Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

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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 Ni0 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 Ni0 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 Ni0 = H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.

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