First steps in the geography of scale Hilbert structures
aa r X i v : . [ m a t h . S G ] O c t First steps in the geography of scale Hilbertstructures
Urs Frauenfelder ∗ May 29, 2018
Abstract
Scale structures were introduced by H. Hofer, K. Wysocki, and E. Zehnder.In this note we define an invariant for scale Hilbert spaces modulo scaleisomorphism and use it to distinguish large classes of scale Hilbert spaces.
Contents
Scale structures on a Banach space were introduced by H. Hofer, K. Wysocki,and E. Zehnder, see [1, 2, 3]. They observed that on a scale Banach space a newnotion of smoothness can be defined which still meets the chain rule. Thereforescale structure give rise to new smooth structures in infinite dimensions. Mani-folds modelled on this new smooth structures are called scale manifolds. Theseprovide the first step in the construction of polyfolds which in turn can be usedto deal with transversality issues in Symplectic Field theory, Gromov-Wittentheory, or Floer theory. The author’s interest in this new smooth structuresin infinite dimensions is based on the following guiding principle. He believesthat the various Floer homologies should be interpretable as
Morse homology onscale manifolds . Such a unified framework would lead to various simplificationsof the existing theory, since gluing and transversality issues could be referred tothe general set-up currently developed by H. Hofer, K. Wysocki, and E. Zehnder,and have not be checked anymore individually for each Floer homology. ∗ Department of Mathematics and Research Institute of Mathematics, Seoul National Uni-versity
1n this note we introduce a first invariant to distinguish different Hilbertscale structures and we construct various examples of nonisomorphic Hilbertscale structures. The restriction to the Hilbert case instead of the more generalBanach case is justified by our intension to apply scale structures to Floerhomology. In Floer homology one need to have metrics since one has to beable to define a gradient.We first recall the definition of a Hilbert scale structure which is due toH. Hofer, K. Wysocki, and E. Zehnder.
Definition 1.1 A scale Hilbert space is a tuple H = (cid:8)(cid:0) H k , h· , ·i k (cid:1)(cid:9) k ∈ N where for each k ∈ N the pair (cid:0) H k , h· , ·i k (cid:1) is a real Hilbert space and the vectorspaces H k build a nested sequence H = H ⊃ H ⊃ H ⊃ . . . such that thefollowing two axioms hold. (i) For each k ∈ N the inclusion (cid:0) H k , h· , ·i k (cid:1) ֒ → (cid:0) H k − , h· , ·i k − (cid:1) is compact. (ii) For each k ∈ N the subspace H ∞ = T ∞ n =0 H n is dense in H k with respectto the topology induced from h· , ·i k . Remark 1.2 If H is finite dimensional, then the second axiom in Definition 1.1implies that H k = H for every k ∈ N . On the other if H is infinite dimensional,then the first axiom implies that H k = H for every k ∈ N .We next introduce the notion of isomorphism between two scale Hilbert spaces.Hence suppose that H = { ( H k , h· , ·i k ) } and H ′ = { ( H ′ k , h· , ·i ′ k ) } are two scaleHilbert spaces. We denote for k ∈ N by || · || k and || · || ′ k the norms on H k ,respectively H ′ k , induced from the scalar products h· , ·i k and h· , ·i ′ k . Definition 1.3 A scale isomorphism Φ from H to H ′ is a bijective linear map Φ : H → H ′ satisfying the following two axioms. (i) For each k ∈ N the map Φ restricts to a bijection Φ k : H k → H ′ k , Φ k = Φ | H k . (ii) For each k ∈ N there exists a constant c k > , such that c k || h || k ≤ || Φ( h ) || ′ k ≤ c k || h || k , ∀ h ∈ H k . Definition 1.4
Two scale Hilbert spaces H and H ′ are called scale isomorphic ,if there exists a scale isomorphism from H to H ′ . S = (cid:8) H scale Hilbert space , dim( H ) = ∞ (cid:9)(cid:14) ∼ where the equivalence relation is given by scale isomorphism. Geography ofHilbert scale structures refers to the description of the set S .To construct examples as well as invariants for scale Hilbert spaces we in-troduce the notion of a scale Hilbert n -tuple for n ∈ N . Definition 1.5 A scale Hilbert n -tuple is a tuple H = (cid:8)(cid:0) H k , h· , ·i k (cid:1)(cid:9) ≤ k ≤ n − where for each k ∈ { , . . . , n − } the pair (cid:0) H k , h· , ·i k (cid:1) is a real Hilbert space andthe vector spaces H k build a nested sequence H = H ⊃ H ⊃ H ⊃ . . . ⊃ H n − such that the following two axioms hold. (i) For each k ∈ { , . . . , n − } the inclusion (cid:0) H k , h· , ·i k (cid:1) ֒ → (cid:0) H k − , h· , ·i k − (cid:1) iscompact. (ii) For each k ∈ { , . . . , n − } the subspace H n − is dense in H k with respectto the topology induced from h· , ·i k .We refer to scale Hilbert -tuples as scale Hilbert pairs and to scale Hilbert -tuples as scale Hilbert triples . The notion of scale isomorphism between scale Hilbert n -tuples is the sameas the one for scale Hilbert spaces and two Hilbert n -tuples are called scaleisomorphic if there exists a scale isomorphism between them. We next introducefor each n ∈ N the set S n = (cid:8) H scale Hilbert n -tuple , dim( H ) = ∞ (cid:9)(cid:14) ∼ . We further denote by e F the space of all functions f : N → (0 , ∞ ) which aremonotone and unbounded. We say that f , f ∈ e F are equivalent if there exists c > c f ( n ) ≤ f ( n ) ≤ cf ( n ) , n ∈ N and we write f ∼ f for equivalent functions. We introduce the quotient F = e F / ∼ . By ℓ we denote as usual the Hilbert space of all square summable sequencestogether with its standard inner product. For f ∈ e F we define ℓ f ⊂ ℓ to bethe vector space of all sequences x = ( x , x , . . . ) satisfying || x || ℓ f = vuut ∞ X n =1 f ( n ) x n < ∞ . h x, y i ℓ f = ∞ X n =1 f ( n ) x n y n , x, y ∈ ℓ f endows ℓ f with the structure of a Hilbert space. We are now in position to stateour first main result. Theorem A
There is a bijection between the sets F and S given by themap [ f ] [( ℓ , ℓ f )] . As an immediate consequence of Theorem A we obtain the following Corollary.
