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Spectra of a class of non-symmetric operators in Hilbert spaces with applications to singular differential operators

Part of: Ordinary differential operators Operator theory: Ordinary differential operators

Published online by Cambridge University Press:  15 February 2019

Huaqing Sun  and
Bing Xie
Huaqing Sun
Affiliation:
Department of Mathematics, Shandong University at Weihai Weihai, Shandong264209, P. R. China (sunhuaqing@email.sdu.edu.cn; xiebing@sdu.edu.cn)
Bing Xie
Affiliation:
Department of Mathematics, Shandong University at Weihai Weihai, Shandong264209, P. R. China (sunhuaqing@email.sdu.edu.cn; xiebing@sdu.edu.cn)
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Abstract

This paper is concerned with a class of non-symmetric operators, that is, 𝒥-symmetric operators, in Hilbert spaces. A sufficient condition for λ ∈ C being an element of the essential spectrum of a 𝒥-symmetric operator is given in terms of the number of linearly independent solutions of a certain homogeneous equation, and a characterization for points of the essential spectrum plus the set of all eigenvalues of a 𝒥-symmetric operator is obtained in terms of the numbers of linearly independent solutions of certain inhomogeneous equations. As direct applications, the corresponding results are obtained for singular 𝒥-symmetric Hamiltonian systems and their special forms of singular Sturm-Liouville equations with complex-valued coefficients, which enable us to study the spectra of singular 𝒥-symmetric differential expressions using numerous tools available in the fundamental theory of differential equations.

Keywords

𝒥-symmetric operatordefect indexeigenvalueessential spectrumsingular Hamiltonian system

MSC classification

Primary: 34L05: General spectral theory
Secondary: 47E05: Ordinary differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 4 , August 2020 , pp. 1769 - 1790
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Copyright
Copyright © Royal Society of Edinburgh 2019

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