Generalized eigenvectors of linear operators and biorthogonal systems

Year 2022, , 60 - 71, 15.06.2022

Ruslan Khats'

https://doi.org/10.33205/cma.1077842

Cited By: 2

Abstract

The notions of a set of generalized eigenvalues and a set of generalized eigenvectors of a linear operator in Euclidean space are introduced. In addition, we provide a method to find a biorthogonal system of a subsystem of eigenvectors of some linear operators in a Hilbert space whose systems of canonical eigenvectors are over-complete. Related to our problem, we will show an example of a linear differential operator that is formally adjoint to Bessel-type differential operators. We also investigate basis properties (completeness, minimality, basicity) of the systems of generalized eigenvectors of this differential operator.

Keywords

Linear operator, generalized eigenvector, Bessel function, complete system, minimal system, biorthogonal system

References

• H. Bateman, A. Erdélyi: Higher transcendental functions, Vol. 2, McGraw-Hill, New York-Toronto-London (1953).
• Yu. M. Berezanskii: Expansions in eigenfunctions of self-adjoint operators, Transl. Math. Monogr., Vol. 17. Amer. Math. Soc., Providence, RI (1968).
• M. Bertola, M. Gekhtman and J. Szmigielski: Cubic string boundary value problems and Cauchy biorthogonal polynomials, J. Phys. A: Math. Theor., 42 (45) (2009), 13 p.
• C. Cesarano: A note on bi-orthogonal polynomials and functions, Fluids, 5 (3) (2020), 105.
• N. Dunford, J. T. Schwartz: Linear operators. Spectral operators, Part III. Wiley-Interscience, New York-London-Sydney-Toronto (1971).
• R. V. Khats’: On conditions of the completeness of some systems of Bessel functions in the space $L^2((0;1);x^{2p} dx)$, Azerb. J. Math., 11 (1) (2021), 3–10.
• R. V. Khats’: Completeness conditions of systems of Bessel functions in weighted $L^2$-spaces in terms of entire functions, Turk. J. Math., 45 (2) (2021), 890–895.
• R. V. Khats’: Integral representation of one class of entire functions, Armen. J. Math., 14 (1) (2022), 1–9.
• S. G. Krein: Functional analysis, Noordhoff, Groningen (1972).
• A. M. Perelomov: On the completeness of a system of coherent states, Theor. Math. Phys., 6 (1971), 156–164.
• A. M. Sedletskii, Analytic Fourier transforms and exponential approximations. I., J. Math. Sci., 129 (6) (2005), 4251–4408.
• O. V. Shavala: On some approximation properties of the Bessel functions of order $-5/2$, Mat. Stud., 43 (2) (2015), 180–184. (in Ukrainian)
• O. V. Shavala: On completeness of systems of functions generated by the Bessel function, Bukovinian Math. J., 5 (3-4) (2017), 168–171. (in Ukrainian)
• A. A. Shkalikov: Boundary problems for ordinary differential equations with parameter in the boundary conditions, J. Math. Sci., 33 (6) (1986), 1311–1342.
• K. Stempak: On convergence and divergence of Fourier-Bessel series, Electron. Trans. Numer. Anal., 14 (2002), 223–235.
• C. Tretter: Linear operator pencils $A-\lambda B$ with discrete spectrum, Integr. Equ. Oper. Theory., 37 (3) (2000), 357–373.
• V. S. Vladimirov: Equations of mathematical physics, Nauka, Moscow (1981). (in Russian)
• B. V. Vynnyts’kyi, V. M. Dilnyi: On approximation properties of one trigonometric system, Russ. Math., 58 (11) (2014), 10–21.
• B. V. Vynnyts’kyi, R. V. Khats’: Some approximation properties of the systems of Bessel functions of index $-3/2$, Mat. Stud., 34 (2) (2010), 152–159.
• B. V. Vynnyts’kyi, R. V. Khats’: Completeness and minimality of systems of Bessel functions, Ufa Math. J., 5 (2) (2013), 131–141.
• B. V. Vynnyts’kyi, R. V. Khats’: On the completeness and minimality of sets of Bessel functions in weighted $L^2$-spaces, Eurasian Math. J., 6 (1) (2015), 123–131.
• B. V. Vynnyts’kyi, R. V. Khats’: A remark on basis property of systems of Bessel and Mittag-Leffler type functions, J. Contemp. Math. Anal. (Armen. Acad. Sci.), 50 (6) (2015), 300–305.
• B. V. Vynnyts’kyi, R. V. Khats’: Complete biorthogonal systems of Bessel functions, Mat. Stud., 48 (2) (2017), 150–155.
• B. V. Vynnyts’kyi, R. V. Khats’ and I. B. Sheparovych: Unconditional bases of systems of Bessel functions, Eurasian Math. J., 11 (4) (2020), 76–86.
• B. V. Vynnyts’kyi, O. V. Shavala: Boundedness of solutions of a second-order linear differential equation and a boundary value problem for Bessel’s equation, Mat. Stud., 30 (1) (2008), 31–41. (in Ukrainian).
• B. V. Vynnyts’kyi, O. V. Shavala: Some properties of boundary value problems for Bessel’s equation, Math. Bull. Shevchenko Sci. Soc., 10 (2013), 169–172.
• G. N. Watson: A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge (1944).