Corollary 1.6
Assume that H = { ( H k , h· , ·i k ) } is a scale Hilbert space, thenfor every k ∈ N the Hilbert space ( H k , h· , ·i k ) is separable. Remark 1.7
Since every separable Hilbert space is actually isometric to ℓ , itfollows that in a scale Hilbert space all Hilbert spaces are isometric to each otherand the geography question for scale Hilbert spaces is reduced to the questionhow these infinitely many ℓ -spaces can be nested into each other.Theorem A can be used to define invariants for scale Hilbert spaces modulo theequivalence relation given by scale isomorphism. Let ∆ ⊂ N × N be the set∆ = (cid:8) ( i, j ) ∈ N × N : i < j (cid:9) . Denote by J : F → S the bijection [ f ] [( ℓ , ℓ f )] given by Theorem A. Nowwe introduce the map K : S → Map(∆ , F )which is given for an infinite dimensional scale Hilbert space H = { ( H k , h· , ·i k ) } by the formula K ([ H ])( i, j ) = J − (cid:16)(cid:2)(cid:0) H i , h· , ·i i (cid:1) , (cid:0) H j , h· , ·i j (cid:1)(cid:3)(cid:17) , ( i, j ) ∈ ∆ . (1)The same kind of invariant can also be used for scale Hilbert n -tuples for every n ∈ N satisfying n ≥
2. Indeed, let ∆ n ⊂ { , . . . , n − } × { , . . . , n − } be theset ∆ n = (cid:8) ( i, j ) ⊂ N × N : i < j < n (cid:9) . Then we define the map K n : S n → Map(∆ n , F )by the same formula (1) as before. That the invariants K and K n are well defined,i.e. independent of the choice of the representative H is a further Corollary ofTheorem A. 4 orollary 1.8 The maps K and K n for n = { , , . . . } are well defined. We can furthermore use Theorem A to construct a class of examples of scaleHilbert spaces which are not scale isomorphic to each other. We first introducethe set e F = Map( N , e F ) . We say that F , F ∈ e F are equivalent, if F ( k ) is equivalent to F ( k ) in e F forevery k ∈ N . We write again F ∼ F for equivalent F , F ∈ e F and set F = e F / ∼ . For F ∈ e F we introduce the nested sequence of Hilbert spaces ℓ ,F = ℓ ,F ⊃ ℓ ,F ⊃ ℓ ,F ⊃ . . . where we set ℓ ,F = ℓ and for each k ∈ N ℓ ,Fk = ℓ Q kj =1 F ( j ) where the product of two functions f , f ∈ e F is defined pointwise as( f · f )( ν ) = f ( ν ) · f ( ν ) , ν ∈ N . An analogous procedure gives us finite nested sequences of Hilbert spaces. For n ∈ N satisfying n ≥ e F n = Map (cid:16) { , . . . n − } , e F (cid:17) . Defining the equivalence relation pointwise as before we set F n = e F n / ∼ and for F ∈ e F n we introduce again the now finite nested sequence of Hilbertspaces ℓ ,F = ℓ ,F ⊃ ℓ ,F ⊃ ℓ ,F ⊃ . . . ⊃ ℓ ,Fn − . As a second Corollary of Theorem A we can draw the following assertion.
Corollary 1.9
For each n ∈ N satisfying n ≥ there is an injective map J n : F n → S n , [ F ] [ ℓ ,F ] . Moreover, there is an injective map J : F → S given by the same formula. Remark 1.10
For n = 2 the above Corollary is just a special case of Theorem Aif one uses the canonical identification of F with F given by [ F ] [ F (1)]. Inthis case the map J is actually surjective. So one might wonder if this continuesto hold for larger n ∈ N . However, surjectivity of J n actually already fails for n = 3 which is the content of Corollary 1.12.5rom Corollary 1.9 we obtain some information about which values of the in-variant K are realizable by scale Hilbert spaces. Given F ∈ e F the invariant K ([ ℓ ,F ]) satisfies for ( i, j ) ∈ ∆ K ([ ℓ ,F ])( i, j ) = " j Y k = i +1 F ( k ) . In particular, by noting that there is a well defined product in F which is givenfor [ f ] , [ f ] ∈ F by [ f ] · [ f ] = [ f · f ] we obtain the relations K ([ ℓ ,F ])( i, j ) = j − Y k = i K ([ ℓ ,F ])( k, k + 1) . We define an embedding ι : F → Map(∆ , F )which is given for F ∈ F by the formula ι ( F )( i, j ) = j Y k = i +1 F ( k ) , ( i, j ) ∈ ∆ . By the same formula we define also an embedding ι n : F n → Map(∆ n , F ) . As a Corollary of Corollary 1.9 we obtain the following statement.
Corollary 1.11
Every A ∈ ι ( F ) ⊂ Map(∆ , F ) is realizable as the invariant ofa scale Hilbert space, i.e. there exists a scale Hilbert space H such that K ([ H ]) = A. Similarly, every A ∈ ι n ( F n ) is realizable as the invariant of a scale Hilbert n -tuple. Our second main result deals with the question if there are other invariants than ι ( F ) which can be realized by scale Hilbert spaces. It actually deals with thequestion of scale Hilbert triples, but these can be used to construct new scaleHilbert spaces. For triples the set ∆ has just cardinality three, namely∆ = { (0 , , (1 , , (0 , } . We identify Map(∆ , F ) with F via the identificationMap(∆ , F ) → F , F (cid:0) F (0 , , F (1 , , F (0 , (cid:1) . For given φ , φ ∈ F we introduce the set B ( φ , φ ) = (cid:8) φ ∈ F : ∃ H ∈ S , K ( H ) = ( φ , φ , φ ) (cid:9) .
6e know from Corollary 1.11 that φ · φ ∈ B ( φ , φ )so that in particular, B ( φ , φ ) is not empty. Our second main result is thefollowing Theorem. Theorem B
For any φ , φ ∈ F , the set B ( φ , φ ) has infinite cardinality. As a Corollary of Theorem B we get the following assertion which we alreadydiscussed in Remark 1.10.
Corollary 1.12
The map J : F → S is not surjective, as neither the maps J n : F n → S n for every n ≥ . The proof of Theorem A is based on three Lemmas which we prove first.
