Module Information

Module Identifier
Module Title
Spectral Theory
Academic Year
Semester 2
Exclusive (Any Acad Year)
Reading List
Other Staff

Course Delivery



Assessment Type Assessment length / details Proportion
Semester Exam 2 Hours   Written Examination  100%
Supplementary Assessment 2 Hours   Written Examination  100%

Learning Outcomes

On successful completion of this module students should be able to:

Demonstrate knowledge of examples of Hilbert spaces and inner products

Make use of orthogonality relations to manipulate inner products

Expand elements of a Hilbert space in terms of an orthonormal basis

Determine the norm of bounded linear operators

Determine adjoints of operators and check for selfadjointness and compactness

Define the spectrum and resolvent set

Describe spectrum of compact and selfadjoint operators

Demonstrate a knowledge of advanced developments/applications of Spectral Theory

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 study more advanced topics via mini-courses.


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.


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 mini-courses 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

Skills Type Skills details
Application of Number Required throughout the course
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.
Information Technology Students will be encouraged to research topics on the internet and can use technology in their presentation
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.
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.
Subject Specific Skills Broadens exposure of student to topics in mathematics
Team work Students will be encouraged to work on problems in groups during exercise classes.


This module is at CQFW Level 7