Decomposition of Spectral flow and Bott-type Iteration Formula

Let $A(t)$ be a continuous path of Fredhom operators, we first prove that the spectral flow $sf(A(t))$ is cogredient invariant. Based on this property, we give a decomposition formula of spectral flow if the path is invariant under a matrix-like cogredient. As applications, we give the generalized Bott-type iteration formula for linear Hamiltonian systems.


Introduction
In this paper, we consider the decomposition of spectral flow for a path of self-adjoint Fredholm operators. Let H be a separable Hilbert space, and we denote by FS(H) be the set of all densely defined self-adjoint Fredholm operator on H. We always equipped FS(H) with the gap topology. For a continuous path A(s) ∈ FS(H), t ∈ [a, b]. The spectral flow sf (A(t); t ∈ [a, b]) is an integer that counts the net number of eigenvalues that change sign. This notation is first introduced by Atiyah-Patodi-Singer [2] in their study of index theory on manifolds with boundary, since then it had found many significant applications, see [27,4] and reference therein.
Some basic property of spectral flow such as homotopy invariant, path additivity, direct sum e.t. are well known, please refer the Appendix. Our first result is another basic property which is called cogredient invariant property of spectral flow. For convenience, we first introduce some notations. Let H 1 , H 2 be separable Hilbert space, we denote by L(H 1 , H 2 ) and C(H 1 , H 2 ) the set of bounded and closed operators from H 1 → H 2 . We let S(H) be the set of self-adjoint operators on H. For convenience, we denote by L * (H 1 , H 2 ), C * (H 1 , H 2 ), S * (H), FS * (H) be the invertible subsets. In a preprint paper [13], Fitzpatrick-Stuar-Pejsachowicz proved (1.2) in the case that M s is constant, the domain of A s is fixed and both A a , A b are invertible. Theorem 1.1 can be consider as a generalization of their result.
Our second main result is the decomposition formula based on the cogredient invariant property. Let H i be closed subspace of H for i = 1, · · · , m, then we define 1≤i≤m H i = H 1 + · · · H m which is the subspace spanned by H i , i = 1, · · · , m. Suppose g ∈ L(H), we call g is a matrix-like operator if σ(g) = {λ 1 , · · · , λ n } is finite and there exist m > 0, such that Moreover, letF = span{F λ , λ ∈ σ(g) ∩ U c }, we have the next theorem. In [18], by assume g is unitary and σ(g) is finite, Hu-Sun proved the decomposition formula sf (A s ) = sf (A s | ker(g−λ 1 ) + · · · + sf (A s | ker(g−λ j ) ) (1. 7) under the condition Obviously, we give a generalization of (1.7). In fact, there is a significant difference is we are not assume g is unitary in Theorem 1.2, hence the subspace H λ is not orthogonal. To overcome this difficulty, we develop a new technique (Lemma 2.2) to prove the equality of spectral flow.
As an applications of Theorem 1.2, we give generalization for the Bott-type iteration formula which is a powerful tool in study the multiplicity and stability of periodic orbits in Hamiltonian systems. In 1956, Bott got his celebrated iteration formula for the Morse index of closed geodesics [5], and it was generalized by [3,10,9,11]. The precise iteration formula of the general Hamiltonian system was established by Long [22,23]. In fact, the iteration could be regarded as a unitary group action. Motivated by the symmetry orbits in n-body problem [12], Hu-Sun [18] use this opinion to give generalization of Bott-type iteration formula to the system under a circle-type symmetry or brake symmetry group action, and prove the stability of Figure Eight orbit [8]. The case of the brake symmetry was deeply studied in [24,20,21,16].
Based on Theorem 1.2, we prove the Bott-type iteration formula which cover all the previous cases and moreover give some new generalizations. Our generalized formula could be applied to the closed geodesics on Semi-Riemanian manifold and heteroclinic orbits with brake symmetry. Now we consider the linear Hamiltonian systeṁ where J = 0 −I n I n 0 , I ⊂ R is a connected subinterval, B(t) ∈ C(I, S(R 2n )). In the case I is finite, the boundary conditions is given by the Lagrangian subspaces. Let (C 2n , ω) be the standard symplectic space with ω(x, y) = (Jx, y). A Lagrangian subspace V is a n-dimensional subspace with ω| V = 0. We denote the set of Lagrangian subspace by Lag(2n). It is obvious (C 2n ⊕ C 2n , −ω ⊕ ω) is a 4n-dimensional symplectic space, then for I = [a, b], the boundary condition is given by In the case I = R, we always assume B(±∞) = lim t→±∞ B(t) exist and JB(±∞) is hyperbolic, that is Let H = L 2 (I, C 2n ) and E is W 1,2 (I, C 2n ) which satisfied some boundary conditions. We denote by and B ∈ H be the multiplicity operator of B(t). Let A s = A − sB for s ∈ R, then A s ∈ FS(H).