Lemma 2.1 If f ∈ e F , then the tuple ( ℓ , ℓ f ) is a scale Hilbert pair. Proof:
Abbreviate by I : ℓ f → ℓ the inclusion. We first observe that theinclusion is continuous. Indeed, by the assumption that f is monotone weobtain || Ix || ℓ ≤ p f (1) || x || ℓ f . To show that I is compact, we denote for n ∈ N byΠ n : ℓ → ℓ the orthogonal projection of a sequence to its first n entries,( x , . . . , x n , x n +1 , . . . ) ( x , . . . x n , , . . . ) . The operators I n = Π n ◦ I : ℓ f → ℓ have finite dimensional image and are therefore compact. Since f is monotonewe obtain || I − I n || L ( ℓ f ,ℓ ) = 1 p f ( n + 1)where || · || L denotes the operator norm. Because f is unbounded, we concludethat I is the uniform limit of compact operators and therefore itself compact.This proves condition (i) in the Definition of a scale Hilbert pair.It remains to check condition (ii) in the Definition of a scale Hilbert pair, i.e. that ℓ f is dense in ℓ . To see that let x ∈ ℓ and define x n = Π n x for n ∈ N . Wenote that x n ∈ ℓ f and the sequence x n converges to x in the ℓ -norm as n goesto infinity. This proves (ii) and hence the Lemma. (cid:3) emma 2.2 Assume that f , f ∈ e F . Then the scale Hilbert pairs ( ℓ , ℓ f ) and ( ℓ , ℓ f ) are scale isomorphic iff f ∼ f . Proof:
We first prove the implication ” ⇒ ”. Assume that ( ℓ , ℓ f ) and( ℓ , ℓ f ) are scale isomorphic. Then there exists a scale isomorphismΦ : ( ℓ , ℓ f ) → ( ℓ , ℓ f )with inverse Ψ : ( ℓ , ℓ f ) → ( ℓ , ℓ f ) . We abbreviate c = max n || Φ || L ( ℓ ,ℓ ) , || Φ || L ( ℓ f ,ℓ f ) , || Ψ || L ( ℓ ,ℓ ) , || Ψ || L ( ℓ f ,ℓ f ) o where || · || L is the operator norm. As in the proof of Lemma 2.1 we denote for n ∈ N by Π n : ℓ → ℓ the orthogonal projection of a sequence to its first n entries,( x , . . . , x n , x n +1 , . . . ) ( x , . . . , x n , , . . . ) . For n, m ∈ N we introduce the map A nm : Π n ℓ → Π n ℓ , A nm = Π n ◦ Ψ ◦ Π m − ◦ Φ . We first prove the following Claim.
Claim:
Assume that n, m ∈ N satisfy f ( n ) < f ( m ) c , then the map A nm isa bijection. To prove the Claim assume that ξ is in the kernel of A nm , i.e. ξ ∈ ℓ satis-fies Π n ξ = ξ, A nm ξ = 0 . It follows that ξ = Π n ξ = Π n ΨΦ ξ = Π n Ψ(id − Π m − )Φ ξ. We estimate || ξ || ℓ = || Π n Ψ(id − Π m − )Φ ξ || ℓ ≤ c || (id − Π m − )Φ ξ || ℓ ≤ c p f ( m ) || (id − Π m − )Φ ξ || ℓ f ≤ c p f ( m ) || ξ || ℓ f ≤ c p f ( n ) p f ( m ) || ξ || ℓ . c p f ( n ) p f ( m ) < || ξ || ℓ = 0and hence ξ = 0 . This proves that A nm is injective and since it is an endomorphism of a finitedimensional vector space we conclude that A nm is a bijection. This finishes theproof of the Claim.We next show how the Claim can be used to deduce the implication ” ⇒ ”of the Lemma. We continue assuming the hypothesis of the Claim. Since themap A nm factors as A nm = (Π n ◦ Ψ)(Π m − ◦ Φ)and A nm is bijective by the Claim, we deduce that Π n ◦ Ψ | Π m − ℓ is surjective.Hence we obtain n = dim (cid:0) Π n ℓ (cid:1) = dim (cid:0) imΠ n Ψ | Π m − ℓ (cid:1) ≤ dim (cid:0) Π m − ℓ (cid:1) = m − < m. Hence we have shown the implication f ( n ) < f ( m ) c = ⇒ n < m. We conclude from this that the inequality f ( n ) ≥ f ( n ) c has to hold for each n ∈ N . Interchanging the roles of Ψ and Φ we obtain thereverse inequality f ( n ) ≥ f ( n ) c . This proves that f and f are equivalent.We are left with showing the inverse implication ” ⇐ ” of the Lemma. Hence weassume that f ∼ f . But under this assumption id | ℓ gives a scale isomorphismbetween ( ℓ , ℓ f ) and ( ℓ , ℓ f ). Hence ( ℓ , ℓ f ) and ( ℓ , ℓ f ) are scale isomorphic.This finishes the proof of the Lemma. (cid:3) Before stating the next Lemma we first introduce another equivalence relationfor scale Hilbert spaces different from scale isomorphism.9 efinition 2.3 A scale isometry Φ from a scale Hilbert space H = { ( H k , h· , ·i k ) } to a scale Hilbert space H ′ = { ( H ′ k , h· , ·i k ) } is a linear map Φ : H → H ′ whichrestricts for all k ∈ N to an isometry Φ | k : H k → H ′ k . Two scale Hilbert spacesare called scale isometric , if there exists a scale isometry between them. Note that a scale isometry is a special case of a scale isomorphism, so that twoscale isometric scale Hilbert spaces are also scale isomorphic. Moreover, thesame definition also applies to scale Hilbert n -tuples for any n ∈ N . Lemma 2.4
Let ( H, W ) be an infinite dimensional scale Hilbert pair. Thenthere exists a unique f ∈ e F such that ( H, W ) is scale isometric to ( ℓ , ℓ f ) . Proof:
By the Riesz representation theorem there exists a bounded linearoperator A : W → W such that h w , w i H = h w , Aw i W , w , w ∈ W. The operator A is symmetric and we next show that it is compact. Choose asequence w ν in the unit ball of W , i.e. || w ν || W ≤ , ν ∈ N . Since the inclusion
