Note however that Theorem 1.6 is a sense more precise than Corollary 1.7,as it almost says that to obtain the bipolar of B from B , it is enough to applythe operations (i), (ii), (iii), (iv) and (v) only once, and in that order. Almostbecause we obtain in this way all operators of the form T ⊗ id L p ([0 , with domain { ( ω, s ) f ( ω ) | f ∈ dom( T ) } for T ∈ ( ◦ B ) ◦ , so one needs to apply one lasttime (iii) to obtain all of ( ◦ B ) ◦ . This improvement is not minor. For a longtime, the author was only able to prove Corollary 1.7, and actually expectedthat to construct ( ◦ B ) ◦ out of B , it was necessary to iterate these operations(and in particular (iv) and (v)) a large number of times (even an arbitrarilylarge countable ordinal of times), and this ordinal number was a measurementof the difficulty of computing the bipolar of a B . This is closely related to theclassical fact, essentially due to Banach, that, to obtain the weak-* closure of aconvex subset in the dual of a separable Banach space, the number of times oneneeds to take limits of weak-* convergent sequences can be an arbitrary countableordinal. That this is not the case will rely on a particularily strong form of thebipolar theorem (in the linear setting) for the weak-* topology that we prove inProposition 3.3. See the discussion in subsection 3.1. DUALITY OPERATORS/BANACH SPACES 5
As for the usual bipolar theorem, the main content of the theorem is theinclusion ◦ B ◦ ⊂ B ′ . The reverse inclusion is rather obvious because it is ratherclear that ◦ B ◦ contains B and satisfies all the properties (i-v).So one can reformulate the non-trivial part of Corollary 1.7 as follows : if T / ∈ B ′ , then there is a Banach space X such that X ∈ ◦ B but k T X k > X explicitly,but we let the Hahn-Banach theorem construct it for us. This is achieved bysuitably encoding the class of Banach spaces in a locally convex topological vectorspace H and the class of operators between subspaces of L p spaces in its dual H ∗ ,in such a way that the polarity between X and T corresponds to the usual polarityin topological vector spaces. So, once these two encodings are well understood,both Theorems 1.3 and 1.6 are just an application of the bipolar theorem in H and H ∗ . When X is a Banach space of dimension n , X will be encoded inside the realBanach space C ( KP n − ) of real-valued continuous functions on the projectivespace of dimension n −
1. Similarly an operator T with a domain of dimension n will be encoded inside the dual of C ( KP n − ). The space H evoked would thenbe the projective limit of a suitable system of the spaces C ( KP n − ). But sincethe study of the polarity between X and T readily reduces to finite dimensionalBanach spaces and operators with finite dimensional domains, we prefer to workdirectly with C ( KP n − ) and never even formally introduce H . Notation.
To avoid any set-theoretical problem (of T not being a set), all themeasure spaces appearing here will be standard measure spaces taken in somefixed set containing [0 ,
1] with the Lebesgue measure and that is stable by takingequivalent measures, measurable subsets with restriction of the measure, andfinite direct sums. By direct sum of a finite sequence (Ω , m ) , . . . , (Ω n , m n ) wemean the space (Ω ∪ · · · ∪ Ω n , m ⊕ · · · ⊕ m n ) where Ω ∪ · · · ∪ Ω n is the disjointunion and the measure is A P i m i ( A ∩ Ω i ). None of the results depend on thechoice.The ℓ p -direct sum of a finite family T , . . . , T n of operators from dom( T i ) ⊂ L p (Ω i , m i ) to ran( T i ) ⊂ L p (Ω ′ i , m ′ i ) is the operator T ⊕ · · · ⊕ T n from dom( T ) ⊕· · · ⊕ dom( T n ) ⊂ L p (Ω ∪ · · · ∪ Ω n , m ⊕ · · · ⊕ m n ) to ran( T ) ⊕ . . . ran( T n ) ⊂ L p (Ω ′ ∪ · · · ∪ Ω ′ n , m ′ ⊕ · · · ⊕ m ′ n ).An operator T ∈ T is called regular if k T X k < ∞ for every Banach space X ,or equivalently if k T ℓ ∞ k < ∞ . In that case the quantity k T ℓ ∞ k = sup X k T X k iscalled the regular norm of T and denoted k T k r . We will denote by REG the setof operators T ∈ T such that k T k r ≤ Organization of the paper.
The first section presents some necessary back-ground and some motivation for studying this polarity. It also contains a discus-sion of variants of the duality presented in the introduction. Section 3 containsvarious preliminaries, including basic reminders on measure theory and on the
MIKAEL DE LA SALLE linear bipolar theorem, as well as one result on which the rest will rely: Proposi-tion 3.3. Section 4 contains the proof of the main theorem. It starts by definingthe encoding of spaces and operators in a linear duality, and then studies thisencoding. In an appendix we present a new proof and a generalization, in thecontext of Section 4, of Hardin’s theorem [9]. Hardin’s theorem appears as adirect corollary of the study of the invariant subspaces for some families of rep-resentations of GL n ( K ) on C ( KP n − ).Some of the results have been announced in the report Group actions on Ba-nach spaces and a duality spaces/operators [23, pp 2304–2307].2.
Background and motivation
Reminders on Banach space geometry. If n is an integer, the set Q ( n )of all n -dimensional normed space up to isometry, equipped with the Banach-Mazur distance d ( E, F ) = inf {k u kk u − k | u : E → F linear invertible } , becomes a compact metric space, the Banach-Mazur compactum . Beware thatit is not d but log d which is a distance in the usual way ( d is submultiplicative d ( E, G ) ≤ d ( E, F ) d ( F, G ) rather subadditive, and two isometric spaces are atBanach-Mazur distance 1), but following the tradition we still call d the Banach-Mazur distance.We say that a Banach space X is finitely representable in another Banachspace Y if for every finite-dimensional space E ⊂ X and every ε > F ⊂ Y of same dimension as E such that d ( E, F ) ≤ ε . In otherwords, if for every n , the closure in Q ( n ) of the space of n -dimensional subspacesof X is contained in the same closure but for Y . This is equivalent to X beingisometrically a subspace of an ultraproduct of Y .More generally, we say that a Banach space X is finitely representable in aclass B of Banach spaces if for every finite-dimensional space E ⊂ X and every ε > F of a space in B of same dimension as E such that d ( E, F ) ≤ ε . We can therefore define a class of Banach spaces up to finiterepresentability as a collection A n of closed subsets of Q ( n ) such that for every n > m , every m -dimensional subspace of every E ∈ A n belongs to A m . In thisrepresentation, finite representability corresponds to inclusion.The ℓ p -direct sum of a finite family X , . . . , X n of Banach spaces is the space X ⊕ X ⊕ · · · ⊕ X n for the norm k ( x , . . . , x n ) k = ( k x k p + · · · + k x n k p ) p .2.2. Motivation.
Estimating k T X k in terms of the properties of T and the geo-metric properties of X is a central aspect in the geometry of Banach spaces.Most natural geometric classes of Banach spaces are characterized in terms ofsuch quantities, and most celebrated results can be expressed in the form “ T belongs to the bipolar of B ” for specific T and B ⊂ T . We list a few historicalimportant examples for illustration. See [22, Section 4] for other examples. DUALITY OPERATORS/BANACH SPACES 7 • Hilbert spaces are characterized by the parallelogram inequality, i.e. theproperty k T X k ≤ T : ℓ → ℓ has matrix √ (cid:18) − (cid:19) . • [6, 5] A Banach space has the UMD property (for Unconditional Martin-gale Differences) if and only if the Hilbert transform H : L ( R ) → L ( R )satisfies k H X k < ∞ . • Let (Ω , µ ) be a probability space and ε i : Ω → {− , } , i ∈ N be iidcentered (Bernoulli) random variables. A Banach space X has type p ifthere is a constant T p such that k X ε i x i k L p (Ω; X ) ≤ T p ( X k x i k p ) p for every x i ∈ X . Equivalently if k T X k ≤ T : span( ε i ) ⊂ L p (Ω) → ℓ p ( N ) is the linear map sending ε i to T p (1 k = i ) k ∈ N . • A Banach space X has cotype p if there is a constant C p such that k X ε i x i k L p (Ω; X ) ≥ C p ( X k x i k p ) p for every x i ∈ X . Equivalently if k S X k ≤ where S : ℓ p → L p (Ω) is thelinear map sending ( a i ) i ∈ N to C p P i a i ε i . • A Banach space X has type > k T X k < ∞ , where T ∈ B ( L ( {− , } N )) is the orthogonal projection on the spacespanned by the coordinates ε i : ω = ( ω n ) n ∈ N ω i . • Denote by d n ( X ) the supremum over all n -dimensional subspaces E of X subspaces of the Banach-Mazur distance from U to ℓ n . Then (thisis due to Pisier but written in [12]) up to a factor 2, d n ( X ) is equal tosup k T X k , where the sup is taken over all T : L → L of norm 1 and rank n . We can therefore express the Milman-Wolfson Theorem [18] as follows:a Banach space X has type p > k T X k = o ( k T k rk( T ) ) asrk( T ) → ∞ .We now move to a more detailed discussion of two of the author’s main moti-vations.2.3. Group representations on Banach spaces.
Another motivation comesfrom the study of representations of groups on Banach spaces. Let G be a locallycompact topological group with a fixed left Haar measure. We recall that everystrong-operator-topology (SOT) continuous representation π of G on a Banachspace X extends to a representation of the convolution algebra C c ( G ) of com-pactly supported continuous functions on G by setting π ( f ) x = R f ( g ) π ( g ) xdg for every x ∈ X .For example, if λ p denotes the left-regular representation on L p ( G ) λ p ( g ) f = f ( g − · ), then λ p ( f ) is the convolution operator ξ f ∗ ξ . MIKAEL DE LA SALLE
When A is a class of Banach spaces, denote by C A ( G ) the completion of C c ( G )for the norm k f k C A ( G ) = sup k π ( f ) k B ( X ) , where the supremum is over all SOT-continuous continuous isometric represen-tations π of G on a space X in A .The following result, which generalizes the classical fact that, for amenablegroups, the full and reduced C ∗ -algebras coincide, reduces the understanding therepresentation theory of G on a Banach space X to the understanding of k T X k for convolution operators T . This known fact has already appeared in severalunpublished texts (for example in the author’s habilitation thesis), but seems tobe missing from the published literature. Proposition 2.1. If G is amenable and π is an isometric representation of G on a Banach space X , then for every f ∈ C c ( G ) , k π ( f ) k B ( X ) ≤ k λ p ( f ) X k . In particular, if a class of Banach spaces A has the property that L p ( G ; X ) ∈ A for every X ∈ A , then k f k C A ( G ) = sup X ∈ A k λ p ( f ) X k . Proof.
