7CCMMS08 Operator Theory
7CCMMS08T (MSc Programme)/ 7CCMMS08U (MSci Programme)
Lecturer: Professor Alexander Pushnitski
Credit level: 7 Credit Value: 15
Mathematics MSci Fourth Year
Mathematics and Physics MSci Fourth Year
MSc Theoretical Physics
Advice/Help for this module is also available from: Professor Eugene Shargorodsky
This module will introduce you to the terminology, notation and the basic results and concepts of Banach and Hilbert spaces. The goal is to establish one major theoretical result (the spectral theorem for compact self-adjoint operators) and demonstrate some applications. The relation of the subject with other branches of mathematics (Fourier analysis, complex functions, differential equations) will be indicated. This module should prepare you for reading the literature of a wide variety of subjects in which Hilbert space ideas are used.
Elementary properties of Hilbert and Banach spaces. Orthonormal bases. Fourier expansions. Riesz representation theorem. The adjoint. Orthogonal projections. Spectral theory of bounded linear operators. The spectral theorem for compact self-adjoint operators. Applications to differential and integral equations. Further topics as time permits chosen from: the spectral theorem for bounded selfadjoint operators; comments on unbounded operators and applications; Fredholm operators.
Two hours of lectures per week, together with a half hour informal tutorial.
7CCMMS05 Metric and Banach Spaces. 6CCM321A and 5CCM222A or equivalents (that is, a module in analysis using normed spaces and a module in linear algebra).
| Type||Weight||Marking Model|
|2 hr written examination
Exercise sheets will be given out. Solutions handed in will be marked and difficulties discussed in class. In addition, it is essential that you work through the theory as the module progresses.
Suggested reading/resource (link to My Reading Lists)
16 August 2017