W ֒ → H is compact we deduce that w ν has a convergentsubsequence w ν j in H . In particular, w ν j is a Cauchy sequence in H . We claimthat Aw ν j is a Cauchy sequence in W . Denote by || A || > A : W → W . Since w ν j is a Cauchy sequence in W there exists for given ǫ > j = j ( ǫ ) ∈ N such that || w ν j − w ν j ′ || H ≤ ǫ p || A || , j, j ′ ≥ j . We further abbreviate v = w ν j − w ν j ′ . We estimate 0 ≤ (cid:28) v − || A || Av, v − || A || Av (cid:29) H = || v || H − || A || h Av, v i H + 1 || A || h Av, Av i H = || v || H − || A || h Av, Av i W + 1 || A || h Av, A v i W ≤ ǫ || A || − || A || || Av || W + 1 || A || || Av || W || A v || W ≤ ǫ || A || − || A || || Av || W + 1 || A || || Av || W = ǫ || A || − || A || || Av || W || Aw ν j − Aw ν j ′ || W = || Av || W ≤ ǫ. This proves that Aw ν j is a Cauchy sequence in W and since W is complete ithas to converge. We deduce that A is a compact operator.We next apply the spectral theorem to the compact symmetric operator A .Since A is further positive we conclude that there exists an orthogonal Schauderbasis { e n } n ∈ N of W with the following properties. (i) For each n ∈ N the vector e n is an eigenvector of A to a real eigenvalue λ n > (ii) The eigenvalues λ n build a monotone decreasing sequence.Since { e n } n ∈ N is an orthogonal Schauder basis of W we can represent each w ∈ W in the form w = ∞ X n =1 x n e n , x = ( x , x , · · · ) ∈ ℓ . (2)We next construct an orthogonal basis for H . For n ∈ N define¯ e n := 1 √ λ n e n . Denoting for n, m ∈ N by δ mn the Kronecker symbol we compute h ¯ e n , ¯ e m i H = h ¯ e n , A ¯ e m i W = λ m √ λ m λ n h e n , e m i W = λ m √ λ m λ n δ mn = δ mn and hence the vectors ¯ e n are orthogonal to each other. To see that they form aSchauder basis of H define H ′ = span(¯ e , ¯ e , · · · ) ||·|| H to be the || · || H -closure of the vector space spanned by { ¯ e n } n ∈ N . We observethat H ′ is a closed subspace of H and W is dense in H ′ . Since W is dense in H by assumption we conclude that H ′ = H. We now define an isometry Φ : H → ℓ in the following way. By the reasoning above each element h ∈ H has a uniquerepresentation h = ∞ X n =1 y n ¯ e n , y = ( y , y , . . . ) ∈ ℓ h ) = y. We next study the restriction of Φ to W . Define f ∈ e F by f ( n ) = 1 λ n , n ∈ N . Since λ n is a monotone decreasing zero sequence, the function f is actuallymonotone and unbounded. We claim that the restriction of Φ to W gives anisometry Φ | W : W → ℓ f ⊂ ℓ . To prove that assertion let w , w ∈ W . By (2) there exist x = ( x , x , · · · ) ∈ ℓ and x = ( x , x , · · · ) ∈ ℓ such that for i ∈ { , } we have w i = ∞ X n =1 x in e n = ∞ X n =1 p λ n x in ¯ e n . In particular, we get Φ( w i ) = ( √ λ x i , √ λ x i , · · · ) . Hence we compute h Φ( w ) , Φ( w ) i ℓ f = ∞ X n =1 f ( n ) λ n x n x n = ∞ X n =1 x n x n = h w , w i W . This proves that Φ | W interchanges the two scalar products. In particular, Φ | W is injective. To see that it is surjective we note that if y = ( y , y , · · · ) ∈ ℓ f ,then (cid:18) y √ λ , y √ λ , · · · (cid:19) ∈ ℓ and hence w = ∞ X n =1 y n √ λ n e n ∈ W. But Φ( w ) = y which shows that Φ | W : W → ℓ f is surjective. This finishes the proof that Φ | W is an isometry from W to ℓ f . In particular,Φ : ( H, W ) → ( ℓ , ℓ f )defines a scale isometry.It finally remains to show that f ∈ e F is unique with this property. To seethis assume that f , f ∈ e F such that there exist scale isometriesΦ : ( H, W ) → ( ℓ , ℓ f ) , Φ : ( H, W ) → ( ℓ , ℓ f ) . ◦ Φ − : ( ℓ , ℓ f ) → ( ℓ , ℓ f )is also a scale isometry. Let { ε n } n ∈ N be the standard basis of ℓ given by ε n = ( δ n , δ n , · · · ) . For f ∈ e F and n ∈ N we set ε fn = 1 p f ( n ) ε n for the standard ℓ -basis of ℓ f . We further define by A f : ℓ f → ℓ f the linear map which is given on basis vectors by A f ε fn = 1 f ( n ) ε fn . (3)With this convention we have for vectors w , w ∈ ℓ f the equality h w , w i ℓ = h w , A f w i ℓ f . Now using that Ψ is a scale isometry we compute for w , w ∈ ℓ f (cid:10) Ψ w , A f Ψ w (cid:11) ℓ f = (cid:10) Ψ w , Ψ w (cid:11) ℓ = (cid:10) w , w (cid:11) ℓ = (cid:10) w , A f w (cid:11) ℓ f = (cid:10) Ψ w , Ψ A f w (cid:11) ℓ f implying that A f Ψ = Ψ A f . This shows that A f and A f have the same eigenvalues and by (3) we deducethe following equality of sets { f ( n ) : n ∈ N } = { f ( n ) : n ∈ N } . Since f and f are monotone we get f ( n ) = f ( n ) , n ∈ N . This proves the uniqueness part and hence the Lemma follows. (cid:3)
Proof of Theorem A:
By Lemma 2.1 the map b J : e F → S , f [( ℓ , ℓ f )]13s well defined. By Lemma 2.2 this map induces a map J : F → S . By Lemma 2.4 the map J is surjective and again by Lemma 2.2 it is also in-jective. This finishes the proof of Theorem A. (cid:3) Proof of Corollary 1.6:
