Optimal decomposable witnesses without the spanning property

One of the unsolved problems in the characterization of the optimal entanglement witnesses is the existence of optimal witnesses acting on bipartite Hilbert spaces H-m,H-n = C-m circle times C-n such that the product vectors obeying < e, f vertical bar W vertical bar e, f > = 0 do not span H-m,H-n. So far, the only known examples of such witnesses were found among indecomposable witnesses, one of them being the witness corresponding to the Choi map. However, it remains an open question whether decomposable witnesses exist without the property of spanning. Here we answer this question affirmatively, providing systematic examples of such witnesses. Then, we generalize some of the recently obtained results on the characterization of 2 circle times n optimal decomposable witnesses [R. Augusiak et al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H-m,H-n with m,n >= 3.