MMA140 Spectral Theory and Operator Algebras Spring 25

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, Thusday are reserved for lectures, while Friday is the day of the exercise session. However we may also use Fridays 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
24/3	1.1, 1.2	Spectrum and invertibility
25/3	1.3	Banach algebras: definitions, examples. The spectrum of an element of a Banach algebra.
27/3		Exercises, review, discussion
31/3	1.6, 1.7	General properties of the spectrum. Spectral radius.
3/4	1.8, 1.9, 1.10	Gelfand's theory of commutative Banach algebras: the Gelfand transform and spectrum.
4/4		Exercises, review, discussion
7/4	2.1	Operators on Hilbert spaces. Adjoint. Types of operators and their spectrum.
10/4	2.2	Commutative C*-algebras: definition, examples, special elements and their spectrum.
11/4		Exercises, review, discussion
25/4	2.3	Continuous calculus for normal elements in a C*-algebra.
28/4	2.4	Spectral Theorem and diagonalization
2/5		Exercises, review, discussion
5/5	2.8, 3.2	Compact operators. Riesz theory of compact operators.
8/5	3.2, 3.3	Riesz theory (continuation), Fredholm operators and index
9/5		Exercises, review, discussion
12/5	3.4	Fredholm operators and index
15/5	4.2, 4.3	Applications: Toeplitz Operators and Index. Exercises, review, discussion
16/5		Exercises, review, discussion
19/5	4.4, 4.6	Applications: Toeplitz Operators and Index.
22/5	4.7, 4.8	GNS construction. The Gelfand Naimark theorem
23/5		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 Thusdays (the dates are in the table below, where I also refer to Recommended Exercises above) and should be sent to turowska@chalmers.se or downloaded via Canvas. Any format is allowed  (typed or hand-written).

Day	Assignments
1/4	Assignment I: 7,8,9 from Exercises I
10/4	Assignment II: 5,9,11 from Exercises II
24/4	Assignment III: 7,9 from Exercises III
8/5	Assignment IV: 11 a, b, 12 a,b
15/5	Assignment V: 5,8,11
22/5	Assignment VI: 4,5,6

 

 

 

 

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