Note that { ( H k , h· , ·i k ) , ( H k − , h· , ·i k − ) } is a scaleHilbert pair. Hence Theorem A implies the Corollary. (cid:3) Proof of Corollary 1.8:
Assume that we have given two infinite dimensionalscale Hilbert spaces H = { ( H k , h· , ·i k ) } and H ′ = { ( H ′ k , h· , ·i ′ k ) } which are scaleisomorphic to each other. In particular, there exists a scale isomorphismΦ : H → H ′ . If ( i, j ) ∈ ∆ then by restricting Φ we obtain a scale isomorphism for scale Hilbertpairs Φ i,j : (cid:0) ( H i , h· , ·i i ) , ( H j , h· , ·i j ) (cid:1) → (cid:0) ( H ′ i , h· , ·i ′ i ) , ( H ′ j , h· , ·i ′ j ) (cid:1) . In particular, we conclude (cid:2) ( H i , h· , ·i i ) , ( H j , h· , ·i j ) (cid:3) = (cid:2) ( H ′ i , h· , ·i ′ i ) , ( H ′ j , h· , ·i ′ j ) (cid:3) ∈ S . Hence the map K is well defined, since the map J is well defined by Theorem A.The same reasoning also applies to K n for n an integer greater than 1. Thisfinishes the proof of the Corollary. (cid:3) Proof of Corollary 1.9:
For the proof of the Corollary we use the con-vention S ∞ = S , F ∞ = F , J ∞ = J , K ∞ = K and we assume that n ∈ { , , . . . , ∞} . We first show that the map J n iswell defined, i.e. that [ ℓ ,F ] ∈ S n for every F ∈ e F n . We claim that if k is apositive integer less than n that the inclusion ℓ ,Fk ֒ → ℓ ,Fk − is compact. If k = 1this inclusion corresponds to the inclusion ℓ F (1) ֒ → ℓ which is compact byLemma 2.1. Now assume that k >
1. Let { ε ν } ν ∈ N be the standard orthogonalbasis of ℓ . Let { e ε ν } ν ∈ N the orthogonal basis of ℓ ,Fk − = ℓ Q k − j =1 F ( j ) defined by e ε ν = 1 qQ k − j =1 F ( j )( ν ) ε ν . Denote by I : ℓ ,Fk − → ℓ the isometry which is given on basis vectors by I ( e ε ν ) = ε ν , ν ∈ N . I to ℓ ,Fk gives an isometry I | ℓ ,Fk : ℓ ,Fk → ℓ F ( k ) . Hence we conclude that the pair ( ℓ ,Fk − , ℓ ,Fk ) is scale isometric to the pair( ℓ , ℓ F ( k ) ) and the compactness of the embedding follows again from Lemma 2.1.We next show that the intersection T n − j =0 ℓ ,Fj is dense in ℓ ,Fk for every nonneg-ative integer k less than n . To see that let f = span { ε ν : ν ∈ N } be the subspace of ℓ consisting of finite linear combinations of the standardbasis vectors of ℓ . We note that f ⊂ ℓ ,Fk is a dense subspace for every nonnegative integer k less than n . In particular, f ⊂ n − \ j =0 ℓ ,Fj . This shows that T n − j =0 ℓ ,Fj is dense in ℓ ,Fk . We conclude that [ ℓ ,F ] ∈ S n andhence the map J n is well defined.We are left with showing injectivity of the map J n . Hence assume that F , F ∈ e F n such that [ F ] = [ F ] ∈ F . This implies that there exists a positive integer k less than n such that[ F ( k )] = [ F ( k )] ∈ F . We noted before that the pairs ( ℓ ,F i k − , ℓ ,F i k ) and ( ℓ , ℓ F i ( k ) ) are scale isometricfor i ∈ { , } . In particular, these pairs are scale isomorphic so that we obtain J − (cid:0) ℓ ,F i k − , ℓ ,F i k (cid:1) = [ F i ( k )] , i ∈ { , } . Combining the above two facts we conclude K ([ ℓ ,F ])( k − , k ) = [ F ( k )] = [ F ( k )] = K ([ ℓ ,F ])( k − , k ) . In particular, K ([ ℓ ,F ]) = K ([ ℓ ,F ])implying that [ ℓ ,F ] = [ ℓ ,F ]which proves that J n is injective. This finishes the proof of the Corollary. (cid:3) Proof of Corollary 1.11: If A ∈ ι ( F ), then there exists F ∈ e F such that A = ι ([ F ]) . H = ℓ ,F . Then K ([ H ]) = A. The same reasoning applies to scale Hilbert n -tuples. (cid:3) Remark 2.5
The uniqueness statement in Lemma 2.4 was actually not usedin the proof of Theorem A. However, it can be used to describe the set of scaleHilbert pairs modulo the equivalence given by scale isometry instead of scaleisomorphism. Introduce the set f S = (cid:8) H scale Hilbert pair , dim( H ) = ∞} / ∼ ′ where ∼ ′ is the equivalence relation given by scale isometry. Then the map e J : e F → f S , f ( ℓ , ℓ f )gives a bijection between f S and e F . The proof of Theorem B roots on the following idea. Choose representatives f , f ∈ e F for φ respectively φ . The separable Hilbert space ℓ f is isometricto ℓ and we have a canonical isometry I : ℓ f → ℓ . Let Φ : ℓ → ℓ be an isometry of ℓ to itself. Now consider the scale Hilbert triple H = (cid:0) ℓ , ℓ f , I − Φ( ℓ f ) (cid:1) . Applying Φ − ◦ I to ℓ f we get a scale isomorphismΦ − ◦ I : (cid:0) ℓ f , I − Φ( ℓ f ) (cid:1) → (cid:0) ℓ , ℓ f (cid:1) . In particular, K ([ H ])(1 ,
2) = [ f ] = φ . Moreover, we have K ([ H ])(0 ,
1) = [ f ] = φ . On the other hand K ([ H ])(0 ,
2) depends on Φ and we show that by varying Φwe can achieve infinitely many values for K ([ H ])(0 ,
