Uporaba grafov v problemih ohranjevalcev / Graph Theory in Preserver Problems

Predlagani projekt je v grobem razdeljen na 4 dele in raziskuje grafe, inducirane z relacijami ter splošne ohranjevalce na

 -normiranih prostorih,

-Hilbertovih C*-modulih,

-matričnih algebrah,

-množicah (ne nujno zveznih) operatorjev, in

-mnogoterostmi povezanimi s specialno teorijo relativnosti.

Bolj natančno, znotraj predlaganega projekta nameravamo:

 (1) Preučevati lastnosti grafov, ki jih inducirajo Birkhoff-Jamesova ortogonalnost, Robertsova ortogonalnost ter  močna Birkhoff-Jamesova  ortogonalnost, in klasificirati njihove (linearne) ohranjevalce.

 (2) Klasificirati  splošne bijektivne bi-ohranjevalce leve-zvezdica urejenosti na B(H).

 (3) Klasificirati aditivne preslikave, ki ohranijo spekter neomejenih linearnih operatorjev na kompleksnih Banachovih prostorih.

 (4) Klasificirati preslikave na de Sitter-jevem prostoru, ki geodetke svetlobnega tipa slikajo zopet v geodetke svetlobnega tipa. Injektivnost tovrstnih preslikav bomo privzeli le na vsaki svetlobni geodetki posebej.

ANG

The proposed project is roughly divided into 4 parts and belongs to the theory of graphs induced by relations and general preservers on:

 -normed spaces,

-Hilbert C*-modules,

-matrix algebras, 

-sets of (possibly unbounded) operators, and 

-manifolds related to special relativity.

In particular we aim to:

 (1) Study the properties of graph induced by Birkhoff-James and Roberts orthogonality and also induced by strong Birkhoff-James orthogonality and study their (linear) preservers.

 (2) Classify general bijective bi-preservers of left-star partial order on B(H).

 (3) Classify additive maps which preserve the spectrum of unbounded linear operators on a complex Banach space.

 (4) Classify maps on the de Sitter space, which maps light-like geodesics into light-like geodesics. Injectivity  of the maps will be assumed only on each light-like geodesic separately