Fix a norm 1 element ξ ∈ L p ( G ) and define an isometric linear map α : X → L p ( G ; X ) by α ( x )( g ) = ξ ( g ) π ( g − ) x .Then for h ∈ G , ( α ( π ( h ) x ) − λ ( h ) α ( x ))( g ) = ( ξ ( g ) − ξ ( h − g )) π ( g − h ) x , and k α ( π ( h ) x ) − λ ( h ) α ( x ) k = k x kk ξ − λ ( h ) ξ k L p ( G ) . By the triangle inequality k α ( π ( f ) x ) − λ ( f ) α ( x ) k ≤ k f k L ( G ) k x k sup h ∈ supp( f ) k ξ − λ ( h ) ξ k L p ( G ) , and using that α is isometric we obtain k π ( f ) x k ≤ k λ ( f ) X kk x k + k f k L ( G ) k x k sup h ∈ supp( f ) k ξ − λ ( h ) ξ k L p ( G ) . We deduce k π ( f ) k ≤ k λ ( f ) X k + k f k L ( G ) sup h ∈ supp( f ) k ξ − λ ( h ) ξ k L p ( G ) . When G is amenable, the last term can be made arbitrarily small, which provesthe proposition. (cid:3) In the particular case of a compact group, this result lies at the heart of theproofs of Lafforgue’s strong property (T) for higher-rank algebraic groups. Forexample, thanks to the techniques of strong property (T), the conjecture [2] thatany action by isometries of a lattice in a connected higher-rank simple Lie groupon a super-reflexive Banach space has been reduced to the following conjecture,see [14, 15, 11], see also [26]. Denote, for any δ ∈ [ − , T δ the operator DUALITY OPERATORS/BANACH SPACES 9 on L ( S ) mapping f to the fonction ( T δ f )( x ) = the average of f on the circle { y ∈ S | h x, y i = δ } . For any θ ∈ R / π Z , denote by S θ the operator on L ( S ) mapping f to the fonction ( S θ f )( z ) = the average of f on the circle { √ ( e iθ + e iϕ j ) z | ϕ ∈ R / π Z } (where we identify S with the norm 1 quaternionsin the usual way). The conjecture is that for every super-reflexive Banach space,there exist α > C ∈ R + such that for every δ ∈ [ − ,
1] and θ ∈ R , k ( T δ − T ) X k ≤ C | δ | α and k ( S θ − S π/ ) X k ≤ C | θ − π/ | α . Super-expanders and embeddability of graphs in Banach spaces.
Another motivation for studying the quantity k T X k is its well-known connectionwith Poincar´e inequalities and embeddability of expanders in X . If G = ( V, E )is a finite connected graph, we may define its X -valued p -Poincar´e constant π p, G ( X ) as the smallest constant π such that for every f : V → X satisfying P v ∈ V deg( v ) f ( v ) = 0, X v ∈ V deg( v ) k f ( v ) k p ! p ≤ π X ( v,w ) ∈ E k f ( v ) − f ( w ) k p p . Note that π p, G ( X ) = k T X k for T the inverse of the linear map f ∈ ℓ p ( V, deg) ( f ( v ) − f ( w )) ( v,w ) ∈ E ∈ ℓ p ( E ).A sequence G n = ( V n , E n ) of bounded degree graphs is called a sequence ofexpanders with respect to X if lim n | V n | = ∞ and sup n π p, G n ( X ) < ∞ . This doesnot depend on p [19, 20, 7], see also [13, Proposition 3.9].For example, if p = 2 and X = K (or a Hilbert space), then π p, ( K ) is equalto (2 − λ ) − , for λ the second largest eigenvalue of the random walk operatoron G . So being a sequence of expanders with respect to K , or to an L p space forsome p < ∞ , is the same as the usual definition of expander graphs.According to [17], a sequence G n is called a sequence of super-expanders if theyare expanders with respect to all uniformly convex Banach spaces. The existenceof super-expanders is a difficult result. Essentially two classes of examples havebeen obtained, by Lafforgue [15] and by Mendel and Naor [17]. Lafforgue’s exam-ples are even expanders with respect to all Banach spaces of type >
1. All theseresults are therefore results of the norm “ T belongs to be bipolar of S ”, where S is any of the operators quantifying the fact that a Banach space has nontrivialtype or is super-reflexive, and T are correctly scaled operators in the definitionof the p -Poincar´e constant. Many intriguing questions remain open, which canall be formulated in the same way. For example, Question 2.2. [17] Are all expander sequences super-expanders? Expanderswith respect to all spaces of non-trivial type? There are many small variants of the definition. But they do not matter for the discussionhere, though they do matter for other issues, see for example [13].
Question 2.3. [17, 15] Does there exist a sequence of super-expanders of girthgoing to infinity? And of logarithmic girth in the number of vertices? Are theexpanders coming from higher-rank simple Lie groups super-expanders?
Question 2.4. [17, 15] Does there exist a sequence of expanders with respect toall Banach spaces of nontrivial coptype?A positive answer to this question is conjectured in [17], and Lafforgue evensuggests that the super-expanders coming from lattices in SL ( Q p ) (or otherhigher-rank simple algebraic groups over non-archimedean local fields) as in [15]are such examples. But this is wide open, as is the following. Question 2.5. [22] Are all expander sequences expanders with respect to allspaces of non-trivial cotype?One of the reasons for the interest in expanders with respect to Banach spacesis the well-known fact, which essentially goes back to Gromov, that a sequenceof expanders with respect to X does not coarsely embed into X . See for example[22, Section 3]. Being an expander with respect to X is much stronger thannon coarse embeddability (a striking example is given in [1]), but by [29] thereis equivalence between non-coarse embeddability into families of Banach spacesunder closed finite representability and ℓ p direct sums and some other forms ofPoincar´e inequalities.2.5. Comparing different notions of polarity.
The duality defined in theIntroduction was implicit in many early work on the geometry of Banach spaces,and was essentially present in [22], where Pisier explicitly considered a dualitythat is very close to ours. He defines the polars by the same formulas as inDefinition 1.1 and 1.2, but he only considers for T the operators between L p spaces, instead of subspaces of L p spaces. In particular, the polar of a set B ofBanach spaces is smaller for Pisier’s duality, and therefore its bipolar is larger.Indeed, Hernandez proved that for this duality, the bipolar of B is the set ofBanach spaces that are finitely representable in subspaces of quotients of finite ℓ p direct sums of spaces in B . This is quite different from Theorem 1.3. Forexample for the duality considered here, every Banach space in the bipolar of ℓ has cotype max( p,
2) (this is immediate from Hernandez’s Theorem 1.3 and thefact that ℓ p ( ℓ ) has cotype max( p, ℓ contains every space finitely representable in a quotient of ℓ , i.e. everyBanach space.However, as far as the bipolar of a set of operators is concerned, the twodualities are very related : if B is a set of operators between L p spaces, then itsbipolar for the polarity in [22] is the set of operators between L p spaces whichbelong to ◦ B ◦ (for our polarity). So our Theorem 1.6 also provides an answer to[22, Problem 4.1].The dualities discussed so far are isometric variants of two other isomorphicforms of the duality in [22], where A ◦ is the class of operators such that k T X k < ∞ DUALITY OPERATORS/BANACH SPACES 11 for all X ∈ A , and ◦ B is the class of Banach space such such k T X k < ∞ forall T ∈ B . But, if B is finite, the bipolar of B for this “isomorphic” dualitycoincides with ∪ R> R ◦ ( R − B ) ◦ . If B is infinite, the “isomorphic” bipolar of B is ∪ R> ∪ B ′ R ( ◦ B ′ ) ◦ , where B ′ = {{ c T T | T ∈ B } | c ∈ (0 , B } . So our bipolarTheorem 1.6 also allows to describe the bipolar for the isomorphic forms of theduality. 3. Preliminaries
On the bipolar in a dual Banach space.
In the whole paper, for a subset C of a real Banach space E with dual E ∗ , we denote its polar C ◦ = { x ∗ ∈ E ∗ , h x ∗ , x i ≥ − x ∈ C } . When C ⊂ E is a cone (that is x ∈ C implies { tx | t ∈ [0 , ∞ ) } ⊂ C ), then itspolar C ◦ coincides with { x ∗ ∈ E ∗ , h x ∗ , x i ≥ x ∈ C } . It is also a cone.Similarily, when C ⊂ E ∗ we denote its polar for the weak-* topology by ◦ C = { x ∈ E, h x ∗ , x i ≥ − x ∗ ∈ C } . Again, if C is a cone, ◦ C coincides with { x ∈ E, h x ∗ , x i ≥ x ∗ ∈ C } andis again a cone.It should be always clear from the context whether the polarity is consideredin this linear setting of two vector spaces in duality or between X and T as inDefinition 1.1 and 1.2.The classical bipolar theorems in this setting take the following forms: Theorem 3.1.
Let E be a real Banach space.If C ⊂ E , then its bipolar ◦ ( C ◦ ) is equal to the norm closure of the convex hullof C ∪ { } .If C ⊂ E ∗ , then its bipolar ( ◦ C ) ◦ is equal to the weak-* closure of the convexhull of C ∪ { } . The second statement is not so useful for our purposes because taking theweak-* closure can be quite complicated, as we shall soon recall. Fortunately,there is an interesting consequence of the Krein-Smulian theorem [8, TheoremV.12.1], which asserts that a convex subset of E ∗ for a separable Banach space E is weak-* closed if and only if it is sequentially weak-* closed, see [8, TheoremV.12.10]. This allows to significantly strengthen the result for separable Banachspaces as follows.If C is a subset of a dual E ∗ , let us define an increasing family of subsets C α ⊂ E ∗ indexed by the ordinals α by letting C = C , C α be the set of allweak-* limits of sequences in C α − if α is a successor and C α = ∪ β<α C β if α is alimit ordinal. The smallest ordinal α such that C α = C α +1 (that is C α is weak-*sequentially closed) is sometimes called the order of C . When E is separable, theorder of C is countable, see for example the argument in the proof of [8, TheoremV.12.10]. Moreover, if C is convex, then so is C α for every α . It follows from [8, Theorem V.12.10] that, for the order of C , C α coincides with the weak-* closureof C . Let us summarize this discussion. Proposition 3.2.
