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8.1 Pseudodifferential Operators on Boundary Manifolds 419

m
Definition 8.1.2. Let A ∈L (Γ) and let A = {(O r ,U r ,χ r ) | r ∈ I} be an
c
(χ r (x),ξ) be the corresponding family of principal
atlas on Γ and let σ A χ r ,m
−m
symbols. Then A is called elliptic on Γ if there exists an operator B ∈L (Γ)
such that
n
σ Aχ r ,m (χ r (x),ξ)σ Bχ r ,−m (χ r (x),ξ) − 1 ∈ S S S −1 (U r × IR ) (8.1.17)
Clearly, Lemma 6.2.1 remains valid for an elliptic pseudodifferential operator
on Γ.If A =((A jk )) p×p is given as a system of pseudodifferential operators
on Γ then also Definition 6.2.3 of ellipticity in the sense of Agmon–Douglis–
Nirenberg can be carried over in the same manner to ((A jk )) p×p on Γ.
m
Since for every elliptic operator A ∈L (Γ) there exists a parametrix
c
−m
Q 0 ∈L (Γ), as a consequence we have the following theorem.
c
m
Theorem 8.1.3. Every elliptic operator A ∈L (Γ) is a Fredholm operator,
c
s
A : H 1 = H s+m (Γ) →H 2 := H (Γ). The parametrix Q 0 ∈L −m is also
c
elliptic and a Fredholm operator Q 0 : H 2 →H 1 . Moreover,
index(A)= −index(Q 0 ) .
Proof: Since Γ is compact, the operators C 1 and C 2 in (6.2.6) now are com-
s
pact linear operators in H s+m (Γ)and H (Γ), respectively, and the results
follow from Theorem 6.2.4 with Q 1 = Q 2 = Q 0 .
Remark 8.1.1: As a consequence, the equation
Au = f on Γ (8.1.18)

m
s
with A ∈L (Γ) is solvable if and only if f ∈ H (Γ) satisfies the compatibility
c
∗
conditions (5.3.27) with the adjoint pseudodifferential operator A . In accor-
dance with Stephan et al. [296] we call the system ((A jk )) p×p on Γstrongly
0
elliptic if to the principal symbol matrices a (x, ξ)= a jk0 (χ r (x),ξ)
s j +t k p×p
on the charts (O r ,U r ,χ r ) of the atlas A there exists a C ∞ matrix–valued
function Θ(x)=((Θ j )) p×p on Γ, and a constant β 0 > 0 such that for all
p

x ∈ Γ, all ζ ∈ C and all ξ ∈ IR n−1 with |ξ | =1
0 2
Re ζ Θ(x)a (x, ξ )ζ ≥ β 0 |ζ| (8.1.19)
is satisfied. In terms of the Bessel potential on Γ defined by
α
Λ =(−∆ Γ +1) α/2 where ∆ Γ is the Laplace–Beltrami operator (3.4.64) for
Γ
the Laplacian on Γ, Theorem 6.2.7 implies the following G˚arding inequality
on the whole of Γ since Γ here is a compact manifold.
Theorem 8.1.4. If A =((A jk )) p×p is a strongly elliptic system of pseudodif-
ferential operators on Γ then there exist constants β 0 > 0 and β 1 ≥ 0 such
that G˚arding’s inequality holds in the form