General preservers of invariant subspace lattices

Let B(X) be the space of all bounded linear operators on a Banach space X and let LatA be the lattice of invariant subspaces of the operator A∈B(X). We characterize some maps Φ:B(X)→B(X) with one of the following preserving properties: Lat(Φ(A)+Φ(B))=Lat(A+B), or Lat(Φ(A)Φ(B))=Lat(AB), or Lat(Φ(A)Φ(B)+Φ(B)Φ(A))=Lat(AB+BA), or Lat(Φ(A)Φ(B)Φ(A))=Lat(ABA), or Lat([Φ(A),Φ(B)])=Lat([A,B]).