## Archive for **September 2012**

## Symbols

This post contains various definitions of the symbol of a differential operator. We will state a local version, then a global version and then we will finally view the symbol in its most abstract form: a section of a bundle over the total space of a cotangent bundle.

**Review of Local Definitions**

Let’s start by recalling that a differential operator of order on the manifold is is defined by:

where is smooth and if , then

Let and . The Symbol of , denoted by is then

A differential operator is said to be elliptic if for all and every we have that is invertible.

**Global definition of the Symbol**

Consider a globally defined differential operator

for and we want to define a linear map

in a coordinate free way.

With this in mind let and choose:

1. such that

2. such that

Then we define

Notice that even though this is a coordinate free definition of the symbol, it is still unclear how it changes in and . We will later see that is actually smooth on . Before this, we should prove that this definition is in fact independent on the choice of and .

* does not depend on *

**Claim 1** If is a smooth function such that and , then

*Proof:* For any differential operator , any section and any function ,

Setting we have

Induction on the order of and (3) will give us the result:

Let , then by definition

and so

Now assume the claim is true for every differential operator of order less than and suppose . By definition,

Thus, by induction

and notice that (3) gives us

so that

* does not depend on *

**Claim 2** Let be such that , then

*Proof:* It is easier if we use the easy direction of Peetre’s Theorem so that we can use the fact that is local, that is

equivalently

equivalently

So, since , we have as sought.

Let us finish the section with a short remark:

is homogeneous of degree in . That is, for every ,

Proof: Simply take instead of in the definition for .

**Local=Global**

**Lemma 1** For a differential operator of order , the two definitions of symbol coincide under the identification given by

*Proof:* Let . The function satisfies the conditions stated in the coordinate free definition of .
Let be the constant section , that is, for every .

Then (2) reads

where by (1)

Notice that here

since is a constant section.

Also notice that

1. for every :
This is because there is always a factor of in the expression for whenever .

2. :
This is a simple calculation.

Consolidating all the information we conclude

**Symbol as a section**

By consolidating definitions (*) and (1) of we get . Here is the bundle map and we are just looking at the diagram

To be explicit, if , then with . So

that is, and we are using the identification .

Smoothness follows from the smoothness of the local definition and the fact that both definitions coincide locally.

Finally, let

then we have

**Proposition 2** There is an exact sequence

Notice that this proposition (re)captures the fact that the symbol of an operator only `sees’ the `top’ degree of the operator.

**Fundamental Theorem of Elliptic Operators**

Now that we have a global definition of the symbol of a differential operator, we can state what it means for a differential operator to be elliptic. Namely, is elliptic if for every (i.e is in the complement of the zero section of the cotangent bundle), the map is invertible. The most important result involving elliptic operators is the following theorem:

**Theorem 3** Fundamental Theorem of Elliptic Operators

If is an elliptic differential operator over a compact manifold , then both and are finite dimensional vector spaces.

## Two beautiful theorems about C(X)

Let be a topological space. Let be the set of continuous functions from to . can also be thought of as set of smooth sections of the trivial bundle . Anyway, we get a contravariant functor

where is the category of . is the category of with and every sends to . The two beautiful to be discussed here are the following. *(Hewitt)*For , compact Hausdorff there is a bijection

*(Swan)*If is compact Hausdorff then taking sections gives a bijection from

These two beautiful *theorems*have some remarkable consequences If , are compact and Hausdorff then,

\textup{2} leads to the following result in –

is a consequence of the following Let be compact, Hausdorff topological space

- For ,
is a maximal ideal,

- If is a maximal ideal, then such that ,
- where is the set of all maximal ideals of a ring equipped with topology. The isomorphism takes to

*Proof:*

- Clearly is maximal as which is a field.
- Notice, if , then If is an such that for all in then for every , such that . Each there exists such that . Since is compact cover . Using which do not vanish on respectively, define
. Observe, . Define . Clearly and . Thus . Thus the only maximal ideals of is of the form for some .

- For any ideal of a ring define,
is the basis for all sets in the space under topology. The map

which sends

is already a bijection. All we need to show is

IF closed then define and \vspace{5pt}

IF be a basic closed set in , ie, for some then, define. Then is clearly a closed set and clearly .

*Proof:* *(of Theorem 1)* In fact the gives the map between the sets of the respective category.

**One-one**

Let . Then

, where . If then

by using bump functions near each point

**Onto**

Given a map , we induce a map

By \textup{5} we get a map

It is clear that . *Proof:* *( sketch of proof of theorem 2)*

Notice that Let be the map

G: isomorphism class of vector bundles over finitely generated C(X)-modules

where given a vector bundle

= smooth sections of .

Since is compact, any vector bundle is a of a trivial bundle of finite dimension, ie . Hence is a sub-module of due to the following isomorphism.

smooth sections on the trivial bundle

Thus is a finitely generated module. Moreover every bundle of finite dimension over a compact space has a complement, say , hence . Hence its projective. Given a finitely generated projective module over , say , find and a module , such that

Then define . This is a over . The proof is non-trivial and is a of .