Strictly semi-transitive operator algebras

An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every A-invariant linear subspace is norm-closed. Moreover, Lat A is totally and well ordered by reverse inclusion. If X is complex and A is transitive and strictly semi-transitive, then A is WOT-dense in L(X). It is also shown that if A is an operator algebra on a complex Banach space with no invariant operator ranges, then A is WOT-dense in L(X). This generalizes a similar result for Hilbert spaces proved by Foias.