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Induced and coinduced Banach Lie–Poisson spaces and integrability

Induced and coinduced Banach Lie–Poisson spaces and integrability

Journal of Functional Analysis 255 (2008) 1225–1272 www.elsevier.com/locate/jfa Induced and coinduced Banach Lie–Poisson spaces and integrability Ana...

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Journal of Functional Analysis 255 (2008) 1225–1272 www.elsevier.com/locate/jfa

Induced and coinduced Banach Lie–Poisson spaces and integrability Anatol Odzijewicz a , Tudor S. Ratiu b,∗ a Institute of Mathematics, University of Bialystok, Lipowa 41, PL-15424 Bialystok, Poland b Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne,

Switzerland Received 22 December 2007; accepted 2 June 2008 Available online 9 July 2008 Communicated by D. Voiculescu

Abstract The Poisson induction and coinduction procedures are used to construct Banach Lie–Poisson spaces as well as related systems of integrals in involution. This general method applied to the Banach Lie–Poisson space of trace class operators leads to infinite Hamiltonian systems of k-diagonal trace class operators which have infinitely many integrals. The bidiagonal case is investigated in detail. © 2008 Elsevier Inc. All rights reserved. Keywords: Banach Lie–Poisson space; Induction; Coinduction; Momentum map; Coadjoint orbit; Integrable Hamiltonian system

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Induced and coinduced Banach Lie–Poisson spaces . 3. Induction and coinduction from L1 (H) . . . . . . . . . 4. Dynamics generated by Casimirs of L1 (H) . . . . . . 5. The bidiagonal case . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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* Corresponding author.

E-mail addresses: [email protected] (A. Odzijewicz), [email protected] (T.S. Ratiu). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2008.06.001

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. 1226 . 1227 . 1235 . 1246 . 1253 . 1271 . 1271

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1. Introduction This paper continues the investigation of Banach Lie–Poisson spaces introduced in [12] and also studied in [4,13]. The theory of Banach Lie–Poisson spaces gives a natural generalization in the functional analytical context of the Poisson geometry of finite-dimensional integrable Hamiltonian systems. It gives also a solid mathematical foundation for the theory of Hamiltonian systems with infinitely many degrees of freedom. The interest in these system was initiated in [3] and [8] and from then on they have played an important role in mathematics and physics. With few notable exceptions, for infinite-dimensional systems, the Lie–Poisson bracket formulation is mostly formal. It is our belief that these formal approaches can be given a solid functional analytic underpinning. The present paper formulates such an approach for various families of integrable systems which arise in a natural way when one investigates Banach Lie–Poisson spaces of trace class operators. The paper is organized as follows. Section 2 presents the general theory of induced and coinduced Banach Lie–Poisson structures (Propositions 2.1–2.4) and gives a method of construction of systems of integrals in involution. The associated involution theorem (Proposition 2.2 and Corollary 2.3) is an analogue of the classical R-matrix method for Banach Lie–Poisson spaces. Section 3 investigates the Banach Lie–Poisson geometry of several classes of spaces of trace class operators. The general constructions of Section 2 are implemented explicitly to these spaces. The multi-diagonal Banach Lie group, its Lie algebra, and its dual are introduced and studied (Propositions 3.1 and 3.2). The naturally induced and coinduced Poisson structures on the preduals of their Banach Lie algebras are presented. Section 4 formulates the equations of motion induced by the Casimir functions of the Banach Lie–Poisson space of trace class operators relative to the various induced and coinduced Poisson brackets discussed previously. These systems represent a k-diagonal version of the semi-infinite Toda system which is obtained from this point of view if k = 2. The solution of the systems associated to two different splittings of the space of trace class operators in terms of group decompositions are also presented. Section 5 emphasizes the important particular case of bidiagonal operators. The Banach Lie group of upper bidiagonal bounded operators is studied in detail and the topological and symplectic structure of the generic coadjoint orbit is presented (Proposition 5.1). The Banach space analogue of the Flaschka map (defined for the first time in [7]) is analyzed and its relationship to the coadjoint orbits is pointed out (Propositions 5.2 and 5.3). There are new, typical infinite-dimensional, phenomena that appear in this context. For example, as opposed to the finite-dimensional case, the Banach space of lower bidiagonal trace class operators does not form a single coadjoint orbit and there are non-algebraic invariants for the coadjoint orbits. As an example of the theory, the semi-infinite Toda lattice is rigorously investigated using the method of orthogonal polynomials first introduced, to our knowledge, in [5]. The explicit solution of this system is obtained, both in action–angle as well as in the original variables, thereby extending the formulas in [10] from the finite to the semi-infinite Toda lattice. Conventions. In this paper all Banach manifolds and Lie groups are real. The definition of the notion of a Banach Lie subgroup follows Bourbaki [6], that is, a subgroup H of a Banach Lie group G is necessarily a submanifold (and not just injectively immersed). In particular, Banach Lie subgroups are necessarily closed.

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2. Induced and coinduced Banach Lie–Poisson spaces In this section we shortly review some material from [12] and present constructions that are necessary for the development of the ideas in the rest of the paper. Preliminaries. Let us recall how a given Banach Lie–Poisson structure induces and coinduces similar structures on other Banach spaces. All the proofs of the statements below can be found in [12]. Throughout this paper, unless specified otherwise, all objects are over R. A Banach Lie algebra (g, [·,·]) is a Banach space g that is also a Lie algebra such that the Lie bracket is a bilinear continuous map g × g → g. Thus the adjoint and coadjoint maps adx : g → g, adx y := [x, y], and ad∗x : g∗ → g∗ are also continuous for each x ∈ g. Here g∗ denotes the dual of g, that is, the Banach space of all linear continuous functionals on g. A Banach Lie–Poisson space (b, {·,·}) is defined to be a real Poisson manifold such that b is a Banach space and the dual b∗ ⊂ C ∞ (b) is a Banach Lie algebra under the Poisson bracket operation. We need to explain what does it mean for b to be a Banach Poisson manifold. The Poisson bracket induces the derivation h → {·, h} on C ∞ (b) which defines a map Xh : b → b∗∗ by Xh (b), Df (b) = {f, h}(b) for any b ∈ b and f a smooth real-valued function defined in an open subset of b containing b. Thus, Xh (b) ∈ b∗∗ ∼ = Tb∗∗ b and therefore Xh (b) is not a tangent vector to b at b. The requirement that b be a Banach Poisson manifold is that Xh (b) ∈ b ∼ = Tb b for all b ∈ b. Denote by [·,·] the restriction of the Poisson bracket {·,·} from C ∞ (b) to the Lie subalgebra b∗ . The following criterion characterizes the Banach Lie–Poisson structure. The Banach space b is a Banach Lie–Poisson space (b, {·,·}) if and only if its dual b∗ is a Banach Lie algebra (b∗ , [·,·]) satisfying ad∗x b ⊂ b ⊂ b∗∗ for all x ∈ b∗ . Moreover, the Poisson bracket of f, h ∈ C ∞ (b) is given by    {f, h}(b) = Df (b), Dh(b) , b ,

(2.1)

where b ∈ b and Df (b) ∈ b∗ denotes the Fréchet derivative of f at the point b. If h is a smooth function on b, the associated Hamiltonian vector field is given by Xh (b) = − ad∗Dh(b) b ∈ b

(2.2)

for any b ∈ b. Therefore Hamilton’s equations are d b(t) = − ad∗Dh(b(t)) b(t). dt

(2.3)

In particular, h ∈ C ∞ (b) is a Casimir function, that is, {f, h} = 0 for all f ∈ C ∞ (b) if and only if ad∗Dh(b) b = 0

for all b ∈ b.

(2.4)

Given two Banach Lie–Poisson spaces (b1 , {,}1 ) and (b2 , {,}2 ), a smooth map ϕ : b1 → b2 is said to be canonical or a Poisson map if {f, h}2 ◦ ϕ = {f ◦ ϕ, h ◦ ϕ}1

(2.5)

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for any two smooth locally defined functions f and h on b2 . Like in the finite-dimensional case, (2.5) is equivalent to 1 Xh2 ◦ ϕ = T ϕ ◦ Xh◦ϕ

(2.6)

for any smooth locally defined function h on b2 . Therefore, the flow of a Hamiltonian vector field is a Poisson map and Hamilton’s equations f˙ = {f, h} in Poisson bracket formulation are valid. If the Poisson map ϕ is, in addition, linear, then it is called a linear Poisson map. Given the Banach Lie–Poisson spaces (b1 , {,}1 ) and (b2 , {,}2 ) there is a unique Banach Poisson structure {,} on the product space b1 × b2 such that: (i) the canonical projections π1 : b1 × b2 → b1 and π2 : b1 × b2 → b2 are Poisson maps; (ii) π1∗ (C ∞ (b1 )) and π2∗ (C ∞ (b2 )) are Poisson commuting subalgebras of C ∞ (b1 × b2 ). This unique Poisson structure on b1 × b2 is called the product Poisson structure and its bracket is given by the formula {f, g}(b1 , b2 ) = {fb2 , gb2 }1 (b1 ) + {fb1 , gb1 }2 (b2 ),

(2.7)

where fb1 , gb1 ∈ C ∞ (b2 ) and fb2 , gb2 ∈ C ∞ (b1 ) are the partial functions given by fb1 (b2 ) := fb2 (b1 ) := f (b1 , b2 ) and gb1 (b2 ) := gb2 (b1 ) := g(b1 , b2 ). In addition, this formula shows that this unique Banach Poisson structure is Lie–Poisson and that the inclusions ι1 : b1 → b1 × b2 , ι2 : b2 → b1 × b2 given by ι1 (b1 ) := (b1 , 0) and ι2 (b2 ) := (0, b2 ), respectively, are also linear Poisson maps. Induced structures. Let b1 be a Banach space, (b, {·,·}) a Banach Lie–Poisson space, and ι : b1 → b an injective continuous linear map with closed range. Then ker ι∗ is an ideal in the Banach Lie algebra (b∗ , [·,·]) if and only if b1 carries a unique Banach Lie–Poisson bracket {·,·}ind 1 such that {F ◦ ι, G ◦ ι}ind 1 = {F, G} ◦ ι

(2.8)

for any F, G ∈ C ∞ (b); see [12, Proposition 4.10]. This Poisson structure on b1 is said to be induced by the mapping ι and it is given by  ∗  ∗ −1    −1     {f, g}ind ι Df (b1 ) , ι∗ Dg(b1 ) 1 , b1 1 (b1 ) = ι

(2.9)

for any f, g ∈ C ∞ (b1 ) and b1 ∈ b1 , where [ι∗ ] : b∗ / ker ι∗ → b∗1 is the Banach space isomorphism induced by ι∗ : b∗ → b∗1 and [·,·]1 denotes the Lie bracket on the quotient Lie algebra b∗ / ker ι∗ . Let us assume now that the range ι(b1 ) is a closed split subspace of b, that is, there exists a projector R = R 2 : b → b such that ι(b1 ) = R(b). Taking in (2.8) F := f ◦ ι−1 ◦ R, G := g ◦ ι−1 ◦ R ∈ C ∞ (b) for f, g ∈ C ∞ (b1 ) and noting that ι−1 ◦ R ◦ ι = idb1 , we get    −1 {f, g}ind ◦ R, g ◦ ι−1 ◦ R ι(b1 ) 1 (b1 ) = f ◦ ι         = D f ◦ ι−1 ◦ R ι(b1 ) , D g ◦ ι−1 ◦ R ι(b1 ) , ι(b1 ) . We shall make use of this formula in Section 3.

(2.10)

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We return now to the general case, that is, we consider an arbitrary quasi-immersion ι : b1 → b of Banach spaces which means that the range ι(b1 ) is closed but does not necessarily possess a closed complement. Proposition 2.1. Let ι : b1 → b be a quasi-immersion of Banach Lie–Poisson spaces (so range ι is a closed subspace of b and ker ι∗ is an ideal in the Banach Lie algebra b∗ ). Assume that there is a connected Banach Lie group G with Banach Lie algebra g := b∗ . Then the G-coadjoint orbit Oι(b1 ) := Ad∗G ι(b1 ) is contained in ι(b1 ) for any b1 ∈ b1 . In addition, if N ⊂ G is a closed connected normal Lie subgroup of G whose Lie algebra is ker ι∗ , then the N -coadjoint action restricted to ι(b1 ) is trivial. Therefore the Banach Lie group G/N := {[g] := gN | g ∈ G} naturally acts on ι(b1 ) and the orbit of ι(b1 ) under this action coincides with Oι(b1 ) for any b1 ∈ b1 . Proof. Since ker ι∗ is an ideal in g = b∗ , it follows that [x, y] ∈ ker ι∗ for all x ∈ g and y ∈ ker ι∗ . Therefore, since ker ι∗ is closed in g, it follows that Adexp x y = eadx y ∈ ker ι∗ for any x ∈ g and y ∈ ker ι∗ . This shows that for any g ∈ G in an open neighborhood of the identity element of G we have Adg ker ι∗ ⊂ ker ι∗ . Since G is connected, it is generated by a neighborhood of the identity and we conclude that Adg ker ι∗ ⊂ ker ι∗ for any g ∈ G. The upper index ◦ on a set denotes the annihilator of that set relative to a duality pairing; the annihilator of a set is always a vector subspace. Let b1 ∈ b1 and g ∈ G. Since ker ι∗ = ι(b1 )◦ , closedness of ι(b1 ) in b implies that (ker ι∗ )◦ = ι(b1 ). Thus, for any g ∈ G and x ∈ ker ι∗ , we have 

   Ad∗g ι(b1 ), x = ι(b1 ), Adg x = 0

which proves that Ad∗G ι(b1 ) ⊂ ι(b1 ). Now let N ⊂ G be a closed connected normal Lie subgroup of G with Banach Lie algebra ker ι∗ ⊂ g. For any b1 ∈ b1 , x ∈ g = b∗ , y ∈ ker ι∗ , we have  ∗    ady ι(b1 ), x = ι(b1 ), [y, x] = 0 since ker ι∗ is an ideal in g and ker ι∗ = ι(b1 )◦ . Since this is valid for all x ∈ g, it follows that ad∗y ι(b1 ) = 0 for all y ∈ ker ι∗ and b1 ∈ b1 . Using the exponential map, this shows that Ad∗n ι(b1 ) = ι(b1 ) for any n in a neighborhood of the identity in N . Since N is connected, it is generated by a neighborhood of the identity and we conclude that Ad∗n ι(b1 ) = ι(b1 ) for all n ∈ N. The quotient G/N := {[g] := gN | g ∈ G} is a Banach Lie group and the projection G → G/N is a smooth surjective submersive Banach Lie group homomorphism [6, Chapter III, §1.6]. Since the coadjoint action of N on ι(b1 ) is trivial, the Banach Lie group G/N acts smoothly on ι(b1 ) by [g] · ι(b1 ) := Ad∗g −1 ι(b1 ). The orbit of a fixed element ι(b1 ) ∈ ι(b1 ) by this group action is obviously equal to the G-orbit Oι(b1 ) . 2 Coinduced structures. Let (b, {,}) be a Banach Lie–Poisson space and π : b → b1 a continuous linear surjective map onto the Banach space b1 . Then b1 carries a unique Banach Lie–Poisson bracket {,}coind making π into a linear Poisson map, that is, 1 ◦π {f ◦ π, g ◦ π} = {f, g}coind 1

(2.11)

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for any f, g ∈ C ∞ (b1 ) if and only if π ∗ (b∗1 ) ⊂ b∗ is closed under the Lie bracket [·,·] of b∗ ; see [12, Proposition 4.8]. This unique Poisson structure on b1 is said to be coinduced by the Banach Lie–Poisson structure on b and the linear continuous map π . It should be noted that im π ∗ is a closed subspace of b∗ since im π ∗ = (ker π)◦ . To determine the coinduced bracket on b1 note that π ∗ : b∗1 → b∗ is an injective linear continuous map whose closed range is a Banach Lie subalgebra of b∗ . Thus, on im π ∗ we can invert π ∗ . The coinduced bracket on b1 has then the form   −1  ∗     π Df (b1 ) , π ∗ Dg(b1 ) , b1 (b1 ) = π ∗ {f, g}coind 1

(2.12)

for any f, g ∈ C ∞ (b1 ) and b1 ∈ b1 . Let us assume that ker π admits a closed complement. This is equivalent to the existence of a linear continuous injective map ι : b1 → b with closed range such that π ◦ ι = idb1 . Thus (2.11) implies that = {f ◦ π, g ◦ π} ◦ ι {f, g}coind 1

(2.13)

for any f, g ∈ C ∞ (b1 ). Assume now that the Banach Lie–Poisson space b splits into a direct sum b = b1 ⊕ b2 of closed Banach subspaces. Denote by Rj : b → b the projection onto bj , for j = 1, 2. So we have the following relations: R1 + R2 = idb , R12 = R1 , R22 = R2 , R2 R1 = R1 R2 = 0, b1 := im R1 , and b2 := im R2 . Dualizing we get the projectors R1∗ , R2∗ : b∗ → b∗ satisfying R1∗ + R2∗ = idb∗ , (R1∗ )2 = R1∗ , (R2∗ )2 = R2∗ , R2∗ R1∗ = R1∗ R2∗ = 0. The relationship between these spaces is given by ker R1 = im R2 = b2 ker R1∗

= im R2∗

b = b1 ⊕ b 2



= (im R1 )

∼ = b∗2

and

ker R2 = im R1 = b1 ,

and

ker R2∗

= im R1∗

(2.14) ◦

= (im R2 )

and b∗ = b◦2 ⊕ b◦1 ∼ = b∗1 ⊕ b∗2 .

