Which linear transformations have isomorphic hyperinvariant subspace lattices? (Q1187391)

If \(A\) is a linear transformation of an \(n\)-dimensional complex vector space then a subspace is said to be hyperinvariant if it is invariant under all linear mappings which commute with \(A\). The author discusses the following question: To what extent does the structure of the lattice of hyperinvariant subspaces of \(A\) determine the Jordan structure of \(A\)? It is shown that except for some special cases the lattice of hyperinvariant subspaces of \(A\) determines, firstly, the number of distinct eigenvalues of \(A\) and, secondly, for each eigenvalue the different sizes of the Jordan-blocks belonging this eigenvalue (disregarding repeated appearence in the Jordan normal form).