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

Boyko O., Martynyuk O., Pivovarchik V.

Opuscula Mathematica

2018

Vol. 38, no. 4

483--500

The spectrum of a selfadjoint quadratic operator pencil of the form [formula] is investigated where M ≥ 0, G ≥ 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: q

A nonstandard difference Sturm-Liouville operator

Javorskij J., Ljance V.

Bulletin of the Polish Academy of Sciences. Mathematics

1998

Vol. 46, no 3

285-290

After Nelson's Radically Elementary Probability Theory [1] a natural question arises: whether a hyperfinite-dimensional space is sufficiently rich to be used for the same goal as an infinite-dimensional one. Here a hyperfinite 3-diagonal matrix is investigated, which spectral properties are simular to the Naimarks's singular nonselfadjoint Sturm-Liouville differential operator on semi-axis [2, 3].