Projects / Programmes
Applications of algebra in analysis
| Code |
Science |
Field |
Subfield |
| 1.01.00 |
Natural sciences and mathematics |
Mathematics |
|
| Code |
Science |
Field |
| P120 |
Natural sciences and mathematics |
Number theory, field theory, algebraic geometry, algebra, group theory |
| P140 |
Natural sciences and mathematics |
Series, Fourier analysis, functional analysis |
functional analysis, operator theory, algebra, multiparameter spectral analysis, invariant subspaces, semigroups, groups, matrix varieties
Researchers (19)
Organisations (1)
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.
Next research goal is to develop representation theory for ordered associative rings with involution. In the commutative case the represenentations are just homomorphism. A noncommutative analogue of rings of continous functions are Cx-algebras and a noncommutative analogue of homomorphisms are irreducible representations.The open problems are to find a good definition of the enveloping Cx-algebra of an ordered associative ring with involution, to characterize Cx-algebras within the class of ordered associative rings with involution, to generalize Kadison-Dubois and Jacobi''s Representation Theorem to ordered associative rings with involution.
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.
