It follows from the above definition that the space Ψd(E, F) transforms naturally under isomorphisms of vector bundles. More precisely, let ϕ : M1 → M2 be an isomorphism of smooth manifolds, and letϕE :E1 →E2 be a compatible isomorphism of smooth complex vector bundles πE1 :E1 → M1 and πE2 :E2 →M2. Here the requirement of compatibility means that the pair (ϕE, ϕ) is an isomorphism of E1 and E2 in the sense of Lecture 2,
§3. The isomorphism ϕE induces a linear isomorphism ϕE∗ : Γ∞c (M1, E1) →Γ∞c (M2, E2), given by ϕE∗f =ϕE◦f◦ϕ−1.
Likewise, let ϕF : F1 → F2 be an isomorphism of vector bundles Fj → Mj which is compatible with ϕ. Then we have an induced linear isomorphism ϕF∗ : Γ∞(M1, F1) → Γ∞(M2, F2). Moreover, the map
ϕ∗ : Hom(Γ∞c (M1, E1),Γ∞(M1, F1))→Hom(Γ∞c (M2, E2),Γ∞(M2, F2)) given by ϕ∗(Q) =ϕF∗ ◦Q◦ϕ−1E∗ restricts to a linear isomorphism
ϕ∗ : Ψd(M1, E1, F1)−→' Ψd(M2, E2, F2).
We note that it also follows from the above definition that ifE and F admit trivializations τE :E →E0 =U×Ck and τF :F →F0 =U×Ck, respectively, thenτ∗ maps Ψd(U, E, F) linearly isomorphically onto Ψd(U, E0, F0) 'Ml,k(Ψd(U)). Indeed, by the previous remark and the remark below (8.1.1), it suffices to prove this for E =M ×Ck and F =M ×Cl. In this case
Hom(Γ∞c (E), C∞(F))'Ml,k(Hom(Cc∞(M), C∞(M))).
If P ∈ Hom(Γ∞c (E), C∞(F)), then P ∈ Ψd(E, F) in the sense of Definition 8.1.1 if and only if all its components Pij belong to Ψd(M) in the sense of Definition 7.3.1.
Remark 8.1.2 The straightforward analogues of Exercise 7.3.4 and 7.3.5, Lemma 7.3.6, Exercise 7.3.7, Lemma 7.3.8, Exercise 7.3.10 are valid for operators from Ψd(E, F), by reduction to trivial bundles and the scalar case, along the lines discussed above. We leave it to the reader to check the details.
It follows from Definition 8.1.1 and the corresponding fact for scalar operators that Ψ−∞(E, F) equals the intersection of the spaces Ψd(E, F), for d∈R.
There is a natural identification of this space with the space of sections of the pull-back π∗H of the vector bundle H under the map π : T∗M → M, so that Γ∞(T∗M, H) ' Γ∞(T∗M, π∗H), but we shall not need this. If H is trivial of the form H =M×CN,then the elements of Γ∞(T∗M, H) are precisely the functions of the form ξx 7→ (x, f(ξx)) with f ∈C∞(T∗M,CN). It follows that Γ∞(T∗M, H)'C∞(T∗M,CN).Accordingly, we define
Sd(M, H) :={f ∈Γ∞(T∗M, V)| ∀j : fj ∈Sd(M)} 'Sd(M)N.
Let τ :H1 →H2 be an isomorphism of two vector bundles on M. Then the map τ∗ : Γ∞(T∗M, H1)→Γ∞(T∗M, H2),
defined by
τ∗(f)(ξx) =τx(f(ξx)), (x∈M, ξx∈Tx∗M),
is a linear isomorphism. Assume now that τ is a bundle automorphism of the trivial bundleH =M×CN.Thenτ has the formτ(x, v) = (x, τx(v)),withx7→τx a smooth map M →GL (N,C).It follows that the linear automorphismτ∗ of Γ∞(T∗M, H)'C∞(T∗M)N is given by
τ∗(f)(ξx) =τx(f(ξx)).
It is readily checked that this map restricts to a linear automorphism of the symbol space Sd(T∗M, H).
Now assume the bundle H is trivializable and let τ0 : H → H0 = M × CN be a trivialization. Then we define Sd(M, H) := τ∗−1Sd(M, H0). This definition is independent of the particular choice of the trivialization τ,in view of the preceding discussion.
Definition 8.2.1 LetH →M be a complex vector bundle. Ford∈R∪ {−∞},we define the symbol space Sd(M, H) to be the space of sections p ∈ Γ∞(T∗M, H) such that for every open neighborhood U on whichH admits a trivialization, the restriction pU :=p|T∗U belongs toSd(U, HU).
Clearly, for p ∈ Γ∞(T∗M, H) to belong to Sd(M, H) is suffices that for every a ∈ M there exists an open trivializing neighborhood U such that pU ∈Sd(U, HU).
