Functional analysis

Involving the study of normed, Banach and Hilbert spaces, the operators on them and generalizations of these concepts, such as Banach algebras, this key area of mathematics underpins much of the research and applications in areas of analysis such as measure and probability theory, financial mathematics, quantum field theory in theoretical physics, approximation theory and differential and integral equations. Our group are involved with one of the active research areas, which concerns spectral theory of operators and, more generally, in Banach algebras. Spectral theory is one of the main branches of modern functional analysis and its applications. Roughly speaking, it is concerned with certain inverse operators, which arise quite naturally in connection with the problem of solving equations (e.g. differential and integral equations). Spectral theory can also be considered a generalization of matrix eigenvalue theory. Some research also involves the theory of strongly continuous semigroups of operators with a particular focus on multiplication operators on vector-valued function spaces and stability concepts. This is also branching out to operator theoretic aspects of ergodic theory. The members of our group are Dr R Benjamin, Dr R Heymann and Prof Sonja Mouton. We work individually and together. In addition, we have collaborators elsewhere in South Africa, as well as in Germany, Ireland, Netherlands, the UK and Zambia. M.Sc. and Ph.D. theses written by our recent students include the following topics: spectrum preserving mappings, finite rank elements, Fredholm theory, Drazin inversion and, in particular, aspects of spectral theory in ordered Banach algebras.

The following list contains some of our recent papers. (Nr. 7 is a survey paper.)

1. R. Heymann: Eigenvalues and stability properties of multiplication operators and multiplication semigroups. Mathematische Nachrichten 287, 2014, 574-584.
2. S. Mouton: Applications of the scarcity theorem in ordered Banach algebras. Studia Mathematica 225, 2014, 219–234.
3. S. Mouton and K. Muzundu: Domination by ergodic elements in ordered Banach algebras. Positivity 18, 2014, 119–130.
4. R. Benjamin and S. Mouton: Fredholm theory in ordered Banach algebras. Quaestiones Mathematicae 39, 2016, 643–664.
5. R. Benjamin and S. Mouton: The upper Browder spectrum property. Positivity 21, 2017, 575-592.
6. S. Mouton and R. Harte: Linking the boundary and exponential spectra via the restricted topology. Journal of Mathematical Analysis and Applications 454, 2017, 730-745.
7. S. Mouton and H. Raubenheimer: Spectral theory in ordered Banach algebras. Positivity 21, 2017, 755-786.
8. R. Benjamin and S. Mouton: A note on the lower Weyl and Lozanovsky spectra of a positive element. Positivity 22, 2018, 533-549.
9. R. Benjamin: Spectral mapping theorems for the upper Weyl and upper Browder spectra, to appear in Quaestiones Mathematicae.
10. R. Benjamin, N. Laustsen and S. Mouton: r-Fredholm theory in Banach algebras, to appear in Glasgow Mathematical Journal.