2) in the set F .16e now start with the preparations for the proof of Theorem B. We denoteby U the set of all functions u : N → (0 , ∞ ) satisfying lim n →∞ u ( n ) = ∞ .Obviously, e F ⊂ U . We further introduce S = { σ : N → N : σ bijective } the group of permutations of N . Lemma 3.1
The group S acts on U by σ ∗ u ( n ) = u ( σ ( n )) , σ ∈ S , u ∈ U , n ∈ N . Proof:
We prove that the action is well defined, i.e. that σ ∗ u ∈ U . We have toshow that lim n →∞ σ ∗ u ( n ) = ∞ . (4)Pick r ∈ R . Since u ∈ U there exists n = n ( r ) such that u ( n ) ≥ r, ∀ n ≥ n . (5)Since σ is bijective the set { n ∈ N : σ − ( n ) < n } is finite. Hence we can set N := max { n ∈ N : σ − ( n ) < n } . In particular, we have the implication n ≥ N = ⇒ σ ( n ) ≥ n . Hence using (5) we conclude σ ∗ u ( n ) = u ( σ ( n )) ≥ r, ∀ n ≥ N . This proves (4) and hence the Lemma. (cid:3)
Lemma 3.2 If u ∈ U there exists σ ∈ S such that σ ∗ u ∈ e F . Moreover, if σ ′ ∈ S is another element with this property, than σ ∗ u = σ ′∗ u . Remark 3.3
Although σ ∗ u in Lemma 3.2 is canonical, the permutation σ neednot be. It is only canonical if σ ∗ u is strictly monotone. Proof of Lemma 3.2:
Pick u ∈ U . We first note that since u converges toinfinity it follows that for each finite subset B ⊂ N the infimum of the restrictionof u to N \ B is attained so that we are allowed to put a B := min (cid:8) u ( n ) : n ∈ N \ B (cid:9) . We set B = ∅ k ∈ N a k := a B k − , σ ( k ) := min (cid:8) n ∈ N \ B k − : u ( n ) = a k (cid:9) , B k := B k − ∪ { σ ( k ) } . We claim that σ ∈ S , (6)i.e. that σ is bijective. We first show injectivity. We assume by contradictionthat there exist k, k ′ ∈ N such that σ ( k ) = σ ( k ′ ) , k = k ′ . We can assume without loss of generality that k < k ′ . It follows from the definition of B k that B k = { σ ( j ) : 1 ≤ j ≤ k } . We deduce from the definition of σ ( k ′ ) that σ ( k ′ ) ∈ N \ { σ ( j ) : 1 ≤ j ≤ k ′ − } ⊂ N \ { σ ( k ) } = N \ { σ ( k ′ ) } which is absurd. Therefore injectivity of σ has to hold. We next show surjec-tivity again by contradiction. We assume that there exists m ∈ N such that σ ( k ) = m, ∀ k ∈ N . It follows that m ∈ N \ B k , ∀ k ∈ N . Therefore a k ≤ u ( m ) , ∀ k ∈ N . We conclude u ( σ ( k )) ≤ u ( m ) , ∀ k ∈ N . Since σ is injective as we have already shown we deduce that { n ∈ N : u ( σ ( n )) ≤ u ( m ) } = ∞ . But this contradicts the fact that u converges to infinity. Hence σ has to besurjective and (6) is proved.We next check that σ ∗ u ∈ e F , i.e. σ ∗ u is monotone. To see that we estimate for k ∈ N σ ∗ u ( k + 1) = u ( σ ( k + 1))= a k +1 = a B k = min (cid:8) u ( n ) : n ∈ N \ B k (cid:9) = min (cid:8) u ( n ) : n ∈ N \ { σ ( j ) : 1 ≤ j ≤ k } (cid:9) ≥ min (cid:8) u ( n ) : n ∈ N \ { σ ( j ) : 1 ≤ j ≤ k − } (cid:9) = σ ∗ u ( k ) . k that σ ∗ u ( k ) = σ ′∗ u ( k ) . (7)Using the monotonicity of σ ∗ u and σ ′∗ u and the bijectivity of σ and σ ′ wecompute σ ∗ u (1) = min { σ ∗ u ( n ) : n ∈ N } = min { u ( σ ( n )) : n ∈ N } = min { u ( n ) : n ∈ N } = σ ′∗ u (1) . which is (7) for k = 1. Assuming (7) for all j ≤ k we obtain σ ∗ u ( k + 1) = min { σ ∗ u ( n ) : n ≥ k + 1 } = min (cid:16) { u ( n ) : n ∈ N } \ { σ ∗ u ( j ) : 1 ≤ j ≤ k } (cid:17) = min (cid:16) { u ( n ) : n ∈ N } \ { σ ′∗ u ( j ) : 1 ≤ j ≤ k } (cid:17) = σ ′∗ u ( k + 1) . We have proved the induction step and hence (7) follows for all k ∈ N . Thisfinishes the proof of uniqueness and hence of the Lemma. (cid:3) By the previous Lemma we obtain a well defined map P : U → e F . Namely, let u ∈ U and choose σ ∈ S such that σ ∗ u ∈ e F and set P ( u ) = σ ∗ u. The uniqueness statement of the Lemma assures that P is well defined, i.e. in-dependent of the choice of σ . Moreover, we have the following Corollary. Corollary 3.4
The map P : U → e F is a projection, i.e. P = P . Proof:
Since
P u ∈ e F , we have (id) ∗ ( P u ) ∈ e F and hence P u = P ( P u ) =
P u.
This proves the Corollary. (cid:3)
For u , u ∈ U the product is defined pointwise by( u · u )( n ) = u ( n ) · u ( n ) , n ∈ N . u · u ∈ U . For σ ∈ S we define a map ℘ σ : e F × e F → e F by ℘ σ ( f , f ) = P ( f · σ ∗ f ) , f , f ∈ e F . Note that ℘ id ( f , f ) = f · f , f , f ∈ e F . If f ∈ e F we denote by [ f ] the equivalence class of f in F . Definition 3.5
Given f , f ∈ e F , a subset S ⊂ S is called ( f , f )-wild , if [ ℘ σ ( f , f )] = [ ℘ σ ′ ( f , f )] , ∀ σ, σ ′ ∈ S , σ = σ ′ . Proposition 3.6
Given f , f ∈ e F , there exists an ( f , f ) -wild subset S ⊂ S of infinite cardinality. We prove the Proposition with the help of the following Lemma.
Lemma 3.7
Given f , f ∈ e F and a finite ( f , f ) -wild subset S ⊂ S , thenthere exists σ ∈ S \ S such that S ∪ { σ } is still an ( f , f ) -wild subset of S . Proof:
We prove the Lemma in six steps.