∼ = b∗1 ,

(2.15) (2.16)

Let ιj : bj → b be the inclusion determined by the splitting b = b1 ⊕ b2 for j = 1, 2. Denote by πj : b → bj the projection determined by the projector Rj : b → b, that is, ιj ◦ πj = Rj and note that πj ◦ ιj = idbj . We summarize these notations in the following diagram.

From (2.13) we get = {f ◦ πj , g ◦ πj } ◦ ιj {f, g}coind j or, explicitly

(2.17)

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{f, g}coind (bj ) j       = D(f ◦ πj ) ιj (bj ) , D(g ◦ πj ) ιj (bj ) , ιj (bj ) ,

where bj ∈ bj .

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(2.18)

The following proposition presents some properties of the induced and coinduced structures on b1 and b2 . Proposition 2.2. Assume that im R1∗ and im R2∗ are Banach Lie subalgebras of b∗ . Then: (i) bj has a Banach Lie–Poisson structure coinduced by πj and the expression of the coinduced on bj is given by (2.17). The Hamiltonian vector field of h ∈ C ∞ (bj ) at bracket {,}coind j bj ∈ bj is given by   Xh (bj ) = −πj ad∗π ∗ Dh(bj ) ιj (bj ) , j

j = 1, 2,

(2.19)

where Dh(bj ) ∈ b∗j and adx is the adjoint action of x ∈ b∗ on b∗ . (ii) The Banach space isomorphism R := 12 (R1 −R2 ) : b → b defines a new Banach Lie–Poisson structure      {f, g}R (b) := R ∗ Df (b), Dg(b) + Df (b), R ∗ Dg(b) , b

(2.20)

on b, f, g ∈ C ∞ (b), that coincides with the product structure on b1 × b2 , where b1 carries and b2 denotes b2 endowed with the Lie–Poisson bracket the coinduced bracket {,}coind 1 coind −{,}2 . ) → (b, {,}R ) and ι2 : (b2 , {,}coind ) → (b, {,}R ) are lin(iii) The inclusion maps ι1 : (b1 , {,}coind 1 2 ear injective Poisson maps with closed range. (iv) The map ιj induces from (b, {,}R ) a Banach Lie–Poisson structure on bj which coincides with the coinduced structure described in (i), for j = 1, 2. Proof. (i) By hypothesis, the range im Rj∗ of the map Rj∗ : b∗ → b∗ is a Banach Lie subalgebra of b∗ . Thus πj coinduces a Banach Lie–Poisson structure on b∗j . Let h ∈ C ∞ (bj ) and note that for any function f ∈ C ∞ (bj ) and bj ∈ bj we have 

       Df (bj ), Xh (bj ) = {f, h}coind (bj ) = D(f ◦ πj ) ιj (bj ) , D(h ◦ πj ) ιj (bj ) , ιj (bj ) j    = πj∗ Df (bj ), πj∗ Dh(bj ) , ιj (bj )   = πj∗ Df (bj ), − ad∗π ∗ Dh(bj ) ιj (bj ) j   = Df (bj ), −πj ad∗π ∗ Dh(bj ) ιj (bj ) , j

which proves formula (2.19). (ii) Let b = b1 + b2 ∈ b1 ⊕ b2 . Then Rj (b) = bj , for j = 1, 2. A direct verification shows then that      {f, g}R (b) = R ∗ Df (b), Dg(b) + D(b), R ∗ Dg(b) , b =

  1  ∗ R Df (b) − R2∗ Df (b), R1∗ Dg(b) + R2∗ Dg(b) , b 2 1

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  1  ∗ R1 Df (b) + R2∗ Df (b), R1∗ Dg(b) − R2∗ Dg(b) , b + 2       = R1∗ Df (b), R1∗ Dg(b) , R1 b + R2 b − R2∗ Df (b), R2∗ Dg(b) , R1 b + R2 b       = R1∗ Df (b), R1∗ Dg(b) , R1 b − R2∗ Df (b), R2∗ Dg(b) , R2 b = {fb2 , gb2 }coind (b1 ) − {fb1 , gb1 }coind (b2 ), 1 2 where in the third equality we have used the fact that [R1∗ Df (b), R1∗ Dg(b)] ∈ im R1∗ = (im R2 )◦ and [R2∗ Df (b), R2∗ Dg(b)] ∈ im R2∗ = (im R1 )◦ and b = b1 + b2 with bj ∈ bj . To prove the last equality above it suffices to note that D1 fb2 (b1 ) · δb1 = Df (b) · δb1 = Df (b) · R1 δb1

and

D2 fb1 (b2 ) · δb2 = Df (b) · δb2 = Df (b) · R2 δb2 for any δbj ∈ bj , where Dj is the Fréchet derivative on bj , for j = 1, 2. The last expression is that of the product Banach Lie–Poisson structure on b1 × b2 (see (2.7)). (iii) This is an immediate consequence of (ii) and the general fact, recalled earlier for products of Banach Lie–Poisson spaces, that these inclusions are Poisson maps with closed range. coind be the induced and coinduced brackets on b from (b, {·,·} ) and (iv) Let {,}ind j R j and {,}j (b, {·,·}), respectively. Therefore, {F, G}R ◦ ιj = {F ◦ ιj , G ◦ ιj }ind j

(2.21)

for any F, G ∈ C ∞ (b) and, by (2.17), {f, g}coind = (−1)j −1 {f ◦ πj , g ◦ πj } ◦ ιj j

(2.22)

for any f, g ∈ C ∞ (bj ). Apply relation (2.21) to the functions F := f ◦ πj , G := g ◦ πj and use πj ◦ ιj = idbj , πj ◦ R = 12 (−1)j −1 πj , and (2.22) to get for any bj ∈ bj   {f, g}ind j (bj ) = {f ◦ πj , g ◦ πj }R ιj (bj )       = R ∗ D(f ◦ πj ) ιj (bj ) , D(g ◦ πj ) ιj (bj ) , ιj (bj )       + D(f ◦ πj ) ιj (bj ) , R ∗ D(g ◦ πj ) ιj (bj ) , ιj (bj )    = R ∗ πj∗ Df (bj ), πj∗ Dg(bj ) , ιj (bj )    + πj∗ Df (bj ), R ∗ πj∗ Dg(bj ) , ιj (bj )    = (−1)j −1 πj∗ Df (bj ), πj∗ Dg(bj ) , ιj (bj )       = (−1)j −1 D(f ◦ πj ) ιj (bj ) , D(g ◦ πj ) ιj (bj ) , ιj (bj )   = (−1)j −1 {f ◦ πj , g ◦ πj } ιj (bj ) = {f, g}coind (bj ). j

2

This proposition implies the following involution theorem.

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Corollary 2.3. In the notations and hypotheses of Proposition 2.2 we have: (i) The Casimir functions on (b, {·,·}) are in involution on (b, {·,·}R ) and restrict to functions in involution on bj , j = 1, 2. (ii) If H is a Casimir function on b, then its restriction H ◦ ιj to bj has the Hamiltonian vector field   XH ◦ι1 (b1 ) = π1 ad∗R ∗ DH (ι1 (b1 )) ι1 (b1 ) , 2   XH ◦ι2 (b2 ) = π2 ad∗R ∗ DH (ι2 (b2 )) ι2 (b2 ) 1

(2.23)

for any b1 ∈ b1 and b2 ∈ b2 , where ιj : bj → b is the inclusion, j = 1, 2. Proof. (i) Let F, H ∈ C ∞ (b) be Casimir functions for the Lie–Poisson bracket {,}, that is, ad∗DF (b) b = 0 and ad∗DH (b) b = 0 for any b ∈ b (see (2.4)). Therefore      {F, H }R (b) = R ∗ DF (b), DH (b) + DF (b), R ∗ DH (b) , b     = − R ∗ DF (b), ad∗DH (b) b + R ∗ DH (b), ad∗DF (b) b = 0 which shows that F and H are in involution relative to {,}R . Then statements (iii) and (iv) of Proposition 2.2 show that F ◦ ιj , H ◦ ιj are also in involution on bj , j = 1, 2. (ii) Since H is a Casimir function on b, we have ad∗DH (b) b = 0 for any b ∈ b (see (2.4)). Therefore, since R1∗ + R2∗ = idb∗ , we get for any b1 ∈ b1 0 = ad∗DH (ι1 (b1 )) ι1 (b1 ) = ad∗R ∗ DH (ι1 (b1 )) ι1 (b1 ) + ad∗R ∗ DH (ι1 (b1 )) ι1 (b1 ). 1

2

A similar relation holds for any b2 ∈ b2 . So, we have − ad∗R ∗ DH (ιj (bj )) = ad∗R ∗

j +1 DH (ιj (bj ))

j

,

(2.24)

where j is taken modulo 2. Since ιj ◦ πj = Rj we get   πj∗ D(H ◦ ιj )(bj ) = D(H ◦ ιj )(bj ) ◦ πj = DH ιj (bj ) ◦ ιj ◦ πj     = DH ιj (bj ) ◦ Rj = Rj∗ DH ιj (bj ) , so (2.19) and (2.24) yield       ιj XH ◦ιj (bj ) = −(ιj ◦ πj ) ad∗π ∗ D(H ◦ιj )(bj ) ιj (bj ) = −Rj ad∗R ∗ DH (ιj (bj )) ιj (bj ) j j   (2.25) = Rj ad∗R ∗ DH (ιj (bj )) ιj (bj ) = ad∗R ∗ DH (ιj (bj )) ιj (bj ). j +1

j +1

The last equality follows from the fact that ad∗R ∗

j +1 x

ιj (bj ) ∈ im Rj = im ιj for any x ∈ b∗ and

bj ∈ bj . Indeed, for any y ∈ (im Rj )◦ = im Rj∗+1 we have 

ad∗R ∗

j +1 x

    ιj (bj ), y = ιj (bj ), Rj∗+1 x, y = 0

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because [Rj∗+1 x, y] ∈ im Rj∗+1 = (im Rj )◦ by hypothesis (the image of Rj∗+1 is a Banach Lie subalgebra of b∗ ) and ιj (bj ) ∈ im Rj . Therefore, ad∗R ∗ x ιj (bj ) ∈ (im Rj )◦◦ = im Rj = im Rj . Finally, applying πj to (2.25) yields (2.23).

2

j +1

Taken together, Proposition 2.2 and Corollary 2.3 give a construction of integrals in involution. This construction is an infinite-dimensional version of the classical R-matrix method. However, as was seen, the extension of this construction to the Banach Lie–Poisson space context is not direct and needs additional functional analytic hypotheses. Proposition 2.4. Let (b, {,}) be a Banach Lie–Poisson space and let R1 , R3 : b → b be projectors. Assume that im R21 = im R23 =: b2 , where R21 := idb −R1 , R23 := idb −R3 , and denote b1 := im R1 , b3 := im R3 . We summarize this situation in the diagram

where π1 , π21 , π23 , π3 are the projections onto the ranges of R1 , R21 , R23 , and R3 , respectively, according to the splittings b = b1 ⊕ b2 = b2 ⊕ b3 , and ι1 : b1 → b, ι3 : b3 → b are the inclusions. Then one has: (i) If b◦2 is a Banach Lie subalgebra of b∗ , then Φ31 := π3 ◦ ι1 : (b1 , {,}coind ) → (b3 , {,}coind ) 1 3 coind coind and Φ13 := π1 ◦ ι3 : (b3 , {,}3 ) → (b1 , {,}1 ) are mutually inverse linear Poisson isomorphisms. (ii) If b◦1 and b◦3 are Banach Lie subalgebras of b∗ , then b2 has two coinduced Banach Lie– and {,}coind which are not isomorphic in general. Poisson brackets {,}coind 21 23 Proof. (i) Since b◦2 = (im R21 )◦ = im R1∗ (see (2.15)) is a Banach Lie subalgebra of b∗ it follows on b1 . Similarly, the relation b◦2 = that R1 coinduces a Banach Lie–Poisson bracket {,}coind 1 (im R23 )◦ = im R3∗ implies that R3 coinduces a Banach Lie–Poisson bracket {,}coind on b3 . 3 Let us notice that Φ31 ◦ Φ13 = π3 ◦ ι1 ◦ π1 ◦ ι3 = π3 ◦ R1 ◦ ι3 = π3 ◦ (idb −R21 ) ◦ ι3 = π3 ◦ ι3 − π3 ◦ R21 ◦ ι3 = idb3 since π3 ◦ R21 = 0. One proves similarly that Φ13 ◦ Φ31 = idb1 . From ker π1 = ker π3 = b2 and b − (ι3 ◦ π3 )(b) ∈ ker π3 for any b ∈ b, it follows that π1 ◦ ι3 ◦ π3 = π1 . Therefore, if f, g ∈ C ∞ (b1 ) we get from (2.17) and the fact that π1 : b → b1 is a Poisson map {f ◦ Φ13 , g ◦ Φ13 }coind = {f ◦ π1 ◦ ι3 , g ◦ π1 ◦ ι3 }coind 3 3 = {f ◦ π1 ◦ ι3 ◦ π3 , g ◦ π1 ◦ ι3 ◦ π3 } ◦ ι3 = {f ◦ π1 , g ◦ π1 } ◦ ι3 = {f, g}coind ◦ π1 ◦ ι3 = {f, g}coind ◦ Φ13 . 1 1 It is shown in a similar way that Φ31 : b1 → b3 is a Poisson map.

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1235

∗ and b◦ = im R ∗ which, by hypothesis, are Banach Lie sub(ii) By (2.14) we have b◦1 = im R21 3 23 ∗ and {,}coind on b2 . 2 algebras of b . Therefore, π21 and π23 coinduce Poisson brackets {,}coind 21 23

Remark. A concrete counterexample where the two coinduced Poisson structures are not iso1 which is the predual of two different Banach Lie algebras I ∞ morphic is obtained for b2 = I+,1 −,1 ∞ and LA . For the definition of these Banach spaces see Section 3. The two different coinduced brackets are given by (3.45) and (3.46). 3. Induction and coinduction from L1 (H) In this section we introduce several Banach spaces that will be used later on and implement the constructions presented in Section 2 in these concrete cases. The Banach Lie–Poisson space L1 (H). The Banach space of trace class operators (L1 (H),

· 1 ) on a separable Hilbert space H has a canonical Banach Lie–Poisson bracket defined by    {f, g}(ρ) = Tr ρ Df (ρ), Dg(ρ) ,

(3.1)

where f, g ∈ C ∞ (L1 (H)) and the Fréchet derivatives Df (ρ), Dg(ρ) are regarded as elements of the Banach Lie algebra (L∞ (H), · ∞ ) of bounded operators on H, identified with the dual of L1 (H) by the strongly nondegenerate pairing for ρ ∈ L1 (H), x ∈ L∞ (H).

