# MMA140, Spectral Tehory and Operator Algebras, Spring 19

## 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, e-mail: 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 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, 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

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 hand-in 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

Hand-in 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.