Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-28T06:27:24.807Z Has data issue: false hasContentIssue false
  • English
  • Français

On the (sub)logarithmic property of the pole-, zero- and algebraic multiplicity of operator functions

Published online by Cambridge University Press:  20 January 2009

G. Philip A. Thijsse
G. Philip A. Thijsse
Affiliation:
Abt. Mathematik-Lehrstuhl IUniversität DortmundPostfach 50 05 00 4600 Dortmund 50Fed. Rep. of Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper contains an extension of a result obtained by H. Bart, M. A. Kaashoek and D. C. Lay in (2). These authors studied the reduced algebraic multiplicity RM(A; λ0) of a meromorphic operator function at a point λ0 ∈ C. They proved that under certain conditions this quantity has logarithmic behaviour, i.e.,

For more restricted cases such results had been proved by others, notably I. C. Gohberg and E. I. Sigal (see (4) and (5)). Here we shall prove that such a result also holds for a larger class of operator functions than the diagonable functions considered in (2).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 23 , Issue 2 , June 1980 , pp. 207 - 217
Copyright
Copyright © Edinburgh Mathematical Society 1980

References

REFERENCES

(1)Bart, H., Kaashoek, M. A. and Lay, D. C., Stability properties of finite meromorphic operator functions, Nederl. Akad. Wetensch. Proc. Ser. A 77 (1974), 217–259.CrossRefGoogle Scholar
(2)Bart, H., Kaashoek, M. A. and Lay, D. C., The integral formula for the reduced algebraic multiplicity of meromorphic operator-functions, Proc. Edinburgh Math. Soc. 21 (1978), 65–72.CrossRefGoogle Scholar
(3)Gleason, A. M., Finitely generated ideals in Banach algebras, J. Math, and Mech. 13 (1964), 125–132.Google Scholar
(4)Gohberg, I. C. and Sigal, E. I., Operator generalizations of the theorem on the logarithmic difference and of the theorem of Rouché, Mat. Sb. 84 (1971), 607–630 [Russian] = Math. USSR-Sb. 13 (1971), 603–625.Google Scholar
(5)Gohberg, I. C. and Sigal, E. I., On the root multiplicity of the product of meromorphic operator functions, Mat. Issled. 6 (1971), 34–50 [Russian].Google Scholar
(6)Kato, T., Perturbation theory for linear operators (Springer, Berlin, Heidelberg, New York, 1966).Google Scholar
(7)Keldysh, M. V., On the eigenvalues and eigenvectors of some classes of nonselfadjoint equation, Dokl. Akad. Nauk SSSR (1951), 11–14 [Russian].Google Scholar
(8)Thijsse, G. Ph. A., Decomposition theorems for finite-meromorphic operator-functions (Thesis Vrije Universiteit, Amsterdam, 1978).Google Scholar
(9)Vandewalle, J. and Dewilde, P., A local I/O structure theory for multivariable systems and its application on minimal cascade realization, preprint 1977.CrossRefGoogle Scholar
(10)Youla, D. C. and Tissi, P., An explicit formula for the degree of a rational matrix (Poly tech. Inst. of Brooklyn, Report PI BRI-1273–65, 06 1965).Google Scholar