Step 1:
We can assume without loss of generality that S is nonempty. This follows since { id } is an ( f , f )-wild subset of S . Step 2:
The function g = g f ,f : N → (0 , ∞ ) which is defined for n ∈ N by the formula g f ,f ( n ) = g ( n ) = min (cid:8) f ( k ) f ( n + 1 − k ) : 1 ≤ k ≤ n (cid:9) lies in e F , i.e. g is monotone and unbounded. To prove Step 2 we first show that g is monotone. Let k ∈ { , . . . , n + 1 } such that g ( n + 1) = f ( k ) f ( n + 2 − k ) . We first treat the case where k ≤ n . In this case we estimate using the mono-tonicity of f g ( n ) ≤ f ( k ) f ( n + 1 − k ) ≤ f ( k ) f ( n + 2 − k ) = g ( n + 1) . If k = n + 1 we estimate using the monotonicity of f g ( n ) ≤ f ( n ) f (1) ≤ f ( n + 1) f (1) = g ( n + 1) .
20e have shown that g is monotone. We next show that g is unbounded. Since f and f are unbounded there exists for given r ∈ R a positive integer n = n ( r )with the property that f ( n ) ≥ r min { f (1) , f (1) } , f ( n ) ≥ r min { f (1) , f (1) } , ∀ n ≥ n . Using the above inequality and the monotonicity of f and f we estimate for k ∈ { , . . . , n } f ( k ) f (2 n + 1 − k ) ≥ f (1) f ( n ) ≥ r. (8)Similarly, we estimate for k ∈ { n + 1 , . . . , n } f ( k ) f (2 n + 1 − k ) ≥ f ( n ) f (1) ≥ r. (9)Inequalities (8) and (9) imply g (2 n ) ≥ r. This proves that g is unbounded and hence Step 2 follows. Step 3:
Definition of σ ∈ S . For ℓ ∈ N we introduce the shift map s ℓ : e F → e F which is given for f ∈ e F by the formula s ℓ ( f )( n ) = f ( n + ℓ ) , n ∈ N . Note that s ℓ is well defined, i.e. s ℓ ( f ) is still monotone and unbounded. Since S is finite and nonempty by Step 1, we can set for ℓ ∈ N b ℓ = max (cid:8) ℘ σ ( f , f )( ℓ ) : σ ∈ S (cid:9) . Again for ℓ ∈ N we further introduce the set A ℓ = (cid:8) n ∈ N : g s ℓ − ( f ) ,s ℓ − ( f ) ( n ) ≥ ℓb ℓ (cid:9) . Applying Step 2 to g s ℓ − ( f ) ,s ℓ − ( f ) we conclude that the set A ℓ is nonempty.Hence we can set a ℓ = min { n : n ∈ A ℓ } . We put ℓ = 1and define recursively for ν ∈ N ℓ ν +1 = a ℓ ν + ℓ ν . ν ∈ N ℓ ν < ℓ ν +1 . We define σ by the formula σ ( k ) = ℓ ν + ℓ ν +1 − k − , ℓ ν ≤ k ≤ ℓ ν +1 − , ν ∈ N . To show that σ ∈ S we have to check that σ is a bijective map from N to N .But σ | { ℓ ν ,...,ℓ ν +1 − } : { ℓ ν , . . . , ℓ ν +1 − } → { ℓ ν , . . . , ℓ ν +1 − } are bijections for every ν ∈ N . This proves that σ is a bijection and finishesStep 3. Step 4:
For every ν ∈ N we have the inequality ( f · σ ∗ f )( k ) ≥ ℓ ν b ℓ ν , ∀ k ≥ ℓ ν . (10)We first consider the case where k ∈ { ℓ ν , . . . , ℓ ν +1 − } and estimate( f · σ ∗ f )( k ) = f ( k ) f ( σ ( k ))= f ( k ) f ( ℓ ν + ℓ ν +1 − k − (cid:0) s ℓ ν − ( f )( k − ℓ ν + 1) (cid:1)(cid:0) s ℓ ν − ( f )( ℓ ν +1 − k ) (cid:1) ≥ g s ℓν − ( f ) ,s ℓν − ( f ) ( ℓ ν +1 − ℓ ν )= g s ℓν − ( f ) ,s ℓν − ( f ) ( a ℓ ν ) ≥ ℓ ν b ℓ ν . Now let us consider the case where k ≥ ℓ ν +1 . In this case there exists ν ′ > ν such that k ∈ { ℓ ν ′ , · · · ℓ ν ′ +1 − } . Using the monotonicity of f and f we estimate in this case( f · σ ∗ f )( k ) = f ( k ) f ( σ ( k ))= f ( k ) f ( ℓ ν ′ + ℓ ν ′ +1 − k − ≥ f ( ℓ ν ) f ( ℓ ν +1 − (cid:0) s ℓ ν − ( f )(1) (cid:1)(cid:0) s ℓ ν − ( f )( ℓ ν +1 − ℓ ν ) (cid:1) ≥ g s ℓν − ( f ) ,s ℓν − ( f ) ( ℓ ν +1 − ℓ ν ) ≥ ℓ ν b ℓ ν . Hence (10) and therefore Step 4 are proved.
Step 5:
For every ν ∈ N we have the inequality ℘ σ ( f , f )( ℓ ν ) ≥ ℓ ν b ℓ ν . (11)22e assume by contradiction that ℘ σ ( f , f )( ℓ ν ) < ℓ ν b ℓ ν . (12)By construction of ℘ σ there exists σ ′ ∈ S such that ℘ σ ( f , f ) = σ ′∗ ( f · σ ∗ f ) ∈ e F . (13)Since ℘ σ ( f , f ) is monotone, we deduce from (12) ℘ σ ( f , f )( k ) < ℓ ν b ℓ ν , ∀ k ∈ { , · · · , ℓ ν } . (14)We define A = σ ′ (cid:0) { , . . . , ℓ ν } (cid:1) ⊂ N . By (13) and (14) we conclude( f · σ ∗ f )( k ) < ℓ ν b ℓ ν , ∀ k ∈ A. (15)Since σ ′ is a bijection we have A = ℓ ν and hence it follows from (15) that there exists k ≥ ℓ ν with the property ( f · σ ∗ f )( k ) < ℓ ν b ℓ ν . But this contradicts Step 4 and hence Step 5 follows.
