Module Identifier MA38010  
Module Title SPECTRAL THEORY  
Academic Year 2007/2008  
Co-ordinator Dr Ian Wood  
Semester Semester 2  
Other staff Dr Robert J Douglas  
Pre-Requisite MA30210  
Mutually Exclusive MAM8020  
Course delivery Lecture   19 X 1 HOUR LECTURES  
  Seminars / Tutorials   3 X 1 HOUR EXERCISE CLASSES  
  Workload Breakdown   Lectures: 19, Exercise classes: 3, Private study: 48, Assignments: 12, exam preparation: 16, exam: 2  
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Assessment2 Hours TWO HOUR EXAMINATION  100%
Supplementary Assessment2 Hours TWO HOUR EXAMINATION  100%

Learning outcomes

On successful completion of this module students should be able to:
1. demonstrate knowledge of examples of Hilbert spaces and inner products

2. make use of orthogonality relations to manipulate inner products

3. expand elements of a Hilbert space in terms of an orthonormal basis

4. determine the norm of bounded linear operators

5. determine adjoints of operators and check for selfadjointness and compactness

6. define the spectrum and resolvent set

7. describe spectrum of compact and selfadjoint operators

Aims

Spectral Theory is one of the main current areas of research in mathematical analysis with applications in many sciences, in particular quantum mechanics. It underpins much of the modern theory of solutions of partial differential equations.

Brief description

Spectral theory deals with solvability of equations of the form (T-z)x=y, where T is a linear, but not necessarily bounded operator on a Banach or Hilbert space and z is a complex number. This module aims to introduce the basic concepts needed from Hilbert space theory and the theory of linear operators and give some first results on the spectrum of operators.

Content

Introduction to Hilbert spaces: inner products, orthogonality, orthogonal complements, orthonormal systems, basis of a Hilbert space, examples
Linear operators on Hilbert spaces: bounded and unbounded operators, adjoint operators, selfadjoint operators, compact operators, examples
Spectral theory: spectrum, resolvent, Neumann series, spectral radius for bounded operators, spectrum of compact operators, spectrum of selfadjoint operators, the spectral theorem for compact selfadjoint operators, examples

Module Skills

Problem solving The assignments will give the students opportunities to show creativity in finding solutions and develop their problem solving skills.  
Research skills N/A  
Communication Written answers to exercises must be clear and well-structured.  
Improving own Learning and Performance Students are expected to develop their own approach to time-management regarding completion of assignments on time and preparation between lectures.  
Team work Students will be encouraged to work on problems in groups during exercise classes.  
Information Technology N/A  
Application of Number Required throughout the course  
Personal Development and Career planning Completion of tasks (assignments) to set deadlines will aid personal development. The course will give indications of whether a student wants to further pursue mathematical analysis and its applications.  
Subject Specific Skills Broadens exposure of student to topics in mathematics  

Reading Lists

Books
** Recommended Text
Kreyszig, Erwin (1978.) Introductory functional analysis with applications /Erwin Kreyszig. Wiley 047103729X
Yosida, K (1995) Functional Analysis Springer
Young, N. (July 1988) An Introduction to Hilbert Spaces Cambridge University Press 9780521330718
** Supplementary Text
Dunford, N & Schwartz J T (1963) Linear Operators I & II Wiley
Reed, M & Simon B (1972) Methods of Modern Mathematical Physics I: Functional Analysis Academic Press

Notes

This module is at CQFW Level 6