For g ∈ H, gE = E and satisfied then g * A s g = A s . By construct g, we get the spectral flow decomposition of sf (A s ). We list 6-cases which common in applications of Hamiltonian systems. Our results generalization all the previous results, especially for the brake symmetry of heteroclinic orbits (Case 5 and 6), our result is new. Please see Section 4 for the detail. It is well known that spectral flow is equal to Maslov index, and this is also true for the unbounded domain, see [6,27,26,7,15] and reference therein. The Maslov index is associated integer to a pair of continuous path f (t) = (L 1 (t), L 2 (t)), t ∈ I, in Lag(2n) × Lag(2n) [6]. From the decomposition of spectral flow, we get the decomposition of Mslov index, please refer Section 5 for the detail. For reader's convenience, we give a brief describe for the Maslov index and spectral flow in the Appendix.
This paper is organized as follows. We proved Theorem 1.1 in Section 2 and Theorem 1.2 in Section 3. In Section 4, we list 6-cases of decompositions in Hamiltonian systems. In Section 5, we give the some case of the Bott-type iteration formulas. At last, we briefly review the basic property of spectral flow and Maslov index in the Section 6.

Spectral flow is preserved under cogredient
Let V be a closed space of H, and P V be the orthogonal projection from H to V . For A ∈ C(H), we denote the operator (2.1) Let A ∈ FS(H) and B ∈ L(H) ∩ S(H), then A + tB ∈ FS(H) with t ∈ R. Note that it is positive if B| ker(A+tB)>0 for any t ∈ {t| ker(A + tB) = 0}. For example, A + tI is a positive curve with A ∈ FS(H) for t ∈ R.
It follows that dim ker(h k (δ, t 0 )) = 0 for 1 ≤ k ≤ n. Note that A(t 0 ) + δK is a Fredholm operator, so there is δ 1 > 0 such that By homotopy invariance of spectral flow, we have and Note that A(t 0 ± δ 1 ) + sK, h k (s, t 0 ± δ 1 ), 1 ≤ k ≤ n are positive paths. It follows that This complete the proof.
Please note that Lemma 2.2 can be consider as a generalization of Direct sum property of spectral flow.
In the next, we will prove the spectral flow is invariant under the cogredient. The next Lemma is contained in [13], but for reader's convenience, we give details here. where The gap distance have the following properties.
Lemma 2.4. Let X, Y be two Hilbert spaces. Let M, N be two closed linear subspaces of X.
Proof. Without loss of generality, we assume that M, N = {0}, and let This conclude the proof.
Proof. We only need to show that M * s A s M s is a continuous curve with the gap topology. Let E s be the domain of A s . Note that By the continuity of A s and Q s , we see that for any ǫ > 0 there is δ 1 > 0 such that for any s ∈ (s 0 − ǫ, s 0 − ǫ), we haveδ(Gr(A s 0 ), Gr(A s )) < ǫ/(2C 1 ) and Q s − Q s 0 < ǫ/(2C 2 ). Then we This complete the proof.
Now we give the proof of Theorem 1.1.
Proof of Theorem 1.1. Please note that (1.1) is from Lemma 2.5. We first prove the case M s ≡ M .
By the homotopy invariant property of spectral flow, we have . This complete the proof.
As an example, we consider the one parameter family of linear Hamiltonian systemṡ The boundary condition is given by where we assume Λ s is continuous depend on s.
which implies Gr(γ s (T )) ∈ Lag(4n). The following formula which gives the relation of spectral flow and Maslov index (please refer Theorem 6.1) From (6.6), we can express the left of (2.6) as Maslov index. In fact, the fundamental solution is P s (t)γ s (t), and the boundary conditions is given by (P s (T )Λ s . Hence we have which is just the symplectic invariant property (6.4) of Maslov index.

Decomposition of Spectral flow under cogredient invariant
In this section, we will prove the decomposition formula for spectral flow.
where H λ i := ker(g −λ i ) m for large enough m. Note that (λ−λ 1 ) m and (λ−λ 2 ) m are coprime, then there are polynomials p 1 , p 2 such that Then we can conclude that We We have the following lemmas.
It follows that g(ker A) ⊂ ker A. So ker A is a invariant subspace of g. Similarly we have ker A = 1≤i≤n ker A ∩ H i . This complete the proof.
