Yosra Barkaoui's doctoral dissertation in mathematics at the University of Vaasa, Finland, has successfully generalised a fundamental theorem that has been limited to the bounded case. The research provides new mathematical tools for unbounded operators, which are essential in physics for describing concepts like kinetic energy, momentum, and time.
Yosra Barkaoui's doctoral dissertation focuses on closed operators that are nonnegative, which are essentially the mathematical tools used to describe real-world quantities, like energy, speed, or time, that can never go below zero. Her research extends Sebestyén's theorem, a key rule for how these operators behave.
– The Sebestyén theorem has been around since 1983, but it was only explored in the bounded case. This is the first time the theorem has been extended to the unbounded case and to linear relations, says Barkaoui.
The bounded case in mathematics refers to systems where an operator's norm, or its "size", is finite. Barkaoui's work generalises these rules to unbounded operators, where the norm can go to infinity.
– Many models in physics are based on unbounded systems. What is new in our work is that we found a link between two types of inequalities that describe how the underlying operators relate to each other, Barkaoui explains.
A mentorship dream come true
Barkaoui's research is theoretical, focusing on abstract mathematical structures rather than direct applications. While the immediate applications are in foundational research, the work provides a new, more solid basis for mathematicians to build upon.
– Our results give mathematicians the tools to work more confidently with unbounded operators. When the theoretical foundation is clear, it becomes easier to explore new questions and make further discoveries, says Barkaoui.
For Barkaoui, this dissertation marks not only a scientific achievement but also a personal milestone. It is her second PhD in mathematics, following her first doctorate completed in Tunisia. She decided to pursue a second doctorate largely because of the opportunity to work under the supervision of Professor Seppo Hassi.
– It was a dream of mine to work with Professor Hassi. I truly admire him, both as a mathematician and as a person. Working with him has been a real pleasure and a privilege, and his guidance has meant a great deal to me, says Barkaoui.
Doctoral dissertation
Barkaoui, Yosra (2025) Products of nonnegative selfadjoint operators, linear relations, and their local spectral theory. Acta Wasaensia 573. Doctoral dissertation. University of Vaasa.