ρ, x = Tr(ρx),

(3.2)

Hamilton’s equations defined by the Poisson bracket (3.1) are easily verified to be given in Lax form (see [12] for details)  dρ  = Dh(ρ), ρ . dt

(3.3)

The orthonormal basis {|n }∞ n=0 of H, that is, n|m = δnm for n, m ∈ N ∪ {0}, induces the Schauder basis {|n m|}∞ of L1 (H) since it is orthonormal in the Hilbert space L2 (H) of n,m=0 1 Hilbert–Schmidt operators and L (H) ⊂ L2 (H). Thus, every trace class operator ρ ∈ L1 (H) can be uniquely expressed as ρ=



ρnm |n m|,

(3.4)

n,m=0

where the series is convergent in the · 1 topology. The coordinates ρnm ∈ R are given by ρnm = Tr(ρ|m n|). The rank one projectors |l k| thought of as elements of L∞ (H), by giving their values on the Schauder basis of L1 (H) as Tr(|l k| |n m|) = δkn δlm , form a biorthogonal family of functionals (see [9]) in L∞ (H) associated to the given Schauder basis {|n m|}∞ n,m=0 of L1 (H). Therefore, each bounded operator x ∈ L∞ (H) can be uniquely expressed as x=



l,k=0

xlk |l k|,

(3.5)

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where the series is convergent in the w∗ -topology. The coordinates xlk ∈ R are also given by xlk = Tr(x|k l|). Recall that w∗ -convergence of the series (3.5) means that the numerical series ∞

∞  

xlk Tr ρ|l k| = xlk ρkl = Tr(xρ)

l,k=0

l,k=0

is convergent for any ρ ∈ L1 (H). Since the separable Hilbert space H is fixed throughout this paper we shall simplify the notation by writing L1 := L1 (H) and L∞ := L∞ (H). Shift operator notation. The shift operator S :=



|n n + 1|,

(3.6)

n=0

and its adjoint S T :=



|n + 1 n|,

(3.7)

n=0

turn out to give a very convenient coordinate description of various objects that we shall study in this paper. Note that the matrix of S has all entries of the upper diagonal equal to one and all other entries equal to zero whereas the matrix of S T has all entries of the lower diagonal equal to one and all other entries equal to zero. To facilitate various subsequent computations, we note that  k S k S T = I,

k−1

 T k k S S =I− pi ,

for k = 1, 2, . . . ,

(3.8)

i=0

where pi = |i i| : H → H are the orthogonal projectors on R|i ⊂ H for any i ∈ N ∪ {0}. ∞ and L1 ⊂ L1 denote the closed subspaces of diagonal operators and define the Let L∞ 0 ⊂L 0 1 bounded linear operators s, s˜ on both L∞ 0 and L0 by Sx = s(x)S S x = s˜ (x)S T

T

or

xS T = S T s(x),

or

xS = S s˜ (x)

and (3.9)

1 for x ∈ L∞ 0 or x ∈ L0 . The effect of the map s is that the ith coordinate of s(x) equals the (i + 1)st coordinate of x, that is, s(x0 , x1 , x2 , . . . , xn , . . .) := (x1 , x2 , . . . , xn , . . .) for any (x0 , x1 , x2 , . . . , xn , . . .) ∈ ∞ ∼ = L∞ 0 . Similarly, the effect of the map s˜ is that the ith coordinate of s˜ (x) equals the (i − 1)st coordinate of x and the zero coordinate of s˜ (x) is zero, that is, s˜ (x0 , x1 , x2 , . . . , xn , . . .) := (0, x0 , x1 , x2 , . . . , xn , . . .). Thus

s k ◦ s˜ k = id

and s˜ k ◦ s k = MI− k−1 pi , i=0

k = 1, 2, . . . ,

(3.10)

A. Odzijewicz, T.S. Ratiu / Journal of Functional Analysis 255 (2008) 1225–1272

1237

∞ ∞ where My : L∞ 0 → L0 is defined by My (x) := yx for any y ∈ L0 . The following identities are useful in several computations later on:         Tr ρs(x) = Tr s˜ (ρ)x and Tr s(ρ)x = Tr ρ s˜ (x) (3.11)

for any ρ ∈ L10 and x ∈ L∞ 0 , which means that s and s˜ are mutually adjoint operators. Any x ∈ L∞ and ρ ∈ L1 can be written as x=

∞ ∞



 T j S x−j + x0 + xi S i , j =1

ρ=

(3.12)

i=1





j =1

i=1

 T j S ρj + ρ0 +

ρ−i S i ,

(3.13)

1 where xi , x0 , x−j ∈ L∞ 0 and ρj , ρ0 , ρ−i ∈ L0 . Note the different conventions: the indices of the lower diagonals for the bounded operators are negative whereas for the trace class operators they are positive. This convention simplifies many formulas later on. The expressions (3.12) and (3.13) suggest the introduction, for every k ∈ Z, of the Banach subspaces

 ∞ ρnm = 0 for m = n + k ⊂ L∞ , L∞ k := ρ ∈ L  L1k := ρ ∈ L1 ρnm = 0 for m = n + k ⊂ L1

(3.14) (3.15)

consisting of operators whose only non-zero elements lie on the kth diagonal. We have the following Schauder decompositions L∞ =

L∞ k

and L1 =

k∈Z

L1k .

(3.16)

k∈Z

See [14, Chapter III, §15], namely Definition 15.1 (p. 485), Definition 15.3 (p. 487), and Theorem 15.1 (p. 489) for a detailed discussion of this concept and generalizations. The duality 1 relations between the various spaces L∞ n and Lk are given by Tr(ρk xn ) = δkn Tr(ρk xk )

if ρk ∈ L1k and xn ∈ L∞ −n .

(3.17)

∞ T k Finally, note that if k  0 then S k ∈ L∞ k , (S ) ∈ L−k , and

  j S l−j , Sl ST = (S T )j −l ,

if l  j, if l  j

(3.18)

which implies ρ, x =



Tr ρi xi

k∈Z

if ρ and x are expressed in the form (3.13) and (3.12).

(3.19)

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1 Banach subspaces of L1 (H) and L∞ (H). Given the Schauder basis {|n m|}∞ n,m=0 of L (or biorthogonal family of L∞ ) inducing the direct sum splitting (3.16), define the transposition operator T : L1 → L1 (or T : L∞ → L∞ ) by (ρ T )ij := ρj i for any i, j ∈ N ∪ {0}. We construct the following Banach subspaces of L1 :

∞ 1 1 1 k=−∞ Lk and L+ := k=0 Lk ; 1 1 1 T LS := {ρ ∈ L | ρ = ρ } and LA := {ρ ∈ L1 | ρ = −ρ T };   1 L1−,k := 0i=−k+1 L1i and L1+,k := k−1 i=0 Li , for k  1; −k ∞ 1 1 := 1 1 I−,k i=k Li , for k  1; i=−∞ Li and I+,k := L1S,k := L1S ∩ (L1+,k + L1−,k ) and L1A,k := L1A ∩ (L1+,k + L1−,k ),

• L1− := • • • •

0

for k  1.

1 is an ideal in L1 , I 1 is an ideal in L1 , but neither is Relative to operator multiplication, I−,k − +,k + an ideal in L1 . Therefore, relative to the commutator bracket, the same is true in the associated Banach Lie algebras. ∞ Similarly, using the biorthogonal family of functionals {|l k|}∞ l,k=0 in L inducing the direct ∞ sum splitting (3.16), we construct the following Banach subspaces of L :

• • • •

0

∞ k=−∞ Lk ∞ ∈ L | xT

∞

∞ k=0 Lk ; ∞ ∞ = x} and LA := {x ∈ L∞ | x T = −x}; LS := {x  k−1 ∞ 0 ∞ ∞ L∞ i=−k+1 Li and L+,k := i=0 Li , for k  1; −,k :=   −k ∞ ∞ := ∞ ∞ ∞ I−,k i=k Li , for k  1; i=−∞ Li and I+,k := ∞ ∞ ∞ ∞ ∞ ∞ ∞ LS,k := LS ∩ (L+,k + L−,k ) and LA,k := L∞ A ∩ (L+,k + L−,k ),

• L∞ − :=

and L∞ + :=

for k  1.

The following splittings of Banach spaces of trace class operators 1 L1 = L1− ⊕ I+,1 ,

1 L1 = L1S ⊕ I+,1 ,

1 L1− = L1−,k ⊕ I−,k

(3.20)

and of bounded operators ∞ L∞ = L∞ + ⊕ I−,1 ,

∞ L∞ = L∞ + ⊕ LA ,

∞ ∞ L∞ + = L+,k ⊕ I+,k

(3.21)

will be used below. The strongly nondegenerate pairing (3.2) relates the splittings (3.20) and (3.21) by  1 ∗  1 ◦ L− ∼ = I+,1 = L∞ +,  1 ∗  1 ◦ ∞ , I+,1 ∼ = L− = I−,1

 1 ∗  1 ◦ LS ∼ = I+,1 = L∞ +,  1 ∗  1 ◦ I+,1 ∼ = LS = L∞ A,

 1 ∗  1 ◦ L−,k ∼ = I−,k = L∞ +,k ,  1 ∗  1 ◦ ∞ I−,k ∼ (3.22) = L−,k = I+,k

where, as usual, ◦ denotes the annihilator of the Banach subspace in the dual of the ambient space. The splittings (3.20) and (3.21) define six projectors of L1 and L∞ , respectively. Let 1 , L1 , and I 1 1 P− , P01 , P+1 : L1 → L1 be the projectors whose ranges are I−,1 0 +,1 defined by the 1 1 1 1 1 1 1 1 splitting L = I−,1 ⊕ L0 ⊕ I+,1 . In particular P− + P0 + P+ = I. Let P−,k : L1− → L1− be the

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1239

1 . Define the six projecprojector whose range is L1−,k defined by the splitting L1− = L1−,k ⊕ I−,k tors

R− := P−1 + P01 ,

RS := P−1 + P01 + T ◦ P−1 ,

1 R−,k := P−,k ,

R+ := P+1 ,

RS,+ := P+1 − T ◦ P−1 ,

Rik := R− |L1 − R−,k −

(3.23)

associated to the splittings (3.20). The order of presentation of these projectors corresponds to the order of the splittings in (3.20). Similarly, the six projectors associated to the dual splittings (3.21) are given by ∗ R− := P+∞ + P0∞ ,

RS∗ := P+∞ + P0∞ + T ◦ P−∞ ,

∗ ∞ R−,k := P+,k ,

∗ := P−∞ , R+

∗ RS,+ := P−∞ − T ◦ P−∞ ,

∗ ∗ ∞ ∞ Rik := R− |L+ − P+,k ,

(3.24)

∞ , L∞ , I ∞ defined by where P−∞ , P0∞ , P+∞ : L∞ → L∞ are the projectors whose ranges are I−,1 0 +,1 ∞ ∞ ∞ ∞ ∞ ∞ ∞ the splitting L = I−,1 ⊕ L0 ⊕ I+,1 and P+,k : L+ → L+ is the projector with range L∞ +,k ∞ ∞ defined by the splitting L∞ + = L+,k ⊕ I+,k . ∞ All Banach spaces appearing in (3.21), with the exception of L∞ k and L+,k , are Banach sub∞ ∞ ∞ algebras of L or L+ whereas I+,k , for k ∈ N, are ideals of the Banach algebra L∞ + (but not ∞ define a filtration of L∞ and hence L∞ ∼ L∞ /I ∞ inherits the strucof L∞ ). Therefore, I+,k = + + +,k +,k ture of an associative Banach algebra. Thus all these associative Banach algebras are naturally ∞ ⊂ L∞ . Banach Lie algebras. The same considerations apply to the Banach ideals I−,k − It will be useful in our subsequent development to distinguish between the projectors defined in (3.23) and (3.24) and the corresponding maps onto their ranges. We shall denote by π− , π+ , πS , and πS,+ the maps on L1 equal to R− , R+ , RS , and RS,+ but viewed as taking 1 , im R = L1 , and im R 1 values in im R− = L1− , im R+ = I+,1 S S,+ = I+,1 , respectively. Similarly, S denote by π−,k and πik the maps on L1− equal to R−,k and Rik , but viewed as having values 1 , respectively. For the projectors on L∞ we shall denote in im R−,k = L1−,k and im Rik = I−,k ∞ ∞ ∞ ∞ ∗ , R ∗ , R ∗ , and R ∗ by π+ , π− , πS , and πA the maps equal to R− + S S,+ viewed as having values in ∞ ∗ ∞ ∗ = L∞ , respectively. Finally, let π ∞ and ∗ ∞ ∗ im R− = L+ , im R+ = I−,1 , im RS = LS , and im RS,+ A +,k ∞ denote the maps on L∞ equal to R ∗ and R ∗ viewed as having values in im R ∗ = L∞ πik + −,k ik −,k +,k ∗ = I ∞ , respectively. and im Rik +,k

Associated Banach Lie groups. Note that the Banach Lie group  GL∞ := x ∈ L∞ | x is invertible

(3.25)

has Banach Lie algebra L∞ and is open in L∞ . Define the closed Banach Lie subgroup of upper triangular operators in GL∞ by ∞ ∞ GL∞ + := GL ∩ L+ .

(3.26)

∞ ∞ Since GL∞ + is open in L+ , we can conclude that its Banach Lie algebra is L+ . Define the closed ∞ Banach Lie subgroup of orthogonal operators in GL by

 O ∞ := x ∈ L∞ xx T = x T x = I .

(3.27)

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∞ consists of all bounded skew-symmetric operators. The Banach Lie algebra L∞ A of O Denote by

   ∞ ∞ ∞ ∞ GI ∞ +,k := I + I+,k ∩ GL+ = I + ϕ ϕ ∈ I+,k , I + ϕ is invertible in GL+

(3.28)

∞ formed by the group of all bounded invertible upper triangular opthe open subset of I + I+,k erators whose strictly upper (k − 1)-diagonals are identically zero and whose diagonal is the identity. This is a closed normal Banach Lie subgroup of GL∞ + whose Lie algebra is the closed ∞ . ideal I+,k ∞ does not consist Remark. Unlike the situation encountered in finite-dimensions, the set I + I+,k ∞ that is only of invertible bounded linear isomorphisms. An example of an operator in I + I+,2 ∞ 1 2 not onto is given by I − S , where S is the shift operator defined in (3.6), since n=0 n+1 |n ∈ / 2 im(I − S ). Returning to the general case, define the product

x ◦k y :=

 l k−1

l=0

 i

xi s (yl−i ) S l

(3.29)

i=0

k−1 ∞ i i of the elements x = k−1 i=0 xi S and y = i=0 yi S ∈ L+,k , where xi , yi are diagonal op∞ erators. Relative to ◦k , the Banach space L+,k is an associative Banach algebra with unity. ∞ : L∞ → (L∞ , ◦ ) is an associative Banach It is easy to see that the projection map π+,k + +,k k ∞ algebra homomorphism whose kernel is I+,k . So, it defines a Banach algebra isomorphism ∞ ] : L∞ /I ∞ → (L∞ , ◦ ) of the factor Banach algebra L∞ /I ∞ with (L∞ , ◦ ). [π+,k + +,k + +,k +,k k +,k k The associative algebra L∞ with the commutator bracket +,k [x, y]k := x ◦k y − y ◦k x =

k−1

l

 i  xi s (yl−i ) − yi s i (xl−i ) S l

(3.30)

l=0 i=0

is the Banach Lie algebra of the group  GL∞ +,k

= g=

k−1

g i S g i ∈ L∞ 0 , |g0 |  ε(g0 )I for some ε(g0 ) > 0 i

 (3.31)

i=0

of invertible elements in (L∞ +,k , ◦k ). In (3.31), the inequality |g0 |  ε(g0 )I means the component wise inequalities for the diagonal operators, that is, |g0p |  ε(g0 ) for all p ∈ N ∪ {0}, where g0 := diag(g00 , g01 , . . . , g0p , . . .). Remark. It is important to note that invertibility in the Banach algebra (L∞ +,k , ◦k ) does not mean , that is, I − S 2 is an invertible invertibility of the operator on H. For example, I − S 2 ∈ GL∞ +,3 ∞ 2 element in (L+,3 , ◦3 ), but I − S is not an invertible operator, as noted in the previous remark.

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∞ ∞ ∞ ∞ Note that (L∞ +,k , [·,·]k ) is not a Banach Lie subalgebra of L+ . Since π+,k : L+ → L+,k is also a Banach Lie algebra homomorphism one has

  ∞ [x, y] [x, y]k = π+,k

for x, y ∈ L∞ +,k .

(3.32)

∞ (GL∞ ) ⊂ GL∞ , since every invertible operator in L∞ is mapped by the hoNote that π+,k + + +,k ∞ to an invertible element of L∞ . Moreover, if x ∈ π ∞ (GL∞ ) ⊂ GL∞ , then momorphism π+,k + +,k +,k +,k

−1   ∞ (x) = g(I + ψ) I + ψ ∈ GI ∞ π+,k |GL∞ +,k +

 ∞ −1 for some g ∈ π+,k |GL∞ (x). +

∞ | ∞ )−1 (x), then there exists some gψ ∈ I ∞ , since g is invertible, such Indeed, if g  ∈ (π+,k GL+ +,k ∞ ∞ that g −1 g  = I + ψ ∈ GI ∞ +,k . The next proposition shows that the restriction of π+,k to GL+ has ∞ range equal to GL+,k . ∞ | ∞ : GL∞ → GL∞ is surjective Proposition 3.1. The Banach Lie group homomorphism π+,k GL+ + +,k ∞ ∞ ∞ ∞ and induces a Banach Lie group isomorphism π : GL /GI → GL for any k = 1, 2, . . . . + +,k +,k +,k ∞ : GL∞ → GL∞ is surjective is equivalent to proving that for any Proof. To show that π+,k + +,k ∞ such that there exists ϕk ∈ I+,k g0 + g1 S + · · · + gk−1 S k−1 ∈ GL∞ +,k

g0 + g1 S + · · · + gk−1 S k−1 + ϕk ∈ GL∞ +.

