Eigenvalue Expansion of Nonsymmetric Linear Compact Operators in Hilbert Space

Year 2021, Volume: 4 Issue: 2, 55 - 74, 30.06.2021

Ludwig Kohaupt

https://doi.org/10.33434/cams.817490

Cited By: 1

Abstract

For a symmetric linear compact resp. symmetric densely defined linear operator with compact inverse, expansion theorems in series of eigenvectors are
known. The aim of the present paper is to generalize the known expansion theorems to the case of corresponding operators without the symmetry property. For this, we replace the set of orthonormal eigenvectors in the symmetric case by a set of biorthonormal eigenvectors resp. principal vectors in the case of simple eigenvalues resp. general eigenvalues. The results for the operators without the symmetry property are all new. Furthermore, if the operators are symmetric, the generalized results deliver the known expansions. As an application of the results for nonsymmetric operators with simple eigenvalues, we obtain a known expansion in a series of eigenfunctions for a non-selfadjoint Boundary Eigenvalue
Problem with ordinary differential operator discussed in a book of Coddington/Levinson. But, we obtain a new result if the eigenvalues are general, that
is, not necessarily simple. In addition, for a differential operator of 2nd order with constant coefficients, the eigenfunctions and Green’s function are explicitly determined. This result is also new, as far as the author is aware.

Keywords

Boundary eigenvalue problem, Eigenvalue expansion, Green's function, Nonsymmetric copact operator

Supporting Institution

No supporting Institution

Project Number

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References

• [1] N.I. Achieser, I.M. Glasmann, Theorie der linearen Operatoren im Hilbert-Raum (Theory of Linear Operators in Hilbert Space; German Translation of the Russian Original), Akademie- Verlag, Berlin, 1968.
• [2] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill Book Company, New York Toronto London, 1955.
• [3] D.K. Crane, M.S. Gockenbach, M.J. Roberts, Approximating the Singular Value Expansion of a Compact Operator, SIAM J. Numer. Anal. 58(2)(2020)1295-1318.
• [4] R.D. Grigorieff, Diskrete Approximation von Eigenwertproblemen. III: Asymptotische Entwicklungen (Discrete Approximation of Eigenvalue Problems. III: Asymptotic Expansions), Num. Math. 25(1975)79-97.
• [5] H. Heuser, Funktionalanalysis (Functional Analysis), B.G. Teubner, Stuttgart, 1975.
• [6] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
• [7] K. Knopp, Theory of Functions, Parts I and II (English Translation of the German Original), Dover Publications, 1975.
• [8] L. Kohaupt, Construction of a Biorthogonal System of Principal Vectors for Matrices A and A* with Applications to dx/dt = Ax; x(t_0) = x_0, Journal of Computational Mathematics and Optimization 3(3)(2007)163-192.
• [9] L. Kohaupt, Biorthogonalization of the Principal Vectors for the Matrices A and A* with Applications to the Computation of the Explicit Representation of the Solution x(t) of dx/dt = Ax; x(t_0) = x_0, Applied Mathematical Sciences 2(20)(2008)961-974.
• [10] W. Luther, K. Niederdrenk, F. Reutter, H. Yserentant, Gewoehnliche Differentialgleichungen, Analytische und numerische Behandlung (Ordinary Differential Equations, Analytic and Numerical Treatment), Vieweg, Braunschweig Wiesbaden, 1987.
• [11] S.G. Michlin, Variationsmethoden der Mathematischen Physik (Variational Methods of Mathematical Physics; German Translation of the Russian Original), Akademie-Verlag, Berlin, 1962.
• [12] E.C. Pestel, F.A. Leckie, Matrix Methods in Elastomechanics, McGraw-Hill Book Company, Inc., New York San Francisco Toronto London, 1963.
• [13] F. Stummel, Diskrete Konvergenz linearer Operatoren II (Discrete Convergence of Linear Operators, Part II), Mathematische Zeitschrift 120 (1971)231-264.
• [14] A.E. Taylor, Introduction to Functional Analysis, John Wiley & Sons, New York London, 1958.