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

Alexander V. Kiselev Department of Mathematical Physics, Institute of Physics, St. Petersburg State University Serguei Naboko Department of Mathematical Physics, Institute of Physics, St. Petersburg State University

TBD mathscidoc:1701.333140

Arkiv for Matematik, 47, (1), 91-125, 2007.3
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.
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@inproceedings{alexander2007nonself-adjoint,
  title={Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure},
  author={Alexander V. Kiselev, and Serguei Naboko},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203624054958949},
  booktitle={Arkiv for Matematik},
  volume={47},
  number={1},
  pages={91-125},
  year={2007},
}
Alexander V. Kiselev, and Serguei Naboko. Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure. 2007. Vol. 47. In Arkiv for Matematik. pp.91-125. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203624054958949.
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