Collineations preserving the lattice of invariant subspaces of a linear transformation

Given a linear transformation $A$ on a finite-dimensional complex vector space $\text{eV}$, in this paper we study the group $\text{Col}(A)$ consisting of those invertible linear transformations $S$ on $\text{eV}$ for which the mapping ${\mathrm{\Phi }}_S$ defined as ${\mathrm{\Phi }}_S\text{\hspace{0.17em}⁣}:\text{eM}\mapsto S\text{eM}$ is an automorphism of the lattice $\text{Lat}(A)$ of all invariant subspaces of $A$. By using the primary decomposition of $A$, we first reduce the problem of characterizing $\text{Col}(A)$ to the problem of characterizing the group $\text{Col}(N)$ of a given nilpotent linear transformation $N$. While $\text{Col}(N)$ always contains all invertible linear transformations of the commutant $(N{)}^{\mathrm{\prime }}$ of $N$, it is always contained in the reflexive cover $\text{Alg}\text{Lat}(N{)}^{\mathrm{\prime }}$ of $(N{)}^{\mathrm{\prime }}$. We prove that $\text{Col}(N)$ is a proper subgroup of $(\text{Alg}\text{Lat}(N{)}^{\mathrm{\prime }}{)}^{-1}$ if and only if at least two Jordan blocks in the Jordan decomposition of $N$ are of dimension $2$ or more. We also determine the group $\text{Col}({\text{bdJ}}_2\oplus {\text{bdJ}}_2)$.