Let E be a real separable Banach space and C be a subsetof E ∗ . There is a countable ordinal α such that the bipolar of C coincides with (conv( C )) α . The smallest ordinal α in the previous proposition measures the difficulty toconstruct the bipolar of C out of C .The order has been more studied for linear subspaces C . It is known that formany cases such as E ∗ = ℓ = ( c ) ∗ , ℓ ∞ , H ∞ [16, 28, 27], every countable ordinalappears as the order of a linear subspace of E ∗ .It turns out that, for our applications, the order will always be equal to 1. Thiswill follow from the following result. Proposition 3.3.
Let E be a real Banach space and C ⊂ E ∗ . Assume that thereis a convex subset A ⊂ E such that A ∩ { x ∈ E | k x k ≤ r } is norm-compact forevery r > and A ◦ ⊂ C .Then the bipolar ( ◦ C ) ◦ of C is equal to the norm closure of the convex hull of C .Proof. Note that our assumptions implies that 0 ∈ C (as 0 ∈ A ◦ ). Let C ′ bethe norm closure of the convex hull of C . We know from the bipolar theorem(Theorem 3.1) that ( ◦ C ) ◦ is equal to the weak-* closure of conv( C ), so the inlusion C ′ ⊂ ( ◦ C ) ◦ is obvious. To prove the converse inclusion, consider x ∈ E ∗ \ C ′ . Wehave to prove that x does not belong to the weak-* closure of the convex hull of C .Let j : E → E ∗∗ be the canonical inclusion of E in its bidual. By the Hahn-Banach separation theorem in the Banach space E ∗ , there is ϕ ∈ E ∗∗ such thatinf C ϕ ≥ − ϕ ( x ) < −
1. In particular, we have inf A ◦ ϕ ≥ −
1, that is ϕ ∈ ( A ◦ ) ◦ = ( ◦ ( j ( A ))) ◦ . By Theorem 3.1 again, ( ◦ ( j ( A ))) ◦ is equal to the weak-*closure of (the convex set) j ( A ). But the assumption on A implies that j ( A )is already weak*-closed. Indeed, by the Krein-Smulian theorem, it is enough toshow that j ( A ) ∩ B E ∗∗ (0 , r ) is weak-* closed for every r >
0. This is true as j ( A ) ∩ B E ∗∗ (0 , r ) = j ( A ∩ B E (0 , r )) is even norm-compact as a continuous imageof a norm-compact set, and norm-compact subsets of E ∗∗ are weak-* closed. So j ( A ) being weak-* closed, we have proved that ϕ ∈ j ( A ). In particular, ϕ is σ ( E ∗ , E )-continuous, and we obtain, as announced, that x does not belong to theweak-* closure of the convex hull of C . (cid:3) Reminders on the Jordan decomposition of measures.
Recall thatany signed measure m on a Borel space has a unique decomposition m = m + − m − for two positive measures satisfying k m k = k m + k + k m − k (where the norm is thetotal variation norm). This is the Jordan decomposition of m . If m = m − m isany other decomposition with m , m positive measures, then m − m + = m − m − is a positive measure. We will use the following elementary fact. DUALITY OPERATORS/BANACH SPACES 13
Lemma 3.4.
Let m and m ′ be any signed measure, and let m , m be any positivefinite measures such that m = m − m . There is a decomposition m ′ = m ′ − m ′ with k m − m ′ k + k m − m ′ k = k m − m ′ k . Proof.
Let m = m + − m − and m ′ = m ′ + − m ′− be the Jordan decompositions. Asmall computation gives that k m − m ′ k = k m + − m ′ + k + k m − − m ′− k .By the property of the Jordan decomposition just recalled, m ′′ := m − m + = m − m − is a positive measure. Define m ′ = m ′ + + m ′′ and m ′ = m ′ + m ′′ , sothat m ′ = m ′ − m ′ and k m − m ′ k + k m − m ′ k = k m + − m ′ + k + k m − − m ′− k = k m − m ′ k . (cid:3) On (iv) in Corollary 1.7.
This short subsection is not needed anywhereelse in the paper, but it hopefully illustrates some basic things about Theorem 1.6and Corollary 1.7. We start by a lemma which clarifies in which situation anoperator T is of the form (iv) in Corollary 1.7. Lemma 3.5.
Let T be a norm ≤ operator between subspaces dom( T ) , ran( T ) ⊂ L p (Ω , m ) and A ⊂ Ω measurable. The following are equivalent. • T f ( x ) = f ( x ) for almost every x ∈ Ω \ A and every f ∈ dom( T ) . • If we write L p (Ω , m ) = L p ( A, m ) ⊕ p L p (Ω \ A, m ) , then there is an oper-ator S with domain equal to the image of dom( T ) by the first coordinateprojection such that T ( f , f ) = ( Sf , f ) for all ( f , f ) ∈ dom( T ) .In that case, S is unique, dom( S ) = { f | A , f ∈ dom( S ) } and S ( f | A ) = ( T f ) | A for all f ∈ dom( T ) .Proof. Clearly, the assumption that
T f ( x ) = f ( x ) for almost every x ∈ Ω \ A andevery f ∈ dom( T ) is equivalent to the existence of a linear map S : dom( T ) → L p ( A, m ) such that T ( f , f ) = ( S ( f , f ) , f ). So to prove the equivalence statedin the lemma, we have to observe that, in this situation, S ( f , f ) depends onlyon f , i.e. (by linearity) that S ( f , f ) = 0 if f = 0. For (0 , f ) ∈ dom( T ) wehave k T (0 , f ) k pp = k S (0 , f ) k pp + k f k p , which (by the assumption that k T k ≤ k f k p . This proves that k S (0 , f ) k pp = 0, as requested.The last assertion is a tautology. (cid:3) Finally, we provide an example that illustrates the main result.
Example . The inequality k ( T ◦ S ) X k ≤ k T X kk S X k is clear for every Banachspace X and every operators T, S such that T ◦ S makes sense. So it follows fromCorollary 1.7 that, with the notation therein, if S, T ∈ B then T ◦ S belongs to B ′ . We prove this directly, because it illustrates the subtle property (iv).So let S, T ∈ B such that ran( S ) ⊂ dom( T ). By (ii) the operator S ⊕ T : dom( S ) ⊕ dom( T ) → ran( S ) ⊕ ran( T ) belongs to B ′ . By composing bythe spatial isometry ( f, g ) ∈ ran( S ) ⊕ ran( T ) ( g, f ) ∈ ran( T ) ⊕ ran( S ) (which is allowed by (iii)) and restricting to the subspace D = { ( f, Sf ) | f ∈ dom( S ) } ⊂ dom( S ) ⊕ dom( T ) (which is allowed by (v)), we obtain that the map( f, Sf ) ∈ D ( T ◦ Sf, Sf ) belongs to B ′ . By (iv), we conclude that T ◦ S belongs to B ′ as required.4. The space of degree p homogeneous functions on K n Let n be a positive integer. Denote by | z | the ℓ p -norm on K n | z | = ( | z | p + · · · + | z n | p ) p . In the rare occasions when we want to insist on p , we write | z | p for this quantity.A function ϕ : K n → R is called homogeneous of degree p if ϕ ( λz ) = | λ | p ϕ ( z )for all z ∈ K n and λ ∈ K . The space H n of continuous homogeneous of degree p functions on K n is a Banach space over the field of real numbers for the topologyof uniform convergence on compact subsets on K n . A particular choice of norm is k ϕ k = sup | z |≤ | ϕ ( z ) | , so that for this norm H n is isometrically isomorphic to thespace of real-valued continuous functions on KP n − through the identification of ϕ ∈ H n with the function K z ∈ KP n − ϕ ( z | z | ). An equivalent definition of thenorm of ϕ ∈ H n is the smallest number such that for every z ∈ K n (4.1) | ϕ ( z ) | ≤ ( | z | p + · · · + | z n | p ) k ϕ k . We encode a class A ⊂ X of Banach spaces by the cone N ( A, n ) ⊂ H n (N fornorms) of functions of the form z
7→ k P ni =1 z i x i k p for X ∈ A and x , . . . , x n ∈ X .When (Ω , m ) is a measure space and f = ( f , . . . , f n ) is an n -uple of elementsof L p (Ω , m ), we can define a continuous linear form µ f on H n by(4.2) h µ f , ϕ i = Z ϕ ( f ( ω ) , . . . , f n ( ω )) dm ( ω ) . Indeed, it follows from (4.1) that the integral is well-defined and that µ f ∈ H ∗ n with norm equal to k f k pp + · · · + k f n k pp (the inequality ≤ is immediate from (4.1),and the equality follows by evaluating µ f at the norm 1 element z
7→ | z | p in H n ).We encode a class B ⊂ T of operators by the cone P ( B, n ) ⊂ H ∗ n P ( B, n ) = { µ f − µ T f , T ∈ B and f ∈ dom( T ) n } where for f = ( f , . . . , f n ) ∈ dom( T ) n , we denote T f = (
T f , . . . , T f n ). It is acone because for every t ≥ t ( µ f − µ T f ) = µ t p f − µ T t p f .The crucial but obvious property motivating these definitions is that, if ϕ ( z ) = k P ni =1 z i x i k pX for elements x , . . . , x n in a Banach space X , then h µ f , ϕ i = k P i f i x i k pL p (Ω ,m ; X ) . As a consequence, h µ f − µ T f , ϕ i = k X i f i x i k pL p (Ω ,m ; X ) − k X i ( T f i ) x i k pL p (Ω ,m ; X ) . In particular, we have
DUALITY OPERATORS/BANACH SPACES 15
Lemma 4.1.
Let A ⊂ X be a class of Banach spaces and B ⊂ T a class ofoperators.(1) B ⊂ A ◦ if and only if for every n , P ( B, n ) ⊂ N ( A, n ) ◦ .(2) A ⊂ ◦ B if and only if for every n , N ( A, n ) ⊂ ◦ P ( B, n ) . Polarity in H n . We start by improving Lemma 4.1. The next result ex-presses that the polarity in hX , T i (see Definition 1.1 and 1.2) is well encodedby the polarity h H n , H ∗ n i (see Subsection 3.1). Recall that REG the class of alloperators T ∈ T with regular norm k T k r := sup X ∈X k T X k ≤ Proposition 4.2.
Let A ⊂ X be a class of Banach spaces and B ⊂ T a class ofoperators. Then(1) P ( A ◦ , n ) = N ( A, n ) ◦ .(2) N ( ◦ B, n ) ⊂ ◦ P ( B ∪ REG, n ) . In the proof, we need a description of the dual of H n : Lemma 4.3.
