Linear maps preserving the isomorphism class of lattices of invariant subspaces

Abstract

Let V {\mathcal {V}} be an n n -dimensional complex linear space and L ( V ) {\mathcal {L}}({\mathcal {V}}) the algebra of all linear transformations on V {\mathcal {V}} . We prove that every linear map on L ( V ) {\mathcal {L}}({\mathcal {V}}) , which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds if we replace the lattice of invariant subspaces by the lattice of hyperinvariant subspaces or the set of reducing subspaces. Some of these results are extended to linear transformations of finite-dimensional linear spaces over fields other than the complex numbers. We also characterize linear bijective maps on the algebra of linear bounded operators on an infinite-dimensional complex Hilbert space which have similar properties with respect to the lattice of all invariant subpaces (not necessarily closed).