(3.33)

Assume for the moment that (3.33) holds. We shall draw a consequence from it. By (3.31), k g0 + g1 S + · · · + gk−1 S k−1 is in GL∞ +,k if and only if g0 is invertible. Decompose ϕk = αk S g0 + ∞ . Choosing N ∈ N large enough so that I − N1 αk S k ∈ GL∞ αk+1 , where αk+1 ∈ I+,k+1 + , we obtain N    1 k GL∞ α g0 + g1 S + · · · + gk−1 S k−1 + αk S k g0 + αk+1  I − S k + N = g0 + g1 S + · · · + gk−1 S k−1 + ϕk+1 ,

(3.34)

where  ϕk+1 =

N  

N j =2

j

    1  k j (−1) j αk S g0 + g1 S + · · · + gk−1 S k−1 + αk S k g0 + αk+1 N j

  ∞ . + αk+1 − αk S k g1 S + · · · + gk−1 S k−1 + αk S k g0 + αk+1 ∈ I+,k+1

(3.35)

∞ Therefore, if g0 + g1 S + · · · + gk−1 S k−1 + ϕk ∈ GL∞ + for some ϕk ∈ I+,k , then there exists some ∞ such that g0 + g1 S + · · · + gk−1 S k−1 + ϕk+1 ∈ GL∞ ϕk+1 ∈ I+,k+1 +. Now we prove the proposition by induction on k. If k = 1, then g0 ∈ GL∞ + by definition. Next, let us assume that (3.33) holds. As we just saw, it follows that (3.34) holds. Consider then g0 + g1 S + · · · + gk−1 S k−1 + gk S k ∈ GL∞ +,k and k−1 + g S k = (I + g S k g −1 ) ◦ as g + g S + · · · + g S decompose it in the group GL∞ 0 1 k−1 k k k +,k 0 (g0 + g1 S + · · · + gk−1 S k−1 ). Let us assume, that gk < min(1, g0 ) which implies that

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gk Sg0−1 < 1 and hence that I + gk S k g0−1 ∈ GL∞ + . By (3.34) there exists ϕk+1 ∈ I+,k+1 such ∞ k−1 + ϕk+1 ∈ GL+ . Thus we get that g0 + g1 S + · · · + gk−1 S

   I + gk S k g0−1 g0 + g1 S + · · · + gk−1 S k−1 + ϕk+1 = g0 + g1 S + · · · + gk S k + ψk+1 ∈ GL∞ + for   ∞ ψk+1 = I + gk S k g0−1 ϕk+1 + gk S k g0−1 (g1 S + · · · + gk−1 S k−1 ) ∈ I+,k+1 which proves the assertion (3.33) for any element in the connected component of GL∞ +,k . Since k ∞ {I + g1 S + · · · + gk S | g1 , . . . , gk diagonal operators in L } is a connected Banach Lie subgroup ∞ of the connected component of the identity in GL∞ +,k and any element of GL+,k can be written as of diagonal operators, a product of an element of this group and the Banach Lie subgroup GL∞ +,1 . 2 it follows that (3.33) holds for any element in GL∞ +,k Remark. There is a shorter proof of this proposition based on the following general remark, pointed out by the referee. If ϕ : G → H is a homomorphism of Banach Lie groups whose tangent map Te ϕ : g → h is surjective, then ϕ(G) contains the identity component of H because it is generated (as a group) by the image of the exponential map. In the concrete situation of the proposition, the surjectivity of the induced Banach Lie algebra homomorphism is very easy to show. −1 = g −1 + h S + · · · + h k−1 of In the Banach Lie group (GL∞ 1 k−1 S +,k , ◦k ), the inverse g 0 ∞ k−1 g = g0 + g1 S + · · · + gk−1 S ∈ GL+,k is given for all p = 1, . . . , k − 1 by

 hp = −g0−1

p

(−1)r−1 r=1

···s

i1 +···+iq−1



  gi1 s i1 g0−1 gi2 s i1 +i2 (g0 gi3 )

i1 +···+ir =p, i1 ,...,iq 1

  −1    p  −1  i1 +···+ir−1 −1 g0 gi q · · · s g0 gi r s g0 .

(3.36)

For example, here are the first elements:   h1 = −g0−1 g1 s g0−1 ,      h2 = −g0−1 g2 − g1 s g0−1 g1 s 2 g0−1 ,            h3 = −g0−1 g3 − g2 s 2 g0−1 g1 − g1 s g0−1 g2 + g1 s g0−1 g1 s 2 g0−1 g1 s 3 g0−1 ,            h4 = −g0−1 g4 − g3 s 3 g0−1 g1 − g2 s 2 g0−1 g2 − g1 s g0−1 g3 + g2 s 2 g0−1 g1 s 3 g0−1 g1         + g1 s g0−1 g1 s 2 g0−1 g2 + g1 s g0−1 g2 s 3 g0−1 g1         − g1 s g0−1 g1 s 2 g0−1 g1 s 3 g0−1 g1 s 4 g0−1 ,          h5 = −g0−1 g5 − g4 s 4 g0−1 g1 − g3 s 3 g0−1 g2 − g2 s 2 g0−1 g3 − g1 s g0−1 g4             + g3 s 3 g0−1 g1 s 4 g0−1 g1 + g2 s 2 g0−1 g2 s 4 g0−1 g1 + g2 s 2 g0−1 g1 s 3 g0−1 g2

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            + g1 s g0−1 g3 s 4 g0−1 g1 + g1 s g0−1 g2 s 3 g0−1 g2 + g1 s g0−1 g1 s 2 g0−1 g3             − g2 s 2 g0−1 g1 s 3 g0−1 g1 s 4 g0−1 g1 − g1 s g0−1 g2 s 3 g0−1 g1 s 4 g0−1 g1             − g1 s g0−1 g1 s 2 g0−1 g1 s 3 g0−1 g2 − g1 s g0−1 g1 s 2 g0−1 g2 s 4 g0−1 g1           + g1 s g0−1 g1 s 2 g0−1 g1 s 3 g0−1 g1 s 4 g0−1 g1 s 5 g0−1 . Coinduced Banach Lie–Poisson structures. After these preliminary remarks and notations let us apply the results of the previous section to the Banach Lie–Poisson space L1 . We shall drop the upper indices “ind” and “coind” on the Poisson brackets because it will be clear from the context which brackets are induced and coinduced on various subspaces. We start with points (i) of Proposition 2.2 and Proposition 2.4. So let us consider the diagram

where we recall that πS , πS,+ , π+ and π− are the projections onto the ranges of RS , RS+ , R+ , and R− , respectively, and ιS , ιS,+ , ι+ , and ι− are inclusions. We see from the above that the 1 )◦ = L∞ is a Banach Lie assumptions in part (i) of Proposition 2.4 are satisfied because (I+,1 + 1 ∗ ∞ subalgebra of (L ) = L . Thus we can conclude the following facts. (i) By Proposition 2.4(i) it follows that L1S and L1− are isomorphic Banach Lie–Poisson spaces with the Poisson brackets defined by formula (2.18). They are given, respectively, by       {f, g}S (σ ) = Tr ιS (σ ) D(f ◦ πS ) ιS (σ ) , D(g ◦ πS ) ιS (σ )

(3.37)

for σ ∈ L1S and f, g ∈ C ∞ (L1S ) and       {f, g}− (ρ) = Tr ι− (ρ) D(f ◦ π− ) ι− (ρ) , D(g ◦ π− ) ι− (ρ)

(3.38)

for ρ ∈ L1− and f, g ∈ C ∞ (L1− ). The linear continuous maps Φ−,S := π− ◦ ιS : L1S → L1− and ΦS,− := πS ◦ ι− : L1− → L1S are mutually inverse isomorphisms of the Banach Lie–Poisson spaces (L1S , {,}S ) and (L1− , {,}− ). 1 1 The coadjoint actions of the Banach Lie group GL∞ + on L− and LS are given by    + ∗ Ad g −1 ρ = π− gι− (ρ)g −1 for ρ ∈ L1− ,    S ∗ Ad g −1 σ = πS gιS (σ )g −1 for σ ∈ L1S and g ∈ GL∞ + . Differentiating these formulas relative to g at the identity, we get

(3.39) (3.40)

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 + ∗   ad x ρ = −π− x, ι− (ρ) for ρ ∈ L1− ,  S ∗   ad x σ = −πS x, ιS (σ ) for σ ∈ L1S

(3.41) (3.42)

1 1 1 1 for x ∈ L∞ + . The isomorphisms Φ−,S : LS → L− and ΦS,− : L− → LS are equivariant relative to these coadjoint actions, that is,

 S ∗ ∗  Ad g −1 ◦ ΦS,− = ΦS,− ◦ Ad+ g −1 , ∗   + ∗ Ad g −1 ◦ Φ−,S = Φ−,S ◦ AdS g −1

(3.43) (3.44)

for any g ∈ GL∞ +. 1 ∞ and L∞ . Thus (3.20)– (ii) By (3.22), I+,1 is the predual of the two Banach Lie algebras I−,1 A (3.24) and point (ii) of Proposition 2.4 imply that I+,1 carries two different Lie–Poisson brackets, namely by (2.18) we have       {f, g}+ (ρ) = Tr ι+ (ρ) D(f ◦ π+ ) ι+ (ρ) , D(g ◦ π+ ) ι+ (ρ)

(3.45)

      {f, g}S,+ (ρ) = Tr ιS+ (ρ) D(f ◦ πS,+ ) ιS,+ (ρ) , D(g ◦ πS,+ ) ιS,+ (ρ) ,

(3.46)

and

1 , f, g ∈ C ∞ (I 1 ). where ρ ∈ I+,1 +,1 1 ∞ The coadjoint actions (Ad− )∗ and (AdA )∗ of the groups GI ∞ −,1 and O , respectively on I+,1 , are given by

 − ∗   Ad h−1 ρ = π+ hι+ (ρ)h−1

for h ∈ GI ∞ −,1

(3.47)

  ρ = πS+ gιS,+ (ρ)g −1 for g ∈ O ∞ ,

(3.48)

and 

AdA

∗

g −1

1 . We shall not pursue the investigation of this interesting case in this paper. where ρ ∈ I+,1

Induced Banach Lie–Poisson structures. We begin with the study of the lower triangular case. 1 1 Denote by ι−,k : L1−,k → L1− the inclusion and let ι−1 −,k : ι−,k (L−,k ) → L−,k be its inverse (de∞ ∗ ∞ fined, of course, only on the range of ι−,k ). Then ι∗−,k : L∞ + → L+,k . Since ker ι−,k = I+,k is an + ∗ 1 1 ∞ ideal in L+ , by Proposition 2.1 we have (Ad )g −1 ι−,k (L−,k ) ⊂ ι−,k (L−,k ) for any g ∈ GL∞ +. 1 ∞ Therefore there are GL∞ + and L+ coadjoint actions on L−,k defined by

    +,k ∗ −1 for ρ ∈ L1−,k and g ∈ GL∞ ρ := ι−1 Ad +, −,k π− g(ι− ◦ ι−,k )(ρ)g g −1  +,k ∗    ad ρ := ι−1 for ρ ∈ L1−,k and x ∈ L∞ +. −,k π− x, (ι− ◦ ι−,k )(ρ) x

(3.49) (3.50)

Since the action (3.49) is trivial for all elements of the closed normal Lie subgroup GI ∞ +,k , ∞ ∞ ∼ /GI given by (3.49) that will be it induces the coadjoint action of the group GL∞ GL = + +,k +,k

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also denoted by (Ad+,k )∗ . Similarly, the Lie algebra action (3.50) is trivial for all elements in the ∞ so it induces the coadjoint action of the Lie algebra L∞ ∼ L∞ /I ∞ on L1 closed ideal I+,k +,k = +,k −,k denoted also by (ad+,k )∗ . One can express (3.49) and (3.50) in terms of the expansions ρ = ρ0 + S T ρ1 + · · · + T (S )k−1 ρk−1 ∈ L1−,k , x = x0 + x1 S + · · · + xk−1 S k−1 ∈ L∞ +,k , and g = g0 + g1 S + · · · + in the following way: gk−1 S k−1 ∈ GL∞ +,k  +,k ∗ ρ= Ad g −1

k−1

 T j −i−l l  j  i   S s˜ s s˜ (gi ) ρj hl ,

(3.51)

i,j,l=0, j i+l

where the diagonal operators hl are expressed in terms of the gi in (3.36), and (using (3.18)) k−1 k−1

 +,k ∗  T j

 i−j  ad S s˜ (ρi xi−j ) − ρi s j (xi−j ) . ρ = x j =0

(3.52)

i=j

By (3.30) and (3.19), the Lie–Poisson bracket on L1−,k is given by     {f, g}k (ρ) = Tr ρ Df (ρ), Dg(ρ) k       k−1

l

δf δg δf δg = Tr ρl (ρ)s i (ρ) − (ρ)s i (ρ) δρi δρl−i δρi δρl−i

(3.53)

l=0 i=0

for f, g ∈ C ∞ (L1−,k ), where defined by Df (ρ) =

δf δρi (ρ) denotes the partial functional δf δf δf k−1 . δρ0 (ρ) + δρ1 (ρ)S + · · · + δρk−1 (ρ)S

derivative of f relative to ρi

If in the previous formulas we let k = ∞ one obtains the Lie–Poisson bracket on L1− . Indeed, the Lie–Poisson bracket {f, g}− on L1− given by (3.38) expressed in the coordinates {ρi }∞ i=0 equals (3.53) for k = ∞. Proposition 3.2. The Lie–Poisson bracket (3.53) on L1−,k coincides with the induced bracket (2.10) determined by the inclusion ι−,k : L1−,k → L1− and the Lie–Poisson bracket (3.38) on L1− . Proof. We need to prove that the induced bracket (2.10) evaluated on two linear function∞ 1 ∼ 1 ∗ als x, y ∈ L∞ +,k = (L−,k ) ⊂ C (L−,k ) coincides with [x, y]k . To see this we note that −1 ∞ ∞ D(x ◦ ι−,k ◦ R−,k )(ι−,k (ρ)) = ι+,k x ∈ L∞ + , where ι+,k : L+,k → L+ is the inclusion. Then, a direct verification shows that for any ρ ∈ L1−,k we have       {x, y}ind (ρ) = [ι+,k x, ι+,k y], ι−,k ρ = Tr [x, y]ρ = Tr [x, y]k ρ by (3.30).

2

Let us study now the symmetric representation of (L1−,k , {·,·}−,k ) for k ∈ N ∪ {∞}. This will be done by using the Banach Lie–Poisson space isomorphism ΦS,− := πS ◦ ι− : L1− → L1S . Let π−,k : L1− → L1−,k and πS,k : L1S → L1S,k be the projections with the indicated ranges and

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ιS,k : L1S,k → L1S the inclusion. Define ΦS,−,k := πS,k ◦ ΦS,− ◦ ι−,k : L1−,k → L1S,k . The following commutative diagram illustrates these maps: L1− ι−,k

ΦS,−

π−,k

L1−,k

L1S ιS,k

ΦS,−,k

πS,k

L1S,k .

Pushing forward the Poisson bracket {·,·}k on L1−,k by the Banach space isomorphism ΦS,−,k endows L1S,k with an isomorphic Poisson structure denoted by {·,·}S,k . From Propositions 2.4 and 3.2, all the maps in the diagram above are linear Poisson maps, with the exception of π−,k 1 1 and πS,k which are not Poisson. Recall that GL∞ + acts on L− and LS by (3.39) and (3.40), ∞ ∞ 1 respectively, and that GL+ (and hence GL+,k ) acts on L−,k by (3.49). Using the isomorphisms ΦS,− and ΦS,−,k to push forward these actions to L1S and L1S,k , respectively, all the maps in the ∞ diagram above are also GL∞ + -equivariant. Consequently, one has the GL+ -invariant filtrations         ι−,1 L1−,1 → ι−,2 L1−,2 → · · · → ι−,k L1−,k → ι−,k+1 L1−,k+1 → · · · → L1− , (3.54)         ιS,1 L1S,1 → ιS,2 L1S,2 → · · · → ιS,k L1S,k → ιS,k+1 L1S,k+1 → · · · → L1S (3.55) of Banach Lie–Poisson spaces predual to the sequence ∞ ∞ ∞ ∞ L∞ + −→ · · · −→ L+,k −→ L+,k−1 −→ · · · −→ L+,2 −→ L+,1

(3.56)

∞ ∞ of Banach Lie algebras in which each arrow is the surjective projector π+,k,k−1 : L∞ +,k → L+,k−1 that maps k-diagonal upper triangular operators to (k − 1)-diagonal upper triangular operators ∞ ∞ = π∞ ◦ π+,k by eliminating the kth diagonal. We have π+,k,k−1 +,k−1 .

4. Dynamics generated by Casimirs of L1 (H) We begin by presenting Hamilton’s equations on L1− and L1S given by arbitrary smooth functions h and f defined on the relevant Banach Lie–Poisson spaces. Using formula (2.19) of Proposition 2.2, one obtains Hamilton’s equations       d ρ = π− D(h ◦ π− ) ι− (ρ) , ι− (ρ) for ρ ∈ L1− and h ∈ C ∞ L1− , dt       d σ = πS D(f ◦ πS ) ιS (σ ) , ιS (σ ) for σ ∈ L1S and f ∈ C ∞ L1S , dt

(4.1) (4.2)

on the isomorphic Banach Lie–Poisson spaces (L1− , {·,·}− ) and (L1S , {·,·}S ); from Section 3 we ∼ know that this isomorphism is ΦS,− := πS ◦ ι− : (L1− , {·,·}− ) − → (L1S , {·,·}S ). Therefore, if f ◦ ΦS,− = h then Eqs. (4.1) and (4.2) give the same dynamics. Recall that π− : L1 → L1− and πS : L1 → L1S are, by definition, the projectors P−1 + P0 : L1 → L1 and πS := P−1 + P0 + T ◦ P−1 : L1 → L1 considered as maps on their ranges (see (3.23) and the subsequent comments) and ι− : L1− → L1 , ιS : L1S → L1 are the inclusions.