Every continuous linear form l on H n is of the form µ f − µ g forsome measure spaces (Ω , m ) and (Ω ′ , m ′ ) and n -uples f ∈ L p (Ω , m ) n and g ∈ L p (Ω ′ , m ′ ) n . Moreover Ω , m, f and Ω ′ , m ′ , g can be chosen so that f and g takealmost surely their values in { z ∈ K n , | z | = 1 } and so that m (Ω) + m ′ (Ω ′ ) is equalto the norm of l .Proof. By the identification of H n with C ( KP n − ) and by the Riesz representa-tion theorem, every continuous linear form l on H n is of the form ϕ Z KP n − ϕ (cid:18) z | z | (cid:19) dν ( K z )for a unique signed measure ν on KP n − , and the norm of l is the total variationof ν . Let ν = ν + − ν − be the Jordan decomposition of ν and s : KP n − →{ z ∈ K n , | z | = 1 } a measurable section. Define (Ω , m ) = ( KP n − , ν + ) and f ∈ L p (Ω , m ) n by s ( ω ) = ( f ( ω ) , . . . , f n ( ω )). Similarly define (Ω ′ , m ′ ) = ( KP n − , ν − )and g ∈ L p (Ω ′ , m ′ ) n by s ( ω ) = ( g ( ω ) , . . . , g n ( ω )). Then we have Z KP n − ϕ (cid:18) z | z | (cid:19) dν ( K z ) = h µ f − µ g , ϕ i . This proves the lemma, because by construction f, g both take values in { z ∈ K n , | z | = 1 } and m (Ω) + m ′ (Ω) = ( ν + + ν − )( KP n − ) is the norm of l . (cid:3) Proof of Proposition 4.2.
We start by (1). If every space in A is trivial (of di-mension 0), we have N ( A, n ) ◦ = H ∗ n , A ◦ = T , and the result is easy. We cantherefore assume that A contains a space of dimension ≥
1. Let f, g be n -uplesin L p spaces. Note that if ϕ ( z ) = k P ni =1 z i x i k p then h µ f − µ g , ϕ i = k X i f i x i k pL p ( X ) − k X i g i x i k pL p ( X ) . So the linear form µ f − µ g ∈ H ∗ n belongs to N ( A, n ) ◦ if and only if for every X ∈ A and x , . . . , x n ∈ X , k P f i x i k pL p ( X ) ≥ k P g i x i k pL p ( X ) . Using that there isa nonzero X ∈ A , this holds if and only if there is a linear map T sending f i to g i such that T ∈ A ◦ . This shows that µ f − µ g belongs to N ( A, n ) ◦ if and only ifit belongs to P ( A ◦ , n ). By Lemma 4.3 every element of H ∗ n is of this form, whichproves (1).We move to (2). Denote by C n the closed convex cone C n = N ( X , n ). Wefirst prove that N ( ◦ B, n ) = ◦ P ( B, n ) ∩ C n . By definition N ( ◦ B, n ) ⊂ C n . So wehave to prove that for ϕ ∈ C n , ϕ ∈ N ( ◦ B, n ) if and only if ϕ ∈ ◦ P ( B, n ). But if ϕ ( z ) = k P ni =1 z i x i k p and X = span( x , . . . , x n ), then we have that ϕ ∈ N ( B ◦ , n )if and only if k T ⊗ id X k ≤ T ∈ B , if and only if for all T ∈ B and f , . . . , f n ∈ dom( T ), k P i T f i x i k p ≤ k P i f i x i k p , if and only if ϕ ∈ P ( B, n ) ◦ .We can now conclude with (2). By (1) for A = X , we have C ◦ n = P ( REG, n ).On the other hand, since C n is a closed convex cone, the bipolar theorem impliesthat C n = ◦ C ◦ n , and hence C n = ◦ P ( REG, n ). We therefore get N ( ◦ B, n ) = ◦ P ( B, n ) ∩ ◦ P ( REG, n )= ◦ ( P ( B, n ) ∪ P ( REG, n ))= ◦ P ( B ∪ REG, n ) . This proves (2). (cid:3)
By the bipolar theorem in H n and H ∗ n , we obtain Corollary 4.4.
Let A ⊂ X be a class of Banach spaces and B ⊂ T a class ofoperators. Then(1) N ( ◦ A ◦ , n ) = conv N ( A, n ) .(2) P ( ◦ B ◦ , n ) = conv w ∗ P ( B ∪ REG, n ) . The rest of this section consists in understanding the closed convex hulls of N ( A, n ) and P ( B, n ).4.2.
Understanding the encoding of Banach spaces in H n . The followingeasy fact will be important later.
Lemma 4.5.
For every integer n , bounded subsets of N ( X , n ) are relativelynorm-compact.Proof. By the Arzel`a-Ascoli theorem, we have to prove that bounded subsets of N ( X , n ) are equicontinuous, seen in C ( KP n − ). This follows from the triangleinequality. For example for p = 1 and ϕ ( z ) = k P i z i x i k , then we have | ϕ ( z ) − ϕ ( z ′ ) | ≤ k X i ( z i − z ′ i ) x i k ≤ X i | z i − z ′ i | ϕ ( e i ) . The case of arbitrary p is similar. Alternatively, it follows from the case p = 1by continuity of the map t t p . (cid:3) DUALITY OPERATORS/BANACH SPACES 17
Lemma 4.5 allows to considerably strengthen the second statement in Corol-lary 4.4, replacing weak-* closure by norm closure.
Corollary 4.6.
Let B ⊂ T a class of operators. Then P ( ◦ B ◦ , n ) = conv k·k P ( B ∪ REG, n ) . Proof.
The set N ( X , n ) is a closed convex cone in H n , so by Lemma 4.5 N ( X , n ) ∩{ ϕ ∈ H n | k ϕ k ≤ r } is norm-compact for every r . Moreover, we have that N ( X , n ) ◦ = P ( REG, n ) by Proposition 4.2. So, since P ( B ∪ REG, n ) contains N ( X , n ) ◦ , Proposition 3.3 implies that its bipolar is equal to the norm closure ofits convex hull. (cid:3) Let us list elementary properties of N . Lemma 4.7.
Let
A, A , A ⊂ X be classes of Banach spaces.(1) N ( A , n ) ⊂ N ( A , n ) if and only if, for every X ∈ A , every subspace ofdimension ≤ n of X is isometric to a subspace of a space in A .(2) The convex hull of N ( A, n ) is equal to N ( ⊕ ℓ p A, n ) , where ⊕ ℓ p A denotesthe set of all finite ℓ p -direct sums of Banach spaces in A .(3) The norm closure of N ( A, n ) in H n coincides with N ( A, n ) where A de-notes the set of Banach spaces finitely represented in A . As a consequence of (1) and (3), if two classes of Banach spaces A , A are closedunder finite representability, then A = A if and only if N ( A , n ) = N ( A , n )for all n . Proof.
The first point is obvious from the following observation : if x , . . . , x n (respectively y , . . . , y n ) are elements in a Banach space X (respectively in aBanach space Y ), then the functions z
7→ k P ni =1 z i x i k p and z
7→ k P ni =1 z i y i k p coincide if and only if there is an isometry from the linear span of { x , . . . , x n } to the linear span of { y , . . . , y n } sending x i to y i .If ϕ , . . . , ϕ k ∈ N ( A, n ) are given by ϕ j ( z ) = k P ni =1 z i x ( j ) i k pX j then by thedefinition of the ℓ p -direct sum X ⊕ p · · · ⊕ p X k we can write k X j =1 ϕ j ( z ) = k n X i =1 z i ( x ( j ) i ) ≤ j ≤ k k pX ⊕ p ···⊕ p X k . This shows that N ( ⊕ ℓ p A, n ) coincides with { ϕ + · · · + ϕ k , k ∈ N , ϕ j ∈ N ( A, n ) } . This is the convex hull of N ( A, n ) because N ( A, n ) is a cone.We move to (3). If a sequence ϕ k ∈ N ( A, n ) converges uniformly on compactsubsets to ϕ ∈ H n , then ϕ p is the uniform limit on compact sets of the seminorms ϕ p k , so it is a seminorm on K n . This means that there is a Banach space X ∈ X and x , . . . , x n spanning X such that ϕ ( z ) = k P ni =1 z i x i k p . The family x , . . . , x n might not be linearly independant, so we extract from it a basis of X . Withoutloss of generality we can assume that this basis is x , . . . , x m for some m ≤ n .Write ϕ k ( z ) = k n X i =1 z i x ( k ) i k pX k for some X k ∈ A and x ( k )1 , . . . , x ( k ) n ∈ X k . From the assumption that ϕ k convergesuniformly on compacta to ϕ and the assumption that x , . . . , x m is linearly inde-pendant, we get that for every ε > k such that(1 − ε ) ϕ ( z, ≤ ϕ k ( z, ≤ (1 + ε ) ϕ ( z, z ∈ K m . This means that the linear map u : X → X k sending x i to(1 − ε ) − p x ( k ) i for i ≤ m satisfies k x k ≤ k u ( x ) k ≤ ( 1 + ε − ε ) p k x k for all x ∈ X. Since ε > X is finitely representable in A , i.e. that ϕ ∈ N ( A, n ). This proves that N ( A, n ) ⊂ N ( A, n ). The converseinclusion is proved by reading the preceding argument backwards. (cid:3)
We can conclude our proof of Hernandez’ theorem.
Proof of Theorem 1.3.
Let A ′ be the class of Banach spaces which are finitely rep-resentable in the class of ℓ p -direct sums of spaces in A . It follows from Corollary4.4 and Lemma 4.7 that for every integer n , N ( ◦ A ◦ , n ) = N ( A ′ , n ) . Since both ◦ A ◦ and A ′ are closed under finite representability, we get the equality ◦ A ◦ = A ′ by the remark following Lemma 4.7. (cid:3) Proof of Theorem 1.4. If X is at Banach-Mazur ≤ C from ◦ A ◦ ; then theinequality k T X k ≤ C for every T ∈ A ◦ is clear.For the converse, we will need the following consequence of the Hahn-Banachtheorem. Lemma 4.8.