Let X = 1≤i≤k F λ i , we define an inner product on X: ((x 1 , x 2 , · · · , x k ), (y 1 , y 2 , · · · , y k )) = where (x i , y i ) is the inner product in H. Then X is a Hilbert space and the map Note that X = 1≤i≤k F λ i is an orthogonal decomposition. By the Direct sum property of spectral flow, we have This complete the proof.
It follows that −A s | F λ = Q(A s | F λ )Q. Then by Theorem 1.1, we have The lemma then follows.
Proof of Theorem 1.2. By Proposition 3.4 and Lemma 3.5 , we only need to show that In fact ker A 1 |F = ker A 1 ∩F . By Lemma 3.3, we see that ker . It is also true for A 0 . The theorem then follows.

Applications to Hamiltonian systems
In this section, we will give the applications for Hamiltonian systems. We list 6 cases which are common in applications. For Λ ∈ Lag(4n), we consider the solution of the flowing linear Hamiltonian systemṡ In order to make gE Λ = E Λ , g is always assumed to preserve the boundary condition, that is gΛ = Λ which means Hence we have and get the decomposition formula (1.6). It is well known that for P ∈ Sp(2n), if λ ∈ σ(P ), thenλ, λ −1 ,λ −1 ∈ σ(P ) and possess the same geometric and algebraic multiplicities [23]. Case 1 is given by symplectic matrix.
Let V λ = ker(P − λ) 2n , then H λ = L 2 ([0, T ], V λ ). Case 2. For S ∈ Sp(2n), we consider the S-periodic solution of (4.1), that is and moreover we assume We assume (4.1) with S-periodic boundary conditions admits a Z k symmetry. More precisely, let P ∈ Sp(2n) and P S = SP , the group generator g is defined by (4.6) Easy computation show that g ∈ L(H) and gE = E. By direct computation, we get the adjoint operator g * .
We assume (4.1) admits a generalized brake symmetry. More exactly, for N ∈ Sp a (2n), let (gx)(t) = N x(T − t). (4.10) We assume gΛ = Λ, that is (4.11) then gE = E. Obviously, (g * x)(t) = N * x(T − t). We assume then (4.2) is satisfied. Please note that for the S-periodic boundary conditions, N S −1 = SN implies gΛ = Λ. Separated boundary conditions is another kind of important boundary conditions. More preciselly, we consider solution of (4.1) under the boundary conditions where V 0 , V 1 ∈ Lag(2n). In this case g is defined by (4.10), for N ∈ Sp a (2n) which satisfied hence g ∈ M(H) and For λ ∈ σ(g), H λ = ker(g − λ) 2n . From Theorem 1.2 we get the decomposition of spectral flow. Since on the finite interval B is relative compact with respect to A, then from Remark 3.7, we have All the above discussions can be applied to Sturm-Liouville systems, so we not give the detail in all cases, instead we only consider the following two cases which have clearly background.
Case 4. We consider the one parameter family Sturm-Liouville system where S ∈ L * (R n ). We suppose G s (t), R s (t) ∈ S(n), instead the Legender convex condition we only assume G s (t) is invertible. let P ∈ L * (R n ) and P S = SP , the group generator g is defined as same form of (4.6). We assume (4.16) Then and we could give the decomposition of spectral flow from Theorem 1.2. This case include the Bott-type formula of Semi-Riemann manifold [17]. Let c be a space-like or time-like closed geodesic on n + 1 dimension Semi-Riemann manifold (M, g) with period T . We choose a parallel g-orthonormal frame e i (t) alone c, and satisfied g(e i (t),ċ(t)) = 0. Assume and (e 1 (0), · · · , e n (0)) = (e 1 (T ), · · · , e n (T ))P, Writing theċ g-orthogonal Jacobi vectorfield alone c as J(t) = n i=1 u i (t)e i (t), then we get the linear second order system of ordinary differential equations where R is symmetry matrices which is get by the curvature. A period solution is satisfied For ω ∈ U, let Let S = P m , G s = G, R s = R(t) + sG, (gu)(t) = P u(t + T ), then from Case 4. we get the decomposition of spectral flow. Since g m = ω, then Let ω j be the m-th root of ω, then We have Hence we get the Bott-type iteration formula [17] i ω spec (c (m) ) = Obviously, we can consider the case of brake symmetry, since it is similar, we omit the detail. Case 5. Now we consider the case of heteroclinic orbits, for the one parameter family linear Hamiltonian systemẋ Let B s (±∞) = lim t→±∞ B s (t) exist and satisfied the hyperbolic condition, i.e.
