Intersection theory and the Horn inequalities for invariant subspaces

Series: 
Analysis Seminar
Wednesday, February 25, 2015 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Georgia Institute of Technology
Organizer: 
It is well known that the Horn inequalities characterize the relationship of eigenvalues of Hermitian matrices A, B, and A+B. At the same time, similar inequalities characterize the relationship of the sizes of the Jordan models of a nilpotent matrix, of its restriction to an invariant subspace, and of its compression to the orthogonal complement. In this talk, we provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_0 (such operator can be thought of as the infinite dimensional generalization of matrices, that is an operator will be annihilated by an H-infinity function), of its restriction to an invariant subspace, and of its compression to the orthogonal complement, satisfy a multiplicative form of the Horn inequalities, where ‘inequality’ is replaced by ‘divisibility’. When one of these inequalities is saturated, we show that there exists a splitting of the operator into quasidirect summands which induces similar splittings for the restriction of the operator to the given invariant subspace and its compression to the orthogonal complement. Our approach also explains why the same combinatorics solves the eigenvalue and the Jordan form problems. This talk is based on the joint work with H. Bercovici.