Let K be a compact Hausdorff topological space, and C ( K ) thespace of real-valued continuous functions on K . Let A be a closed convex cone inthe positive cone of C ( K ) such that A ∩ B (0 , is compact. Let s ≥ . Then forevery ψ ∈ C ( K ) , the following are equivalent • ∃ ϕ ∈ A, ψ ≤ ϕ ≤ sψ . • h sµ − ν, ψ i ≥ for every positive measures µ, ν on K such that h µ − ν, ϕ i ≥ for all ϕ ∈ A .Proof. = ⇒ is easy because the inequality ψ ≤ ϕ ≤ sψ implies h sµ − ν, ψ i ≥h µ − ν, ϕ i . DUALITY OPERATORS/BANACH SPACES 19
For the converse, since A is a convex cone, the set B of ψ satisfying ∃ ϕ ∈ A, ψ ≤ ϕ ≤ sψ is a convex cone. Moreover, the compactness assumption on A implies that B is also closed. Assume that ψ / ∈ B . By Hahn-Banach there isa linear form on C ( K ) which is nonnegative on B and negative at ψ . By theRiesz representation theorem and the Hahn decomposition, this linear form canbe written as f R f d ( µ − ν ) for positive measures µ, ν such that there is aBaire measurable subset E ⊂ K satisfying ν ( E ) = 0 and µ ( K \ E ) = 0.Let ϕ ∈ A . Let f n : K → [0 ,
1] be a sequence of continuous functions convergingin L ( K, µ + ν ) to the indicator function of E . Then for every n , the function( s + (1 − s ) f n ) ϕ belongs to B so h µ − ν, ( s + (1 − s ) f n ) ϕ i >
0. By making n → ∞ we get h µ − ν, s ϕ K \ E + ϕ E i ≥
0, which can be written as h s µ − ν, ϕ i ≥ h s µ − ν, ϕ i ≥ ϕ ∈ A , whereas h µ − ν, ψ i <
0. Thisproves the lemma. (cid:3)
We can now prove the converse implication in Theorem 1.4. Assume that k T X k ≤ C for every T ∈ A ◦ . Let x , . . . , x n ∈ X . Define ψ ∈ H n by ψ ( z ) = k P ni =1 z i x i k p , and view ψ in C ( KP n − ). The assumption that k T X k ≤ C forevery T ∈ A ◦ implies that h C p µ − ν, ψ i ≥ µ, ν on KP n − such that h µ − ν, ϕ i ≥ ϕ ∈ A . By Lemma 4.8 (rememberLemma 4.5) this implies that there is ϕ in the closed convex hull of N ( A, n ) suchthat ψ ≤ ϕ ≤ C p ψ . By the proof of Theorem 1.3, there is a space Y ∈ ◦ A ◦ and y , . . . , y n ∈ Y such that ϕ ( z ) = k P i z i y i k p . By taking the 1 /p -th power in theinequality ψ ≤ ϕ ≤ C p ψ we get that k P z i x i k ≤ k P z i y i k ≤ C k P z i x i k forevery y ∈ K n . This means that the linear span of x , . . . , x n is at Banach-Mazurdistance ≤ C from the linear span on { y , . . . , y n } and concludes the proof.4.4. Understanding the encoding of operators in H ∗ n .Lemma 4.9. Let f, g, ˜ f , ˜ g be n -uples of elements of L p spaces. Then µ f − µ g = µ ˜ f − µ ˜ g if and only if there is h ∈ L p (Ω , m ) n , ˜ h ∈ L p ( ˜Ω , ˜ m ) n such that µ ( f i ⊕ h i ) ni =1 = µ ( ˜ f i ⊕ ˜ h i ) ni =1 and µ ( g i ⊕ h i ) ni =1 = µ (˜ g i ⊕ ˜ h i ) ni =1 .Proof. The if direction is easy, because µ ( f i ⊕ h i ) ni =1 = µ f + µ h .For the converse, assume that µ f − µ g = µ ˜ f − µ ˜ g . Let ν f be the positive measureon KP n − such that, for every ϕ ∈ H n (4.3) h µ f , ϕ i = Z KP n − ϕ (cid:18) z | z | (cid:19) dν f ( K z ) . Define similarly ν g , ν ˜ f , ν ˜ g . Then ν f − ν g = ν ˜ f − ν ˜ g is a signed measure on KP n − .Let ν + − ν − be its Jordan decomposition. By the properties of the Jordan de-composition, ν f − ν + = ν g − ν − is a positive measure on KP n − , and thereforeby the proof of Lemma 4.3 it is of the form to ν ˜ h for some n -uple ˜ h ∈ L p ( ˜Ω , ˜ m ).Similarly, there is a h ∈ L p (Ω , m ) n such that ν ˜ f − ν + = ν ˜ g − ν − = ν h . We can rewrite these equalities as ν + = ν f − ν ˜ h = ν ˜ f − ν h and ν − = ν f − ν ˜ h = ν ˜ f − ν h . This implies that µ f + µ h = µ ˜ f + µ ˜ h and µ g + µ h = µ ˜ g + µ ˜ h and proves thelemma. (cid:3) Lemma 4.10.
For two families f ∈ L p (Ω , m ) n and g ∈ L p (Ω ′ , m ′ ) n , µ f = µ g if and only if there is a spatial isometry span { f , . . . , f n } → span { g , . . . , g n } sending f i to g i .Proof. The if direction is easy : firstly if there is a measurable function h : Ω → K \ { } , if (Ω ′ , m ′ ) = (Ω , | h | − p m ) and g i = hf i for all i , then for every ϕ ∈ H n , ϕ ( g , . . . , g n ) = | h | p ϕ ( f , . . . , f n ) and therefore h µ g , ϕ i = h µ f , ϕ i . Secondly if f , . . . , f n and g , . . . , g n are equimeasurable outside of 0 in the sense of Defini-tion 1.5, then R ϕ ( f , . . . , f n ) dm = R ϕ ( g , . . . , g n ) dm ′ for every Borel function ϕ vanishing at 0 and such that the integrals are defined. In particular µ f = µ g .For the converse, assume that µ f = µ g . Take a measurable section s : KP n − → K n with values in { z ∈ K n , | z | = 1 } . Then there are measurable nonvanishingfunctions h : Ω → K ∗ and h ′ : Ω ′ → K ∗ such that f ( ω ) = h ( ω ) s ( K f ( ω )) for every ω ∈ Ω such that f ( ω ) = 0, and similarly g ( ω ′ ) = h ′ ( ω ′ ) s ( K g ( ω ′ )) if g ( ω ′ ) = 0.By replacing m by | h | − /p m and f i by h i f i and similarly for g we can assume that h = 1 and h ′ = 1, and we shall prove that f and g are equimeasurable outside of0. By this we mean that for every Borel E ⊂ K n \{ } , m ( { ω, ( f ( ω ) , . . . , f n ( ω )) ∈ E } ) = m ′ ( { ω ′ , ( g ( ω ′ ) , . . . , T g n ( ω ′ )) ∈ E } ). It is clear that this implies that, forevery matrix A ∈ M m,n ( K ), Af and Ag are equimeasurable outside of 0, andtherefore that the linear map sending f i to g i is well-defined and is as in thedefinition of equimeasurability outside of 0.By the identification of H n with C ( KP n − ), using that | f | ∈ { , } we have Z Ω \ f − (0) ψ ( K f ) dm = Z Ω ′ \ g − (0) ψ ( K g ) dm ′ for every continuous function ψ : KP n − → K , and therefore also for everybounded Borel function ϕ : KP n − → K . This implies, since f and g take valuesin { }∪ s ( KP n − ), that R ϕ ( f ) dm = R ϕ ( g ) dm ′ for every Borel function K n → K vanishing at 0. Equivalently, f and g are equimeasurable outside of 0. (cid:3) Remark . The same proof shows actually a bit more : if a linear map T : E ⊂ L p (Ω , m ) → L p (Ω ′ , m ′ ) satisfies µ f = µ T f for every n and every f ∈ E n , then T is a spatial isometry. Indeed, since by our standing assumption E (as everyother space considered in this paper) is separable, we can find a sequence ( f i ) i ≥ generating a dense subspace of E and satisfying P i k f i k p < ∞ , and in particular( f i ( ω )) i ≥ belongs to ℓ p for almost every ω . Then the same proof applies, exceptthat we replace K n by ℓ p and KP n − by its projectivization ℓ p / K ∗ . DUALITY OPERATORS/BANACH SPACES 21
We shall also need the following variant :
Lemma 4.12.
For two families f ∈ L p (Ω , m ) n and g ∈ L p (Ω ′ , m ′ ) n and ε > , k µ f − µ g k < ε if and only if there are spatial isometries U : span { f , . . . , f n } → L p (Ω ′′ , m ′′ ) and V : span { g , . . . , g n } → L p (Ω ′′ , m ′′ ) such that Z Ω ′′ ( | U f | p + | V g | p ) χ Uf = V g < ε.
Proof.
We prove the slightly stronger statement with < ε replaced by ≤ ε .The if direction is easy : by Lemma 4.10 we have µ f = µ Uf and µ g = µ V g , andtherefore for every ϕ ∈ H n , h µ f − µ g , ϕ i = Z Ω ′′ ϕ ( U f ) − ϕ ( g ) ≤ Z Ω ′′ ( | ϕ ( U f ) | + | ϕ ( V g ) | ) χ f = Ug ≤ Z Ω ′′ ( | U f | p + | V g | p ) k ϕ k ≤ ε k ϕ k . Taking the supremum over ϕ we get k µ f − µ g k ≤ ε .The converse follows from a coupling argument. Assume that k µ f − µ g k ≤ ε .Let ν f and ν g be the measures on KP n − given by (4.3), so that the total variationnorm of ν f − ν g is at most ε . This means that we can decompose ν f = ν + ν and ν g = ν + ν for positive measures with ( ν + ν )( KP n − ) ≤ ε . As in the proof ofLemma 4.3, each ν k corresponds by (4.3) to µ h k for an n -uple h k ∈ L p (Ω k , m k ) n with P i k h ki k pp = ν k ( KP n − ). In particular, we have µ f = µ h + µ h and µ g = µ h + µ h .Let us define Ω ′′ as the disjoint union Ω ∪ Ω ∪ Ω , m ′′ as m + m + m , and f ′ = h ⊕ h ⊕ g ′ = h ⊕ ⊕ h , so that µ f ′ = µ h + µ h = µ f and µ g ′ = µ g .By Lemma 4.10, there are spatial isometries U and V sending f to f ′ and g to g ′ respectively, and we have Z Ω ′′ ( | f ′ | p + | g ′ | p ) χ f ′ = g ′ = Z Ω | h | p + Z Ω | h | p ≤ ε. This proves the lemma. (cid:3)
There is also an asymetric variant of the preceding lemma, that can be useful.
Remark . In Lemma 4.12, we can moreover assume that (Ω ′′ , m ′′ ) = (Ω × [0 , , m ⊗ dλ ) (for λ the Lebesgue measure), and that the spatial isometry U issimply U ξ ( ω, s ) = ξ ( ω ). Proof.