We assume then g * B s g = g. Easy computation show that g * Ag = A, then we have Obviously, we have (g 2 x)(t) = N 2 x(t), hence g ∈ M(H) and then For λ ∈ σ(g), H λ = ker(g − λ) 2n , then we get the decomposition formula from Theorem 1.2.
In the case N 2 = I, let we have Now we consider the case of Homoclinics. For the linear Hamiltonian systeṁ assume lim t→±∞ B(t) = B * and JB * is hyperbolic. In this case, B − B * is relative compact with respect to A − B * , where A = −J d dt . The relative index is defined by In the case (4.22) is a linear system of Homoclinic orbits z, the index of z is defined by [7] i(z) = I(A − B * , A − B).
In the case N 2 = I, we have (4.26)

Relation with the Maslov index
In this section, we will give some Bott-type iteration formulas of Maslov-index. In the what follows g pointed as the Matrix-like operator appear in Case 2, 3, 5. To avoid discuss too many technique details, we only consider the case g m = ωI for some ω ∈ U. Let ω 1 , · · · , ω m be the m-th roots of ω, and let H i = ker(g − ω i ). In Case 2, 3, 5, H(I) = L 2 (I, C 2n ) where I is some finite interval or R, and E is W 1,2 (I, C 2n ) which satisfied some boundary conditions. We choose a subintervalÎ ⊂ I, and let T be the restricition map from H to H(Î) := L 2 (Î, C 2n ), that is I is called a fundamental domain if for any i = 1, · · · , m, T is a bijection from H i to L 2 (Î, C 2n ).
We chooseÎ = [0, T m ] be the fundamental domain, then Remark 5.2. In the case P = I 2n , (5.3) is the standard Bott-type iteration formula for Hamiltonian systems, please refer [22], [23] for the detail. In the case P ∈ Sp(2n) ∩ O(2n), (5.3) is established by Hu and Sun [18], the general case is proved by Liu and Tang [19].
For Case 3. We assume N ∈ Sp a (2n) and N 2 = I. Since N is anti-symplectic, we have (Jx, x) = −(JN x, N x) = −(Jx, x) = 0 with x ∈ ker(N − I). So ker(N − I) is a Lagrange subspace of the symplectic space (R 2n , J). Recall that (gx)(t) = N x(T − t) and gΛ = Λ. Let For the S-periodic boundary conditions, that is x(0) = Sx(T ), then We have Similarly if the boundary condition is given by Remark 5.3. To our knowledge, in the case S = I 2n , N 2 = I, (5.5) is first established by Long Zhang and Zhu [24]. A deep study is given by Liu and Zhang [20] [21]. Hu and Sun had established the case of S, N ∈ O(2n), for the case of dihedral group please refer [16]. Now we consider Case 5. For λ ∈ [0, 1], let γ λ (τ, t) be the fundamental solution of (4.19), that isγ λ (τ, t) = JB λ (t)γ λ (τ, t), γ λ (τ, τ ) = I 2n . (5.6) be the stable and unstable paths, then V s λ (τ ), V u λ (τ ) ∈ Lag(2n). Recall that in this case, H = L 2 (R, R 2n ) and Let R − be the fundamental domain, then Let A ± λ be the restricted operators on H(R − ) with domain T E ± . From Prop 3.7 of [15], we have Then we have In the case of homoclinics, let A λ = A − B * − λ(B − B * ), then from [7] or [15] the index satisfied Compare (5.8) and (5.9), we have Obviously, we can use R + as the fundamental domain, for reader's convenience, we list the formulas below. Here we let A ± λ be the restricted operators on H(R + ) with domain T (E ± ). In case of the homoclinic, In study the stability problem of homographic solution in planar n-body problem, Hu and Ou [14] use the McGehee blow up method to get linear heteroclinic system, this system with brake symmetry if the corresponding central configurations with brake symmetry. Please refer [14] for the detail.
where * denotes the usual catenation between the two paths.
(Direct sum) If for i = 1, 2, H i are Hilbert space, and A i ∈ C [a, b]; FS(H i )) , then The spectral flow is related to Maslov index in Hamiltonian systems. We now briefly reviewing the Maslov index theory [1,6,25]. Let (R 2n , ω) be the standard symplectic space and Lag(2n) the Lagrangian Grassmanian. For two continuous paths L 1 (t), L 2 (t), t ∈ [a, b] in Lag(2n), the Maslov index µ(L 1 , L 2 ) is an integer invariant. Here we use the definition from [6]. We list several properties of the Maslov index. The details could be found in [6].
From the homotopy invariance of Maslov index, we have Corollary 6.2.