Operators in finite distributive subspace lattices. III

It is proved that the free distributive lattice on three generators is the smallest lattice among all finite distributive lattices which have a subspace (in a normed space) realization and an operator T in Alg which cannot be written as a sum of rank T rank one operators from Alg . Also some information concerning the form of such a lattice is given. If, furthermore, T has rank two, then contains a sublattice with two Boolean lattices with three atoms each, and one of them is below the other.