Let µ f , µ g , ν f = ν + ν , ν g = ν + ν be as in the proof of Lemma 4.12, where k ν + ν k < ε . We can even assume that ν = 0 (this is where the strict inequality < ε is used). Denote by dν dν f : KP n − → [0 ,
1] the Radon-Nikodym derivative.Define A ⊂ Ω × [0 ,
1] = Ω ′′ by A = { ( ω, s ) | s ≤ ≤ s ≤ dν dν f ( K ∗ f ( x )) } , so that µ fχ A = ν and µ fχ Ω ′′\ A = ν . In particular, Ω ′′ \ A has positive measure and istherefore an atomless standard measure space, and we can find h ∈ L p (Ω ′′ , m ′′ ) n that vanishes on A such that µ h corresponds to ν . We then have µ g = µ h + µ fχ A = µ h + fχ A . The last equality is because h and f χ A are disjointly supported. ByLemma 4.10, there is a spatial isometry V sending g to h + f χ A . Moreover, wehave Z Ω ′′ ( | f | p + | h + f χ A | p ) χ f = h + fχ A ≤ Z Ω ′′ \ A ( | f | p + | h | p ) < ε. (cid:3) If B ⊂ T , we define new (larger) classes as follows : • Λ ( B ) is the set of operators ( T, id) : dom( T ) ⊕ p L p (Ω , µ ) → ran( T ) ⊕ L p (Ω , µ ) for T ∈ B and a measure space (Ω , µ ). • Λ ( B ) = { U ◦ T ◦ V | U, V spatial isometries, T ∈ B } . • Λ ( B ) is the set of all S : dom( S ) ⊂ L p (Ω , m ) → L p (Ω , m ) suchthat there is T ∈ B where dom( T ) ⊂ L p (Ω , m ) ⊕ L p (Ω , m ), ran( T ) ⊂ L p (Ω , m ) ⊕ L p (Ω , m ), dom( S ) is the image of dom( T ) by the first coor-dinate projection and T ( f ⊕ g ) = Sf ⊕ g for every f ⊕ g ∈ dom( T ). • Λ ( B ) is the set of all S : dom( S ) ⊂ L p (Ω , m ) → L p (Ω ′ , m ′ ) such thatfor every finite family f , . . . , f n in the domain of T and every ε > T ∈ B with domain contained in L p (Ω , m ) and range containedin L p (Ω ′ , m ′ ) and elements g , . . . , g n ∈ D ( S ) such that k f i − g i k ≤ ε and k T f i − Sg i k ≤ ε .To save place, we denote Λ ( B ) = Λ (Λ (Λ ( B ))). Corollary 4.14.
For every T ∈ T and B ⊂ T , the following are equivalent: • for every n , P ( T, n ) ⊂ P ( B, n ) . • The restriction of T to every finite dimensional subspace of dom( T ) be-longs to Λ ( B ) .Proof. Assume that, for a fixed n , P ( T, n ) ⊂ P ( B, n ). This means that, for every f ∈ dom( T ) n , there is S ∈ B and g ∈ dom( S ) n such that µ f − µ T f = µ g − µ S g .By Lemma 4.9 and Lemma 4.10, there are h ∈ L p (Ω , m ) n and h ∈ L p (Ω ′ , m ′ ) n and spatial isometries U : span { Sg i ⊕ h i } → span { T f i ⊕ h i } sending Sg i ⊕ h i to T f i ⊕ h i and V : span { f i ⊕ h i → g i ⊕ h i } sending f i ⊕ h i to g i ⊕ h i . Theoperator S = ( S, id) on dom( T ) ⊕ L p (Ω ′ , m ′ ) belongs to Λ ( B ), so the operator S = U ◦ S ◦ V , which sends f i ⊕ h i to T f i ⊕ h i belongs to Λ (Λ ( B )), andtherefore the restriction of T to span { f , . . . , f n } belongs to Λ (Λ (Λ ( B ))). Thisproves one direction.The converse is simpler: it follows from the easy directions in Lemma 4.9and Lemma 4.10 that P (Λ i ( B ) , n ) = P ( B, n ) for i = 1 , ,
3. In particular, if therestriction of T to every ≤ n -dimensional subspace of dom( T ) belongs to Λ ( B ),then P ( T, n ) ⊂ P ( B, n ). (cid:3) DUALITY OPERATORS/BANACH SPACES 23
Convergences in H ∗ n . This section is devoted to the understanding of theencoding of both weak-* sequential convergence and norm convergence in H ∗ n .Our first result asserts that weak-* convergence of sequences corresponds to theoperation Λ we just defined. Proposition 4.15.
Let B ⊂ T . The smallest class containing B and stable byall operations Λ , Λ , Λ , Λ coincides with the set of T ∈ T such that for every n , P ( T, n ) is contained in the sequential weak-* closure of P ( B, n ) .Proof. We define by transfinite induction, for every ordinal α , a class B α asfollows. B is Λ ( B ). If α is a successor ordinal, B α = Λ (Λ ( B α − )). If α isa limit ordinal we set B α = ∪ β<α B β .Similarly, we define, for every integer n and every ordinal α , a subset C nα ⊂ H ∗ n by C n = P ( B, n ), for a successor ordinal C nα is the set of all limits of weak-*converging sequences of elements of C nα − . If α is a limit ordinal we set C nα = ∪ β<α C nβ .We claim that, for every T ∈ T with dom( T ) finite-dimensional, P ( T, n ) ⊂ C nα for every n if and only if T belongs to B α . We prove it by transfinite induction.If α = 0, this is Corollary 4.14. Let α > β < α . If α is a limit ordinal, the claim is clear.So assume that α is a successor. Assume first that P ( T, n ) ⊂ C nα for every n .Let n be the dimension of dom( T ) and f = ( f , . . . , f n ) a basis. Then µ f − µ T f is alimit of a weak-* converging sequence ν k of elements of C nα − . By Lemma 4.3 thereare f ( k ) ∈ L p (Ω k , m k ) n and g ( k ) ∈ L p (Ω ′ k , m ′ k ) n with values in { z ∈ K n , | z | = 1 } such that ν k = µ f ( k ) − µ g ( k ) and m k (Ω k ) + m ′ k (Ω ′ k ) is the norm of the corre-sponding linear form, which is bounded by Banach-Steinhaus. For simplicity ofthe exposition assume that m k (Ω k ) + m ′ k (Ω ′ k ) ≤
1. By the induction hypothe-sis, there is an operator S k ∈ B α − such that f ( k ) ∈ D ( S k ) n and S k f ( k ) = g ( k ) .We have two sequences of probability measures, f ( k ) ∗ m k + (1 − m k (Ω k )) δ and g ( k ) ∗ m ′ k + (1 − m ′ k (Ω ′ k )) δ , on { } ∪ { z ∈ K , | z | = 1 } ⊂ K n . By compactness, up toan extraction we can assume that both sequences converge weak-*, and by Sko-rohod’s representation theorem we can assume that (Ω k , m k ) does not depend on k and that f ( k ) converges almost surely to some f ( ∞ ) ∈ L np and similarly g ( k ) con-verges almost surely, and in particular in L p , to g ( ∞ ) (this modifies the operators S k , but they still satisfy ν k = µ f ( k ) − µ Sf ( k ) and therefore still belong to B α − ).In particular, the operator S ( ∞ ) from dom( S ( ∞ ) ) = span { f ( ∞ )1 , . . . , f ( ∞ ) n } → span { g ( ∞ )1 , . . . , g ( ∞ ) n } sending f ( ∞ ) i to g ( ∞ ) i belongs to Λ ( B α − ) and it satisfies µ f ( ∞ ) − µ S ( ∞ ) f ( ∞ ) = lim k µ f ( k ) − µ S ( k ) f ( k ) = µ f − µ T f . By Corollary 4.14 again, the restriction of T to span { f , . . . , f n } = E belongs toΛ ( S ( ∞ ) ) ⊂ B α . This concludes the proof that P ( T, n ) ⊂ C nα for all n impliesthat the restriction of T belongs to B α . The converse is similar but easier andleft to the reader. (cid:3) Similarly, norm convergence is well encoded.
Lemma 4.16.
Let B ⊂ T , T : dom( T ) ⊂ L p (Ω , m ) → L p (Ω , m ) a linearmap with domain of finite dimension n , and ( f , . . . , f n ) be a basis of dom( T ) .Then P ( T, n ) is contained in the norm-closure of P ( B, n ) if and only if for every ε > , there is S ∈ Λ ( B ) with dom( S ) ⊂ L p (Ω × [0 , , m ⊗ dλ ) and ran( S ) ⊂ L p (Ω × [0 , , m ⊗ dλ ) , there are g , . . . , g n ∈ dom( S ) such that Z Ω × [0 , ( | f ( ω ) | p + | g ( ω, s ) | p ) χ f ( ω ) = g ( ω,s ) dm ( ω ) ds ≤ ε and Z Ω × [0 , ( | T f ( ω ) | p + | Sg ( ω, s ) | p ) p χ T f ( ω ) = Sg ( ω,s ) dm ( ω ) ds ≤ ε. Proof.
Assume that P ( T, n ) is contained in the norm closure of P ( B, n ). Thismeans that for every ε >
0, there µ ′ ∈ P ( B, n ) such that k µ f − µ T f − µ ′ k < ε .By Lemma 3.4, we can write µ ′ = µ g − µ h for n -uples of elements of L p spaces g, h where k µ f − µ g k + k µ T f − µ h k < ε . Since we have some room ( < ε ), we caneven assume that { g , . . . , g n } are linearly independant, so that we can define alinear map S sending g i to h i . By Corollary 4.14, S belongs to Λ ( B ), and sodoes S composed with any spatial isometry. So the only if direction follows fromLemma 4.12 and its improvement in Remark 4.13 .The converse is proved the same way. (cid:3) Proof of the main Theorem.
We are also ready to prove our main The-orem 1.6. Before we do so, we only need to understand the operation of takingconvex hulls.
Lemma 4.17.
Let B ⊂ T and n ∈ N . The convex hull of P ( B, n ) is equal to P ( ⊕ ℓ p ( B ) , n ) where ⊕ ℓ p ( B ) is the class of all finite ℓ p -direct sums of operators in B .Proof. This is clear: if T , . . . , T k ∈ B and f ( j ) ∈ D ( T j ) n for all j , then X j µ f ( j ) − µ T f ( j ) = µ f − µ ( T ⊕···⊕ T k ) f where f i = f (1) i ⊕ · · · ⊕ f ( k ) i ∈ D ( T ) ⊕ . . . D ( T k ) and f = ( f , . . . , f n ) ∈ ( D ( T ) ⊕ . . . D ( T k )) n . (cid:3) We can conclude.
Proof of Theorem 1.6.
