# Spectral Theory in Hilbert Spaces, Spring Semester 09, ETH

Update (August 15, 2011): I have just made a few more changes; these are mostly cosmetic corrections, but I’ve also added a full discussion of Hilbert-space valued integrals, as they are needed in the proof of Stone’s theorem given in Chapter 6.

Update (August 2, 2009): a more complete version is now available than the one last mentioned on this page.

The lecture notes are always available here.
The ETH web site of the course is there.

Here are links to posts which are related to spectral theory:
Euler for another day (using the trace to compute ζ(2)).
A post pointing out the fact that the spectrum of a multiplication operator is the support of the image measure of the multiplier.

Any comments are welcome…

The approximate outline and dates of updates of the various versions are indicated below (the last item is usually not yet complete…). The same section may appear multiple times if serious changes are made (e.g., corrections, indicated by a * when they are of some importance):

* February 26, 2009:
1 – Introduction and motivation
2 – Chapter 2: Abstract spectral theory and compact operators
2.1 – Review of general spectral theory in Banach algebras (elementary facts)
2.2 – Review of spectral theorem for compact operators
2.3 – Square root and standard form for general compact operators
2.4 – Variational characterizations of eigenvalues and applications
2.5 – Trace class operators

* March 9, 2009:
2.5 * Trace class operators (quite a few corrections)
3 – Chapter 3: Spectral theorem for bounded operators
3.1 – Continuous functional calculus for self-adjoint operators

* March 25, 2009:
3.2 – Spectral measures
3.3 – The spectral theorem for self-adjoint operators
3.4 – Projection valued measures
3.5 – The spectral theorem for normal operators
4 – Chapter 4: Unbounded operators on a Hilbert space
4.1 – Basic definitions

* April 2, 2009:
4.2 – The graph, closed and closable operators
4.3 – The adjoint
4.4 – Criterion for self-adjointness and for essential self-adjointess
4.5 – Basic spectral theory for unbounded operators

* April 7, 2009:
4.6 – The spectral theorem for unbounded self-adjoint operators
5 – Chapter 5: Applications I, the Laplace operator
5.1 – Definition

* May 5, 2009:
5.2 – Two basic examples
5.3 – Survey of more general cases

* August 2, 2009:
6 – Introduction to Quantum Mechanics

## 4 thoughts on “Spectral Theory in Hilbert Spaces, Spring Semester 09, ETH”

1. A.Matteï says:

Dear Sir,

I am a French student from the Ecole Polytechnique and I am actually your notes.

Page 38, while demonstrating the spectral theorem for bounded self-adjoint operators, you are introducing a measure “mu”. According to you, “it is easily checked that this is indeed a measure”.

Could you be kind enough to explain me why it is so easy ?

1. Il suffit ici d’utiliser la définition d’une mesure ; pour vérifier l’additivité dénombrable, il faut juste échanger l’ordre de deux séries à terms positifs. (Un exemple à garder en tête est le cas où chaque mu_n est la même mesure mu_0 sur sigma(T), et dans ce cas la mesure mu est la mesure produit de mu_0 et de la mesure de comptage).

2. A.Matteï says:

Oui mais la mesure mu n’est pas définie sur la tribu produit de IN x Sp(T). (Elle est juste définie sur P(IN)x B(Sp(T)).

Comment faire pour prolonger mu à cette tribu ?

3. Raphael says:

Dear Kowalski,

I am studying your lecture notes and on page 77, when you prove the uniqueness of the map f-> f (T), you argue that \$ L ^ {\ infty} (R) \$ consists of functions that are pontual limit of sequences(uniformly limited) of functions in \$ C_0 (R) \$. I don’t understanding this argument. Could you explain in more detail, please?