Non-reflexive double triangles

Abstract

AbstractA double triangle subspace lattice in a Hilbert space H is a 5-element set of subspaces of H, containing (0) and H, with each pair of non-trivial elements intersecting in (0) and spanning H. It is shown that if any pair of non-trivial elements has a closed vector sum the double triangle is both non-reflexive and non-transitive. A double triangle in H⊕H is an operator double triangle if each non-trivial elements is the graph of an operator acting on H. A sufficient condition is given for any operator double triangle to be non-reflexive.