Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure

Alexander V. Kiselev, Serguei Naboko

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Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy-Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.

Original languageEnglish
Pages (from-to)91-125
Number of pages35
JournalArkiv for Matematik
Issue number1
Publication statusPublished - 1 Apr 2009

Bibliographical note

Funding Information:
The authors acknowledge support received from RFBR (grant no. 06-01-00249), IRCSET and INTAS grant 05-1000008-7883. (1) The theory developed in this paper can be generalized to the case of operators with not necessarily additive imaginary part, that is, to the class of operators with non-empty resolvent set. However, this would lead to purely technical difficulties that would complicate the reading of the paper. Due to this, we have elected not to include this generalization into the present paper.

ASJC Scopus subject areas

  • General Mathematics


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