Latest news
Welcome to the course! The schedule for the course can be found in TimeEdit.
Questions to exam
Teachers
Course coordinator:Lyudmila Turowska, Department of Mathematical Sciences, room L3025, tel: 7725341, email: turowska ("vid") chalmers.se
Course literature
We will be mainly use the book by W. Arveson "A short Course on Spectral Theory", Springer.
Description of course
In this course you will see a comprehensive treatment of the theory of linear operators on infinitedimensional spaces. Our fundamental problem is to calculate spectra of specific operators. The spectrum of bounded operators on Banach spaces is best studied within the context of Banach and in particular C*algebras, and a part of the course will be devoted to the theory of these algebras. You will also learn about the spectral theorem for normal operators, the Riesz theory of compact operators and index of Fredholm operators with applications.
Program
Lectures
Week 
Sections  Contents 

1 
Linear operators on Banach spaces, spectrum and invertibility of a bounded operator. 
1.1, 1.2 
Banach algebras: definitions, examples. The spectrum of an element of a Banach algebra. 
1.3, 1.5, 1.6 

2 
General properties of the spectrum. Spectral radius. 
1.6, 1.7 

Gelfand's theory of commutative Banach algebras: the Gelfand transform and spectrum. 
1.8, 1.9, 1.10 
3 
Operators on Hilbert spaces. Adjoint. Types of operators and their spectrum. 
2.1 

Commutative C*algebras: definition, examples, special elements and their spectrum. 
2.2 
4 
The continuous calculus for normal elements in a C*algebra. Spectral Theorem and diagonalization 
2.3, 2.4 
5 
Compact operators. Riesz theory of compact operators. 
2.8, 3.2 
6 
Fredholm operators and index 
3.3, 3.4 
7 
Applications: Toeplitz Operators and Index. (General C*algebras) 
4.2, 4.3, 4.4 



Recommended exercises
Week 
Exercises 

1 
Exercises I 
2 
Exercises II 
3 
Exercises III 
4 
Exercises IV 
5 
Exercises V 
6 
Exercises VI 
7 
Exercises VII 
Assignments
Deadlines for the handin exercises are on Thursday, the dates in the table below.
Day 
Exercises 

4/4 
6,8,9(Exercises I) 
11/4 
8, 10, 11(Exercises II) 
2/5 
3, 6, 8(Exercises III) 
9/5 
11 a,b, 12 a,b(Exercises IV) 
16/5 
8, 10, 11(Exercises V) 
23/5 
(Exercises VI) 
Examination
Handin exercises and oral exam.
Examination procedures
In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.
Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.
At the exam, you should be able to show valid identification.
After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.
At the annual (regular) examination:
When it is practical, a separate review is arranged. The date of the
review will be announced here on the course homepage. Anyone who can not
participate in the review may thereafter retrieve and review their exam
at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.
At reexamination:
Exams are reviewed and retrieved at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.
Old exams