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5.23. Theorem. Assume that graphCΦ − (1) ∩ N ∗ W0 = (0) and graphCΦ + (1) ∩ N ∗ Wk = (0).
Then for every p ∈]1, +∞[ theÊ-linear operator
is bounded and Fredholm of index
∂A : X 1,p
S ,W (Σ,�n ) → X p
S (Σ,�n ), u ↦→ ∂u + Au,
ind∂A = µ(N ∗ W0, graphCΦ − ) − µ(N ∗ Wk, graphCΦ + )
− 1
k�
(dimWj−1 + dimWj − 2 dimWj−1 ∩ Wj).
2
j=1
Proof. Given u : Σ →�n define ũ : Σ →�2n by
ũ(z) := (u(z/2), u(z/2 + i)).
The map u ↦→ ũ determines a linear isomorphism
F : X 1,p
S ,W (Σ,�n ) ∼ =
−→ X 1,p
S ′ ,W ,W ′(Σ,�2n ),
where S ′ = {2s1, . . . , 2sk, 2s1 + i, . . .,2sk + i}, W ′ is the (k + 1)-uple (∆Ên, . . . , ∆Ên), and we
have used the identity
The map v ↦→ ˜v/2 determines an isomorphism
The composition G ◦ ∂A ◦ F −1 is the operator
where
Since
∂ Ã
we easily see that the solutions ˜ Φ ± of
are given by
The above formula implies
For t = 1 we get
N ∗ ∆Ên = graphC = {(w, w) | w ∈�n }.
G : X p
S (Σ,�n ) ∼ =
−→ X p
S ′(Σ,�2n ).
: X1,p
S ′ ,W ,W ′(Σ,�2n ) −→ X p
S ′(Σ,�2n ), u ↦→ ∂u + Ãu,
Ã(z) := 1
(A(z/2) ⊕ CA(z/2 + i)C).
2
Ã(±∞, t) = 1
(A(±∞, t/2) ⊕ CA(±∞, 1 − t/2)C),
2
d
dt ˜ Φ ± (t) = i Ã(±∞, t)˜ Φ ± (t), ˜ Φ ± (0) = I,
˜Φ ± (t) = Φ ± (t/2) ⊕ CΦ ± (1 − t/2)Φ(1) −1 C.
˜Φ ± (t) −1 N ∗ ∆Ên = graphCΦ ± (1)Φ ± (1 − t/2) −1 Φ ± (t/2). (122)
˜Φ − (1)N ∗ W0 ∩ N ∗ ∆Ên = ˜ Φ − (1)[N ∗ W0 ∩ graphCΦ − (1)] = (0),
˜Φ + (1)N ∗ Wk ∩ N ∗ ∆Ên = ˜ Φ + (1)[N ∗ Wk ∩ graphCΦ + (1)] = (0),
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