Let be a Hilbert Space and be a Linear Operator. Then is Hermetian or self-Adjoint if i.e. if Proposition For a complex Hilbert Space and a Linear Operator , there exist unique Hermetian with . Proof Properties Eigenvalues of a Hermetian operator are real. Two eigenvectors with different eigenvalues of a Hermetian operator are orthogonal. The set of all eigenvectors of a Hermetian operator forms a complete orthogonal set.