## The Algebraic Hodge Theorem and the Fundamental Theorem of Elliptic Operators

** Statement of the Fundamental Theorem of Elliptic Operators**

**Definition. **Let and be inner product spaces. Let be a linear map. Then a linear map is called the formal adjoint of if for any and any .

**Lemma. ** (1) If a formal adjoint exists, it is unique. (2) If , then exists.

**Example.** The map defined by summing the coodinates has no formal adjoint, where is the colimit of .

If and are Hilbert spaces, then we have

**Theorem.** Any continuous linear map of Hilbert spaces has a formal adjoint.

**Example. ** Let be a differential operator. Suppose and have smooth inner product structures, then we have the “-inner products” on and , given by . Then has a formal adjoint . If we write locally as , then .

An elliptic operator is *self-adjoint* if .

We can now state the Fundamental Theorem of Elliptic Operators. Later in this entry we will give some corollaries and much later in the course we will outline a proof using the method of elliptic regularity.

**Fundamental Theorem of Elliptic Operators.** For a self-adjoint elliptic operator , there is an orthogonal decomposition with finite-dimensional.

It is important here that the manifold is closed.

** The algebraic Hodge theorem **

Suppose now we have (co)chain complex over or :

Give each an inner product. Assume each has a formal adjoint . Define the Laplacian . Then we have

**Lemma.** iff and .

*Proof. *Suppose . Then we have

and hence .

**Theorem. **Let be a (co)chain complex over a field . Then there exist decompositions such that the (co)chain complex can be written as

When or and is finite dimensional for each , setting and , the theorem above becomes a corollary of the following Algebraic Hodge Theorem:

**Algebraic Hodge Theorem.** Let be a (co)chain complex over or . Suppose that has inner product for each and that formal adjoint exists for each . Let , then

(1) TFAE: (a) , (b) and , (c) .

(2) .

(3) If is finite dimensional, then .

(4) If for any , then there are orthogonal decompositions

*Proof.* (1) (a) (b) (c) (a).

(2) Let , then .

(3) Show the inclusion in (2) is an equality by counting dimensions.

(4) It suffices to show the following orthogonal decomposition:

However easily we have

By checking the decomposition diagram above, we can obtain:

**Corollary.** is an isomorphism, where

**Corollary.** iff is an isomorphism for any .

** Wrapping up **

**Corollary. ** *Algebraic Wrapping up. For as above: * is an isomorphism. Hence is an isomorphism for all p.

This corollary “wraps up” a (co)chain complex into a single map.

Next, we consider wrapping up an ellliptic complex.

**Definition.** An e*lliptic complex of differential operators* is a cochain complex of differential operators

so that for all the associated symbol complex is exact.

If we define the symbol of differential operator of order by , then we have .

**Proposition.** Let be an elliptic complex of differential operators. Give and metrics for each . Then is an elliptic operator.

*Proof.* For any , since the complex is elliptic, we have the exact sequence

Thus, is an isomorphism.

Finally note that .

** Consequences of the fundamental theorem.**

We deduce the following corollaries of the Fundamental Theorem, the Algebraic Hodge Theorem, and Wrapping Up.

**Corollary.** Let be an elliptic complex of differential operators.

(1) For any , is finite dimensional.

(2) For any , .

(3) .

**Corollary.** If is an elliptic differential operator, then we have isomorphisms . Hence the kernel and the cokernel of an elliptic differential operator are finite dimensional.

**Corollary.** If is self-adjoint, then .

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