We start with the easy direction. Assume that for every n and ε , the assumption in the second bullet point holds. Let X be Banach spacesuch that sup S ∈ B k S X k ≤
1. We have to prove that k T X k ≤
1. That is, for everyinteger n and every x , . . . , x n ,(4.4) k X i ( T f i ) x i k L p (Ω ; X ) ≤ k X i f i x i k L p (Ω ,X ) . DUALITY OPERATORS/BANACH SPACES 25
Let ε >
0, and S = S ⊕ S . . . S k , U , V , g i , g ′ i , h i given by the assumption. In thefollowing computation, we view T f i as an element of L p (Ω × [0 , , f i . We denote simplyby k · k p the norm in L p (Ω i × [0 , X ) or L p (Ω i × [0 , k X i ( T f i ) x i k L p (Ω ; X ) ≤ X i k T f i − g ′ i k p k x i k + k X i g ′ i x i k p ≤ ε X i k x i k + k X i g ′ i x i k p = ε X i k x i k + k X i ( g ′ i , h i ) x i k pp − k X i h i x i k pp ! p . The quantity inside the parenthesis is equal to k X i S ( g i , h i ) x i k pp − k X i h i x i k pp , so using that k S X k = max ≤ i ≤ k k ( S i ) X k ≤
1, we obtain that it is bounded aboveby k X i ( g i , h i ) x i k pp − k X i h i x i k pp = k X i g i x i k pp . We can therefore go on with our computation and get k X i ( T f i ) x i k L p (Ω ; X ) ≤ ε X i k x i k + k X i g i x i k p ≤ ε X i k x i k + X i k f i − g i k p k x i k + k X i f i x i k p ≤ ε X i k x i k + k X i f i x i k p . Making ε →
0, we obtain (4.4) as required.The converse direction relies on everything we have obtained so far. Assumethat T ∈ ( ◦ B ) ◦ . We know from Corollary 4.6 that for every integer n , P ( T, n ) ⊂ conv k·k P ( B ∪ REG, n ), which is the same as the norm-closure of P ( ⊕ ℓ p ( B ∪ REG ))by Lemma 4.17. So by Lemma 4.16, for every ε > S ∈ Λ ( B ∪ REG )with dom( S ) ⊂ L p (Ω × [0 , , m ⊗ dλ ) and ran( S ) ⊂ L p (Ω × [0 , , m ⊗ dλ ),there are g , . . . , g n ∈ dom( S ) such that Z Ω × [0 , ( | f ( ω ) | p + | g ( ω, s ) | p ) χ f ( ω ) = g ( ω,s ) dm ( ω ) ds ≤ ε p and Z Ω × [0 , ( | T f ( ω ) | p + | Sg ( ω, s ) | p ) p χ T f ( ω ) = Sg ( ω,s ) dm ( ω ) ds ≤ ε p . In particular, using that R ( | a | p + | b | p ) χ a = b ≥ R | a − b | p = P i R | a i − b i | p for every a, b ∈ ( L p ) n , we have for every i , (cid:18)Z Ω × [0 , | f i ( ω ) − g i ( ω, s ) | p dm ( ω ) ds (cid:19) p ≤ ε and (cid:18)Z Ω × [0 , | T f i ( ω ) − Sg i ( ω, s ) | p dm ( ω ) ds (cid:19) p ≤ ε. Also, using that
REG contains the identity and is stable by ℓ p -direct sums,Λ ( ⊕ ℓ p ( B ∪ REG )) is the set of all operators of the form S ⊕ S ⊕ · · · ⊕ S k for S ∈ REG and S , . . . , S k ∈ B . So the fact that S belongs to Λ ( B ∪ REG )means that there exist S ∈ REG , S , . . . , S k ∈ B , spatial isometries U, V and ameasure space (Ω , m ) such that V ◦ ( S ⊕ · · · ⊕ S k ) ◦ V contains elements of theform ( g i , h i ) in its domain for all 1 ≤ i ≤ n and some h i ∈ L p (Ω , m ) and so that V ◦ ( S ⊕ · · · ⊕ S k ) ◦ V ( g i , h i ) = ( Sg i , h i ). We can of course replace (Ω , m ) byany standard measure space as this amounts to conjugating by another spatialisometry. So we have obtained the conclusion of the theorem. (cid:3) Appendix A. On the GL ( n, K ) invariant subspaces of the space ofhomogeneous functions Let 0 < p < ∞ . We recall some definition that already appeared in the bodyof the paper for p ≥ n be a positive integer. Denote by | z | the ℓ p -”norm” on K n | z | = ( | z | p + · · · + | z n | p ) p . A function ϕ : K n → R is called homogeneous of degree p if ϕ ( λz ) = | λ | p ϕ ( z )for all z ∈ K n and λ ∈ K . The space H n of real continuous homogeneous of degree p functions on K n is a Banach space for the topology of uniform convergence oncompact subsets on K n . A particular choice of norm is k ϕ k = sup | z |≤ | ϕ ( z ) | ,so that for this norm H n is isometrically isomorphic to the space of continuousfunctions on KP n − through the identification of ϕ ∈ H n with the function K z ∈ KP n − ϕ (cid:16) z | z | (cid:17) . For this identification, the natural action of GL n ( K )on H n corresponds to the action of GL n ( K ) on C ( KP n − ) given by A · ϕ ( K z ) = | A − z | p | z | p ϕ ( A − K z ) . Theorem A.1.
The GL n ( K ) -invariant closed subspaces of the Banach space H n of continuous p -homogeneous functions K n → R are • { } and H n if p is not an even integer. • { } , H n and the subspace of degree p homogeneous polynomials if p is aneven integer. DUALITY OPERATORS/BANACH SPACES 27
Remark
A.2 . This theorem allows to reprove the result [9] that if p is not an eveninteger, then every isometry between subspaces of L p spaces is a spatial isometry.Indeed, if T is such an isometry, n is an integer, f ∈ D ( T ) n and ϕ ( z ) = | z | p ,then we get for every A ∈ GL n ( K ) (with the notation of (4.2)) h µ f − µ T f , ϕ ◦ A i = k X j a ,j f j k p − k X j a ,j T f j k p = 0 . The linear form µ f − µ T f therefore vanishes on the GL n ( K )-invariant subspacespanned by { ϕ ◦ A, A ∈ GL n ( K ) } . By Theorem A.1 this subspace is dense, whichimplies that µ f − µ T f = 0. One concludes by Remark 4.11 that T is a spatialisometry.When p is an even integer, the same argument shows that if X is a Banachspace and x, y ∈ X are so that ( z , z )
7→ k z x + z y k p is not a polynomial in z , z , z , z (for example if X = K with the ℓ q norm for q which is not an evendivisor of p ), then every operator T between subspaces of L p spaces such that k T X k = k T − X k = 1 is a spatial isometry. In particular we have: Corollary A.3.
For any < p < ∞ (even integer or not) a linear map T between subspaces of L p spaces is a spatial isometry if and only if T is a regularisometry. Rudin’s proof in [24] relied on the Wiener Tauberian theorem. In the proof ofTheorem A.1, we shall need the following variant.
Proposition A.4.
Let f, g : R d → C be two measurable functions and C > such that | f ( x ) | ≤ C (1 + | x | ) p and | g ( x ) | ≤ C (1 + | x | ) − p − d − ) for all x ∈ R d .Assume that g ∗ f = 0 . Then the support of the tempered distribution ˆ f iscontained in { ξ ∈ R d , ˆ g ( ξ ) = 0 } .Proof. First observe that the assumption on g implies that g ∈ L ( R d ).If g belongs to D ( R d ) (the space of compactly supported C ∞ functions), thenthe proposition is easy : by taking Fourier transform we have ˆ g ˆ f = 0 (multiplica-tion of a distribution by a C ∞ function), from which the conclusion follows. Thestrategy will be to approximate g by compactly supported C ∞ functions.We have to prove that for every ξ ∈ R d with ˆ g ( ξ ) = 0, there is a neighbour-hood V of ξ such that h ˆ f , ϕ i = 0 for every ϕ ∈ D ( V ). By standard transla-tion/convolution/dilation arguments, we can assume that ξ = 0, g is C ∞ , andthat ˆ g does not vanish on the closure of B (0 , h ˆ f , ϕ i = 0for every ϕ ∈ D ( B (0 , ρ : R d → [0 ,
1] be a compactly supported C ∞ function, equal to 1 on B (0 , g n ∈ D ( R d ) by g n ( x ) = g ( x ) ρ ( xn ). Bythe dominated convergence theorem, k g n − g k L ( R d ) →
0, and so k ˆ g n − ˆ g k L ∞ → n such that ˆ g n does not vanish on B (0 ,
1) for all n ≥ n . Let ϕ ∈ D ( B (0 , ϕ ˆ g n belongs to D ( B (0 , h ˆ f , ϕ i = h ˆ g n ˆ f , ϕ ˆ g n i = h g n ∗ f, F − ( ϕ ˆ g n ) i where F − is the inverse Fourier transform. Using that g n ∗ f ( x ) = ( g n − g ) ∗ f ( x ) = O ( n (1 + | x | n ) p ) (this inequality will be explained below), we get(A.1) |h ˆ f , ϕ i| ≤ Cn Z (1 + | x | n ) p |F − ( ϕ ˆ g n ) | dx. To justify to domination of ( g n − g ) ∗ f ( x ) = R ( g n − g )( y ) f ( x − y ) dy , use that | ( g n − g )( y ) | . (1 + | y | ) − p − d − | y | >n and | f ( x − y ) | . (1 + | x − y | ) p . (1 + max( | x | , | y | )) p to obtain | f ∗ ( g − g n )( x ) | . Z | y | >n (1 + | y | ) − p − d − (1 + max( | x | , | y | )) p dy. If | x | ≤ n , then the preceding inequality becomes | f ∗ ( g − g n )( x ) | . Z | y | >n (1 + | y | ) − d − dy . n . If | x | ≥ n , then we cut the integral as R n< | y |≤| x | + R | x | < | y | and get | f ∗ ( g − g n )( x ) | . Z n< | y |≤| x | (1 + | x | ) p (1 + | y | ) p + d +1 dy + Z | y | > | x | (1 + | y | ) − d − dy . | x | p n p +1 + 1 | x | . | x | p n p +1 . This proves the announced inequality.In view of (A.1), we see that our goal is to prove good integrability propertieson the function F − ( ϕ ˆ g n ), i.e. good regularity properties of its Fourier transform ϕ ˆ g n . To achieve this, we denote by A ( R d ) the Fourier algebra of R d , i.e. theBanach space F ( L ( R d )) for the norm k h k A ( R d ) = kF − h k L ( R d ) . The inequality(A.2) k h h k A ( R d ) ≤ k h k A ( R d ) k h k A ( R d ) is the reason for the term “algebra” and is clear from the usual properties ofconvolution and Fourier transform. We have the following lemmas. Lemma A.5.
