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 .
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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