## MA4OT-Operator Theory

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA3FA1 Functional Analysis I MA3TLA Topology and Linear Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3OT Operator Theory
Module version for: 2016/7

Module Convenor: Dr Jani Virtanen

Summary module description:
Operator theory is a diverse area that has grown out of linear algebra and complex analysis, and is often described as the branch of functional analysis that deals with bounded linear operators and their (spectral) properties. It has developed with strong links to (mathematical) physics and mechanics, and continues to attract both pure and applied mathematicians in its vast area.

Aims:
To introduce the basic theory of Banach algebras and spectral theory, to develop further theory of bounded operators on Hilbert space, to discuss the theory of compact and Fredholm operators, and finally apply the results to the study of concrete operators in function spaces.

Assessable learning outcomes:
By the end of the module students are expected to be able to:
- use Banach algebra techniques to solve problems in mathematics, applied mathematics and mathematical physics;
- demonstrate understanding of the difference between Banach space and Hilbert space operators;
- demonstrate understanding of the geometry of Hilbert space;
- solve problems involving bounded linear operators acting on function spaces.

To understand the fruitful interplay between functional analysis, complex analysis and linear algebra, and how this produces remarkable results in mathematics and its applications.

Outline content:
Banach algebras. Basic geometry of Hilbert space. Introduction to the theory of C*-algebras. Further theory of compact and bounded linear operators on Hilbert space. Fredholm operators. Concrete operators (such as Toeplitz and Hankel operators) acting on function spaces (such as Hardy and Bergman spaces).

Brief description of teaching and learning methods:
Lectures supported by problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 Guided independent study 80 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Other information on summative assessment:
One examination and two assignments.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convenor will apply the following penalties for work submitted late, in accordance with the University policy.

• where the piece of work is submitted up to one calendar week after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for the piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

• The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Length of examination:
Two hours.

Requirements for a pass:
A mark of 50% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).