Singular spectrum of a non-self-adjoint operator

The study of the spectral structure of nondissipative operators in Hilbert space, started in the previous papers of the author, is continued. In a model representation, generalizing the familiar model of B. S.-Nagy–Foias, there is defined the singular subspace $N_i$ of an operator, and the separability of the singular spectrum from the absolutely continuous one is studied. There is given a decomposition of the subspace $N_i$ into spectral subspaces $N_i^{(\pm)}$ corresponding to the singular spectrum in the upper (lower) half-plane, respectively. There is obtained an estimate of the angle between such subspaces in terms of the characteristic function of the operator. Applications are given to the Schrodinger differential operator, for which the latter estimate leads to an effective expression in terms of integrals of the potential. Formulas are given for spectral projectors onto eigen- and root-subspaces of the nonreal discrete spectrum of the operator. At the end of the paper there are studied questions of similarity of operators. The most complete results are obtained under the imposition on the operator of an additional condition, characterizing its “closeness” to being dissipative.