## 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
Non-modular pre-requisites: Either MA3TLA must have been undertaken as a prerequisite or MA4MS must be undertaken as a co-requisite. Similarly, MA3MTI must have undertaken as a prerequisite or MA4MTI must be undertaken as a co-requisite
Co-requisites:
Modules excluded: MA3OT Operator Theory
Current from: 2018/9

Module Convenor: Dr Jani Virtanen

Type of module:

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

This module will be assessed to a greater depth than the excluded module MA3OT

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

Summative assessment- Examinations:
Two hours.

Summative assessment- Coursework and in-class tests:
One examination and two assignments.

Formative assessment methods:
Problem sheets.

Penalties for late submission:
The Module Convener will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day[1] (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.

Assessment 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).