Module Identifier MAM8020
Module Title SPECTRAL THEORY
Co-ordinator Dr Ian Wood
Semester Semester 2
Pre-Requisite MA30210
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour exercise classes
Other   6 hours of tutorials discussing directed reading and giving help in preparation of presentation, feedback on presentations
Workload Breakdown   Lectures: 19, Exercise classes: 3, Tutorials: 6, Private study: 60, Directed reading 39, Assignments: 16, Presentation preparation 34, Presentations, discussions and feedback 3, Exam preparation: 20, Exam: 2
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Assessment2 Hours TWO HOUR EXAMINATION  80%
Semester Assessment PRESENTATION (30 MINUTE PRESENTATION)  20%
Supplementary Assessment2 Hours TWO HOUR WRITTEN EXAMINATION  80%
Supplementary Assessment PRESENTATION (IF FAILED ORIGINALLY)  20%

#### 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

6. define the spectrum and resolvent set

7. describe spectrum of compact and selfadjoint operators

8. explain an advanced development/application of Spectral Theory via a presentation to an audience

#### 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: essential concepts are introduced for any student seeking a deeper understanding of mathematical analysis and its applications.

#### 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. Students will be expected to work on a small project and give a presentation.

#### 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
Topics for projects include:
Fourier series: conditions for convergence, Dirichlet kernel, Fejer kernel, Gibbs¿ phenomenon
Hilbert-Schmidt operators: Hilbert-Schmidt norm, functional calculus, trace of an operator
Integral operators: Fredholm operators, Fredholm alternative, Volterra operators
Banach spaces: duality, reflexivity, functionals
Lax-Milgram Lemma: Riesz representation theorem, forms, Lax-Milgram, applications

#### 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 The project will make the students independently research a mathematical topic. Communication Written answers to exercises must be clear and well-structured. Project will help students develop presentation skills. Improving own Learning and Performance Students are expected to develop their own approach to time-management regarding completion of assignments and projects on time and preparation between lectures. Team work Students will be encouraged to work on problems in groups during exercise classes. Information Technology Students will be encouraged to research topics on the internet and can use technology in their presentation Application of Number Required throughout the course Personal Development and Career planning Completion of tasks (assignments and presentation) 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

Books
** Recommended Text
Kreyszig, E (1978) Introduction to Functional Analyses with Applications Wiley
Yosida, K. (1995) Functional Analysis Springer
Young, N (1988) An Introduction to Hilbert Space Cambridge University Press
** Supplementary Text
Dunford, N; Schwartz (1963) Linear operatiors I & II Wiley
Evans, L C (1998) Partial Differential Equations American Mathematical Society
Reed, M & Simon, B (1972) Methods of Modern Mathematical Physics I: Functional Analysis Academic Press
Reed, M & Simon, B (1972) of Modern Mathematical Physics II: Fourier Analysis, self-Adjointness Academic Press

#### Notes

This module is at CQFW Level 7