For every ϕ ∈ D ( B (0 , , there is a constant C = C ( ϕ ) such that ϕ ˆ g n belongs to A ( R d ) with norm ≤ C for all n ≥ n . Lemma A.6.
There is a constant C ′ such that D α ˆ g n belongs to A ( R d ) with norm ≤ C ′ for all n ∈ N and α ∈ N d , | α | < p + 1 . DUALITY OPERATORS/BANACH SPACES 29
These two lemmas, together with the Leibniz derivation rule and the fact that A ( R d ) is a Banach algebra (A.2), imply that, for every ϕ ∈ D ( B (0 , C such that D α ϕ ˆ g n belongs to A ( R d ) with norm less than C for all n ≥ n and α ∈ N d , | α | < p + 1. Therefore, for every such n and α we have Z | x α F − ( ϕ ˆ g n ) | dx ≤ C. This implies that, if k is the unique integer in the interval [ p, p + 1), then forevery n ≥ n Z (1 + | x | ) p F − ( ϕ ˆ g n ) | dx ≤ Z (1 + | x | ) k F − ( ϕ ˆ g n ) | dx ≤ C ′ . A fortiori, by (A.1) we have |h ˆ f , ϕ i| ≤ C ′ n , so making n → ∞ we obtain h ˆ f , ϕ i = 0. This concludes the proof. (cid:3) We have to prove the two lemmas used above.
Proof of Lemma A.5.
Let ρ ∈ D ( B (0 , ϕ . The fact that ρ ˆ g (and ρ ˆ g n for every n ≥ n ) belongs to A ( R d ) is essentially theWiener tauberian theorem. Indeed, the proof in [25, Theorem 9.3] shows that forevery x ∈ C such that ˆ g ( x ) = 0, there is ε > ρ ˆ g ∈ A ( R d ) for every ρ ∈ D ( B ( x, ε )). The claimed result follows by a partition of unity argument. Toobtain a bound on ρ ˆ g n independant from n , we write ϕ ˆ g n = ϕ ˆ g − ρ ˆ g (ˆ g − ˆ g n ) . Since ρ ˆ g belongs to A ( R d ) and k ˆ g − ˆ g n k A ( R d ) = k g − g n k L ( R d ) →
0, there is n ≥ n such that ρ ˆ g (ˆ g − ˆ g n ) has A ( R d )-norm less than for all n ≥ n . This implies thatfor n ≥ n ϕ ˆ g n = X k ≥ ϕ ˆ g (cid:18) ρ ˆ g (ˆ g − ˆ g n ) (cid:19) k belongs to A ( R d ) with norm less than 2 k ϕ ˆ g k A ( R d ) . The lemma follows with C = max(2 k ϕ ˆ g k A ( R d ) , max n ≤ n We have k D α ˆ g n k A ( R d ) = k x α g n k L ( R d ) ≤ k x α g k L ( R d ) because g n ( x ) = g ( x ) ρ ( xn ) and 0 ≤ ρ ≤ 1. The quantity k x α g k L ( R d ) is finitebecause g ( x ) = O ( | x | − p − d − ) and | α | < p + 1. (cid:3) We can now prove the main result on GL n ( K )-invariant subspaces of H n . Proof of Theorem A.1. For simplicity we write the proof for K = C . The realcase is similar, see Remark A.9. Let f ∈ H n be a nonzero function such that thespace spanned by the functions f ◦ A for A ∈ GL n ( C ) is not dense in H n . We willprove that p is an even integer and that f is a homogeneous polynomial. By theHahn-Banach theorem, there is a nonzero linear form ϕ on H n which vanisheson f ◦ A for all A . By the Riesz representation theorem, there is a uniquenonzero signed measure µ on CP n − such that ϕ ( f ) = R f ( z | z | ) dµ ( C z ). We canassume that µ is absolutely continuous with respect to the Lebesgue measure(=the unique U( n )-invariant probability measure) on CP n − , with a C ∞ Radon-Nykodym derivative. Indeed, if ρ is a C ∞ function on U( n ), then the measure ρ ∗ µ = R ( u ∗ µ ) ρ ( u ) du is absolutely continuous with respect to the Lebesguemeasure on CP n − , has a C ∞ density, and still satisfies R f ◦ A ( z | z | ) d ( h ∗ µ )( C z ) =0 for every A ∈ GL n ( C ). Moreover if ρ ≥ ρ ∗ µ = 0.So in particular, µ has a nonzero bounded Radon-Nykodym derivative h withrespect to the Lebesgue measure. By Lemma A.8 we can write Z CP n − F ( C z ) dµ ( z ) = Z C n − ( F h )( C (1 , z )) c (1 + | z | + . . . | z n − | ) n dz. Taking F ( C z ) = ( f ◦ A )( z | z | ), we get F ( C (1 , z )) = f ◦ A (1 ,z )1+ | z | p and(A.3) 0 = Z C n − f ◦ A (1 , z ) g ( z ) dz for the nonzero function g ( z ) = | z | p dµdλ ( C (1 , z )) c (1+ | z | ) n , which satisfies.(A.4) g ( z ) = O ((1 + | z | ) − p − n ) . Now if we take for A = (cid:18) b − A ′− (cid:19) for A ′ ∈ GL n − ( C ), then (A.3) becomes0 = Z f (1 , b − A ′− z ) g ( z ) dz = | detA ′ | Z ( g ◦ A ′ )( z ) f (1 , b − z ) dz. The second equality is a change of variable. In other words, if f : C n − → R isthe function f ( z ) = f (1 , z ), then f is a continuous function satisfying f ( z ) = O (1 + | z | p ) as z → ∞ , and such that ( g ◦ A ′ ) ∗ f = 0 for every A ′ ∈ GL n − ( C ).Viewing C n − as a real vector space R d with d = 2 n − 2, we see that weare in the setting of Proposition A.4 ( (A.4) indeed implies that ( g ◦ A ′ )( z ) = O ((1 + | z | ) − p − d − )). So the proposition implies that the support of ˆ f is containedin { ξ ∈ C n − , F ( g ◦ A ′ )( ξ ) = 0 } . But, g being nonzero, there exists ξ = 0 suchthat ˆ g ( ξ ) = 0. Since GL n − ( C ) acts transitively on C n − \ { } , we get that thesupport of ˆ f is contained in { } . This implies that f is a polynomial functionin z, z . So we have proved that (A.3) implies that the function z f (1 , z ) is a DUALITY OPERATORS/BANACH SPACES 31 polynomial function in z, z . But since (A.3) for f clearly implies (A.3) for f ◦ A for every A ∈ GL n ( C ), we get that z f ◦ A (1 , z ) is a polynomial for every A .This implies that p is an even integer and that f is a homogeneous polynomial,see Lemma A.7.This shows that if p is not an even integer, then { } and H n are the only closedGL n ( C )-invariant subspaces of H n , and that otherwise all other invariant closedsubspaces are contained in the space of degree p homogeneous polynomials. Itremains to show that for every nonzero degree p homogeneous polynomial, everyother such polynomial belongs to the linear space spanned by its GL n ( C ) orbit.This is not difficult. (cid:3) Lemma A.7. Let f ∈ H n be a nonzero function such that, for every A ∈ GL n ( C ) , z ∈ C n − f ◦ A (1 , z ) is a polynomial in z, z . Then p is an eveninteger and f is a homogeneous polynomial of degree p .Proof. Let P ∈ C [ X , . . . , X n − ] such that f (1 , z ) = P ( z, z ). Using that f ∈ H n , we have that | P ( z, z ) | = O ((1 + | z | ) p ), and in particular deg( P ) ≤ p , so wecan write P ( z, z ) = X α,β ∈ N d , | α | + | β |≤ p a α,β z α z β . Let c ∈ C n − and A = (cid:18) c ∗ (cid:19) . Similarly there is P c ∈ C [ X , . . . , X n − ] ofdegree ≤ p such that f ◦ A (1 , z ) = P c ( z ). Then P c ( z ) = | h z, c i| p f (1 , z h z, c i ) = | h z, c i| p P ( z h z, c i , z h z, c i ) . We can rewrite this quantity as X α,β a α,β (1 + h z, c i ) p −| α | (1 + h z, c i ) p −| β | z α z β . By expanding (1 + t ) l = P n ≥ (cid:0) ln (cid:1) t n , for small z the preceding sum is X α,β,n,m a α,β (cid:18) p − | α | n (cid:19)(cid:18) p − | β | m (cid:19) h z, c i n h z, c i m z α z β . Since P c is a polynomial of degree ≤ p , we get that for every N > p , X | α | + | β | + n + m = N a α,β (cid:18) p − | α | n (cid:19)(cid:18) p − | β | m (cid:19) h z, c i n h z, c i m z α z β = 0 . Since this is valid for every c , we get a α,β (cid:18) p − | α | n (cid:19)(cid:18) p − | β | m (cid:19) = 0for every α, β ∈ N d and n, n ∈ N such that | α | + | β | + n + m > p . Let α, β such that a α,β = 0 (such α, β exist by the assumption that f isnonzero). Then taking n = 0 and m very large, we find that (cid:0) p −| β | m (cid:1) = 0, whichimplies that p − | β | is a nonnegative integer. Similarly p − | α | is a nonnegativeinteger. This proves that p is an even integer and f (1 , z ) = X | α | , | β |≤ p a α,β z α z β . By homogeneity we get f ( z , z ) = X | α | , | β |≤ p z p −| α | z α z p −| β | z β . This is the lemma. (cid:3) Lemma A.8. The Lebesgue measure λ on CP n − is given by Z CP n − F ( C z ) dλ ( z ) = c Z C n − F ( C (1 , z )) 1(1 + | z | + · · · + | z n − | ) n dz for some number c > .Proof. It is a change of variable to compute that the finite measure F ∈ C ( CP n − ) Z C n − F ( C (1 , z )) 1(1 + | z | + · · · + | z n − | ) n dz is invariant by U( n ). (cid:3) Remark A.9 . We did not use the full strength of Proposition A.4 for K = C ,as we used it for a function g satisfying g ( z ) = O ((1 + | z | ) − p − d − ), which isstrictly stronger that the required g ( z ) = O ((1 + | z | ) − p − d − ). The reason for this2 is that the real dimension drops by 2 between C n and CP n − . 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