Abstract
This paper presents a new method to reduce integral operators of various classes to simpler operators, which often are just finite matrices. By this method the problem to find the inverse, generalized inverses, kernel and image of an integral operator is reduced for several cases to the corresponding problem for a finite matrix. The classes of integral operators dealt with include integral operators of the second kind with a finite rank or semi-separable kernel and also, which is more surprising, systems of Wiener-Hopf integral operators and singular integral operators with rational matrix symbols.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Barnett, S.: Introduction to mathematical control theory. Oxford, Clarendon Press, 1975.
Barras, J.S. and Brockett, R.W.: H2-functions and infinite-dimensional realization theory. SIAM J. Control Optimization 13 (1975), 221–241.
Bart, H., Gohberg, I. and Kaashoek, M.A.: Minimal factorization of matrix and operator functions. Operator Theory: Advances and Applications, Vol. 1, Basel-Boston-Stuttgart, Birkhäuser Verlag, 1979.
Bart, H., Gohberg, I. and Kaashoek, M.A.: Wiener-Hopf integral equations, Toeplitz matrices and linear systems. In: Toeplitz Centennial (ed. I. Gohberg), Operator Theory: Advances and Applications, Vol. 4, Basel-Boston-Stuttgart, Birkhäuser Verlag, 1982, 85–135.
Bart, H., Gohberg, I. and Kaashoek, M.A.: Convolution equations and linear systems. Integral Equations and Operator Theory, 5/3 (1982), 283–340.
Bart, H., Gohberg, I. and Kaashoek, M.A.: Wiener-Hopf factorization of analytic operator functions and realization. Wiskundig Semina-rium der Vrije Universiteit, Amsterdam, Rapport nr. 231, 1983.
Bart, H., Gohberg, I. and Kaashoek, M.A.: Wiener-Hopf factorization and realization. To appear in Proceedings MTNS, Beer-Sheva, 1983.
Bart, H. and Kroon, L.S.: An indicator for Wiener-Hopf integral equations with invertible analytic symbol. Integral Equations and Operator Theory, 6/1 (1983), 1–20.
Clancey, K. and Gohberg, I.: Factorization of matrix functions and singular integral operators. Operator Theory: Advances and Applications, Vol. 3, Basel-Boston-Stuttgart, Birkhäuser Verlag, 1981.
Fuhrmann, P.: Linear systems and operators in Hilbert space. New-York, McGraw-Hill, 1981.
Gohberg, I., Kaashoek, M.A. and Lay, D.C.: Equivalence, linearization and decomposition of holomorphic operator functions. J. Funct. Anal. 28 (1978), 102–144.
Gohberg, I. and Leiterer, J.: Factorization of operator functions relative to a contour. I. Finitely meromorphic operator functions. Math. Nachr. 52 (1972), 259–282 [Russian].
Gohberg, I. and Lerer, L.: Fredholm properties of Wiener-Hopf operators. Private communication (1980).
Kaashoek, M.A., Van der Mee, C.V.M. and Rodman, L.: Analytic operator functions with compact spectrum, I. Spectral nodes, linearization and equivalence. Integral Equations and Operator Theory 4 (1981), 504–547.
Kaashoek, M.A., Van der Mee, C.V.M. and Rodman, L.: Analytic operator functions with compact spectrum, III. Hilbert space case: inverse problem and applications. J. of Operator Theory (to appear).
Kaiman, R.E., Fall, P.F. and Arbib, M.A.: Topics in mathematical systems theory. New-York, McGraw-Hill, 1969.
Editor information
Rights and permissions
Copyright information
© 1984 Springer Basel AG
About this chapter
Cite this chapter
Bart, H., Gohberg, I., Kaashoek, M.A. (1984). The Coupling Method for Solving Integral Equations. In: Dym, H., Gohberg, I. (eds) Topics in Operator Theory Systems and Networks. OT 12: Operator Theory: Advances and Applications, vol 12. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5425-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5425-2_2
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5427-6
Online ISBN: 978-3-0348-5425-2
eBook Packages: Springer Book Archive