# Gwybodaeth Modiwlau

Module Identifier
MA38310
Module Title
Operator Algebra
2013/2014
Co-ordinator
Semester
Intended for use in future years
Mutually Exclusive
MAM8310 Module overlap.
Pre-Requisite
Pre-Requisite
Pre-Requisite
Other Staff

#### Course Delivery

Delivery Type Delivery length / details
Lecture 18 x 1 hour
Seminars / Tutorials 4 x 1 hour
Workload Breakdown (Every 10 credits carries a notional student workload of 100 hours.) Lectures and tutorials 22 hours Worksheets (4 x 5 hours) 20 hours Private study 56 hours Examination 2 hours

#### Assessment

Assessment Type Assessment length / details Proportion
Semester Assessment 2 Hours   conventional examination  100%
Supplementary Exam 2 Hours   supplementary examination  100%

### Learning Outcomes

On completion of this module, students should be able to.
1. illustrate the basic theory and applications of operator theory;

2. define basic notions from operator theory and recognize their occurrence and relevance in applied problems;

3. solve specific problems from mathematical physics formulated in operator theoretic terms;

4. perform algebraic and analytic computations based on operator techniques;

### Aims

IMAPS wishes to introduce new level 3 modules reflecting research interests and expertise of new staff, thereby rectifying the previous problem of very limited range of modules for final year students. This module introduces the important field of operator theory. It is intended to offer this module in alternate years. Operator theory is only represented in the curriculum through spectral theory at present and we would like to introduce it as an option. It has critical importance to areas such as spectral analysis and quantum theory, both of which are strong research topics in the department.

### Brief description

Operator Theory is an important subject in modern mathematics, essential in the description of several areas of mathematics and physics.

### Content

Operators on Hilbert Spaces
Introduction to Hilbert spaces, definitions of bounded, unitary, projective and self-adjoint operators. Applications of the spectral theorem and Stone's theorem.

C*-algebras
Introduction of the notion of abstract operator algebras.

Applied Operator Theory
Motivating examples of operators from Mathematical Physics. Mathematical formulation of
Quantum Theory for closed dynamical systems. Introduction to the quantum theory of measurement.

### Module Skills

Skills Type Skills details
Application of Number Throughout the module.
Communication Students will be expected to submit written worksheet solutions
Improving own Learning and Performance Feedback via tutorials
Information Technology Extensive use of spreadsheets.
Personal Development and Career planning Students will be exposed to an area of application that they have not previously encountered.
Problem solving All situations considered are problem-based to a greater or lesser degree.
Research skills Students will be encouraged to consult various books and journals for examples of application.
Subject Specific Skills Using operator theoretic techniques in modeling.

### Notes

This module is at CQFW Level 6