Abstract
Studying spectral properties of operator polynomials is reduced to studying the corresponding spectral properties of operators defined by operator matrices. The results are used to investigate second-order differential operators by associating them with the corresponding first-order differential operators and using their properties related to invertibility.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. G. Baskakov and A. Yu. Duplishcheva, “Difference operators and operator-valued matrices of the second order,” Izv. Math. 79 (2), 217–232 (2015).
A. G. Baskakov and I. A. Kristal, “Spectral properties of an operator polynomial with coefficients in a Banach algebra,” in Frames and Harmonic Analysis, Contemp. Math. (Amer. Math. Soc., Providence, RI, 2018), Vol. 706, pp. 93–114.
A. G. Baskakov, L. Yu. Kabantsova, I. D. Kostrub and T. I. Smagina, “Linear differential operators and operator matrices of the second order,” Differ. Equations 53 (1), 8–17 (2017).
A. I. Perov and I. D. Kostrub, “On bounded solutions to weakly nonlinear vector-matrix differential equations of order \(n\),” Siberian Math. J. 57 (4), 650–665 (2016).
A. G. Baskakov, “Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations,” Izv. Math. 73 (2), 215–278 (2009).
A. G. Baskakov and V. B. Didenko, “On invertibility states of differential and difference operators,” Izv. Math. 82 (1), 1–13 (2018).
A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations,” Russian Math. Surveys 68 (1), 69–116 (2013).
D. B. Didenko, “Spectral Properties of the Operators \(AB\) and \(BA\),” Math. Notes 103 (2), 196–208 (2018).
B. A. Barnes, “Common operator properties of the linear operators \(RS\) and \(SR\),” Proc. Amer. Math. Soc. 126 (4), 1055–1061 (1998).
V. I. Fomin, “On the solution of the cauchy problem for a second-order linear differential equation in a Banach space,” Differ. Equations 38 (8), 1219–1221 (2002).
A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: the instability index formula,” in Recent Developments in Operator Theory and Its Applications, Oper. Theory Adv. Appl. (Birkhäuser, Basel, 1996), Vol. 87, pp. 358–385.
R. O. Griniv and A. A. Shkalikov, “Exponential stability of semigroups related to operator models in mechanics,” Math. Notes 73 (5), 618–624 (2003).
A. A. Shkalikov and R. O. Griniv, “On an operator pencil arising in the problem of beam oscillation with internal damping,” Math. Notes 56 (2), 840–851 (1994).
A. G. Baskakov and V. D. Kharitonov, “Spectral analysis of operator polynomials and higher-order difference operators,” Math. Notes 101 (3), 391–405 (2017).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space (Nauka, Moscow, 1970) [in Russian].
V. G. Kurbatov, in Functional-Differential Operators and Equations, Math. Appl. (Kluwer Acad. Publ., Dordrecht, 1999), Vol. 173.
Funding
The work of the first author was supported by the Russian Foundation for Basic Research under grant 19-01-00732.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baskakov, A.G., Didenko, D.B. Spectral Analysis of Operator Polynomials and Second-Order Differential Operators. Math Notes 108, 477–491 (2020). https://doi.org/10.1134/S0001434620090205
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434620090205