Geodesic
Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators
Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 39-51.

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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.
Keywords: nonself-adjoint operator, Lagrange optimality principle, functional model, annihilator, almost Hermitian spectrum. 
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A. V. Kiselev; S. N. Naboko. Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators. Funkcionalʹnyj analiz i ego priloženiâ, Tome 38 (2004) no. 3, pp. 39-51. http://geodesic.mathdoc.fr/item/FAA_2004_38_3_a3/

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