We also note that forϕ∈C∞(M) multiplication byπ∗ϕ∈C∞(T∗M) mapsSd(T∗M, H) linear isomorphically to itself. Accordingly, Sd(M, H) becomes a C∞(M)-module. It fol-lows that the quotient
Sd/Sd−1(M, H) :=Sd(M, H)/Sd−1(M, H) is a C∞(M)-module as well.
Let U ⊂ M be open and let K ⊂ U be compact. Then we write SKd(U, HU) for the subspace of Sd(U, HU) consisting of p with support in K in the sense that pU\K = 0.
Equivalently, this means that the function p : T∗U → HU vanishes on π∗(U \ K). The
extension of such a function to T∗M by the requirement u(ξx) = 0x ∈ Hx for every ξx ∈T∗M \T∗U belongs toSKd(M, H).Accordingly, we have a linear injection
SKd(U, HU),→SKd(M, H).
Let Scd(U, HU) denote the union of the spaces SKd(U, HU), for K ⊂ U compact. Then Scd(U, HU),→Scd(M, H). Accordingly, we have an induced linear injection
Scd(U, HU)/Scd−1(U, HU),→Scd(M, H)/Scd−1(M, H).
We will now see that the definition of the principal symbol map can be generalized to the context of bundles. Let E, F be two complex vector bundles on M. The principal symbol map associated with Ψd(E, F) will be a map
σd: Ψd(M, E, F)→Sd(M,Hom (E, F))/Sd−1(M,Hom (E, F)).
Here Hom (E, F) is the vector bundle on M whose fiber at x∈M is given by Hom (E, F)x = HomC(Ex, Fx).
IfU ⊂M is an open subset on which bothE andF admit trivializations τU :EU →U×Ck and τF :FU →U ×Cl, then the bundle Hom (E, F) admits the trivialization
τ : Hom (E, F)U →U ×Hom(Ck,Cl) given by
τx(T) = (τF)x◦T◦(τE)−1x .
Let ϕ : E → F be a vector bundle homomorphism. Then the map ϕ : x 7→ ϕx ∈ Hom(Ex, Fx) defines a smooth section of the bundle Hom (E, F). Using trivializations we readily see that the map ϕ7→ϕdefines a linear isomorphism
Hom(E, F)−→' Γ∞(Hom (E, F)).
Initially we will give the definition of principal symbol for trivial bundles. Assume that E =M ×Ck and F =M ×Cl so that Hom (E, F) = M ×Hom(Ck,Cl)'Ml,k(C). Then
Sd(M,Hom (E, F))'Ml,k(Sd(M)) and, accordingly,
Sd(M,Hom (E, F))/Sd−1(M,Hom (E, F))'Ml,k(Sd(M)/Sd−1(M))
In this setting of trivial bundles, we define the principal symbol mapσd=σdE,F component wise by
σd(P)ij :=σd(Pij), (1≤i≤l, 1≤j ≤k).
Assume now thatτE andτF are automorphisms of the trivial bundlesE andF,respectively and letτ be the induced automorphism of Hom (E, F).We denote byτ∗ the induced auto-morphisms of Ψd(E, F) and of the quotient spaceSd(M,Hom (E, F))/Sd−1(M,Hom (E, F)).
Lemma 8.2.2 For every P ∈Ψd(E, F),
σd(τ∗(P)) = τ∗(σd(P)).
Proof We observe that for f ∈Γ∞(E)'C∞(M)k we have (τ∗(P)f)i =X
r,s,j
(τF x)irPrs(τEx−1)sjfj so that by Lemma 7.5.1
σd(τ∗(P)ij) = X
r,s
(τF)ir(τE−1)sjσd(Prs)
= X
r,s
(τF)ir(τE−1)sjσd(P)rs
= (τ∗σd(P))ij.
This implies that σd(τ∗(P))ij =σd(τ∗(P)ij) = (τ∗σd(P))ij. If E and F admit trivializations τE : E → E0 = M ×Ck and τF : F → F0 =M ×Cl we define the principal symbol map σdE,F on Ψd(E, F) by requiring the following diagram to be commutative
Ψd(E, F) −→τ∗ Ψd(E0, F0)
σE,Fd ↓ ↓σEd0,F0
Sd/Sd−1(M,Hom (E, F)) −→τ∗ Sd/Sd−1(M,Hom (E0, F0)).
(8.2.2)
Finally, we come to the case that E → M and F → M are arbitrary complex vector bundles of rank k and l respectively.
