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.

 

 

 

 

 

Program

The schedule of the course is in TimeEdit.

 

Lectures

Remarks:

1.  Monday, Tuesday are reserved for lectures, while Thursday is the day of the exercise session. However we may also use Thursdays to cove some lecture material.

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

 

 

Day	Sections	Content
20/3	1.1, 1.2	Spectrum and invertibility
21/3	1.3	Banach algebras: definitions, examples. The spectrum of an element of a Banach algebra.
23/3		Exercises, review, discussion
27/3	1.6, 1.7	General properties of the spectrum. Spectral radius.
28/3	1.8, 1.9, 1.10	Gelfand's theory of commutative Banach algebras: the Gelfand transform and spectrum.
30/3		Exercises, review, discussion
17/4	2.1	Operators on Hilbert spaces. Adjoint. Types of operators and their spectrum.
18/4	2.2	Commutative C*-algebras: definition, examples, special elements and their spectrum.
20/4		Exercises, review, discussion
24/4	2.3	Continuous calculus for normal elements in a C*-algebra.
25/4	2.4	Spectral Theorem and diagonalization
27/4		Exercises, review, discussion
2/5	2.8, 3.2	Compact operators. Riesz theory of compact operators.
4/5		Exercises, review, discussion
8/5	3.2, 3.3	Riesz theory (continuation), Fredholm operators and index
9/5	3.4	Fredholm operators and index
11/5		Exercises, review, discussion
15/5	4.2, 4.3	Applications: Toeplitz Operators and Index.
16/5	4.4	Applications: Toeplitz Operators and Index.
18/5		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 Tuesdays (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
28/3	Assignment I: 6,8,9 (Exercises I)
18/4	Assignment II: 5,9,11 (Exercises II)
25/4	Assignment III: 8,10
2/5	Assignment IV: 11 a,b, 12 a,b.
12/5	Assignment V: 5,8,11
19/5	Assignment VI: 4,5,6

 

 

 

 

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