Detecting the length of a Jordan chain
Inverse Problems and Applications
22 April 15:30 - 16:30
Let A be a symmetric matrix, then for each eigenvalue of A the algebraic and geometric multiplicities coincide. This is reflected in the fact that the resolvent (and boarded resolvents) have poles of order not larger than 1. In the talk we are discussing the more interesting situation of an operator A in an infinite dimensional space, where eigenvalues can be embeddeed in the continous spectrum, which leads to non-isolated singularities. Instead of self-adjointness we require another kind of symmetry: Let B be a self-adjoint boundedly invertible self-adjoint operator, for which the spectrum on the negative halfline consists of finitely many eigenvalues only. For A, the operator of interest, it is now assumed that BA is selfadjoint. In this case A can have Jordan chains (with finite length). And the question appears how can the length of such a Jordan chain be obtained from analytic properties of boarded resolvents. We will give an overview of the history of the problem (which is about 30 years old) and present the recently obtained solution.