The course Spectral Theory and Operator Algebras will give you a comprehensive treatment of the theory of linear operators on infinite-dimensional 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, one of the deepest, most elegant and important results in mathematics, the Riesz theory of compact operators and index of Fredholm operators with applications. The course can be considered as Functional analysis II and requests the knowledge from the first course in Functional Analysis.

More information on the aim and learning outcomes of the course can be found in a separate course PM

This page contains the program of the course: lectures, exercise sessions and computer labs. Other information, such as learning outcomes, teachers, literature and examination, are in a separate course PM.

 

The Exam takes place the week 22-26/3. Zoom link to the exam: https://chalmers.zoom.us/j/63553511569 Password: 309442

Contact the examiner  if you have not been assigned a time slot for the exam.

 

 

 

Program

The schedule of the course is in TimeEdit.

 

Lectures

Remarks:

1.  All lectures (including the exercise sessions) take place on zoom. Zoom Link to the Lectures: https://chalmers.zoom.us/j/69501112026  Passcode: 816626

2.  Monday, Wednesday are reserved for lectures, while Friday is the day of the exercise session. However we may also use Fridays to cove some lecture material.

3.  Below is a preliminary program and the correspondence between what will be covered and the days is approximate.

 

 

Day	Sections	Content
18/1	1.1, 1.2	Spectrum and invertibility
20/1	1.3	Banach algebras: definitions, examples. The spectrum of an element of a Banach algebra.
22/1		Exercises, review, discussion
25/1	1.6, 1.7	General properties of the spectrum. Spectral radius.
27/1	1.8, 1.9, 1.10	Gelfand's theory of commutative Banach algebras: the Gelfand transform and spectrum.
29/1		Exercises, review, discussion
1/2	2.1	Operators on Hilbert spaces. Adjoint. Types of operators and their spectrum.
3/2	2.2	Commutative C*-algebras: definition, examples, special elements and their spectrum.
5/2		Exercises, review, discussion
8/2	2.3	Continuous calculus for normal elements in a C*-algebra.
10/2	2.4	Spectral Theorem and diagonalization
12/2		Exercises, review, discussion
15/2	2.8, 3.2	Compact operators. Riesz theory of compact operators.
17/2	3.2	Riesz theory, continuation
19/2		Exercises, review, discussion
22/2	3.3	Fredholm operators and index
23/2	3.4	Fredholm operators and index
25/2		Exercises, review, discussion
28/2	4.2, 4.3	Applications: Toeplitz Operators and Index.
2/3	4.4	Applications: Toeplitz Operators and Index.
4/3		Exercises, review, discussion

 

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Recommended exercises

Week	Exercises
1	Exercises I
2	Exercises II
3	Exercises III
4	Exercises IV
5	Exercises V
6	Exercises VI
7

 

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Assignments

Deadlines for the hand-in exercises are on Wednesdays (the dates are in the table below, where I also refer to Recommended Exercises above) and should be sent to turowska@chalmers.se

Day	Assignments
27/1	Assignment I: 6,8,9 (Exercises I)
3/2	Assignment II: 5,9,12 (Exercises II)
10/2	Assignment III: 7,9 (Exercises III)
17/2	Assignment IV: 11 a,b 12 a,b (Exercises IV)
24/2	Assignment V: 5,11,12 (Exercises V)
3/3	Assignment VI: 4,5,6 (Exercises VI)

 

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Piazza

As extra forum for questions and answers we are going to use Piazza. Everyone can ask and answer questions. One can be anonymous if one wants.  If you wish to use this platform  you should go in

piazza.com/chalmers.se/spring2021/mma140
 
create an account and login. Observe that you must use your Chalmers e-mail address.

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