  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,n}=C^m\otimes C^n such that the product vectors
obeying <e,f|W|e,f>=0 do not span H_{m,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\otimes n optimal decomposable witnesses [R. Augusiak et
al., J. Phys. A 44, 212001 (2011)] to finite-dimensional Hilbert spaces H_{m,n}
with m,n\geq 3.
