PDF (1016 kB)

5.20. Lemma. Let (V0, V1, . . . , Vk) be a (k + 1)-uple of linear subspaces ofÊn , with Vj−1 and Vj
partially orthogonal for every j = 1, . . .,k, and let W be a linear subspace ofÊn . Then
c(V0, . . . , Vk; W) = c(W; V0, . . . , Vk) = − 1
2
k�
(dimVj−1 + dimVj − 2 dimVj−1 ∩ Vj).
Proof. Let us start by considering the case W =Ên . By the additivity formula (113),
c(V0, . . . , Vk;Ên ) =
j=1
k�
j=1
c(Vj−1, Vj;Ên ).
j
Since Vj−1 and Vj are partially orthogonal,Ên has an orthogonal splittingÊn = X1 ⊕Xj 2 ⊕Xj 3 ⊕Xj 4
where Vj−1 = X j
1 ⊕Xj 2 and Vj = X j
. By the direct sum formula (114) and by formula (116),
Since
c(Vj−1, Vj;Ên ) = c(X j
1
, Xj
1
; Xj
1
1 ⊕Xj 3
) + c(Xj
2
, (0); Xj
2
) + c((0), Xj
3
; Xj 3 ) + c((0), (0); Xj 4 )
= 0 − 1
2 dimXj
1
2 −
2 dimXj 3 + 0 = −1
2 dimXj 2 ⊕ Xj 3
dimX j
2 ⊕ Xj 3 = dimVj−1 + dimVj − 2 dimVj−1 ∩ Vj,
the formula for c(V0, . . .,Vk;Ên ) follows.
Now let λ :Ê→L (n) be a continuous path of Lagrangian subspaces such that λ(s) =Ên for
s ≤ −1 and λ(s) = N ∗ W for s ≥ 1. By an easy generalization of the additivity formula (113) to
the case of non-constant Lagrangian boundary conditions,
c(N ∗ V0; λ) + c(V0, . . .,Vk; W) = c(V0, . . . , Vk;Ên ) + c(N ∗ Vk; λ). (117)
By (115), c(N ∗ V0; λ) = −µ(λ, N ∗ V0) and c(N ∗ Vk; λ) = −µ(λ, N ∗ Vk), so (117) leads to
c(V0, . . .,V k ; W) = c(V0, . . . , V k ;Ên ) − (µ(λ, N ∗ Vk) − µ(λ, N ∗ V0))
= c(V0, . . . , V k ;Ên ) − h(N ∗ V0, N ∗ Vk;Ên , N ∗ W),
where h is the Hörmander index. By Lemma 5.2, the above Hörmander index vanishes, so we get
the desired formula for c(V0, . . .,Vk; W). The formula for c(W; V0, . . . , Vk) follows by considering
the change of variable v(s, t) = u(s, 1 − t).
The additivity formula (113) leads to
c(V0, . . . , Vk; V ′
0, . . . , V ′
k ′) = c(V0, . . . , Vk; V ′
0) + c(Vk; V ′
0, . . . , V ′
k ′),
and the index formula in the general case follows from (112) and the above lemma. This concludes
the proof of Theorem 5.9.
5.7 Half-strips with jumping conormal boundary conditions
This section is devoted to the analogue of Theorem 5.9 on the half-strips
Σ + := {z ∈�|0≤Im z ≤ 1, Re z ≥ 0} and Σ − := {z ∈�|0≤Im z ≤ 1, Rez ≤ 0}.
In the first case, we fix the following data. Let k, k ′ ≥ 0 be integers, let
0 = s0 < s1 < · · · < sk < sk+1 = +∞, 0 = s ′ 0 < s ′ 1 < · · · < s ′ k ′ < s′ k ′ +1 = +∞,
be real numbers, and let W, V0, . . . , Vk, V ′
0, . . .,V ′
k ′ be linear subspaces ofÊn such that Vj−1 and
Vj, V ′
′
j−1 and V j , W and V0, W and V ′
0 , are partially orthogonal. We denote by V the (k +1)-uple
69