Boundary value problems 343
for π
e
′
: T
∗
∂ M|
Y
∧ ∞
→ Y
∧ ∞
and σ
∂, e
′
✁
: π
∗ ∂,
e
′
E
′
⊗
✁
+
⊕ J
−
−→ π
∗ ∂,
e
′
F
′
⊗
✁
+
⊕ J
+
100 for π
∂, e
′
: T
∗
∂ M \ 0|
Y
∧ ∞
→ Y
∧ ∞
. Note that
✁
+
may be replaced by Sobolev spaces on the half-axis for s d −
1 2
, cf. analogously Section 4.1. Let σ
✁
= σ
ψ
✁
, σ
e
✁
, σ
ψ, e
✁
; σ
∂
✁
, σ
e
′
✁
, σ
∂, e
′
✁
101 for
✁
∈
µ, d;δ
cl
M; v, and set symb
µ, d;δ
cl
M; v = σ
✁
:
✁
∈
µ, d;δ
cl
M; v . We then
have a direct generalization of Remark 13; the obvious details are left to the reader. Note that there are natural compatibility properties between the components of σ
✁
. T
HEOREM
13.
✁
∈
µ, d;δ
cl
M; v, v = E
, F; J
, J
+
, and ∈
ν, e;̺
cl
M; w, w =
E , E ; J
−
, J
, implies
✁
∈
µ+ν, h;δ+̺
cl
M; v ◦ w for h = maxν + d, e and v ◦ w =
E , F; J
−
, J
+
, and we have σ
✁
= σ
✁
σ with componentwise multiplication.
Theorem 13 is the global version of Theorem 5 and, in fact, a direct consequence of this local composition result.
4.3. Ellipticity, parametrices and Fredholm property
D
EFINITION
11. An operator
✁
∈
µ, d;δ
cl
M; v for v = E , F; J
−
, J
+
is called elliptic of order µ, δ if all bundle homomorphisms 95, 96, 97, 98, 99, 100 are isomorphisms.
Similarly to Remark 16, in the conditions for 98, 99, 100 we may replace
✁
+
by H
s +
and H
s−µ +
, respectively, for s maxµ, d −
1 2
. D
EFINITION
12. Given
✁
∈
µ, d;δ
cl
M; v for v = E , F; J
−
, J
+
an operator ∈
−µ,e;−δ cl
M; v
−1
for v
−1
= F, E ; J
+
, J
−
and some e ∈
✁
is called a parametrix of
✁
if
✁
− ∈
−∞,d
l
;−∞
M; v
l
,
✁
− ∈
−∞,d
r
;−∞
M; v
r
for certain d
l
, d
r
∈
✁
, and v
l
= E , E ; J
−
, J
−
, v
r
= F, F; J
+
, J
+
. Note that the Theorem 13 entails σ
✁
−1
= σ with componentwise inversion where is a parametrix of
✁
. T
HEOREM
14. Let
✁
∈
µ, d;δ
cl
M; v be elliptic. Then
✁
: H
s;̺
M, E ⊕
H
s;̺
∂ M, J
−
−→ H
s−µ;̺−δ
M, F ⊕
H
s−µ;̺−δ
∂ M, J
+
102 is a Fredholm operator for every s maxµ, d −
1 2
and every ̺ ∈ , and
✁
has a parametrix ∈
−µ,d−µ
+
;−δ cl
M; v
−1
, where d
l
= maxµ, d and d
r
= d − µ
+
cf. the notation in Definition 12.
344 D. Kapanadze – B.-W. Schulze
The proof of this result can be given similarly to Theorem 8. Alternatively, the methods of Section 3.6 can also be used to first construct σ
✁
−1
and to form e := op σ
✁
−1
∈
−µ,d−µ
+
;−δ
M; v
−1
. Then we get e
✁
− ∈
−1,e;−1
M; v
l
for some e, and we get itself by a formal Neumann series argument.
R
EMARK
21. Let
✁
∈
µ, d;δ
cl
M; v be elliptic. Then we have elliptic regularity of so-
lutions in the following sense.
✁
u = f ∈ H
s−µ;̺−δ
M, F ⊕ H
s−µ;̺−δ
∂ M, J
+
for any s maxµ, d−
1 2
and ̺ ∈ and u ∈ H
r;−∞
M, E ⊕ H
r;−∞
∂ M, J
−
, r maxµ, d−
1 2
, implies u ∈ H
s;̺
M, E ⊕ H
s;̺
∂ M, J
−
. In fact, we can argue in a standard manner. Composing
✁
u = f from the left by we get
✁
u = 1 + u ∈ H
s;̺
M, E ⊕ H
s;̺
∂ M, J
−
and u ∈
✁
M, E ⊕
✁
∂ M, J
−
which yields the assertion.
R
EMARK
22. From Remark 21 we easily obtain that the kernel of
✁
is a finite-dimensional subspace of
✁
M, E ⊕
✁
∂ M, J
−
and as such independent of s and ̺. Moreover, it can easily be shown that there is a finite-dimensional subspace
−
⊂
✁
M, F ⊕
✁
∂ M, J
+
such that im
✁
+
−
= H
s−µ;̺−δ
M, F ⊕ H
s−µ;̺−δ
∂ M, J
+
for all s, where im
✁
means the image in the sense of 102. Thus ind
✁
the index of 102 is independent of s maxµ, d −
1 2
and of ̺ ∈
. R
EMARK
23. Let
✁
i
∈
µ, d;δ
cl
M; v
i
, v
i
= E , F; J
− i
, J
+ i
, i = 1, 2, be elliptic oper- ators where
✁
1
has the same upper left corner as
✁
2
; then there is an analogue of Agranovich- Dynin formula for the indices ind
✁
i
, i = 1, 2: There exists an elliptic operator ∈ L
0;0 cl
∂ M;
J
+ 2
⊕ J
− 1
, J
+ 1
⊕ J
− 2
such that ind
✁
1
− ind
✁
2
= ind .
The idea of the proof is completely analogous to the corresponding result for a compact, smooth manifold with boundary, cf. Rempel and Schulze [16], Section 3.2.1.3. The operator
can be evaluated explicitely by applying reductions of orders and weights cf., also Theorem 22 below and using a parametrix of
✁
2
.
4.4. Construction of global elliptic boundary conditions
