Open Access

# 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.

### Article Information

 Title On similarity of unbounded perturbations of selfadjoint operators Source Methods Funct. Anal. Topology, Vol. 24 (2018), no. 1, 27-33 MathSciNet MR3783815 Milestones Received 13/04/2017; Revised 05/10/2017 Copyright The 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

### 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},
MRNUMBER = {MR3783815},
URL = {http://mfat.imath.kiev.ua/article/?id=1022},
}

Coming Soon.