Projects / Programmes
Algebraic methods in operator theory
January 1, 2004
- December 31, 2008
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P001 |
Natural sciences and mathematics |
Mathematics |
functional analysis, operator theory, algebra, multiparameter spectral analysis, invariant subspaces, semigroups, groups, matrix varieties
Researchers (30)
Organisations (2)
Abstract
We will study bounded linear operators defined on real or complex Banach or Hilbert spaces, as well as operators on finite-dimensional vector spaces over general fields. Various families of linear operators with additional algebraic structure, e. g. families that are semigroups, groups, vector spaces, associative algebras or Lie algebras will be studied. Our central problem will be the existence of a common invariant subspace of the family. Another class of problems comes from the consideration of the sets of operators as algebraic subsets, i.e. varieties in affine or projective spaces. Our next aim is to translate the theory of Banach algebras into the context of Banach modules. We are going to study representations of Banach modules, some spectra of these modules, and structure topologies on them. The next goal is the study of formally real rings. An associative ring is formally real if -1 cannot be expressed as a sum of permuted products of squares. The set of all orderings of such a ring is called its real spectrum. On it, a noncommutative real algebraic geometry can be built. Two basic problems of this theory are construction of new formally real rings that are interesting to physics and to
determine the structure of the real spectra of such rings. We will consider also the theory of preservers. We intend to classify (possibly noninjective or nonsurjective) additive mappings, which either preserve rank-one idempotents, or else annihilate them. Our goal are also elementary operators and operator inequalities. We intend to find some applications in the theory of double operator inegrals. Furthermore we will try to characterize self-adjoint invertible operators with simple operator inequality and generalize this result to unitarily invariant norms as well.
Significance for science
Achieved results are important for the development of the mathematical sciences. Since we have studied the problems that were raised in the international mathematical community, we expect that they will attract a considerable amount of attention. The results are important for the development of algebra and its application to the operator theory. New results will shed light on the structure of certain families of operators. They are also important in the study of invariant subspace problem, study of commuting matrices, symmetries of graphs, real algebraic geometry, abstract theory of groups and semigroups and elswhere. Many of our results have already attracted a considerable amount of attention. We expect that the same will be the case also for our future results. We successfully continued with scientific collaboration with many mathematicians around the world. We have published our results in the refereed scientific journals, presented them at the international scientific meetings and at invited lectures at foreign universities.
Significance for the country
We transfer the newest scientific results to our students and this way we contribute to the social and economic development. Our results are, by our opinion, an important part of Slovenian mathematical research, which is fundamental for many other sciences. Research in the field of financial mathematics will contribute to the transfer of knowledge to the students of the new study program of Financial Mathematics at the University of Ljubljana. In addition, this part of our research will be directly applicable to the financial sector of economy (insurance companies, banks, other financial institutions...). We have already encounetred a considerable interest of the Slovenian financial industry, the central bank, and elsewhere.
Most important scientific results
Final report,
complete report on dLib.si
Most important socioeconomically and culturally relevant results
Final report,
complete report on dLib.si
