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The space ME. Let (x1, x2) ∈ P Λ (H1) × P Λ (H2) and y ∈ P Θ (H1 ⊕ H2) (see section 3.5).
The study of the space of solutions ME(x1, x2; y) reduces to the study of an operator of the form
where
∂A : X 1,p
S ,W (Σ,�2n ) → X p
S (Σ,�2n ),
S = {0, i}, W = (∆Ê2n, ∆ ΘÊn),
the space ∆ ΘÊn being defined in (138). Theorem 5.23 implies that this operator is Fredholm of
index
Arguing as in the study of M Λ Υ
Arguing as in the study of M Θ ∂
We conclude that
ind∂A = µ(N ∗ ∆Ê2n, graphCΦ − ) − µ(N ∗ ∆ ΘÊn, graphCΦ + ) − n
. (140)
2
The ME part of Proposition 3.9 follows.
(identity (136)), we see that
µ(N ∗ ∆Ê2n, graphCΦ − ) = µ Λ (x1) + µ Λ (x1).
(identities (139), we get
µ(N ∗ ∆ ΘÊn, graphCΦ + ) = µ Θ (y) + n
2 .
ind∂A = µ Λ (x1) + µ Λ (x1) − µ Θ (y) − n.
The space MG. Let y ∈ P Θ (H1 ⊕ H2) and z ∈ P Λ (H1#H2) (see section 3.5). The study of
the space of solutions MG(y, z) reduces to the study of an operator of the form
where
∂A : X 1,p
S ,W (Σ,�2n ) → X p
S (Σ,�2n ),
S = {0, i}, W = (∆ ΘÊn, ∆Ên × ∆Ên).
By Theorem 5.23 this operator is Fredholm of index
ind∂A = µ(N ∗ ∆ ΘÊn, graphCΦ − ) − µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ) − n
. (141)
2
As in the study of M Θ ∂ , we have
As in the study of M Λ Υ
and (141) gives us
µ(N ∗ ∆ ΘÊn, graphCΦ − ) = µ Θ (y) + n
2 .
(identity (137)), we see that
This concludes the proof of Proposition 3.9.
µ(N ∗ (∆Ên × ∆Ên), graphCΦ + ) = µ Λ (z), (142)
ind∂A = µ Θ (y) − µ Λ (z).
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