  Entanglement witnesses are observables which when measured, detect
entanglement in a measured composed system. It is shown what kind of relations
between eigenvectors of an observable should be fulfilled, to allow an
observable to be an entanglement witness. Some restrictions on the signature of
entaglement witnesses, based on an algebraic-geometrical theorem will be given.
The set of entanglement witnesses is linearly isomorphic to the set of maps
between matrix algebras which are positive, but not completely positive. A
translation of the results to the language of positive maps is also given. The
properties of entanglement witnesses and positive maps express as special cases
of general theorems for $k$-Schmidt witnesses and $k$-positive maps. The
results are therefore presented in a general framework.
