Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations $L=A+iV$ of a bounded self-adjoint operator $A$ in a Hilbert space $H$. The main goal is to study the properties of the singular spectral subspace $N_i^0$ of $L$ corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.
To some extent, the properties of $N_i^0$ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that $L$ and the adjoint operator $L^*$ are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition $N_i^0=H$. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.