ABSTRACT: We show that mixed states can produce maximal violations in the Bell inequality due to Clauser, Horne, Shimony, and Holt (CHSH). This follows from the degeneracy of the operator which is naturally associated with the Bell inequality (here called the Bell operator). We have calculated the form of all the eigenvalues for the generic CHSH Bell operator. Finally, we consider several examples which demonstrate the utility of studying the eigenvalues and eigenvectors of the Bell operator.