Functional Analysis and Applications Seminar 

22/06/2009, 14:00 — 15:00 — Room P3.10, Mathematics Building
Kostyantyn Yusenko, Institute of Mathematics, National Academy of Science, Kiev, Ukraine

On additive spectral problem. Some historical background and recent results

In 1912 Hermann Weyl posed the following problem: to characterize the possible sets of eigenvalues of the sum of two Hermitian matrices in terms of the sets eigenvalues of given matrices. In 1962 Alfred Horn recasted this problem as a conjectured series of inequalities for the eigenvalues of the sum matrix. The final step in proving this conjecture was made by A. Klyachko, A. Knutson and T. Tao. We consider the following generalization of Weyl's problem: given the finite sets of eigenvalues of hermitian operators, determine all possible scalar operators that can appear as a sum of these operators. This problem could be reformulated in terms of the existence of the *-representations of certain *-algebras connected with star-shaped graphs. We will also discuss some basic properties of such *-algebras related to extended Dynkin graphs.