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Dynamics associated to the restriction of Casimirs. Now let us observe that the family of functions Il ∈ C ∞ (L1 ) defined by Il (ρ) :=

1 Tr ρ l l

for l ∈ N,

(4.3)

are Casimir functions on the Banach Lie–Poisson space (L1 , {·,·}). This follows from (3.1) since one has  ∗ DIl (ρ) = ρ l−1 ∈ L1 ⊂ L∞ ∼ = L1 .

(4.4)

Restricting Il to ι− : L1− → L1 and ιS : L1S → L1 we obtain for all l ∈ N   Il− (ρ) := Il ι− (ρ) for ρ ∈ L1− ,   IlS (σ ) := Il ιS (σ ) for σ ∈ L1S .

(4.5) (4.6)

According to Corollary 2.3(i), (4.5) and (4.6) form two infinite families of integrals in involution  − −  Il , Im − = 0 and IlS , ImS S = 0

for l, m ∈ N.

(4.7)

Since IlS ◦ ΦS,− = Il− , the Hamiltonians Il− and IlS define on (L1− , {·,·}− ) (or (L1S , {·,·}S )) different families of dynamical systems. Firstly, we shall investigate the systems associated to the Hamiltonians Il− given by (4.5). As we shall see, the framework of the Banach Lie–Poisson space (L1− , {·,·}− ) is more natural in this case. Hence, taking into account Corollary 2.3(ii), substituting Il− into (4.1), then applying ι− to (4.1), and using (4.4), yields the family of Hamilton equations on L1−   l−1   ∂ι− (ρ)  1 , ι− (ρ) = P− + P01 P+∞ + P0∞ ι− (ρ) ∂tl

(4.8)

or, equivalently, in Lax form    l−1    l−1   ∂ι− (ρ) , ι− (ρ) = P0∞ ι− (ρ) , ι− (ρ) , = − P−∞ ι− (ρ) ∂tl

(4.9)

where tl denotes the time parameter for the lth flow. Eq. (4.8) implies that its solution is given by the coadjoint action of the group GL∞ + on the satisfying dual L1− of its Lie algebra. Hence, there is some smooth curve R  tl → h+ (tl ) ∈ GL∞ + (Ad+ )∗h (t )−1 ◦ (Ad+ )∗h (s )−1 = (Ad+ )∗h (t +s )−1 such that + l

+

+ l

l

 ∗   ι− ρ(tl ) = Ad+ h

+ (tl )

−1

l

     ρ(0) = P−1 + P01 h+ (tl )ι− ρ(0) h+ (tl )−1

(4.10)

is the solution of (4.8) with initial condition ρ(0) for tl = 0. On the other hand, the solution of (4.9) is given by     ι− ρ(tl ) = h− (tl )−1 ι− ρ(0) h− (tl ),

(4.11)

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for a smooth one-parameter subgroup R  tl → h− (tl ) ∈ GL∞ determined. − that can be explicitly ∞ T i We shall do this by using the decomposition ι− (ρ) = ρ0 + ρ− , where ρ− = i=1 (S ) ρi and ρi ∈ L10 if i ∈ N ∪ {0}. Since P0∞ ([ι− (ρ)]l−1 ) = ρ0l−1 , Eq. (4.9) becomes     ∂ ι− (ρ) = ρ0l−1 , ρ0 + ρ− = ρ0l−1 , ρ− ∂tl which is equivalent to   ∂ ρ− = ρ0l−1 , ρ− and ∂tl

∂ ρ0 = 0. ∂tl

(4.12)

It immediately follows that its solution is given by (4.11) with h− (tl ) = e−tl ρ0 (0)

l−1

,

(4.13)

where ρ(0) = ρ0 (0) + ρ− (0) is the initial value of ρ at time tl = 0. Note that h− (tl ) ∈ GL∞ − is in fact a diagonal operator which can also be obtained from the decomposition etl [ι− (ρ(0))]

l−1

= k− (tl )h− (tl )−1 ,

(4.14)

where k− (tl ) ∈ GI ∞ −,1 . It follows that we can write the solution also in the form      ι− ρ(tl ) = k− (tl )−1 ι− ρ(0) k− (tl ).

(4.15)

Finally, note that in (4.10) we can choose h+ (tl ) = h− (tl ) since also h− (tl ) ∈ GL∞ +. Let us analyze the system (4.9) in more detail. We begin by noting that there is an isome∞ and between 1 and the try between ∞ and the diagonal bounded linear operators L∞ 0 ⊂L 1 1 diagonal trace class operators L0 ⊂ L . Fix a strictly lower triangular element ν− =

k−1

 T i S νi ∈ L1−,k

where k ∈ N ∪ {∞},

(4.16)

i=1

and define the map Jν− : ∞ × 1 → L1−,k by Jν− (q, p) := p + eq ν− e−q ,

(4.17)

where, on the right-hand side, we identify p and q with diagonal operators and eq is the exponential of q. It is easy to see that this map is smooth and that Jν− (q, p) = Jν− (q + αI, p), for any α ∈ R. The weak symplectic manifold (∞ × 1 , ω). In this paper we shall often work with the weak symplectic manifold ( ∞ × 1 , ω), where ∞ and 1 are the Banach spaces of bounded real sequences and absolutely convergent real series, respectively, whose norms are given by

q ∞ := sup |qk |, k=0,1,...

∞ q := {qk }∞ k=0 ∈ ,

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1249

and

p 1 :=



1 p := {pk }∞ k=0 ∈ .

|pk |,

k=0

The strongly nondegenerate duality pairing q, p =



q k pk ,

for q ∈ ∞ , p ∈ 1 ,

(4.18)

k=0

establishes the Banach space isomorphism ( 1 )∗ = ∞ and the weak symplectic form ω is given by   ω (q, p), (q , p ) := q, p − q , p ,

for q, q ∈ ∞ , p, p ∈ 1 .

(4.19)

The differential form ω is conveniently written as ω=



dqk ∧ dpk

(4.20)

k=0 1 in the coordinates qk , pk . Let us elaborate on the notation used in (4.20). If p = {pk }∞ k=0 ∈ , 1 denote by {∂/∂pk }∞ k=0 the basis of the tangent space Tp corresponding to the standard Schauder ∞ 1 ∞ basis {|k }k=0 of . The same basis in has a different meaning: every element a := {ak }∞ k=0 ∈ ∞ ∞ can be uniquely written as a weakly convergent series a = k=0 ak |k . With this notion of ∞ basis in ∞ , given q ∈ ∞ , the sequence {∂/∂qk }∞ k=0 denotes the basis of the tangent space Tq ∞ ∞ 1 corresponding to {|k }k=0 . Thus, any smooth vector field X on × is written as

X(q, p) =

∞ 

Ak (q, p)

k=0

 ∂ ∂ , + Bk (q, p) ∂qk ∂pk

∞ ∞ 1 where {Ak (q, p)}∞ k=0 ∈ and {Bk (q, p)}k=0 ∈ . ∞ ∞ 1 If Y is another vector field whose coefficients are {Ck (q, p)}∞ k=0 ∈ , {Dk (q, p)}k=0 ∈ , employing the usual conventions for the exterior derivatives of coordinate functions to represent elements in the corresponding dual spaces, formula (4.20) gives





k=0

 dqk ∧ dpk (X, Y )(q, p) =



  Ak (q, p)Dk (q, p) − Ck (q, p)Bk (q, p) k=0

which equals (4.19). It is in this sense that the writing in (4.20) represents the weak symplectic form (4.19). 1 the space of bidiagonal trace class operators having non-zero entries only on Denote by B−,k the main and the lower (k − 1)st diagonal. For ν− = (S T )k−1 νk−1 ∈ L1−k+1 ⊂ L1−,k , we shall 1 is a momentum map in the followprove in Proposition 5.2 that the map Jν− : ∞ × 1 → B−,k 1 ), we have {ϕ ◦ J , ing sense: for any two locally defined smooth functions ϕ, ψ ∈ C ∞ (B−,k ν−

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ψ ◦ Jν− }ω = {ϕ, ψ} ◦ Jν− . See [12] for the general definition and properties of momentum maps on Banach Poisson and weak symplectic manifolds. The solution of the system (4.9). We shall argue below, in analogy with the finite-dimensional case, that (q, p) can be considered as angle–action coordinates for the Hamiltonian system (4.9). We begin by recalling that the solution of (4.9) is given by ι− (ρ(tl )) = h− (tl )−1 ι− (ρ(0))h− (tl ), l−1 where h− (tl ) = e−tl ρ0 (0) , ρ(0) = ρ0 (0) + ρ− (0) ∈ L1− is the initial value of the variable ρ at tl = 0, ρ0 ∈ L10 a diagonal operator, and ρ− a strictly lower triangular operator. Therefore, h− (tl )h− (tm ) = h− (tm )h− (tl ) for any l, m ∈ N, and hence the product h− (t) := h− (t1 , t2 , . . .) :=

∞ 

h− (tl )

(4.21)

l=1

is independent on the order of the factors and it exists as an invertible bounded operator if we assume that t := (t1 , t2 , . . .) ∈ ∞ 0 which means that t has only finitely many non-zero elements. One also has   ∞

−1 l−1 tl ρ0 (0) , p for t ∈ ∞ (4.22) h− (t) Jν− (q, p)h− (t) = Jν− q + 0 , l=1

which shows that the flow in the coordinates (q, p) is described by a straight line motion in q with p conserved. If this would be a finite-dimensional system, since (q, p) are also Darboux coordinates (see (4.19) or (4.20)), we would say that they are action–angle coordinates on Jν− ( ∞ × 1 ). In infinite dimensions, even the definition of action–angle coordinates presents problems. First, if the symplectic form is strong, the Darboux theorem (that is, the symplectic form is locally constant) is valid; see the proof of Theorem 3.2.2 in [1]. Second, if the symplectic form is weak, which is our case, the Darboux theorem fails in general, even if the manifold is a Hilbert space; Marsden’s classical counterexample can be found and discussed in Exercise 3.2H of [1]. Third, even if one could show in a particular case that the Darboux theorem holds, there still is the problem of coordinates. In the case presented above, the action–angle coordinates were constructed explicitly. In general, on Banach weak symplectic manifolds this may well be impossible. The solution of the system (4.6). We return now to the systems described by the family of integrals in involution IlS given by (4.6). By Corollary 2.3(ii), substituting IlS into (4.2), applying ιS to (4.2), and using (4.4), yields the family of Hamilton equations on L1S   l−1   ∂ιS (σ )  1 , ιS (σ ) = P− + P01 + T ◦ P−1 P+∞ + P0∞ + T ◦ P−∞ ιS (σ ) ∂tl

(4.23)

or, equivalently, in Lax form   l−1   ∂ιS (σ ) , ιS (σ ) , = − P−∞ − T ◦ P−∞ ιS (σ ) ∂tl where tl denotes the time parameter for the lth flow.

(4.24)

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1251

From (4.23) it follows that the solution of this equation can be written in terms of the coadjoint 1 action of the Banach Lie group GL∞ + on the dual LS of its Lie algebra. More precisely, the solution is necessarily of the form  ∗   ιS σ (tl ) = AdS g¯

+ (tl )

−1

     σ (0) = P−1 + P01 + T ◦ P−1 g¯ + (tl )ιS σ (0) g¯ + (tl )−1 (4.25)

for some smooth curve R  tl → g¯ + (tl ) ∈ GL∞ + and σ (0) the initial condition for tl = 0. On the other hand, the solution of (4.24) is     ιS σ (tl ) = gS (tl )T ιS σ (0) gS (tl ),

(4.26)

where R  tl → gS (tl ) ∈ O ∞ is a smooth curve that will be determined in the next proposition by the same method as in the finite-dimensional case. Proposition 4.1. Assume that we have the decomposition (we set here t = tl ) l−1

et[ιS (σ (0))]

= gS (t)g+ (t)

(4.27)

for gS (t) ∈ O ∞ and g+ (t) ∈ GL∞ + . Then         ιS σ (t) := gS (t)T ιS σ (0) gS (t) = g+ (t) ιS σ (0) g+ (t)−1

(4.28)

is the solution of (4.24) with initial condition ιS (σ (0)). Proof. To prove the first equality in (4.28), use (4.27) to get l−1

gS (t) = et[ιS (σ (0))]

g+ (t)−1

and hence g+ (t)e−t[ιS (σ (0))]

l−1

      l−1 ιS σ (0) et[ιS (σ (0))] g+ (t)−1 = g+ (t) ιS σ (0) g+ (t)−1 l−1

since ιS (σ (0)) commutes with et[ιS (σ (0))] . Let ιS (σ (t)) := gS (t)−1 [ιS (σ (0))]gS (t). Taking the time derivative of (4.27) and multiplying on the right by gS (t)−1 and on the left by g+ (t)−1 we get l−1   = gS (t)−1 g˙ S (t) + g˙ + (t)g+ (t)−1 ιS σ (t) which is equivalent to the equations    l−1  gS (t)−1 g˙ S (t) = P−∞ − T ◦ P−∞ ιS σ (t) ,   l−1   . g˙ + (t)g+ (t)−1 = P+∞ + P0∞ + T ◦ P−∞ ιS σ (t) Therefore

(4.29) (4.30)

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 d  ιS σ (t) dt

      = −gS (t)−1 g˙ S (t)gS (t)−1 ιS σ (0) gS (t) + gS (t)−1 ιS σ (0) g˙ S (t)   l−1        l−1   ιS σ (t) + ιS σ (t) P−∞ − T ◦ P−∞ ιS σ (t) = − P−∞ − T ◦ P−∞ ιS σ (t)  l−1    , ιS (σ ) = − P−∞ − T ◦ P−∞ ιS (σ )

which is (4.24).

2

This proposition shows that the solution (4.28) of the system (4.24) can be expressed using l−1 the Iwasawa decomposition for the operator et[ιS (σ (0))] . Note that since the Gram–Schmidt method is valid in separable Hilbert spaces for a Hilbert basis indexed by N, each operator has ∞ an Iwasawa decomposition. However, an Iwasawa decomposition GL∞ = O ∞ · GL∞ 0 · GI +,1 for ∞ the Banach Lie group GL is not valid. The reason is that the closed Banach Lie subalgebras corresponding to these three factors do not sum up to the whole space L∞ which implies that the map intervening in the Iwasawa decomposition is not smooth. This is in sharp contrast with the polar decomposition theorem; see the appendix in [11]. Note also that (4.28) produces a smooth curve g+ (t) ∈ GL∞ + satisfying (4.25) even without the projection operator in that formula. This follows also directly from (4.28) and (3.40). The previous general considerations involving Proposition 2.1, imply that the families of flows given by (4.1) or (4.2) and, in particular by (4.9) or (4.24), not only preserve the symplectic leaves of L1− and L1S , but also the filtrations (3.54) and (3.55), respectively. This remark has some important consequences which we discussed below. We turn now to the study of Hamiltonian systems induced on the filtrations (3.54) and (3.55). A k-diagonal Hamiltonian system is, by definition, a Hamiltonian system on (L1−,k , {·,·}k ). Since the map ΦS,−,k : (L1−,k , {·,·}k ) → (L1S,k , {·,·}S,k ) introduced at the end of Section 3 is a Banach Lie–Poisson space isomorphism, we can regard k-diagonal Hamiltonian systems as being defined also on (L1S,k , {·,·}S,k ). From (3.52), Hamilton’s equations on (L1−,k , {·,·}k ) defined by an arbitrary function hk ∈ C ∞ (L1−,k ) are given by     k−1 

d δhk δhk s˜ l−j ρl − ρl s j for j = 0, 1, 2, . . . , k − 1. ρj = − dt δρl−j δρl−j

(4.31)

l=j

Note that for all n > k (including n = ∞), any hk ∈ C ∞ (L1−,k ) can be smoothly extended to hn := hk ◦ πkn ∈ C ∞ (L1−,n ), where πkn : L1−,n → L1−,k is the projection that eliminates the last  lower n − k diagonals of an operator in L1−,n := 0i=−n+1 L1i . Conversely, any hn ∈ C ∞ (L1−,n ) gives rise to a smooth function hk := hn ◦ ιnk ∈ C ∞ (L1−,k ), where ιnk : L1−,k → L1−,n is the natural inclusion. Since the flow defined by h ∈ C ∞ (L1− ) preserves the filtration (3.54) (see Proposition 2.1) it follows that if the initial condition ρ(0) ∈ L1−,k its trajectory is necessarily contained in L1−,k . This means that in order to solve the system (4.31) for a given k ∈ N, it suffices to solve the Hamiltonian system given by the extension of hk to (L1− , {·,·}− ) for initial conditions in L1−,k . Let us now specialize the functions hk ∈ C ∞ (L1−,k ) and fk ∈ C ∞ (L1S,k ) to

A. Odzijewicz, T.S. Ratiu / Journal of Functional Analysis 255 (2008) 1225–1272

    Il−,k (ρ) := Il− ι−,k (ρ) = Il (ι− ◦ ι−,k )(ρ) for ρ ∈ L1−,k ,     IlS,k (σ ) := IlS ιS,k (σ ) = Il (ιS ◦ ιS,k )(σ ) for σ ∈ L1S,k ,

1253

(4.32) (4.33)

respectively, where ι−,k : L1−,k → L1− and ιS,k : L1S,k → L1S are the inclusions. Note that since

