Abstract
In this chapter we study the influence of perturbations of a matrix function on its factorization and partial indices. It develops that the partial indices are stable only in two special cases. Namely if they are all equal or if the difference between the larger and the smaller is one. We prove this theorem in detail and study how the partial indices vary when the matrix function depends analytically on a parameter or is a rational matrix function of two variables where one of the variables is considered to be a parameter.
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.
Notes
Section 1. The results in this section are from Gohberg and Krein I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. Uspehi Mat. Nauk, 13 (1958), no. 2(80), 3–72
Section 1. The results in this section are from Gohberg and Krein I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. English transi., Amer. Math. Soc. transi. (2)(1960), 217–287]. The result in Proposition 2.1 has been generalized by Šubin [68] in the case where the partial index tuple of A(•,w) is constant on .
Section 2. The results in Theorem 2.1, Corollaries 2.1 and 2.2 are found in Gohberg and Krein I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. Uspehi Mat. Nauk, 13 (1958), no. 2(80), 3–72
Section 2. The results in Theorem 2.1, Corollaries 2.1 and 2.2 are found in Gohberg and Krein I.C. Gohberg and M.G. Krein: Systems of integral equations on a half line with kernels depending on the difference of arguments. English transi., Amer. Math. Soc. transi. (2)(1960), 217–287] (formulated for factorizations relative to Γ) . Theorem 2.2 and the lemmas of this section are from Widom
H. Widom: Perturbing Fredholm operators to obtain invert-ible operators. Jour, of Functional Analysis 20(1975), 26–31].
Section 4. Theorem 4.1 is due to Heinig G. Heinig: On the inversion and on the spectrum of Wiener-Hopf matrix operators. Math. USSR Sbornik 20(1973), 267–281
G. Heinig: On the inversion and on the spectrum of matrix singular integral operators. Mat. Issled. 8(1973), 106–121. (Russian)]. The method of proof is from Azoff, Clancey and Gohberg [2] .
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer Basel AG
About this chapter
Cite this chapter
Clancey, K.F., Gohberg, I. (1981). Perturbations and Stability. In: Factorization of Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol 3. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5492-4_11
Download citation
DOI: https://doi.org/10.1007/978-3-0348-5492-4_11
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-5494-8
Online ISBN: 978-3-0348-5492-4
eBook Packages: Springer Book Archive