## 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.

Original language | English |
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Pages (from-to) | 192-201 |

Number of pages | 10 |

Journal | Functional Analysis and Its Applications |

Volume | 38 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jul 2004 |

### Bibliographical note

Funding Information:∗The authors acknowledge support received from the RFBR (grant No. 03-01-00090) and IRCSET.

## Keywords

- almost Hermitian spectrum
- annihilator
- functional model
- Lagrange optimality principle
- nonself-adjoint operator

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics