This article is cited in 4 papers

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

A. V. Kiselev^ab, S. N. Naboko<sup>b

a</sup> Dublin Institute of Technology
^b V. A. Fock Institute of Physics, Saint-Petersburg State University

Abstract: 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.

UDC: 517.9

Received: 01.03.2004

DOI: 10.4213/faa116

 English version: Functional Analysis and Its Applications, 2004, 38:3, 192–201

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