Spectral theory in ordered Banach algebras

Around 1900, it was discovered that the spectrum of a square matrix with positive entries has certain special features. During the next several decades, this has led to the more general development of spectral theory of positive operators on Banach lattices, which is widely referred to as the Perron-Frobenius theory. This theory has become an important area of modern operator theory, having many theoretical and practical applications. Yet the even more general notion of a positive element of an ordered Banach algebra (OBA) was only established in the 1990s around the time when Sonja Mouton did her PhD under the supervision of Heinrich Raubenheimer. The three decades since then have seen Sonja and her collaborators, as well as other researchers worldwide, continue with their investigation into several different aspects of spectral theory in OBAs. The results have shown that the application of Banach algebra techniques in ordered (Banach algebra) structures yields new insights into the Perron-Frobenius theory.

The main question in the spectral theory of Banach algebras concerns properties such as continuity of the spectrum and spectral radius functions. It was established in 1951 already that although these functions have certain continuity-type properties, they are generally not continuous. Since then, the amount of literature on the subject has grown steadily. Sonja initiated the fairly recent study of spectral continuity in ordered Banach algebras, which has led her to discover an interesting object that she dubbed the boundary spectrum. This spectrum has proven useful in the study of spectral continuity in OBAs and has shown that, while at many elements the spectral radius function is not continuous, its restriction to the cone of positive elements does have this property. In addition, the study of the boundary spectrum has resulted in the discovery of an unexpected link between this spectrum and another important spectrum, the exponential spectrum. The restricted topology was defined and utilised in this development.

Another highlight of Sonja’s research was the development of the so-called r-Fredholm theory in Banach algebras, which led to a generalisation of Harte’s famous theorem and provided a number of applications in OBAs. She has also developed applications of Aupetit’s Scarcity Theorem and Alekhno’s irreducibility theory in OBAs. In her inaugural lecture, she will elaborate on these aspects of her research in spectral theory in Banach algebras and in OBAs.

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Short biography

Having initially struggled to decide between Mathematics and Music, Sonja Mouton (born Rode) began her studies at the former University of the Orange Free State (UOFS), where she obtained her PhD in Mathematics in 1994. During this time, and for a short while thereafter, she lectured in Mathematics, first at the former Free State Technicon and later at the UOFS, until she was appointed as a lecturer at Stellenbosch University (SU) in 1995. Currently the leader of the Functional Analysis research group in the Mathematics Division, she was promoted to senior lecturer in 2002, associate professor in 2012, and professor in 2023. To date, Sonja has published 27 articles in international journals, of which a sizeable number were single-authored, and currently holds a C1-rating from the National Research Foundation. She has supervised four MSc and three PhD students. In 2014, Sonja helped organise the renowned Workshop on Ordered Banach Algebras in Leiden, Netherlands, where she also presented a three-hour survey lecture on spectral theory in ordered Banach algebras. She has since been invited numerous times to present keynote and plenary lectures at national and international conferences and workshops. This dedicated lecturer also received the Rector’s Award for Excellence in Teaching in 2003. Sonja is married to Toit Mouton, a professor in SU’s Department of Electrical and Electronic Engineering, and she is passionate about caring for animals, especially guinea pigs.

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