Geometric characterization and classification of marked subspaces (Q1911918)

Let \(f\) be a linear endomorphism of a finite-dimensional \(\mathbb{C}\)-vector space \(E\). An \(f\)-invariant subspace \(W \subset E\) is said to be marked, if there exists a Jordan basis of \(f \mid W\) which can be extended to a Jordan basis of \(f\). [See also \textit{R. Bru}, \textit{L. Rodman} and \textit{H. Schneider}, Linear Algebra Appl. 150, 209-225 (1991; Zbl 0726.15005)]. The authors give a characterization and a classification of marked subspaces \(W\) of \(E\) in terms of \(W\) and the family of subspaces \(\ker f^h \cap \text{im} f^d\). However, the details are too involved to be stated here.