where Ja denotes the interval [a, b] with exception of the subintervals I x-xn I < 5; here xn denotes a finite set of points, chosen from among the points for which (3) does not hold. The question, whether a given operator can be represented in the form (1), with condition (2), was investigated by J. von Neumann [2]; he dropped conditions (3) and (4), but essentially restricted the investigation to the case of hermitean operators. von Neumann's work was based on previous results by H. Weyl [3]. More recently, the results of von Neumann were generalized to nonhermitean, especially normal operators and various other results were derived, with a view to application to scattering theory in quantum physics [4]. The present paper describes additional results in this direction. We shall start with several definitions. By "operator" we shall mean a not necessarily bounded linear transformation on the Hilbert function space L2(a, b) of complex-valued functions of one real vari-