Operator equalities related to the quantum E(2) group

The paper deals with normal operators R and S satisfying simple commutation relations that we encounter investigating the quantum deformation of the E(2) group. We show that R+S admits a normal extension if and only if R-1 S satisfies a certain spectral condition. A number of related formulae are derived. In particular, all the functions f satisfying the character equation f(R+S)=f(R)f(S) are found. (orig.)