AEI-DFG - Operator theory on free Banach lattices

• Applicant: Professor Dr. Jochen Glueck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 541471251

Banach lattices are a generalization of function spaces and provide a framework to deal in a unified way with several classical spaces arising in analysis. They are Banach spaces equipped with an additional order and lattice structure that behaves in a similar way to pointwise (or almost everywhere) order in a classical function space. A linear operator between two Banach lattices is called "positive" if it maps positive vectors to positive vectors. The relevance of Banach lattices and positive operators between them stems mainly from two different points of view. On the one hand, these objects possess a deep and extensive structure theory which, after being studied for several decades, is still far from complete. A large variety of open questions continue to stimulate a lot of ongoing research. On the other hand, arguments based on order structures and positive operators play an essential role in applications throughout various fields in mathematical analysis such as partial differential equations, stochastic analysis, or dynamical systems. Banach lattices and positivity present an abstract framework that underlies many arguments used on those fields and provides a rich toolbox that can be employed there. A very recent development is the introduction of "free Banach lattices". They combine the idea of "free objects" (from universal algebra and category theory) with the functional analytic framework of Banach lattices. One of the most remarkable features of this new theory is that it yields a plethora of new examples of Banach lattices which are canonical in a certain way, but which also tend to behave surprisingly different from what one expects from experience with classical function spaces. The development of free Banach lattice theory encompasses a number of concepts and results of positive operators, but so far no approach to develop a comprehensive theory of positive operators on free Banach lattices has been undertaken. The purpose of this project is to fill this gap. This will, on the one hand, lead to a large variety of insights on how positive operators on free Banach lattices behave and thus make a significant contribution to the theory of free Banach lattices themselves. On the other hand, the universal properties of free Banach lattices offer the opportunity to construct new types of counterexamples and to thus give answers to some problems that have been open in the general theory of positive operators for a long time.

• DFG Programme: Research Grants
• International Connection: Spain
• Partner Organisation: Agencia Estatal de Investigación
• Cooperation Partner: Dr. Pedro Tradacete