Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.