IlS,k ◦ ΦS,−,k = Il−,k , the dynamics induced by the functions Il−,k and IlS,k are different in spite of the fact that the Poisson structures on L1−,k and L1S,k are isomorphic. Therefore, we see that on has the family of Hamiltonian systems indexed by k ∈ N which have an infinite number of integrals in involution indexed by l ∈ N. For k = 2 the system is the semi-infinite Toda lattice. Therefore, the k-diagonal semi-infinite Toda systems are defined to be the Hamiltonian systems on L1S,k associated to the functions IlS,k , l ∈ N. An important consequence of the fact that the Poisson brackets on L1−,k and L1S,k are induced

is that the method of solution of the corresponding Hamilton equations for Il−,k and IlS,k , respectively, can be obtained by solving these equations on L1− and L1S , respectively. Namely, it suffices to work with the equations of motion (4.9) and (4.24) with initial conditions ρ(0) ∈ L1−,k and σ (0) ∈ L1S,k , respectively, and use Proposition 4.1. We shall do this in the rest of the paper for a special case related to the semi-infinite Toda system. 5. The bidiagonal case

In this section we shall study in great detail the bidiagonal case consisting of operators that have only two non-zero diagonals: the main one and the lower k − 1 diagonal. As an application of the obtained results we give a rigorous functional analytic formulation of the integrability of the semi-infinite Toda lattice. The coordinate description of the bidiagonal subcase. Due to their usefulness in the study of the Toda lattice, we shall express in coordinates several formulas from Section 3 adapted to the ∞ ⊂ L∞ , k  2, consisting of bidiagonal elements subalgebra B+,k +,k x := x0 + xk−1 S k−1 =



  x0,ii |i i| + xk−1,ii |i i + k − 1| ,

(5.1)

i=0

where x0 , xk−1 are diagonal operators whose entries are given by the sequences {x0,ii }∞ i=0 , ∞ ∞ ∞ ∞ {xk−1,ii }i=0 ∈ , respectively. The subalgebra B+,k of L+,k is hence formed by upper triangular bounded operators that have only two non-zero diagonals, namely the main diagonal and the strictly upper k − 1 diagonal. ∞ is B 1 The predual of B+,k −,k which consists of lower triangular trace class operators having only two non-vanishing diagonals, namely the main one and the strictly lower k − 1 diagonal (k  2), that is, they are of the type ρ = ρ0 + (S k−1 )T ρk−1 =



  ρ0,ii |i i| + ρk−1,ii |i + k − 1 i| , i=0

(5.2)

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where ρ0 and ρk−1 are diagonal operators whose entries are given by the sequences {ρ0,ii }∞ i=0 , ∞ ∞ ∞ 1 {ρk−1,ii }i=0 ∈ , respectively. The Banach Lie subgroup GB+,k of GL+,k whose Banach Lie ∞ has elements given by algebra is B+,k g = g0 + gk−1 S k−1 =



  g0,ii |i i| + gk−1,ii |i i + k − 1| ,

(5.3)

i=0

where g0 and gk−1 are diagonal operators whose entries are given by the sequences {g0,ii }∞ i=0 , ∞ is bounded below by a strictly posi∞ , respectively, and the sequence {g {gk−1,ii }∞ ∈ } 0,ii i=0 i=0 tive number (that depends on g0 ). ∞ The product of g, h ∈ GB∞ +,k in GL+,k is given by   g ◦k h = g0 h0 + g0 hk−1 + gk−1 s k−1 (h0 ) S k−1 ∞ ∞



= g0,ii h0,ii |i i| + (gii hk−1,ii + gk−1,ii h0,i+k−1,i+k−1 )|i i + k − 1| (5.4) i=0

i=0

and the inverse of g in GB∞ +,k is given by   g −1 = g0−1 − g0−1 gk−1 s k−1 g0−1 S k−1 ∞ ∞



1 gk−1,ii = |i i| − |i i + k − 1|. g0,ii g0,ii g0,i+k−1,i+k−1 i=0

(5.5)

i=0

∞ has the expression The Lie bracket of x, y ∈ B+,k

     [x, y]k = xk−1 s k−1 (y0 ) − y0 − yk−1 s k−1 (x0 ) − x0 S k−1 ∞

  xk−1,ii (y0,i+k−1,i+k−1 − y0,ii ) − yk−1,ii (x0,i+k−1,i+k−1 − x0,ii ) |i i + k − 1|. = i=0

(5.6)

1 → B1 k−1 ∈ GB∞ ⊂ The group coadjoint action (Ad+,k )∗g −1 : B−,k +,k −,k for g := g0 + gk−1 S

+,k ∗ 1 → B 1 , for x := x + x k−1 ∈ GL∞ )x : B−,k 0 k−1 S +,k and Lie algebra coadjoint action (ad −,k ∞ ∞ B+,k ⊂ L+,k are given by

  k−2

 +,k ∗   −1 k−1 −1 Ad g0 gk−1 ρk−1 I − ρ = ρ0 + g0 gk−1 ρk−1 − s˜ pj g −1 j =0

 k−1 k−1 + ST s (g0 )g0−1 ρk−1  ∞ 

gk−1,ii gk−1,ii ρ0,ii + ρk−1,ii |i i| = − ρk−1,ii g0,ii g0,i−k+1,i−k+1 i=0

+



i=0

ρk−1,ii

g0,i+k−1,i+k−1 |i + k − 1 i| g0,ii

(5.7)

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1255

and  k−1    +,k ∗ ρ = s˜ k−1 (ρk−1 xk−1 ) − ρk−1 xk−1 + S T ρk−1 x0 − s k−1 (x0 ) ad x =



(ρk−1,ii xk−1,ii − ρk−1,ii xk−1,ii )|i i|

i=0

+



ρk−1,ii (x0,ii − x0,i+k−1,i+k−1 )|i + k − 1 i|,

(5.8)

i=0 1 . where ρ := ρ0 + (S T )k−1 ρk−1 ∈ B−,k 1 ∞ ∗ Since (B−,k ) = B+,k and the duality pairing is given by the trace of the product, it follows 1 are given by that the Lie–Poisson bracket and its associated Hamiltonian vector field on B−,k

{f, h}0,k−1 (ρ)           ∂f ∂h ∂h ∂h ∂f ∂f s k−1 − − s k−1 − = Tr ρk−1 ∂ρk−1 ∂ρ0 ∂ρ0 ∂ρk−1 ∂ρ0 ∂ρ0    ∞

∂f ∂h ∂h = ρk−1,ii − ∂ρk−1,ii ∂ρ0,i+k−1,i+k−1 ∂ρ0,ii i=0   ∂h ∂f ∂f − − ∂ρk−1,ii ∂ρ0,i+k−1,i+k−1 ∂ρ0,ii

(5.9)

and          ∂h ∂h ∂ ∂ ∂ ∂h Xh0,k−1 (ρ) = Tr ρk−1 s k−1 − s k−1 − − ∂ρ0 ∂ρ0 ∂ρk−1 ∂ρk−1 ∂ρ0 ∂ρ0   ∞

∂h ∂ ∂h = ρk−1,ii − ∂ρ0,i+k−1,i+k−1 ∂ρ0,ii ∂ρi+k−1,i i=0   ∂ ∂h ∂ (5.10) − − ∂ρk−1,ii ∂ρ0,i+k−1,i+k−1 ∂ρ0,ii 1 ). Like in Section 4, in (5.10) we have used the standard coordinate confor f, h ∈ C ∞ (B−,k ventions from finite-dimensions to write a vector field. The precise meaning of the symbols ∞ ∂/∂ρk−1 = {∂/∂ρi+k−1,i }∞ i=0 and ∂/∂ρ0 = {∂/∂0,ii }i=0 is that they form the Schauder basis of 1 corresponding to the Schauder basis {|i + k − 1 i|, |i i|}∞ of B 1 . the tangent space Tρ B−,k −,k i=0 Thus Hamilton’s equations in terms of diagonal operators are

  d ∂h ∂h , ρ0 = ρk−1 − s˜ k−1 ρk−1 dt ∂ρk−1 ∂ρk−1     d ∂h ∂h ρk−1 = ρk−1 s k−1 − dt ∂ρ0 ∂ρ0

(5.11) (5.12)

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or, in coordinates, for i ∈ N ∪ {0}, k  2, d ∂h ∂h − ρk−1,i−k+1,i−k+1 , ρ0,ii = ρk−1,ii dt ∂ρk−1,ii ∂ρk−1,i−k+1,i−k+1   ∂h ∂h d . − ρk−1,ii = ρk−1,ii dt ∂ρ0,i+k−1,i+k−1 ∂ρ0,ii

(5.13) (5.14)

Structure of the generic coadjoint orbit. A coadjoint orbit Oν :=

 +,k ∗ Ad ν g ∈ GB∞ +,k , g −1

1 is said to be generic if νk−1,ii = 0 for i = through the element ν = ν0 + (S T )k−1 νk−1 ∈ B−,k 0, 1, 2, . . . . Let us denote by GL0∞,k−1 the Banach Lie subgroup of (k − 1)-periodic elements of GL∞ 0 , that is, g0 ∈ GL0∞,k−1 if and only if s k−1 (g0 ) = g0 . Denote by L0∞,k−1 the Banach Lie algebra of GL0∞,k−1 .

Proposition 5.1. (i) One has the following equalities   ∞   ∞,k−1 , Z GB∞ +,k = GB+,k ν = GL0

(5.15)

∞ ∞ where Z(GB∞ +,k ) is the center of GB+,k and (GB+,k )ν is the stabilizer of the generic element 1 ν ∈ B−,k . (ii) The generic orbit ∞,k−1 Oν ∼ = GB∞ +,k /GL0

(5.16)

Oν = ν0 + O(S T )k−1 νk−1

(5.17)

is a Banach Lie group. (iii) One has the relation

between the coadjoint orbits through ν = ν0 + (S T )k−1 νk−1 and through (S T )k−1 νk−1 . Proof. Part (i) follows from a direct verification. Since GL0∞,k−1 is a normal Banach Lie group ∞,k−1 ∞ is also a Banach Lie group (see [6]). This proves (ii). of GB∞ +,k the quotient GB+,k /GL0 Part (iii) follows from (5.7). 2 We conclude from (5.17) that to describe any Oν it suffices to study coadjoint orbits through the (k − 1)-lower diagonal elements, k  2. 1 Since the Banach Lie group GB∞ +,k and the generic element ν ∈ B−,k satisfy all the hypotheses of Theorems 7.3 and 7.4 in [12] we conclude:

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∞,k−1 1 given by ι ([g]) := (Ad+,k )∗ ν is a weak injective • The map ιν : GB∞ → B−,k ν +,k /GL0 g −1 immersion. This means that its derivative is injective but no conditions on the closedness of its range or the fact that it splits are imposed. The map ιν is not an immersion as we now show by using Theorem 7.5 in [12]. ∞ ) is equal to the center Since the coadjoint stabilizer Lie algebra (B+,k ν

 ∞   ∞ k−1 s (x0 ) = x0 , xk−1 = 0 Z B+,k = x = x0 + xk−1 S k−1 ∈ B+,k it follows that its annihilator is  ∞  ◦   k−1 1 Tr(x0 ρ0 ) = 0, B+,k ν = ρ = ρ0 + S T ρk−1 ∈ B−,k k−1 (x0 ) = x0 . for all x0 ∈ L∞ 0 such that s Because   ∗     ∗    Tr x0 ad+,k x ν 0 = Tr x0 ad+,k x ν = Tr [x0 , x]k ν = 0 ∞ ) and any x ∈ B ∞ , we have S ⊂ ((B ∞ ) )◦ , where for any x0 ∈ Z(B+,k ν +,k +,k ν

Sν :=

 +,k ∗ ∞ ad ν x ∈ B+k x

1 at ν. Moreover, is the characteristic subspace of the Banach Lie–Poisson structure of B−,k +,k ∗ ∞ 1 the bounded operator Kν : x ∈ B+,k → (ad )x ν ∈ B−,k has non-closed range im Kν = Sν ∞ ) )◦ is strict. To see that the range of K is not closed, one and thus the inclusion Sν ⊂ ((B+,k ν ν 1 ∼ 1 × 1 and B ∞ ∼ ∞ × ∞ and shows that the uses the Banach space isomorphisms B−,k = = +,k two components of Kν are both bounded linear operators with non-closed range. Therefore, ∞ ) )◦ , this since Theorem 7.5 in [12] states that ιν is an immersion if and only if Sν = ((B+,k ν argument shows that ιν is only a weak immersion. ∞,k−1 is a weak symplectic Banach manifold relative to the • The quotient space GB∞ +,k /GL0 closed two-form

     ων [g] Tg π(g ◦k x), Tg π(g ◦k y) = Tr ν[x, y]k =



  νk−1,ii xk−1,ii (y0,i+k−1,i+k−1 − yii ) − yk−1,ii (x0,i+k−1,i+k−1 − x0,ii ) , (5.18)

i=0 ∞ , g ∈ GB∞ , [g] := π(g), π : GB∞ → GB∞ /GL∞,k−1 is the canonical where x, y ∈ B+,k +,k +,k +,k 0

∞,k−1 ∞ ) is its derivative at g. In this forprojection, and Tg π : Tg GB∞ +,k → T[g] (GB+,k /GL0 mula we have used the fact that the value at g of the left invariant vector field ξx on GB∞ +,k generated by x is g ◦k x. ∞,k−1 −→ Oν into • Relative to the Banach manifold structure on Oν making ιν : GB∞ +,k /GL0 a diffeomorphism, the push forward of the weak symplectic form (5.18) has the expression

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∗  ∗     ωOν (ρ) ad+,k x ρ, ad+,k y ρ = Tr ρ[x, y]k =



  ρk−1,ii xk−1,ii (y0,i+k−1,i+k−1 − y0,ii ) − yk−1,ii (x0,i+k−1,i+k−1 − x0,ii ) , (5.19)

i=0 ∞ and ρ ∈ O . where x, y ∈ B+,k ν

We shall express the pull back π ∗ ων of the weak symplectic form ων in terms of the diagonal ∞ ∞ ∞ ∞ operators represented by {g0,ii }∞ i=0 ∈ and {gk−1,ii }i=0 ∈ defining the element g ∈ GB+,k . ∞ 1 k−1 k−1 T k−1 If x = x0 + xk−1 S , y = y0 + yk−1 S ∈ B+,k , and ν = ν0 + (S ) νk−1 ∈ B−,k , (5.18) yields       ∗  π ων (g)(g ◦k x, g ◦k y) = ων [g] Tg π(g ◦k x), Tg π(g ◦k y) = Tr ν[x, y]k =



  νk−1,ii xk−1,ii (y0,i+k−1,i+k−1 − y0,ii ) − yk−1,ii (x0,i+k−1,i+k−1 − x0,ii ) ,

(5.20)

i=0 ∞ where νk−1 has the diagonal entries {νk−1,ii }∞ i=0 . The left-invariant vector field ξx on GB+,k generated by x has the expression

ξx =





g0,ii x0,ii

i=0

∂ ∂ + (g0,ii xk−1,ii + gk−1,ii x0,i+k−1,i+k−1 ) . ∂g0,ii ∂gk−1,ii i=0

The symbols {∂/∂g0,ii , ∂/∂gk−1,ii }∞ i=0 denote the biorthogonal family in the tangent space ∞ corresponding to the standard biorthogonal family {|i i|, |i i + k − 1|}∞ in B ∞ . Tg B+,k +,k i=0 We shall use, as in finite-dimensions, the exterior derivative on real-valued smooth functions, in particular coordinates, to represent elements in the dual space. With this convention, we have ∗

π ων =



i=0

 gk−1,ii gk−1,ii , d log g0,ii ∧ d νk−1,ii − νk−1,ii g0,ii g0,i−k+1,i−k+1 

(5.21)

where, as usual, any element with negative index is set equal to zero. To show this, we evaluate the right-hand side of (5.21) on ξx and ξy and observe that it equals the right-hand side of (5.20). Note that the computations make sense since νk−1 ∈ 1 . ∞,k−1 on GB∞ The action of the coadjoint isotropy subgroup (GB∞ +,k )ν = GL0 +,k is given by g0,ii → h0,ii g0,ii , gk−1,ii → h0,ii gk−1,ii , where h0,ii = h0,i+k−1,i+k−1 . As expected, the righthand side of (5.21) is invariant under this transformation and its interior product with any tangent vector to the orbit of the normal subgroup GL0∞,k−1 is zero. This shows, once again, that (5.21) ∞,k−1 . naturally descends to the quotient group GB∞ +,k /GL0 1 1 In order to understand the structure of Oν , define the action α k : GB∞ +,k × L−k+1 → L−k+1 by  k−1   k−1 k−1 αgk S T νk−1 := S T s (g0 )g0−1 νk−1 .

