Operators in finite distributive subspace lattices II

Abstract

It is proved that the free distributive lattice on three generators is the smallest lattice among all finite distributive lattices S a which have a subspace (in a normed space) realization S a' and an operator T in Alg .o~ which cannot be written as a sum of rank T rank one operators from Alg.o~'. Also some information concerning the form of such a lattice is given. If, furthermore, T has rank two, then .9' contains a sublattice with two Boolean lattices with three atoms each, and one of them is below the other. © Elsevier Science Inc., 1997 This paper is a continuation of [8] and [9], with which we assume some familiarity. All the latticos in this paper are finite distributive (unless stated otherwise), and all realizations are formed with (closed) subspaces of a normed space. We recall two definitions (the rest and the notation are as in the mentioned papers). Let .9' be a subspace lattice and T a finite rank operator of Alg S a. We say that T has the FRP (finite rank property) if it can be written as a finite sum of rank one operators from Alg S a. We say that S a has the FRP if every finite rank operator of Alg .9' has the FRP.