Stable subspaces of matrix pairs (Q1970504)

Let \(A:\mathbb{C}^n\to\mathbb{C}^n\) and \(B:\mathbb{C}^m\to\mathbb{C}^n\) be linear maps. A subspace \({\mathcal N}\subseteq \mathbb{C}^n\) is called \((A,B)\)-invariant, if \(A({\mathcal N}) \subseteq {\mathcal N}+ B(\mathbb{C}^m)\). An \((A,B)\)-invariant subspace is called \((A,B)\)-stable if for any pair \((A',B')\)-close to \((A,B)\) there exists an \((A',B')\)-invariant subspace \({\mathcal N}'\) of \(\mathbb{C}^n\) which is close to \({\mathcal N}\) (say in topology of the corresponding Grassmannian). The authors prove several conditions under which an \((A,B)\)-invariant subspace is acutally \((A,B)\)-stable.