Order preserving operators in problems of optimal control and in the theory of partial differential equations
Final Report Abstract
The main objective of this project proposal was to investigate the structure of operators on ordered Banach spaces which are in general not Banach lattices. Examples of such ordered Banach spaces are the spaces of continuously differentiable functions, higher order Sobolev spaces, C∗-algebras, noncommutative Lp spaces and general noncommutative Banach function spaces, certain spaces of absolutely summing operators, spaces of nuclear operators, or spaces of Uryson operators if the range space is not Dedekind complete. The investigations were motivated by applications to evolution equations, that is, abstract differential or integral equations, or differential inequalities in ordered Banach spaces and by applications in optimization. We were in particular able to contribute to the abstract theory of linear and nonlinear operators on pre-Riesz spaces and on ordered Banach spaces. We studied the representability of bi-Riesz homomorphisms in partially ordered vector spaces, the class of atomic operators on vector lattices and of narrow operators in normed vector lattices, and we studied interpolation of order-preserving nonlinear semigroups on interpolation couples of Banach lattices, and the asymptotic behaviour of positive semigroups on ordered Banach spaces with almost interior points. This project was part of a DFG- RSF cooperation. Project partners are located in Moscow and Vladikavkas. The cooperation was established by mutual visits and meetings at various conferences.
Publications
- Narrow and C-compact orthogonally additive operators in latticenormed spaces, Results Math. 74(4) (2019), Paper No. 157, 19 pp.
Pliev, Marat und Polat, Faruk and Weber, Martin
(See online at https://doi.org/10.1007/s00025-019-1075-y) - Almost interior points in ordered Banach spaces and the long-term behaviour of positive operator semigroups. Studia Mathematica 254(3) (2020), 237–263
Glück, Jochen und Weber, Martin
(See online at https://doi.org/10.4064/sm190111-18-10) - Atomic operators on vector lattices. Mediterranean J. Math. 17, Art. no. 138 (2020), 21 Seiten
Chill, Ralph und Pliev, Marat
(See online at https://doi.org/10.1007/s00009-020-01581-9) - Interpolation of nonlinear order preserving operators on Banach lattices. Positivity 24(3) (2020), 507–532
Chill, Ralph und Fiorenza, Alberto und Król, Sebastian
(See online at https://doi.org/10.1007/s11117-019-00688-y) - Multiplicative representation of real-valued bi-Riesz homomorphisms on partially ordered vector spaces. In: Kikianty, E. (ed.), Mabula, M. (ed.), Messerschmidt, M. (ed.), van der Walt, J.H. (ed.), Wortel, M. (ed.): Positivity and its Applications, Proceedings of the Conference Positivity X, Pretoria, South Africa, 8-12 July 2019, Trends in Mathematics, Birkhäuser (2021)
Kalauch, Anke und van Gaans, Onno
(See online at https://doi.org/10.1007/978-3-030-70974-7_10) - Extension and factorisation of Riesz n-morphisms on pre-Riesz spaces, J. Math. Anal. Appl. 507(2) (2022), Art. 125767
Kalauch, Anke und Kusraeva, Zalina
(See online at https://doi.org/10.1016/j.jmaa.2021.125767)
