Intersection theory and the Horn inequalities for invariant subspaces

TL;DR

This paper establishes a new intersection theoretic approach to the Horn inequalities in the context of invariant subspaces of operators of class C_{0}, revealing divisibility relations and operator splittings, even in nonseparable Hilbert spaces.

Contribution

It introduces a direct intersection theoretic proof of multiplicative Horn inequalities for class C_{0} operators and their invariant subspaces, including nonseparable cases.

Findings

Horn inequalities are replaced by divisibility relations.

Existence of operator splittings when inequalities are saturated.

Results extend to nonseparable Hilbert spaces.

Abstract

We provide a direct, intersection theoretic, argument that the Jordan models of an operator of class C_{0}, 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. The result is true even for operators acting on nonseparable Hilbert spaces. For such operators the usual Horn inequalities are supplemented so as to apply to all the Jordan blocks in the model.