Non-hermitian extensions of Schrödinger type uncertainty relations

In quantum mechanics it is well known that observables are represented by Hermitian matrices (or operators). Uncertainty relations are represented as some kinds of trace inequalities satisfied by two observables and one density matrix (or operator). Now we try to release the Hermitian restriction on observables. This is only a mathematical interest. In this case we give several non-Hermitian extensions of the Schrödinger type uncertainty relation for generalized skew information under some conditions.