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April 2009 Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure
Alexander V. Kiselev, Serguei Naboko
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Ark. Mat. 47(1): 91-125 (April 2009). DOI: 10.1007/s11512-007-0068-3


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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Alexander V. Kiselev. Serguei Naboko. "Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure." Ark. Mat. 47 (1) 91 - 125, April 2009.


Received: 12 March 2007; Published: April 2009
First available in Project Euclid: 31 January 2017

zbMATH: 1187.47013
MathSciNet: MR2480917
Digital Object Identifier: 10.1007/s11512-007-0068-3

Rights: 2008 © Institut Mittag-Leffler


Vol.47 • No. 1 • April 2009
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