(5.22)

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∞ 1 → L1 1 1 1 The projector δ k : B−,k −k+1 defined by the splitting B−,k = L−k+1 ⊕ L0 is a GB+,k - equivariant map relative to the coadjoint and the α k -actions of GB∞ +,k , that is, the diagram

1 B−,k

(Ad−,k )∗ −1 g

1 B−,k

δk

L1−k+1

δk αgk

L1−k+1

∞,k−1 commutes for any g ∈ GB∞ of the α k -action does not +,k . We observe that the stabilizer GL0 1 T k−1 depend on the choice of the generic element (S ) νk−1 ∈ Lk−1 . The orbits of the coadjoint action of the subgroup GL0∞,k−1 on (δ k )−1 ((S T )k−1 νk−1 ) are of the form

  k−1  −1  T k−1 1 Δν0 ,νk−1 + S T S νk−1 ⊂ δ k νk−1 ⊂ B−,k , where Δν0 ,νk−1 := ν0 + im Nνk−1 ⊂ L10

(5.23)

1 are affine spaces for each ν0 ∈ L10 and the linear operator Nνk−1 : L∞ 0 → L0 is defined by

 Nνk−1 (gk−1 ) := νk−1 gk−1 + s˜ (νk−1 gk−1 ) I −

k−2

 pj .

j =0 1 The orbits of the α k -action of GB∞ +,k on L−k+1 are

  k−1 k−1  T k−1 νk−1 = S T s (g0 )g0−1 νk−1 g0 ∈ GL∞ GB∞ +,k · S 0 =: Δνk−1 .

(5.24)

Note that if Δνk−1 = Δν  then im Nνk−1 = im Nν  and so Δν0 ,νk−1 = Δν0 ,ν  . These remarks k−1 k−1 k−1 show that the coadjoint orbit Oν is diffeomorphic to the product (ν0 + im Nνk−1 ) × Δνk−1 of the affine space Δν0 ,νk−1 with the α k -orbit Δνk−1 . This diffeomorphism does not depend on the choice  ∈ Δνk−1 . Additionally, one identifies the set of generic coadjoint orbits with the of (S T )k−1 νk−1 ∞ 1 k total space Lk of the vector bundle Lk → L∞ 0 /α (GL0 ), whose fiber at [νk−1 ] is L0 / im Nνk−1 . 1 1 The vector space L0 / im Nνk−1 is not Banach since im Nνk−1 is not closed in L0 because the ∞ 1 ∞ k operator Nνk−1 : L∞ 0 → L0 is compact. Consequently, the bundle Lk → L0 /α (GL0 ) does not have the structure of a Banach vector bundle and does not have fixed typical fiber. The momentum map. Let us now study an important particular case of the map Jν− by taking in (4.17) the element ν− = (S T )k−1 νk−1 ∈ L1−k+1 ⊂ L1−,k . The map (4.17), denoted in this case 1 , becomes Jνk−1 : ∞ × 1 → B−,k  k−1 k−1 Jνk−1 (q, p) = p + S T νk−1 es (q)−q .

(5.25)

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1 T k−1 ν Recall that we identify 1 with L10 and ∞ with L∞ k−1 ∈ L−k+1 , define 0 . Having fixed (S ) ∞ ∞ 1 the action of GB+,k on × by ν

σ gk−1 (q, p)    k−1 k−1 := q + log g0 , p + gk−1 g0−1 νk−1 es (q)−q − s˜ k−1 gk−1 g0−1 νk−1 es (q)−q , (5.26) ∞ 1 where g := g0 + gk−1 S k−1 ∈ GB∞ +,k and (q, p) ∈ × . The coordinate form of the action (5.26) is

qi = qi + log g0,ii , gk−1,ii gk−1,ii νk−1,ii eqk+1 −qk − νk−1,ii eqk −qk−1 pi = pi + g0,ii g0,k−1,k−1

(5.27) (5.28)

for i ∈ N ∪ {0}. Using (5.27) and (5.28) one shows that ∞

pi dqi =

i=0



pi dqi − dQ,

i=0

where the function Q : ∞ → R is given by ∞ 

 gk−1,ii k−1 Q(q) := Tr g0−1 gk−1 νk−1 es (q)−q = νk−1,ii eqk+1 −qk . g0,ii

(5.29)

i=0

Thus we see that ω is invariant relative to the σ νk−1 -action, that is, (σ gk−1 )∗ ω = ω for any g ∈ GB∞ +,k . ν

1 Proposition 5.2. The smooth map Jνk−1 : ∞ × 1 → B−,k given by (5.25) is constant on the

σ νk−1 -orbits of the subgroup GL0∞,k−1 . In addition:

(i) Jνk−1 is a momentum map. More precisely, {f ◦ Jνk−1 , g ◦ Jνk−1 }ω = {f, g}0,k−1 ◦ Jνk−1 , for 1 ), where {·,·} is the canonical Poisson bracket of the weak symplectic all f, g ∈ C ∞ (B−,k ω ∞ 1 Banach space ( × , ω) given by the weak symplectic form (4.20) and {,}0,k−1 is the 1 given by (5.9). Lie–Poisson bracket on B−,k ν ∞ (ii) Jνk−1 is GB+,k -equivariant, that is, Jνk−1 ◦ σ gk−1 = (Ad−,k )∗g −1 ◦ Jνk−1 for any g ∈ GB∞ +,k .

(iii) Jνk−1 ( ∞ × 1 ) = (δ k )−1 (Δνk−1 ), Jνk−1 ( ∞ × {0}) = Δνk−1 , and hence ( ∞ × 1 )/ σ νk−1 (GL0∞,k−1 ) ∼ = Jνk−1 ( ∞ × 1 ) consists of those coadjoint orbits which are projected k k by δ to the α -orbit Δνk−1 . 1 ) and notice that Proof. To prove (i), let f, g ∈ C ∞ (B−,k

∂(f ◦ Jνk−1 )  ∞ ∗ ∈ L ∂q

and

∂(f ◦ Jνk−1 )  1 ∗ ∈ L = L∞ ∂p

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because q ∈ L∞ and p ∈ L1 . However, by (5.25),     ∂(f ◦ Jνk−1 ) ∂f ◦ Jνk−1 (q, p)(ρk−1 ◦ Jνk−1 )(q, p) S k−1 − I ∈ L1 = ∂q ∂ρk−1

(5.30)

since (ρk−1 ◦ Jνk−1 )(q, p) ∈ L1 and ∂(f ◦ Jνk−1 ) = ∂p



 ∂f ◦ Jνk−1 (q, p) ∈ L∞ . ∂ρ0

(5.31)

Note that (5.30) and (5.31) imply that {f ◦ Jνk−1 , h ◦ Jνk−1 }ω makes sense for any f, h ∈ 1 ). C ∞ (B−,k Thus, using the formula for the canonical bracket on the weak symplectic Banach space ( ∞ × 1 , ω) and the fact that the duality pairing (L∞ )∗ × L∞ → R restricted to L1 × L∞ equals to the trace of the product of operators, we get {f ◦ Jνk−1 , g ◦ Jνk−1 }ω (q, p)     ∂(g ◦ Jνk−1 ) ∂(f ◦ Jνk−1 ) ∂(f ◦ Jνk−1 ) ∂(g ◦ Jνk−1 ) , − , = ∂q ∂p ∂q ∂p        ∂g  k−1 ∂f −I ◦ Jνk−1 (q, p) ◦ Jνk−1 (q, p) = Tr (ρk−1 ◦ Jνk−1 )(q, p) S ∂ρ0 ∂ρk−1        ∂f ∂g − S k−1 − I ◦ Jνk−1 (q, p) ◦ Jνk−1 (q, p) ∂ρ0 ∂ρk−1   = {f, g}0,k−1 ◦ Jνk−1 (q, p) by (5.9). Parts (ii) and (iii) are proved by direct verifications.

2

∞ 1 Let us define the map Φ νk−1 (g) : GB∞ +,k → × by ν

Φ νk−1 (g) := σ gk−1 (0, 0),

(5.32)

   Φ νk−1 (g0 , gk−1 ) = log g0 , gk−1 g0−1 νk−1 − s˜ k−1 gk−1 g0−1 νk−1 ,

(5.33)

or, in coordinates,

which shows that Φ νk−1 is smooth and injective. Proposition 5.3. The following diagram 1

GL0∞,k−1

GB∞ +,k

π

L0∞,k−1 × {p}

∞ × 1

1

ινk−1

Φ νk−1

0 × {p}

∞,k−1 GB∞ +,k /GL0

Jνk−1

(δ k )−1 (Δνk−1 )

0

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commutes. The first row is an exact sequence of Banach Lie groups. The second row is also exact in the following sense: the map Jνk−1 is onto and its level sets are all of the form L0∞,k−1 × {p}, where p ∈ L10 . In addition,  ν ∗ Φ k−1 ω = π ∗ ωνk−1 ,

(5.34)

where ω and ωνk−1 are the weak symplectic forms (4.19) and (5.21) on ∞ × 1 and ∞,k−1 GB∞ , respectively. We also have +,k /GL0       ι [g] Φ νk−1 π −1 [g] = Jν−1 k−1 νk−1

(5.35)

for any g ∈ GI ∞ +,0,k−1 . Proof. Commutativity is verified using (5.7), (5.25), and (5.32). The identities (5.34) and (5.35) are obtained by direct verifications. 2 ∞,k−1 through the generic Remarks. (i) The analysis of the coadjoint orbit Oν ∼ = GB∞ +,k /GL0 1 element ν ∈ B−,k carried out in this section shows that it is diffeomorphic to Δν0 ,νk−1 ×Δνk−1 . For   −1 ) ∈ Δν0 ,νk−1 × Δνk−1 , the manifolds ι−1 an arbitrary (ν0 , νk−1 νk−1 ({ν0 } × Δνk−1 ) and ινk−1 (Δν0 ,νk−1 ×  {νk−1 }) are Lagrangian submanifolds in the sense that their tangent spaces are maximal isotropic. ∞ ∞ 1 = L1 (ii) If k = 2 we have B−,2 −,2 and GB+,2 = GL+,2 . If, in addition, we consider the finite1 dimensional case, that is, instead of L−,2 we work with the traceless n × n matrices having non-zero entries only on the main and the first lower diagonals, then Jν1 is a symplectic diffeomorphism of R2(n−1) , endowed with the canonical symplectic structure, with a single coadjoint orbit of the upper bidiagonal group through a strictly lower diagonal element all of whose entries are non-zero. (iii) If k = 2 and we consider the generic infinite-dimensional case, that is, ν1 has all entries different from zero, then the map Jν1 does not provide a morphism of weak symplectic manifolds between ∞ × 1 and a single coadjoint orbit of GL∞ +,2 . The relation between these spaces is more complicated and is explained in the diagram of Proposition 5.3. Each GL∞ +,2 -coadjoint orbit through a generic element S T ν1 is only weakly symplectic and Poisson injectively weakly immersed in L1−,2 but not equal to it. (iv) If k = 2 and we consider the infinite-dimensional case with ν1 having also some vanishing T entries, the structure of the GL∞ +,2 -coadjoint orbit through S ν1 reduces to the two previous cases as we shall explain below. Let i0 be the first index for which the entry ν1,i0 i0 = 0. Formula (5.7) shows that the first i0 × i0 block of OS T ν1 is that of a finite-dimensional orbit of the upper bidiagonal group of matrices of size i0 × i0 and that the coadjoint action preserves this block. Let i1 be the next index for which ν1,i1 i1 = 0. Again by (5.7) it follows that there is an i1 × i1 block of OS T ν1 that is preserved by the coadjoint action and that is equal to a finite-dimensional orbit of the upper bidiagonal group of matrices of size i1 × i1 . Continuing in this fashion we arrive either at an infinite sequence of orbits of finite-dimensional upper bidiagonal groups (in the case that there is an infinity of indices is such that ν1,is is = 0, s ∈ N ∪ {0}) or to a generic infinite dimensional orbit of GL∞ +,2 (if there are only finitely many indices is , s = 0, 1, . . . , r, such that ν1,is is = 0). In the latter case, the last infinite block is preserved by the coadjoint action and we are in the generic case of an orbit of GL∞ +,2 but on the space complementary to the r + 1 finitedimensional blocks of sizes i0 × i0 , . . . , ir × ir . Thus, decomposing the orbit as described, the

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problem of classification of the general GL∞ +,2 -coadjoint orbit is reduced to the finite-dimensional case and to the generic infinite-dimensional case. 1 but these functions are (v) One can restrict the Hamiltonians IlS,k given by (4.33) to B−,k 1 1 not in involution because the inclusion of B−,k in L−,k is not Poisson. Indeed, as recalled in Section 2, the inclusion would be Poisson if and only if the kernel of its dual map is an ideal in L∞ +,k which is easily seen to be false unless k = 2, in which case we have ∞

  S,2 pi I1 ◦ Jν1 (q, p) =

(5.36)

i=0

and ∞ ∞   1 2 2 2(qi+1 −qi ) H2 (q, p) := I2S,2 ◦ Jν1 (q, p) = pi + ν1,ii e . 2 i=0

(5.37)

i=0

The function H2 is, up to a renormalization of constants, the Hamiltonian of the semi-infinite Toda lattice. The first integral I1S,2 ◦ Jν1 is the total momentum of the system which generates the translation action given by the subgroup R+ I. All integrals IlS,2 ◦ Jν1 , l ∈ N, give the full Toda lattice hierarchy on ∞ × 1 as we shall see in the next paragraph. 1 = L1−,2 is an These considerations show that the momentum map Jν1 : ∞ × 1 → I−,0,1 infinite-dimensional analogue of the Flaschka map (see [7]). In fact, as we shall see, Jν1 plays the role of the Flaschka map for the system of integrals in involution (4.33) for k  2.

The semi-infinite Toda lattice. In this example we illustrate the theory of the k-diagonal Hamiltonian systems by the detailed investigation of the semi-infinite Toda lattice which is an example of a bidiagonal system (see Remark (v) at the end of Section 5). We shall follow the method of orthogonal polynomials first proposed in [5], as far as we know. We shall extend below the results in [10] for the finite Toda lattice by explicitly solving the semi-infinite Toda lattice including the construction of action–angle variables. The family of Hamiltonians IlS,2 ∈ C ∞ (L1S,2 ), l ∈ N, leads to the chain of Hamilton equations ∂ ρ = [ρ, Bl ], ∂tl

    T where Bl := P−∞ ρ l − P−∞ ρ l ,

(5.38)

on the Banach Lie–Poisson space (L1S,2 , {·,·}S,2 ) (or on the space (L1−,2 , {·,·}2 ) isomorphic to it) induced from (4.24) by the inclusion ιS,2 : L1S,2 → L1S . The selfadjoint trace class operator ρ ∈ L1S,2 acts on the orthonormal basis {|k }∞ k=0 of H as follows: ρ|k = ρk−1,k |k − 1 + ρkk |k + ρk,k+1 |k + 1 ,

(5.39)

where k ∈ N ∪ {0} and we set ρ−1,0 = 0. Note that if ρ is replaced by ρ  := cρ + bI, where b, c ∈ R, c = 0, then the equations (5.38) remain unchanged by rescaling the time tl := c−l tl . As will be explained later, the norm ρ ∞

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and the positivity ρ  0 are preserved by the evolution defined by (5.38). Taking into account the above facts, we can assume, without loss of generality, that ρ ∞ < 1 and ρ  0. Consequently, from now on we shall work with generic initial conditions ρ(0) for the Hamiltonian system (5.38), i.e., λm (0) = λn (0), λm (0) > 0 and

for n = m,  supm∈N∪{0} λm (0) < 1,

(5.40) (5.41)

where λm (0) are the eigenvalues of ρ(0). This means that ρ(0) has simple spectrum, ρ(0)  0, and ρ(0) ∞ < 1. These hypotheses imply that ρk,k+1 (0) > 0 for all k ∈ N ∪ {0} and are consistent with the physical interpretation of the semi-infinite Toda system. Let us denote by 1 ⊂ L1 the set consisting of operators satisfying (5.40) and (5.41). Ω−,2 S,2 From (5.39), it follows that |k = Pk (ρ)|0 ,

(5.42)

where the polynomials Pk (λ) ∈ R[λ], k ∈ N ∪ {0}, are obtained by solving the three-term recurrence equation λPk (λ) = ρk−1,k Pk−1 (λ) + ρkk Pk (λ) + ρk,k+1 Pk+1 (λ)

(5.43)

with initial condition P0 (λ) ≡ 1. Note that the degree of Pk (λ) is k. We show now that the operator ρ ∈ L1S,2 evolving according to (5.38) also has simple spectrum independent of all times tl . To do this, we write the spectral resolution ρ=



λm Pm ,

Pm Pn = δmn Pn ,

m=0



Pm = I,

(5.44)

m=0

where Pm :=

|λm λm | λm |λm

(5.45)

are the projectors on the one-dimensional eigenspaces spanned by the eigenvector |λm . From (5.38) one obtains 

    ∂ ∂ λk Pn Pk + (λn − λk ) Pn Pk − Pn Bl Pk = 0 ∂tl ∂tl

(5.46)

for any n, k ∈ N ∪ {0} and l ∈ N. Putting n = k in (5.46) one finds ∂ λn = 0 ∂tl

(5.47)

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1265

for any n ∈ N ∪ {0} and l ∈ N. Thus λm = λm (0) = λn for n = m and we can conclude that the coefficients in |λm =



Pl (λm )|l

(5.48)

l=0

are the values Pl (λm ) at the eigenvalue λm of the polynomials Pl (λ) which are orthogonal relative to the L2 -scalar product given by the measure σ in (5.52). Taking n = k in (5.46) and using properties of orthogonal projectors one obtains ∂ Pn = [Pn , Bl ] ∂tl

for any n ∈ N ∪ {0} and l ∈ N.

