Functiones et Approximatio Commentarii Mathematici

Spectral approximations of unbounded operators of the type Normal Plus Compact''

Michael Gil'

Abstract

Let $B$ be a compact operator in a Hilbert space $H$ and $S$ an unbounded normal one in $H$, having a compact resolvent. We consider operators of the form $A=S+B$. Numerous integro-differential operators $A$ can be represented in this form. The paper deals with approximations of the eigenvalues of the considered operators by the eigenvalues of the operators $A_n=S+B_n$ $(n=1,2,...)$, where $B_n$ are $n$-dimensional operators. Besides, we obtain the error estimate of the approximation.

Article information

Source
Funct. Approx. Comment. Math., Volume 51, Number 1 (2014), 133-140.

Dates
First available in Project Euclid: 24 September 2014

https://projecteuclid.org/euclid.facm/1411564619

Digital Object Identifier
doi:10.7169/facm/2014.51.1.7

Mathematical Reviews number (MathSciNet)
MR3263073

Zentralblatt MATH identifier
1314.47029

Citation

Gil', Michael. Spectral approximations of unbounded operators of the type Normal Plus Compact''. Funct. Approx. Comment. Math. 51 (2014), no. 1, 133--140. doi:10.7169/facm/2014.51.1.7. https://projecteuclid.org/euclid.facm/1411564619

References

• H. Abels, M. Kassmann, The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels, Osaka J. Math. 46 (2009), 661–683.
• M.G. Armentano, C. Padra, A posteriori error estimates for the Steklov eigenvalue problem, Applied Numerical Mathematics 58 (2008) 593–601.
• S.A. Buterin, On an inverse spectral problem for a convolution integro-differential operator, Result. Math. 50 (2007), 173–181.
• M. Charalambides, F. Waleffe, Spectrum of the Jacobi tau approximation for the second derivative operator, SIAM J. Numer. Anal. 46, no. 1, (2008) 280–294.
• Chena Huajie, Xingao Gongc,Aihui Zhoua, Numerical approximations of a nonlinear eigenvalue problem and applications to a density functional model, Mathematical Methods in the Applied Sciences 33, Issue 14, (2010) 1723–1742.
• Ding Xiaqi and Luo Peizhu, Finite element approximation of an integro-differential operator, Acta Mathematica Scientia 29B(6), (2009) 1767–1776.
• M.I. Gil', Operator Functions and Localization of Spectra, Lecture Notes in Mathematics, Vol. 1830, Springer-Verlag, Berlin, 2003.
• M.I. Gil', Lower bounds for eigenvalues of Schatten-von Neumann operators, J. Inequal. Pure Appl. Mathem. 8, no 3 (2007) 117–122.
• I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Trans. Mathem. Monographs, v. 18, Amer. Math. Soc., Providence, R. I., 1969.
• J.I.H. Lopez, J.R. Meneghini, F. Saltarab, Discrete approximation to the global spectrum of the tangent operator for flow past a circular cylinder, Applied Numerical Mathematics 58 (2008) 1159–1167.
• M. Malejki, Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvalues of finite submatrices, Opuscula Math. 27 (2007), no. 1, 37–49.
• M. Malejki, Asymptotics of large eigenvalues for some discrete unbounded Jacobi matrices, Linear Algebra and its Applications 431 (2009) 1952–1970.
• M. Malejki, Approximation and asymptotics of eigenvalues of unbounded self-adjoint Jacobi matrices acting in $l^2$ by the use of finite submatrices, Cent. Eur. J. Math. 8(1), (2010) 114–128.
• M. Marcus, H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964.
• A.D. Russo, A.E. Alonso, A posteriori error estimates for nonconforming approximations of Steklov eigenvalue problems, Computers and Mathematics with Applications 62 (2011) 4100–4117.