Projects / Programmes
Operators on Banach spaces
| Code |
Science |
Field |
Subfield |
| 1.01.01 |
Natural sciences and mathematics |
Mathematics |
Analysis |
| Code |
Science |
Field |
| P140 |
Natural sciences and mathematics |
Series, Fourier analysis, functional analysis |
orthogonality, elementary operator, von Neumann-Schatten class, matrix semigroup, semigroup homomorphism, irreducibility, perturbation, maximal eigenvector, norm, preserver, idempotent, additive mapping
Researchers (3)
Organisations (1)
Abstract
We shall study orthogonality of the kernel and the range of an elementary operator, with respect to the operator norm and von Neumann-Schatten norms. This question was first posed by Halmos for an inner derivation albeit under a different name of commutator approximation. We are mainly interested in elementary operators with normal coefficients and we intend to contribute new results as well as new methods to the problem on characterizing the kernel of an elementary operator. We believe that with the use of Frechet differentiability of the von Neumann-Schatten norms we will be able to characterize operators orthogonal to the range of an arbitrary elementary operator and that in some special cases operators orthogonal to the range of an elementary operator are exactly those in the kernel. This is important because we get a geometric (orthogonality) characterization of an algebraic property (kernel). We shall also study semigroup homomorphisms from full matrix semigroup of complex m-by-m matrices to matrix semigroup of n-by-n matrices. We will try to characterize all such homomorphisms for some special cases, for example m=2 and n=4, or under some additional assumptions, for example homomorphisms which map matrices of certain type to matrices of the same type. Furthermore, the family of self-adjoint, compact operators A(t) will be studied. The emphasis will be on the dependence of the maximal eigenvalue, and especially the norm of the corresponding eigenvector, subject to certain fixed normalization strategy, upon the parameter t. As a sample application of this research, we mention the problems of oscillation and of heat transfer, where the behavior of the first eigenvector is very important.Lastly, we will be interested in characterizing all the additive mappings, preserving idempotents on Banach spaces. Recent discoveries suggest such mappings play an important role in the theory of preservers.
