# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

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 Type | Assessment length / details | Proportion |
---|---|---|

Semester Assessment | 2 Hours TWO HOUR EXAMINATION | 80% |

Semester Assessment | PRESENTATION (30 MINUTE PRESENTATION) | 20% |

Supplementary Assessment | 2 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

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

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

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

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. |

### Reading List

**Recommended Text**

Kreyszig, E (1978) Introduction to Functional Analyses with Applications Wiley Primo search Yosida, K. (1995) Functional Analysis Springer Primo search Young, N (1988) An Introduction to Hilbert Space Cambridge University Press Primo search

**Supplementary Text**

Dunford, N; Schwartz (1963) Linear operatiors I & II Wiley Primo search Evans, L C (1998) Partial Differential Equations American Mathematical Society Primo search Reed, M & Simon, B (1972) Methods of Modern Mathematical Physics I: Functional Analysis Academic Press Primo search Reed, M & Simon, B (1972) of Modern Mathematical Physics II: Fourier Analysis, self-Adjointness Academic Press Primo search

### Notes

This module is at CQFW Level 7