The Gershgorin Type Theorem on Localization of the Eigenvalues of Infinite Matrices and Zeros of Entire Functions

Year 2022, Volume: 4 Issue: 1, 9 - 16, 23.09.2022

Michael Gil'

https://doi.org/10.54286/ikjm.1005765

Abstract

The classical Gershgorin theorem on localization of the eigenvalues of finite matrices is extended to infinite Hille-Tamarkin matrices. Applications to finite order entire functions
are also discussed.

Keywords

infinite matrix , compact operator , localization of spectrum , zeros of entire functions

References

• Da Fonseca, C. M. On the location of the eigenvalues of Jacobi matrices. Appl. Math. Lett. 19 , no. 11, (2006) 1168-–1174.
• Reference1\bibitem{Dewan} Dewan, K.K., Govil, N.K. On the location of the zeros of analytic functions. Int. J. Math. Math. Sci. 13(1), (1990) 67-–72.
• Djordjevi\'c, S. V. and Kant\`un-Montiel, G. Localization and computation in an approximation of eigenvalues. Filomat 29 (2015), no. 1, 75–-81.
• Dyakonov, K.M. Polynomials and entire functions: zeros and geometry of the unit ball. Math. Res. Lett. 7(4), (2000) 393-–404.
• Esp\`inola-Rocha, J. A. Factorization of the scattering matrix and the location of the eigenvalues of the Manakov-Zakharov-Shabat system. Phys. Lett. A 372, no. 40, (2008) 6161-–6167.
• Gil’, M.I. Invertibility and spectrum of Hille-Tamarkin matrices, Mathematische Nachrichten , 244, (2002), 1-11
• Gil’, M.I.: Operator Functions and Localization of Spectra. Lectures Notes in Mathematics, vol. 1830, Springer, Berlin 2003.
• Gil', M.I. Localization and Perturbation of Zeros of Entire Functions, Lecture Notes in Pure and Applied Mathematics, 258. CRC Press, Boca Raton, FL, 2010.
• Grammont, L. and Largillier, A. Krylov method revisited with an application to the localization of eigenvalues. Numer. Funct. Anal. Optim. 27 (2006), no. 5-6, 583-–618.
• Ioakimidis, N.I. A unified Riemann–-Hilbert approach to the analytical determination of zeros of sectionally analytic functions. J. Math. Anal. Appl. 129(1), (1988) 134–-141
• Kato, T., Perturbation Theory for Linear Operators, Berlin: Springer-Verlag, 1980.
• Kytmanov A.M. and Khodos O.V., On localization of zeros of an entire function of finite order of growth, Complex Anal. Oper. Theory 11 (2017) 393-–416.