Lemma 8.2.3 Let P ∈Ψd(E, F). Then there exists a unique
σd(P) = σdE,F ∈Sd(M,Hom (E, F))/Sd−1(Hom (E, F))
such that for every open subset U ⊂M on which both E and F admit trivializations, σd(P)U =σdE
U,FU(PU). (8.2.3)
Proof Uniqueness is obvious. We will establish existence. Let {Uj} be an open cover of M consisting of open subsets on which bothEandF admit trivializations. We may assume that {Uj} is locally finite and that {ψj} is a partition of unity on M with suppψj ⊂ Uj for all j. Given P ∈Ψd(E, F) we define σd(P) by
σd(P) = X
j
ψjσdE
Uj,FUj(PUj).
As this is a locally finite sum, it defines an element of Sd(M, H)/Sd−1(M, H). It remains to verify (8.2.3) for an open subset U on which E and F admit trivializations τU. With σd(P) as just defined we have
σd(P)U = X
j
ψj|UσEd
Uj,FUj(PUj)U∩Uj
= X
j
ψj|UσEd
U,FU(PU)U∩Uj
= X
j
ψj|UσEd
U,FU(PU)
= σEd
U,FU
X
j
Mψj|U◦PU
= σEd
U,FU(PU).
Definition 8.2.4 Let P ∈ Ψd(E, F). The d-th order principal symbol of P is defined to be the unique element σd(P) ∈ Sd/Sd−1(M,Hom (E, F)) satisfying the properties of Lemma 8.2.3.
Obviously, P 7→ σd(P) is a linear map. As should be expected, it follows from the above definition that the principal symbol map behaves well under bundle isomorphisms.
Consider isomorphisms τE :E1 →E2 and τF :F1 →F2 of vector bundles on M.Then the definitions have been given in such a way that the following diagram commutes
Ψd(M, E1, F1) −→σd Sd(M,Hom (E1, F1))/Sd−1(M,Hom (E1, F1))
τ∗↓ ↓τ∗
Ψd(M, E2, F2) σ
d
−→ Sd(M,Hom (E2, F2))/Sd−1(M,Hom (E2, F2))
(8.2.4)
The local version of this result is true because of the local requirement (8.2.2). The global validity follows by the uniqueness part of the characterization of the symbol map in Lemma 8.2.3.
Lemma 8.2.5 Let ψ, χ∈C∞(M). Then, for all P ∈Ψd(E, F), σd(Mψ◦P ◦Mχ) =ψχσd(P).
Proof For trivializable bundles E and F the result is a straightforward consequence of the analogous result in the scalar case. Let U be any open subset of M on which bothE and F admit trivializations. Then
σd(Mψ◦P ◦Mχ)U = σUd(Mψ|U◦PU◦Mχ|U)
= ψ|Uχ|UσUd(PU)
= ψχσd(P)
U.
The result follows.
Theorem 8.2.6 The principal symbol map σd induces a linear isomorphism Ψd(E, F)/Ψd−1(E, F) −→' Sd(M,Hom (E, F))/Sd−1(M,Hom (E, F)).
Proof If E, F are trivial, then the result is an immediate consequence of the analogous result in the scalar case. If E, F are trivializable, the result is still true in view of the commutativity of the diagram (8.2.4). Let now E, F be arbitrary complex vector bundles on M and put H = Hom (E, F). We must show that the principal symbol map σd : Ψd(E, F) → Sd(M, H)/Sd−1(M, H) has kernel Ψd−1(E, F) and is surjective. Let P ∈ Ψd(E, F), then σd(P) = 0 if and only if for every open subset U ⊂ M on which both E and F admit a trivialization, σd(P)U = 0. The latter condition is equivalent toσUd(PU) = 0, hence by the first part of the proof to PU ∈ Ψd−1(EU, FU). It follows that kerσd = Ψd−1(E, F).
To establish the surjectivity, letp∈Sd(M, H) and let [p] denote its class in the quotient Sd(M, H)/Sd−1(M, H). Let {Uj} be an open cover of M such that both E and F admit trivializations over Uj, for all j. We may choose the covering such that there exists a partition of one, {ψj}, with suppψj ⊂ Uj for all j. By the first part of the proof, there exists for each j a pseudo-differential operator Pj ∈Ψd(EUj, FUj),such that
σd(Pj) = [p]Uj ∈Sd(Uj, HUj)/Sd−1(Uj, HUj).
For eachj we fixχj ∈Cc∞(Uj) such thatχj = 1 on suppψj.ThenMψj◦Pj◦Mχj is a pseudo-differential operator in Ψd(E, F) with distribution kernel supported by suppψj×suppχj. It follows that the distribution kernels are locally finitely supported. Hence
P :=X
j
Mψj◦Pj◦Mχj
is a well-defined pseudo-differential operator inψd(E, F).LetU be any relatively compact open subset of M on which both E and F admit trivializations, then
σd(P)U =X
j
ψjσd(Pj)
U =X
j
ψj|U[p]U = [p]U,
with only finitely many terms of the sums different from zero. It follows that σd(P) = [p].
We have established the surjectivity of the principal symbol map.