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Projects / Programmes source: ARIS

Teorija operatorjev (Slovene)

Periods
January 1, 1999 - December 31, 2003
Research activity

Code Science Field Subfield
1.01.00  Natural sciences and mathematics  Mathematics   

Code Science Field
P140  Natural sciences and mathematics  Series, Fourier analysis, functional analysis 
Evaluation (rules)
source: COBISS
Researchers (4)
no. Code Name and surname Research area Role Period No. of publicationsNo. of publications
1.  01639  PhD Anton Cedilnik  Mathematics  Researcher  2001 - 2003  111 
2.  18750  PhD Gregor Dolinar  Mathematics  Researcher  2001 - 2003  216 
3.  07680  PhD Tatjana Petek  Mathematics  Researcher  2001 - 2003  129 
4.  05953  PhD Peter Šemrl  Mathematics  Head  2001 - 2003  497 
Organisations (1)
no. Code Research organisation City Registration number No. of publicationsNo. of publications
1.  0101  Institute of Mathematics, Physics and Mechanics  Ljubljana  5055598000  20,258 
Abstract
Let A and B be Banach algebras. Linear preserver is a linear map from A into B that preserves a certain function defined on A and B (example: spectrum preserving maps), or a certain subset of algebras (example: nilpotent preserving maps), or a certain relation defined on algebras (example: commutativity preserving maps). It often turns out that such maps are Jordan homomorphisms sometimes multiplied by a constant and perturbed by a scalar type operator. We will be mainly interested in linear maps preserving spectral properties (maps preserving invertibility, spectral radius, quasinilpotents,…) and commutativity. When studying maps preserving spectral properties we will use the fact that the spectral radius of an analytic function is a subharmonic function. With this approach we will try to extend and simplify some known results. We also expect to simplify some proofs on such maps in the finite-dimensional case. We hope we will be able to solve the problem of the characterization of singular linear maps on matrix algebras preserving commutativity. Such a result would be an extension of the structural result for Lie homomorphisms on matrix algebras. The next topic we will work on is the problem of nonlinear perturbations of homomorphisms and isometries. We expect we will have to improve some results from the theory of functional inequalities in order to obtain the stability results for isometries and algebra homomorphisms. On the matrix spaces the distance between two matrices can be defined as a rank of their difference. Hua studied isometries on matrix spaces with respect to this distance without assuming linearity. His results are important as they unify and extend structural results for homomorphisms and Jordan homomorphisms, results on geometry of matrices and some results in graph theory. The most important are his results on bijective maps preserving the distance one in both directions. We will try to prove the analogues of his results without assuming bijectivity or under the weaker assumption of preserving distance one in one direction only. Finally, we will study the reflexivity problem on operator algebras. These kind of problems are closely related to problems concerning local homomorphisms, local derivations, and locally linearly dependent operators.
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