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On similarity of unbounded perturbations of selfadjoint operators


Abstract

We consider a linear unbounded operator $A$ in a separable Hilbert space with the following property: there is an invertible selfadjoint operator $S$ with a discrete spectrum such that $\|(A-S)S^{-\nu}\|<\infty$ for a $\nu\in [0,1]$. Besides, all eigenvalues of $S$ are assumed to be different. Under certain assumptions it is shown that $A$ is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.

Key words: Similarity, differential operator, spectrum perturbations, operator function.


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Article Information

TitleOn similarity of unbounded perturbations of selfadjoint operators
SourceMethods Funct. Anal. Topology, Vol. 24 (2018), no. 1, 27-33
MilestonesReceived 13/04/2017; Revised 05/10/2017
CopyrightThe Author(s) 2018 (CC BY-SA)

Authors Information

Michael Gil'
Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, BeerSheva 84105, Israel 


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Citation Example

Michael Gil', On similarity of unbounded perturbations of selfadjoint operators, Methods Funct. Anal. Topology 24 (2018), no. 1, 27-33.


BibTex

@article {MFAT1022,
    AUTHOR = {Michael  Gil'},
     TITLE = {On similarity of unbounded perturbations of selfadjoint operators},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {24},
      YEAR = {2018},
    NUMBER = {1},
     PAGES = {27-33},
      ISSN = {1029-3531},
       URL = {http://mfat.imath.kiev.ua/article/?id=1022},
}


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