On Existence of a Biorthonormal Basis Composed of Eigenvectors of   Non-Hermitian Operators
  We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a generalization of the conditions which is applicable to non-diagonalizable operators by considering not only eigenvectors but also all root vectors. 
Toshiaki Tanaka