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We claim that a holomorphic function with the properties listed above is necessarily zero. By
(108), setting z = ρe θi with ρ > 0 and 0 ≤ θ ≤ π,
f(z) = w(0)
√ ρ e −θi/2 + o(|ρ| −1/2 ) for ρ → 0.
Since w(0) is real and not zero, the above expansion at 0 shows that there exists ρ > 0 such that
�
f(z) ∈
e θiÊ, ∀z ∈ (Bρ(0) ∩ Σ) \ {0}. (110)
θ∈]−π/2−α/4,α/4[
If f = 0 onÊ+i, then f is identically zero (by reflection and by analytic continuation), so we
may assume that f(Ê+i) �= {0}. By (107) the set f(Ê+i) is contained inÊe αi/2 . Since f is
holomorphic on Int(Σ), it is open on such a domain, so we can find γ ∈]α/4, α/2[∪]α/2, 3α/4[ such
that f(Int(Σ)) ∩Êe γi �= {0}. By (109) and (110) there exists z ∈ Σ \ Bρ(0) such that
f(z) ∈Êe γi , |f(z)| = sup |f(Σ \ {0}) ∩Êe γi | > 0. (111)
By (106) and (107), z belongs to Int(Σ), but since f is open on Int(Σ) this fact contradicts (111).
Hence f = 0. Therefore v vanishes on Σ, concluding the proof of the invertibility of the operator
∂α.
If we change the sign of α and we invert the boundary conditions onÊwe still get an isomorphism.
Indeed, if we set v(s, t) := u(−s, t) we have
so the operators
∂−αv(s, t) = ∂v(s, t) − αv(s, t) = −∂u(−s, t) + αu(−s, t) = −∂αu(−s, t),
∂α : X 1,p
{0},((0),Ê),(Ê) (Σ,�) → X p
{0} (Σ,�),
∂−α : X 1,p
{0},(Ê,(0)),(Ê) (Σ,�) → X p
{0} (Σ,�)
are conjugated. Therefore Proposition 5.17 implies:
5.18. Proposition. If 0 < α < π/2, the operator
is an isomorphism.
5.6 Computation of the index
∂−α : X 1,p
{0},(Ê,(0)),(Ê) (Σ,�) → X p
{0} (Σ,�)
The computation of the Fredholm index of ∂A is based on the Liouville type results proved in the
previous section, together with the following additivity formula:
5.19. Proposition. Assume that A, A1, A2 ∈ C0 (Ê×[0, 1], L(Ê2n
,Ê2n )) satisfy
A1(+∞, t) = A2(−∞, t), A(−∞, t) = A1(−∞, t), A(+∞, t) = A2(+∞, t), ∀t ∈ [0, 1].
Let V1 = (V0, . . . , Vk), V2 = (Vk, . . .,Vk+h), V ′
1 = (V ′
0, . . . , V ′ ′
k ′), V 2 = (Vk ′, ′ . . . , V k ′ +h ′) be finite
ordered sets of linear subspaces ofÊn such that Vj and Vj+1, V ′ ′
j and V j+1 are partially orthogonal,
for every j. Set V = (V0, . . .,Vk, Vk+1, . . .,Vk+h) and V ′ = (V ′
0, . . .,V ′ ′
k ′, V k ′ ′
+1 , . . .,V k ′ +h ′).
Assume that (A1, V1, V ′
1 ) and (A2, V2, V ′
2 ) satisfy the assumptions of Theorem 5.9. Let S1 be a
set consisting of k points inÊand k ′ points in i +Ê, let S2 be a set consisting of h points inÊ
and h ′ points in i +Ê, and let S be a set consisting of k + h points inÊand k ′ + h ′ points in
i +Ê. For p ∈]1, +∞[ consider the semi-Fredholm operators
∂A1 : X 1,p
S1,V1,V ′
1
(Σ,�n ) → X p
S1 (Σ,�n ), ∂A2 : X 1,p
S2,V2,V ′
2
∂A : X 1,p
S ,V ,V ′(Σ,�n p
) → XS (Σ,�n ).
67
(Σ,�n ) → X p
S2 (Σ,�n )