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so the transversality hypotheses of Theorem 5.9 are fulfilled. By this theorem, the operator
∂A = G−1 ◦ ∂à ◦ F is Fredholm of index
ind∂A = ind∂ à = µ(˜ Φ − N ∗ W0, N ∗ ∆Ên) − µ( ˜ Φ + N ∗ Wk, N ∗ ∆Ên)
− 1
k�
(dim Wj−1 + dim Wj − 2 dimWj−1 ∩ Wj).
2
j=1
(123)
The symplectic paths t ↦→ Φ ± (1)Φ ± (1 − t/2) −1 Φ ± (t/2) and t ↦→ Φ ± (t) are homotopic by means
of the symplectic homotopy
(λ, t) ↦→ Φ ± (1)Φ ±
�
1 + λ
2
�−1 1 − λ
− t Φ
2
±
� �
1 + λ
t ,
2
which fixes the end-points I and Φ ± (1). By the symplectic invariance and the homotopy invariance
of the Maslov index we deduce from (122) that
Similarly,
µ( ˜ Φ − N ∗ W0, N ∗ ∆Ên) = µ(N ∗ W0, ˜ Φ − (·) −1 N ∗ ∆Ên)
= µ(N ∗ W0, graphCΦ − (1)Φ − (1 − ·/2) −1 Φ − (·/2)) = µ(N ∗ W0, graphCΦ − ).
(124)
µ( ˜ Φ + N ∗ Wk, N ∗ ∆Ên) = µ(N ∗ Wk, graphCΦ + ). (125)
The conclusion follows from (123), (124), and (125).
In the case of the right half-strip Σ + , we fix an integer k ≥ 0, real numbers
0 = s0 < s1 < · · · < sk < sk+1 = +∞,
a linear subspace V0 ⊂Ên and a (k + 1)-uple W = (W0, . . . , Wk) of linear subspaces ofÊn ×Ên ,
such that W0 and V0 ×V0 are partially orthogonal, and so are Wj−1 and Wj, for every j = 1, . . .,k.
Set S = {s1, . . . , sk, s1 + i, . . .,sk + i}, and let X 1,p
S ,V0,W (Σ+ ,�n ) be the completion of the space
of maps u ∈ C ∞ S ,c (Σ+ ,�n ) such that
u(it) ∈ V0 ∀t ∈ [0, 1], (u(s), u(s + i)) ∈ N ∗ Wj, ∀s ∈ [sj, sj+1], j = 0, . . . , k,
with respect to the norm �u� X 1,p (Σ + ).
Let A ∈ C 0 ([0, +∞] × [0, 1], L(Ê2n ,Ê2n )) be such that A(+∞, t) ∈ Sym(2n,Ê) for every t ∈
[0, 1], and let Φ + : [0, 1] → Sp(2n) be the solution of the linear Hamiltonian systems
Then we have:
d
dt Φ+ (t) = iA(+∞, t)Φ + (t), Φ + (0) = I.
5.24. Theorem. Assume that graphCΦ + (1) ∩ N ∗ Wk = (0). Then for every p ∈]1, +∞[ the
Ê-linear operator
is bounded and Fredholm of index
∂A : X 1,p
S ,V0,W (Σ+ ,�n ) → X p
S (Σ+ ,�n ), u ↦→ ∂u + Au,
ind∂A = n
2 − µ(N ∗ Wk, graphCΦ + ) − 1
2 (dim W0 + 2 dimV0 − 2 dimW0 ∩ (V0 × V0))
− 1
2
k�
(dimWj−1 + dimWj − 2 dimWj−1 ∩ Wj).
j=1
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