Step 6:
The set S ∪ { σ } is ( f , f ) -wild. Since S is already ( f , f )-wild by assumption we are left with showing that[ ℘ σ ( f , f )] = [ ℘ σ ′ ( f , f )] , ∀ σ ′ ∈ S . (16)We assume by contradiction that there exists σ ′ ∈ S such that[ ℘ σ ( f , f )] = [ ℘ σ ′ ( f , f )] . Hence there exists c > ℘ σ ( f , f )( n ) ≤ c (cid:0) ℘ σ ′ ( f , f )( n ) (cid:1) , ∀ n ∈ N . (17)Now choose ν ∈ N satisfying ν > c . We estimate using Step 5 ℘ σ ( f , f )( ℓ ν ) ≥ ℓ ν b ℓ ν ≥ ν℘ σ ′ ( f , f )( ℓ ν ) > c℘ σ ′ ( f , f )( ℓ ν ) . This contradicts (17) and hence (16) has to hold. This finishes the proof ofStep 6 and hence of the Lemma. (cid:3) roof of Proposition 3.6: We define inductively ( f , f )-wild subsets S n ⊂ S of cardinality n ∈ N in the following way. We set S = { id } . Given S n there exists by Lemma 3.7 σ ∈ S \ S n such that S n ∪ { σ } is ( f , f )-wild. We put S n +10 = S n ∪ { σ } . The sets { S n } n ∈ N build a nested sequence S ⊂ S ⊂ S ⊂ · · · . (18)We define S = ∞ [ k =1 S k ⊂ S . The set S has infinite cardinality. We claim that it is still an ( f , f )-wildsubset of S . Pick σ, σ ′ ∈ S . There exist j, j ′ ∈ N such that σ ∈ S j , σ ′ ∈ S j ′ . We set i = max { j, j ′ } . It follows from (18) that σ, σ ′ ∈ S i . But since S i is an ( f , f )-wild subset of S we deduce that[ ℘ σ ( f , f )] = [ ℘ σ ′ ( f , f )] . This proves that S is ( f , f )-wild and hence we have constructed an ( f , f )-wild subset of infinite cardinality. This finishes the proof of the Proposition. (cid:3) Proof of Theorem B:
For given φ , φ ∈ F we first choose representatives f , f ∈ e F such that [ f i ] = φ i , i ∈ { , } . By Proposition 3.6 there exists an ( f , f )-wild subset S ⊂ S of infinite cardi-nality. We pick σ ∈ S and introduce the triple H = ( ℓ , ℓ f , ℓ f · σ ∗ f ) . Although f · σ ∗ f is only in U and not necessarily in e F we define ℓ f · σ ∗ f as asubset of ℓ in the same way as we do it in the monotone case. We further notethat ℓ f · σ ∗ f ⊂ ℓ f . Let I : ℓ f → ℓ
24e the canonical isometry as explained in the proof of Corollary 1.9. We notethat the restriction of I to ℓ f · σ ∗ f gives rise to an isometry I | ℓ f · σ ∗ f : ℓ f · σ ∗ f → ℓ σ ∗ f . Define a further isometry J σ : ℓ → ℓ which is given on standard basis vectors { ε n } n ∈ N of ℓ by the formula J σ ( ε n ) = ε σ ( n ) . We note that the restriction of J σ to ℓ σ ∗ f gives an isometry J σ | ℓ σ ∗ f : ℓ σ ∗ f → ℓ f . We deduce that the composition of I and J σ gives an isometry of pairs J σ ◦ I : ( ℓ f , ℓ f · σ ∗ f ) → ( ℓ , ℓ f ) . (19)As a first consequence of the isometry (19) and the fact that ( ℓ , ℓ f ) is a scaleHilbert pair we conclude that ( ℓ f , ℓ f · σ ∗ f ) is also a scale Hilbert pair. Since( ℓ , ℓ f ) is a further scale Hilbert pair, we deduce that H is a scale Hilbert triple.As a second consequence of (19) we obtain the formula K ([ H ])(1 ,
2) = [ f ] = φ . (20)By construction of ℘ σ ( f , f ) there exists σ ′ ∈ S such that ℘ σ ( f , f ) = σ ′∗ ( f · σ ∗ f ) . Hence we obtain a scale isometry of scale Hilbert pairs J σ ′ : ( ℓ , ℓ ℘ σ ( f ,f ) ) → ( ℓ , ℓ f · σ ∗ f )from which we deduce K ([ H ])(0 ,
2) = [ ℘ σ ( f , f )] . (21)Furthermore, K ([ H ])(0 ,
1) = [ f ] = φ . (22)Combining (20), (21), and (22) we obtain K ([ H ]) = ( φ , φ , [ ℘ σ ( f , f )])implying that [ ℘ σ ( f , f )] ∈ B ( φ , φ ) . Hence we get a map S → B ( φ , φ ) , σ [ ℘ σ ( f , f )]25hich by definition of ( f , f )-wild is injective. Since S = ∞ we deduce that B ( φ , φ ) = ∞ . This finishes the proof of Theorem B. (cid:3)
Proof of Corollary 1.12:
We only prove that the map J : F → S is notsurjective. The proof that J n : F n → S n is not surjective for n ≥ n .By Theorem B there exists a scale Hilbert triple H = ( H , H , H ) such that K ([ H ])(0 , = K ([ H ])(0 , · K ([ H ])(1 , . Choose an arbitrary scale Hilbert space H ′ = ( H ′ , H ′ , . . . ). By Corollary 1.6the Hilbert spaces H and H ′ are isometric to ℓ and in particular isometric toeach other. Hence let I : H ′ → H be an isometry of Hilbert spaces. We now define a new scale Hilbert space e H = ( e H , e H , . . . ) by setting e H k = (cid:26) H k ≤ k ≤ I ( H ′ k − ) k ≥ . For the scale Hilbert space e H we still have K ([ e H ])(0 , = K ([ e H ])(0 , · K ([ e H ])(1 , . On the other hand if Φ ∈ F , then we necessarily have K ( J (Φ))(0 ,
2) = K ( J (Φ))(0 , · K ( J (Φ))(1 , . This shows that [ e H ] cannot lie in the image of J and hence the Corollary isproved. (cid:3) References [1] H. Hofer,
A general Fredholm theory and applications.
Current develop-ments in mathematics, 2004, 1–71, Int. Press, Somerville, MA, 2006.[2] H. Hofer, K. Wysocki, E. Zehnder,
A general Fredholm theory. I. A splicing-based differential geometry.