On the Bari basis property for even-order differential operators with involution

Main Article Content

Dmitry Polyakov

Abstract

By using the method of similar operators we study even-order differential operators with involution. The domain of these operators are defined by periodic and antiperiodic~boundary conditions. We obtain estimates for spectral projections and we prove the Bari basis property for the system of eigenfunctions and associated functions

Article Details

How to Cite
Polyakov, D. (2022). On the Bari basis property for even-order differential operators with involution. Tamkang Journal of Mathematics, 54(4), 339–351. https://doi.org/10.5556/j.tkjm.54.2023.4899
Section
Papers

References

J. Wu. Theory and applications of partial functional differential equations. Springer-Verlag, New York (1996).

A. Cabada, F.A.F. Tojo. Differential equations with involutions. Atlantis Press, Amsterdam-Paris-Beijing (2015).

V.P. Kurdyumov, A.P. Khromov. Riesz bases formed by root functions of a functional-differential equation with a reflection operator.

Differ. Equ. 44. No. 2, 203--212 (2008).

A.A. Kopzhassarova, A.L. Lukashov, A.M. Sarsenbi. Spectral properties of non-self-adjoint perturbations for a spectral problem with involution. Abstr. Appl. Anal. 2012, Article ID 590781 (2012).

A.G. Baskakov, I.A. Krishtal, E.Yu. Romanova. Spectral analysis of a differential operator with an involution. J. Evol. Equ. 17, 669--684 (2017).

V.P. Kurdyumov. On Riesz bases of eigenfunction of 2-nd order differential operator with involution and integral boundary conditions.

Izv. Saratov Univ. (N. S.) 15, no. 4, 392--405 (2015) [in Russian].

A.A. Kopzhassarova, A.M. Sarsenbi. Basis properties of eigenfunctions of second-order differential operators with involution.

Abstr. Appl. Anal. 2012, Article ID 576843 (2012).

L.V. Kritskov, A.M. Sarsenbi. Riesz basis property of system of root functions of second-order differential operator with involution. Differ. Equ. 53, no. 1, 33--46 (2017).

L.V. Kritskov, V.L. Ioffe. Spectral properties of the Cauchy problem for a second-order operator with involution. Differ. Equ. 57, No. 1, 1--10 (2021).

L.V. Kritskov, A.M. Sarsenbi. Basicity in $L_p$ of root functions for differential equations with involution. Electr. J. Differ. Equ. 278, 1--9 (2015).

N.P. Bondarenko. Inverse spectral problems for functional-differential operators with involution. J. Differ. Equ. V. 318, 169--186 (2022).

A.M. Sarsenbi, A.A. Tengaeva. On the basis properties of root functions of two generalized eigenvalue problems. Differ. Equ. 48, No. 2, 306--308 (2012).

M.A. Sadybekov, A.M. Sarsenbi. Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution. Differ. Equ. 48, No. 8, 1112--1118 (2012).

L.V. Kritskov, A.M. Sarsenbi. Equiconvergence property for spectral expansions related to perturbations of the operator $-u''(-x)$ with initial data. Filomat. 32, No. 3, 1096--1078 (2018).

V.E. Vladykina, A.A. Shkalikov. Spectral properties of ordinary differential operators with involution. Doklady Math. 99, No. 1, 5--10 (2019).

V.E. Vladykina, A.A. Shkalikov. Regular ordinary differential operators with involution. Math. Notes. 106, No. 5, 674--687 (2019).

D.M. Polyakov. Spectral analysis of a fourth order differential operator with periodic and antiperiodic boundary conditions.

St. Petersburg Math. J. 27, No. 5, 789--811 (2016).

A.G. Baskakov, D.M. Polyakov. The method of similar operators in the spectral analysis of the Hill operator with nonsmooth potential.

Sb. Math. 208. No. 1, 1--43 (2017).

A.G. Baskakov, I.A. Krishtal, N.B. Uskova. Similarity techniques in the spectral analysis of perturbed operator matrices. J. Math. Anal. Appl. 477, 930--960 (2019).

I.C. Gohberg, M.G. Krein. Introduction to the theory of linear nonselfadjoint operators in Hilbert space. AMS, Providence, RI. (1969).

D.M. Polyakov. Spectral asymptotics of two-term even order operators with involution. J. Math. Sci. 260, No. 6, 806--819 (2022).