Invariant subspace lattices that complement every subspace (Q1089879)

The following three statements are equivalent for an operator T acting on a complex Hilbert space H: (i) Given a subspace M of H, there exists R in Lat T (the lattice of all invariant subspaces of T) such that \(M\cap R=\{0\}\) and \(M+R=H\); that is, R complements M; (ii) Given a finite chain of subspaces \(M_ 1\subset M_ 2\subset...\subset M_ m\) in H, there exists a finite chain \(R_ 1\supset R_ 2\supset...\supset R_ m\) in Lat T such that \(R_ j\) complements \(M_ j\) for all \(j=1,2,...,m;\) (iii) T is similar to a normal operator with finite spectrum.