2017
Том 69
№ 7

All Issues

On some transformations of integral equation kernels and their influence on the spectra of these equations

Gohberg I. C., Krein S. G.

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Abstract

In recent publications by the authors (1, 2) and V. I. Matsayev (3,4) it was shown that the study of the abstract triangular representation of Volterra operators by the Brodsky integral naturally leads to a series of relations between the eigenvalues of Hermitian components of Volterra operators.
Though some specific properties of the triangular truncation transformation while deducing were used, these results admit in the main a generalization for any transformations (linear continuous operators acting in the Hilbert space of the Hilbert-Schmidt operators). By this generalization both the nature of relations under consideration and their proofs are simplified.
This generalization (§§ 2, 3, 4) is perhaps of interest as it leads to some new applications; it permits us, in particular, to obtain a number of precise estimations for the central stability zone for various Hamiltonian systems of linear differential equations with periodic coefficients (§ §5, 6, 7).

Citation Example: Gohberg I. C., Krein S. G. On some transformations of integral equation kernels and their influence on the spectra of these equations // Ukr. Mat. Zh. - 1961. - 13, № 3. - pp. 12-38.

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