In this lecture, we consider a diffusion operator L which is essentially self-adjoint. Its Friedrichs extension is still denoted by L.

The fact that we are now dealing with a non negative self-adjoint operator allows us to use spectral theory in order to define the semigroup generated by L. We recall the following so-called spectral theorem.

**Theorem:*** Let be a non negative self-adjoint operator on a separable Hilbert space . There exist a measure space , a unitary map and a non negative real valued measurable function on such that for , . Moreover, given , belongs to if only if .
*

We may apply the spectral theorem to the self-adjoint operator in order to define . More generally, given a Borel function and the spectral decomposition of , , we may always define an operator as being the unique operator that satisfies We may observe that is a bounded operator if is a bounded function.

As a particular case, we define the diffusion semigroup on by the requirement

This defines a family of bounded operators whose following properties are readily checked from the spectral decomposition:

- For ,
- and for , .
- For , the map is continuous in .
- For ,

We summarize the above properties by saying that is a self-adjoint strongly continuous contraction semigroup on .

From the spectral decomposition, it is also easily checked that the operator is furthermore the generator of this semigroup, that is for , From the semigroup property, it implies that for , , and that for , the derivative on the left hand side of the above equality being taken in .

It is easily seen that the semigroup is actually unique in the followings sense:

**Proposition:*** Let , , be a family of bounded operators such that:*

- For , .
- For ,

*then for every and , .
*

**Exercise:*** Show that if is the Laplace operator on , then for $t > 0$, *

**Exercise:*** Let be an essentially self-adjoint diffusion operator on . Show that if the constant function and if , then *

**Exercise:*** Let be an essentially self-adjoint diffusion operator on . *

- Show that for every , the range of the operator is dense in .
- By using the spectral theorem, show that the following limit holds for the operator norm on ,

**Exercise:*** As usual, we denote by the Laplace operator on . The Mac-Donald’s function with index is defined for by .*

- Show that for and ,
- Show that for , .
- Show that
- Prove that for and , where (You may use Fourier transform to solve the partial differential equation ).

**Exercise:*** By using the previous exercise, prove that for , the limit being taken in . Conclude that almost everywhere, *

**Exercise:**

- Show the subordination identity
- The Cauchy’s semigroup on is defined as . By using the subordination identity and the heat semigroup on , show that for , where

In the statement of the spectral theorem, is H a comlex Hilbert space, or real Hilbert space?

In lecture 6 you will use this theorem in the case of H=L^2_\mu(R^n,R), the real Hilbert space.

But In many books I have ever seen the famous theorem is treated only in complex Hilbert spaces, not real. Is it also true in the real case?

The theorem is also true in the real case. There is a discussion in the following thread on Math Stack Exchange:

http://math.stackexchange.com/questions/638216/spectral-theorem-for-unbounded-self-adjoint-operators-on-real-hilbert-spaces

The discussion on Math Stack Exchange was difficult for me.

More knowledge seems to be required.

Thank you.