Abstract
To a linear differential operator (respectively, equation) with unbounded periodic coefficients in a Banach space of vector functions defined on the entire real line, we assign a difference operator (respectively, a difference equation) with a constant operator coefficient defined in the corresponding Banach space of two-sided vector sequences. For the differential and difference operators, we prove the coincidence of the dimensions of their kernels and coranges, the simultaneous complementability of kernels and ranges, the simultaneous invertibility, and a relationship between the spectra.
Similar content being viewed by others
References
Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovom prostranstve (Linear Differential Equations in a Banach Space), Moscow: Nauka, 1967.
Howland, J.S., Stationary Scattering Theory for Time-Dependent Hamiltonians, Math. Ann., 1974, vol. 207, no. 4, pp. 315–335.
Baskakov, A.G., Spectral Analysis of Linear Differential Operators, and Semigroups of Difference Operators, Dokl. Akad. Nauk, 1995, vol. 343, no. 3, pp. 295–298.
Baskakov, A.G., Semigroups of Difference Operators in Spectral Analysis of Linear Differential Operators, Funct. Anal. Appl., 1996, vol. 30, no. 3, pp. 149–157.
Hille, E. and Phillips, R., Functional Analysis and Semi-Groups, Providence, 1957. Translated under the title Funktsional’nyi analiz i polugruppy, Moscow: Inostrannaya literatura, 1963.
Baskakov, A.G. and Pastukhov, A.I., Spectral Analysis of a Weighted Shift Operator with Unbounded Operator Coefficients, Siberian Math. J., 2001, vol. 42, no. 6, pp. 1026–1035.
Didenko, V.B., On the Spectral Properties of Differential Operators with Unbounded Operator Coefficients Determined by a Linear Relation, Math. Notes, 2011, vol. 89, no. 1–2, pp. 224–237.
Didenko, V.B., On the Continuous Invertibility and the Fredholm Property of Differential Operators with Multi-Valued Impulse Effects, Izv. Math., 2013, vol. 77, no. 1, pp. 3–19.
Dunford, N. and Schwartz, J., Linear Operators. General Theory, New York, 1958. Translated under the title Lineinye operatory. T. 1. Obshchaya teoriya, Moscow: Inostrannaya Literatura, 1962.
Henry, D., Geometric Theory of Semilinear Parabolic Equations, Heidelberg: Springer-Verlag, 1981. Translated under the title Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii, Moscow, 1985.
Baskakov, A.G. and Sintyaev, Yu.N., Finite-Difference Operators in the Study of Differential Operators: Solution Estimates, Differ. Equ., 2010, vol. 46, no. 2, pp. 214–223.
Baskakov, A.G. and Kobychev, K.S., Estimates for the Embedding Operator of a Sobolev Space of Periodic Functions and for the Solutions of Differential Equations with Periodic Coefficients, Differ. Equ., 2011, vol. 47, no. 5, pp. 609–619.
Perov, A.I., Frequency Criteria for the Existence of Bounded Solutions, Differ. Uravn., 2007, vol. 43, no. 7, pp. 896–904.
Perov, A.I., Frequency Methods in the Theory of Bounded Solutions of Nonlinear nth-Order Differential Equations (Existence, Almost Periodicity, and Stability), Differ. Equ., 2012, vol. 48, no. 5, pp. 670–680.
Baskakov, A.G., Linear Differential Operators with Unbounded Operator Coefficients and Semigroups of Bounded Operators, Math. Notes, 1996, vol. 59, no. 6, pp. 586–593.
Baskakov, A.G., Spectral Analysis of Linear Differential Operators, and Semigroups of Difference Operators. I, Differ. Uravn., 1997, vol. 33, no. 10, pp. 1299–1306.
Baskakov, A.G., Spectral Analysis of Linear Differential Operators, and Semigroups of Difference Operators. II, Differ. Uravn., 2001, vol. 37, no. 1, pp. 3–11.
Baskakov, A.G., On Correct Linear Differential Operators, Sb. Math., 1999, vol. 190, no. 3, pp. 323–348.
Kutateladze, S.S., Osnovy funktsional’nogo analiza (Foundations of Functional Analysis), Novosibirsk, 2000.
Baskakov, A.G., Spectral Analysis of Differential Operators with Unbounded Operator-Valued Coefficients, Difference Relations and Semigroups of Difference Relation, Izv. Math., 2009, vol. 73, no. 2, pp. 215–278.
Bichegkuev, M.S., On the Spectrum of Difference and Differential Operators in Weighted Spaces, Funct. Anal. Appl., 2010, vol. 44, no. 1, pp. 65–68.
Baskakov, A.G., Analysis of Linear Differential Equations by Methods of the Spectral Theory of Difference Operators and Linear Relations, Russian Math. Surveys, 2013, vol. 68, no. 1, pp. 69–116.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.G. Baskakov, V.B. Didenko, 2015, published in Differentsial’nye Uravneniya, 2015, Vol. 51, No. 3, pp. 323–338.
Rights and permissions
About this article
Cite this article
Baskakov, A.G., Didenko, V.B. Spectral analysis of differential operators with unbounded periodic coefficients. Diff Equat 51, 325–341 (2015). https://doi.org/10.1134/S0012266115030052
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266115030052