# MAGIC043: Banach spaces and Fredholm theory

## Course details

### Semester

Spring 2018
Monday, January 22nd to Friday, March 16th; Monday, April 23rd to Friday, May 4th

### Hours

Live lecture hours
10
Recorded lecture hours
0
0

Thursdays
14:05 - 14:55

## Description

The course studies Banach spaces and operators acting on them, thus providing an introduction to an important branch of modern infinite-dimensional linear analysis. To be precise, the starting point of the course is the following classical theorem of F. Riesz. Let T be a compact operator on a Banach space X, and let I be the identity operator on X. Then:
1. the operator I+T has finite-dimensional kernel, and its image is closed and has finite codimension in X;
2. there is a non-negative integer n such that the kernel of (I+T)n is equal to the kernel of (I+T)n+1 and the image of (I+T)n is equal to the image of (I+T)n+1;
3. each non-zero point of the spectrum of T is an eigenvalue for T, and 0 is the only possible accumulation point of the spectrum of T.
The course is devoted to the study of these properties and their interrelationship, starting from a purely algebraic viewpoint.

### Prerequisites

Basic knowledge of functional analysis, up to and including the Open Mapping Theorem and the Hahn-Banach Theorem; MAGIC061 covers this, and much more. No specialist knowledge of Banach spaces or operator theory is assumed. In particular, the course is perfectly suitable for students who haven't seen Riesz's Theorem before; all the ingredients will be carefully introduced and discussed in the lectures.

### Syllabus

1. Course outline and motivation; background results from infinite-dimensional linear algebra; the Index Theorem for pre-Fredholm mappings.
2. Linear mappings with finite ascent and finite descent.
3. Brief review of fundamental background results from functional analysis; operator ideals.
4. Introduction to Fredholm operators and semi-Fredholm operators.
5. Yood's Lemma and Atkinson's Theorem.
6. Continuity of the Fredholm index.
7. Riesz-Schauder operators; introduction to Riesz operators.
8. The holomorphic function calculus and Riesz' Idempotent Theorem.
9. Riesz operators and the essential spectrum.
10. Inessential operators, the Jacobson radical and Kleinecke's characterization of the inessential operators.

## Lecturer

• NL

### Dr Niels Laustsen

University
University of Lancaster

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

• An introduction to Banach space theory (R. E. Megginson, )
• Introduction to functional analysis (R. Meise and D. Vogt, )

## Assessment

The assessment for this course will be released on Monday 7th May 2018 and is due in by Sunday 20th May 2018 at 23:59.

A take-home exam will be set during the official MAGIC exam period (7-20 May 2018). You must submit your answers to it electronically in line with the MAGIC guidelines no later than 20th May 2018. The format of the paper will be as follows. Attempt all questions. The total number of marks is 50, with a minimum pass mark of 25. To obtain full marks, you must give details of your workings and include clear references (for instance by quoting the relevant theorem numbers) to any results from the lecture notes that you use.

Please note that you are not registered for assessment on this course.