(5.49)

Similarly, for the resolvent Rλ := (ρ − λI)−1 =



m=0

1 Pm λm − λ

(5.50)

by (5.49) one has ∞

∂ 1 [Pm , Bl ] = [Rλ , Bl ]. Rλ = ∂tl λm − λ

(5.51)

m=0

Note that (5.42) implies that the vector |0 is cyclic for ρ. Thus, one has a unitary isomorphism of H with L2 (R, dσ ), where the measure dσ (λ) := d0|Pλ 0 =



μm δ(λ − λm ) dλ,

(5.52)

m=0

is given by the orthogonal resolution of the unity P : R  λ → Pλ ∈ L∞ (H) for  ρ = λ dPλ . R

The masses μm in (5.52) are given by μ−1 m = λm |λm =





2 Pl (λm ) .

(5.53)

l=0

Using Pm |0 = μm |λm and μm = 0|Pm 0 , one obtains from (5.49) the differential equation    ∂ μm = 2λm |Bl 0 μm = 2 λlm − 0 ρ l 0 μm ∂tl

(5.54)

for any l ∈ N and m ∈ N ∪ {0}. In order to prove the second equality in (5.54) we notice that

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    T Bl = ρ l − P0∞ ρ l − 2 P−∞ ρ l ,  ∞  l T |0 = 0, P− ρ         P0∞ ρ l |0 = 0 P0∞ ρ l 0 |0 = 0 ρ l 0 |0 ,

(5.55) (5.56) (5.57)

which implies   λm |Bl 0 = λlm − 0 ρ l 0 .

(5.58)

  σ k = 0 ρ k 0

(5.59)

∂ σk = 2(σk+l − σl σk ), ∂tl

(5.60)

Using (5.54) and noticing that

one obtains the system of equations

where σ0 = 1, k ∈ N ∪ {0}, l ∈ N, for the moments  σk =

λk dσ (λ) =



λkm μm

(5.61)

m=0

R

of the measure (5.52). Let us remark here that in the considered case the moment problem is determined, i.e., the moments σk determine the measure (5.52) in a unique way (see, e.g. [2, Chapter 2, Sections 1 and 2]). Let us comment on the formulas obtained above. Introduce the diagonal trace class operators λ, μ, σ ∈ L10 by defining their mth components to be the eigenvalues λm , the masses μm , and 1 one has three naturally defined the moments σm , m ∈ N ∪ {0}, respectively. On the subset Ω−,2 smooth coordinate systems: 1 , (i) ρ ∈ Ω−,2 (ii) (λ, μ) ∈ L10 × L10 , where Tr μ = 1 and μ > 0, (iii) σ ∈ L10 with first component σ0 = 1, σ > 0, and d0 > 0,

where d0 :=



k=0 d0k |k k|,

and ⎡

σ0 ⎢ σ1 ⎢ ⎢ σ2 d0k := det ⎢ ⎢ σ3 ⎢ . ⎣ . . σk

σ1 σ2 σ3 σ4 .. .

σ2 σ3 σ4 σ5 .. .

σ3 σ4 σ5 σ6 .. .

σk+1

σk+2

σk+3

⎤ . . . σk . . . σk+1 ⎥ ⎥ . . . σk+2 ⎥ ⎥ . . . σk+3 ⎥ > 0, .. .. ⎥ ⎦ . . . . . σ2k

(5.62)

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with the convention that d0,−1 = 1. In order to see that σ ∈ L10 we notice that ∞

σk =

k=0

We also define d1 :=

∞ ∞ ∞

% k%

 k 

%ρ %  0 ρ 0 

ρ k = k=0

k=0



k=0 d1k |k k|,



σ0 ⎢ σ1 ⎢ ⎢ σ2 d1k := det ⎢ ⎢ σ3 ⎢ . ⎣ . . σk

k=0

1 < +∞. 1 − ρ ∞

where

σ1 σ2 σ3 σ4 .. .

σ2 σ3 σ4 σ5 .. .

σ3 σ4 σ5 σ6 .. .

σk+1

σk+2

σk+3

... ... ... ... .. .

σk−1 σk σk+1 σk+2 .. .

σk+1 σk+2 σk+3 σk+4

. . . σ2k−1

σ2k+1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.63)

for n ∈ N ∪ {0}. The transformation from ρ-coordinates to σ -coordinates is given by formula (5.59). The inverse transformation to (5.59) has the form ρ = S T ρ1 + ρ0 + ρ1 S  1/2 −1  −1   1/2 −1 d0 + d−1 d0 S, = S T s˜ (d0 )s(d0 ) 0 d1 − s˜ d0 d1 + s˜ (d0 )s(d0 )

(5.64)

or, in components (see, e.g., [2]), −1 −1 ρkk = d0k d1k − d0,k−1 d1,k−1

−1 and ρk,k+1 = (d0,k−1 d0,k+1 )1/2 d0k > 0.

(5.65)

Formula (5.61) gives the transformation from (λ, μ)-coordinates to σ -coordinates. The inverse transformation to (5.61) is obtained by expanding the so-called Weyl function 0|Rλ 0 in a Laurent series 0|Rλ 0 =



m=0



σk μm =− λm − λ λk+1

(5.66)

k=0

for |λ| > supm∈N∪{0} {|λm |} = ρ ∞ . So, one finds (λ, μ) by computing the Mittag-Leffler decomposition of the left-hand side of (5.66). The passage from ρ-coordinates to (λ, μ)-coordinates is obtained by composing the previously described transformations. This can also be done directly constructing the spectral resolution for ρ. After these remarks we present Hamilton’s equations (5.38) in the coordinates (λ, μ)  ∂ λ = λ, IlS,2 S,2 = 0, ∂tl     ∂ μ = μ, IlS,2 S,2 = 2 λl − Tr λl μ μ ∂tl

(5.67) (5.68)

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or, in components, ∂ λm = 0 and ∂tl

  ∞

∂ l l μ m = 2 λm − λn μn μm ∂tl

(5.69)

n=0

and in the coordinates σ    ∂ σ = σ , IlS,2 = 2 s l (σ ) − σl σ ∂tl

(5.70)

whose coordinate expression was already given in (5.60). In deducing equations (5.67), (5.68), and (5.70) we used (5.59) and (5.61). Let us observe now that (5.60) implies that ∂σk ∂σl = ∂tl ∂tk

(5.71)

for k, l ∈ N. Thus there exists a function τ (t1 , t2 , . . .) of infinitely many variables (t1 , t2 , . . .) =: t ∈ ∞ such that σk =

1 ∂ log τ, 2 ∂tk

k ∈ N.

(5.72)

In order to be consistent with the notation assumed in the theory of integrable systems, we have called this function τ -function. Substituting (5.72) into (5.60) we obtain the system of linear partial differential equations ∂ 2τ ∂τ =2 , ∂tl ∂tk ∂tk+l

k, l ∈ N,

(5.73)

on the τ -function. In order to find the explicit form of the τ -function, use (5.61), substitute (5.72) into (5.69), and integrate both sides of the resulting equation to get μm (t1 , t2 , . . . , tl−1 , tl , tl+1 , . . .) = μm (t1 , t2 , . . . , tl−1 , 0, tl+1 , . . .)

τ (t1 , t2 , . . . , tl−1 , 0, tl+1 , . . .) 2λlm tl e . τ (t1 , t2 , . . . , tl−1 , tl , tl+1 , . . .)

(5.74)

Iterating (5.74) relative to l ∈ N yields the final formula for μm (t1 , t2 , . . .), namely μm (t1 , t2 , . . .) = μm (0, 0, . . .) Since



m=0 μm (t1 , t2 , . . .) = 1,

τ (0, 0, . . .) 2 ∞ λlm tl e l=1 . τ (t1 , t2 , . . .)

(5.75)

we get the following expression for the τ -function

τ (t1 , t2 , . . .) = τ (0, 0, . . .)



m=0

μm (0, 0, . . .)e2



l l=1 λm tl

.

(5.76)

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1269

Let us show that the series in (5.76) is convergent if μ(0) ∈ L10 ∼ = 1 and t ∈ ∞ . In order to do this we prove that the linear operator defined by (Λt)m :=



λlm tl

l=1

is bounded on ∞ . This follows from ∞ ∞

l l

Λt ∞ = sup λm tl  t ∞ sup λm m∈N m∈N l=1

l=1

λm

ρ ∞ = t ∞ sup = t ∞ . 1 − ρ ∞ m∈N 1 − λm ∞

Thus the sequence {e2 l=1 λm tl }m∈N ∈ ∞ . Since {μm (0, 0, . . .)}m∈N ∈ 1 , the series in (5.76) converges. Summarizing, we see that the substitution of (5.76) into (5.72) and (5.74) gives the t := (t1 , t2 , . . .)-dependence of the moments σk (t) and the masses μm (t), respectively. The dependence of ρkk (t) and ρk,k+1 (t) on t is given by (5.64), (5.62), and (5.63) which express these quantities in terms of σm (t). From the discussion above we see that the conditions (5.40), (5.41) are preserved by the t-evolution. Next, using (5.72), (5.75), and the formula l

⎡ Pn (λm ) = &

1 d0,n−1 d0,n

⎢ ⎢ ⎢ det ⎢ ⎢ ⎢ ⎣

σ0 σ1 σ2 .. .

σ1 σ2 σ3 .. .

σ2 σ3 σ4 .. .

... ... ... .. .

σn σn+1 σn+2 .. .

σn−1 1

σn λm

σn+1 λ2m

... ...

σ2n−1 λnm

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(5.77)

obtained by orthonormalizing the monomials λn , n ∈ N ∪ {0}, with respect to the measure σ (see, e.g., [2]), we obtain from (5.48) the t-dependence of the eigenvectors |λm (t) and the corresponding projectors Pm (t), m ∈ N ∪ {0}. Formula (5.48) defines the operator O : H → H whose matrix in the basis {|k }∞ k=0 is given by Okl (t) := Pl (t)(λk ). One has the following identities: ρ(t)O(t) = O(t)λ(t),

(5.78)

O(t)μ(t)O(t)T = I

(5.79)

relating the operators ρ(t), λ(t), μ(t), and O(t) for any t. Since λ(t) = λ(0), where 0 := (0, 0, . . .), we obtain from (5.78) and (5.79) −1  ρ(t) = O(t)O(0)−1 ρ(0) O(t)O(0)−1 = Z(t)T ρ(0)Z(t),

(5.80)

 T where Z(t) := O(0)μ(0)1/2 O(t)μ(t)1/2 is an orthonormal operator, i.e., Z(t)T Z(t) = I. As shown in Sections 3 and 4, the flows t → ρ(t) can be expressed in terms of the coadjoint action

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∞ 1 (AdS,2 )∗ : GL∞ +,2 → Aut(LS,2 ) of the bidiagonal group GL+,2 on the Banach Lie–Poisson space L1S,2 ∼ = L1−,2 , namely,

 ∗ ρ(t) = AdS,2 g(t)−1 ρ(0)     = S T s g0 (t) g0 (t)−1 ρ1 (0) + ρ0 (0) + g0 (t)−1 g1 (t)ρ1 (0) − s˜ g0 (t)−1 g1 (t)ρ1 (0)   + s g0 (t) g0 (t)−1 ρ1 (0)S =



ρi,i+1 (0)

i=0

+

∞ 

gi+1,i+1 (t) |i + 1 i| gii (t)

ρii (0) + ρi,i+1 (0)

i=0

+



ρi,i+1 (0)

i=0

 gi+1,i (t) gi+1,i (t) − ρi,i+1 (0) |i i| gii (t) gi+1,i+1 (t)

gi+1,i+1 (t) |i i + 1| gii (t)

(5.81)

(the symmetric version of (5.7)), where ρ0 := diag(ρ00 , ρ11 , . . .), ρ1 := diag(ρ01 , ρ12 , . . .), g0 := (g00 , g11 , . . .), and g1 := (g10 , g21 , . . .) ∈ L10 . In order to find the time dependence t → g(t) = g0 (t) + g1 (t)S for g(t) ∈ GL∞ +,2 let us note that from (5.7) and the three-term recurrence relation (5.43) it follows that ρ00 (t) · · · ρk−1,k−1 (t) ρ00 (0) · · · ρk−1,k−1 (0) ' d0,k−1 (0)d0k (t) Pkk (0) = g00 (t) = g00 (t) Pkk (t) d0k (0)d0,k−1 (t)

gkk (t) = g00 (t)

(5.82)

and    ρ00 (t) + · · · + ρkk (t) − ρ00 (0) − · · · − ρkk (0) ρ00 (t) · · · ρk−1,k−1 (t) gk+1,k (t) = g00 (t) ρ00 (0) · · · ρk−1,k−1 (0) ρkk (0) Pk+1,k (0)Pk+1,k+1 (t) − Pk+1,k (t)Pk+1,k+1 (0) Pkk (t)Pk+1,k+1 (t) & & d1k (t) d0,k+1 (0) − d1k (0) d0,k+1 (t) = g00 (t) & , d0k (0)d0k (t)d0,k−1 (t)d0,k+1 (0) = g00 (t)

(5.83)

where Pkl (t) are the coefficients of the polynomial Pn (t)(λ) = Pnn (t)λn + Pn,n−1 (t)λn−1 + · · · + Pn1 (t)λ + Pn0 (t). The last equalities in (5.82) and (5.83) are obtained using (5.77), (5.62), and (5.63) to get the expressions ' Pkk (t) =

d0,k−1 (t) d0k (t)

−d1k (t) and Pk+1,k (t) = & . d0k (t)d0,k+1 (t)

Recall that d0 (t) and d1 (t) are given by (5.62) and (5.63), respectively.

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Finally, taking in (5.26) (for k = 2) g0 (t) and g1 (t) given by (5.82) and (5.83), we obtain the explicit expression for the time evolution of the position q(t) and the momentum p(t) for all flows in the Toda hierarchy described by the Hamiltonians   Hl (q, p) := IlS,2 ◦ Jν1 (q, p), S,2 where Jν1 : 1 × ∞ → L1−,2 ∼ = L1S,2 is the Flaschka map given by (5.25) for k = 2 and Il = IlS ◦ ιS,2 = Il ◦ ιS ◦ ιS,2 are the restrictions to L1S,2 of the Casimir functions Il of L1 (see (4.33)). Note that the formulas giving the group element g(t) depend on g00 (t). This first component cannot be determined but it does not matter because g00 (t)I is in the center of GL∞ +,2 and hence the coadjoint action defined by it is trivial. Also, in terms of the variables q and p, the action of this group element is a translation in q and has no effect on p. This corresponds to the flow of I1S,2 . To solve the semi-infinite Toda system one takes an initial condition ρ(0) which determines 1 a coadjoint orbit of GL∞ +,2 in LS,2 . In the generic case, when all entries on the upper (hence also lower) diagonal of ρ(0) are strictly positive, the solution of the semi-infinite Toda lattice was given above. If some upper diagonal entries of ρ(0) vanish, see Remark (iv). Then the Toda lattice equations decouple and we get a smaller Toda system for each block. On the infinite block, the solution is as above. On each finite block one obtains a finite dimensional Toda lattice whose solution is known (see, e.g., [10]). The method we used above for the semi-infinite case can be also used in the finite case; one works then with measures σ having finite support and uses finite orthogonal polynomials. If one implements the solution method described in this section to this finite-dimensional case the results in [10] are reproduced.

Acknowledgments This work was begun while both authors were at the Erwin Schrödinger International Institute for Mathematical Physics in the Fall of 2003 during the program The Geometry of the Moment Map and hereby thank ESI for its hospitality. Some of the work on this paper was done during the program Geometric Mechanics at the Bernoulli Center of the EPFL in the Fall of 2004. A.O. thanks the Bernoulli Center for its hospitality and excellent working conditions during his extended stay there. We are grateful to D. Belti¸ta˘ , H. Flaschka, and the referee for several useful discussions and remarks that influenced our presentation. The authors thank the Polish and Swiss National Science Foundations (Polish State Grant P03A 0001 29 and Swiss NSF Grant 200021109111/1) for partial support. References [1] R. Abraham, J.E. Marsden, Foundations of Mechanics, second ed., Addison–Wesley, Reading, MA, 1978. [2] N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, transl. from Russian, Oliver & Boyd, Edinburgh, 1965. [3] V.I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16 (1) (1966) 319–361. [4] D. Belti¸ta˘ , T.S. Ratiu, A.B. Tumpach, The restricted Grassmannian, Banach Lie–Poisson spaces, and coadjoint orbits, J. Funct. Anal. 247 (1) (2007) 138–168. [5] Y.M. Berezanski, The integration of semi-infinite Toda chain by means of inverse spectral problem, Rep. Math. Phys. 24 (1) (1986) 21–47. [6] N. Bourbaki, Groupes et algèbres de Lie, chapitre 3, Hermann, Paris, 1972.

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