Hyperinvariant subspaces of locally nilpotent linear transformations

A subspace X of a vector space over a field K is hyperinvariant with respect to an endomorphism f of V if it is invariant for all endomorphisms of V that commute with f. We assume that f   is locally nilpotent, that is, every x∈Vx∈V is annihilated by some power of f, and that V is an infinite direct sum of f-cyclic subspaces. In this note we describe the lattice of hyperinvariant subspaces of V. We extend a result of Fillmore, Herrero and Longstaff (1977) [2] to infinite dimensional spaces.