Opuscula Math. 38, no. 4 (2018), 483-500
https://doi.org/10.7494/OpMath.2018.38.4.483

Opuscula Mathematica

# On spectra of quadratic operator pencils with rank one gyroscopic linear part

Olga Boyko
Olga Martynyuk
Vyacheslav Pivovarchik

Abstract. The spectrum of a selfadjoint quadratic operator pencil of the form $$\lambda^2M-\lambda G-A$$ is investigated where $$M\geq 0$$, $$G\geq 0$$ are bounded operators and $$A$$ is selfadjoint bounded below is investigated. It is shown that in the case of rank one operator $$G$$ the eigenvalues of such a pencil are of two types. The eigenvalues of one of these types are independent of the operator $$G$$. Location of the eigenvalues of both types is described. Examples for the case of the Sturm-Liouville operators $$A$$ are given.

Keywords: quadratic operator pencil, gyroscopic force, eigenvalues, algebraic multiplicity.

Mathematics Subject Classification: 47A56, 47E05, 81Q10.

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• Olga Boyko
• South-Ukrainian National Pedagogical University, Staroportofrankovskaya Str. 26, Odesa, 65020, Ukraine
• Olga Martynyuk
• South-Ukrainian National Pedagogical University, Staroportofrankovskaya Str. 26, Odesa, 65020, Ukraine
• Vyacheslav Pivovarchik
• South-Ukrainian National Pedagogical University, Staroportofrankovskaya Str. 26, Odesa, 65020, Ukraine
• Communicated by Alexander Gomilko.
• Revised: 2017-08-15.
• Accepted: 2017-08-23.
• Published online: 2018-04-11.