Reflexive subspace lattices in Banach spaces

In this survey paper we present known results about reflexive subspace lattices. We show that every nest and every atomic Boolean subspace lattice in a complex Banach space is reflexive, even strongly reflexive. Our main tool is Ringrose's Lemma about presence of rank-one operators in a reflexive algebra of operators. We consider realizations of small lattices as subspace lattices in Banach spaces. We exhibit some realizations of the pentagon and prove that some of these realizations are reflexive. On the other hand, we show that double triangle subspace lattices in finite-dimensional Banach spaces are non-reflexive.