Arkiv för Matematik

  • Ark. Mat.
  • Volume 47, Number 1 (2009), 91-125.

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

Alexander V. Kiselev and Serguei Naboko

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Abstract

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.

Article information

Source
Ark. Mat., Volume 47, Number 1 (2009), 91-125.

Dates
Received: 12 March 2007
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907064

Digital Object Identifier
doi:10.1007/s11512-007-0068-3

Mathematical Reviews number (MathSciNet)
MR2480917

Zentralblatt MATH identifier
1187.47013

Rights
2008 © Institut Mittag-Leffler

Citation

Kiselev, Alexander V.; Naboko, Serguei. Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure. Ark. Mat. 47 (2009), no. 1, 91--125. doi:10.1007/s11512-007-0068-3. https://projecteuclid.org/euclid.afm/1485907064


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