Abstract
Jordan chains of holomorphic operator functions are studied. It is shown that a “semi-canonical” system of Jordan chains is a linearly independent system if each single nontrivial Jordan chain is a linearly independent system. With the help of local or global linearizations of the operator function it is characterized when nontrivial Jordan chains are linearly independent systems. This is applied to give necessary and sufficient conditions for the existence of roots and spectral roots of regular holomorphic matrix functions.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
H. Bart, Meromorphic Operator Valued Functions. Thesis. Vrije Universiteit, Amsterdam, 1973.
H. Bart, Poles of the Resolvent of an Operator Function. Proc. Roy. Irish Acad. 74A (1974), 169–184.
H. Baumgärtel, Analytic Perturbation Theory for Matrices and Operators, Opera-tor Theory: Advances and Applications, vol. 15, Birkhauser: Basel-BostonStuttgart, 1985.
H. Bart, M.A. Kaashoek and D.C. Lay, Stability Properties of Finite Meromorphic Operator Functions. Proc. Acad. Sci. Amsterdam A77 (1974), 217–259.
G.J. Butler, C.R. Johnson and H. Wolkowicz, Nonnegative Solutions of a Quadratic Matrix Equation Arising from Comparison Theorems in Ordinary Differential Equations. SIAM J. Alg. Disc. Meth. 6(1985), 47–53.
H. Bart and D.C. Lay, Poles of a Generalized Resolvent Operator. Proc. Roy. Irish Acad. 74A (1974), 147–168.
K.-H. Förster and B. Nagy, Some Properties of the Spectral Radius of a Monic Operator Polynomial with Nonnegative Compact Coefficients. Integral Equations and Operator Theory 14(1991), 794–805.
K.-H. Förster and B. Nagy, Spektraleigenschaften von Matrix-and Operatorpoly-nomen. Sitzungsberichte d. Berliner Math. Gesellschaft, 1994,243–262.
I. Gohberg, M.A. Kaashoek and D.C. Lay, Equivalence, Linearization, andDecomposition of Holomorphic Operator Functions. J. Funct. Anal. 28(1978),102–144.
I.Gohberg, M.A.Kaashoek and E van Schagen, On the Local Theory of RegularAnalytic Matrix Functions. Linear Algebra Appl. 182(1993), 9–25.
I. Gohberg, P. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications. Wiley-Interscience: New York, 1986.
M.A. Kaashoek, State-space Theory of Rational Matrix Functions and Applica-tions, Lectures on Operator Theory and its Applications, Fields Institute Monographs vol. 3, Amer. Math. Soc., Providence, 1996,235–333.
P. Lancaster and P.N. Webber Jordan Chains and Lambda Matrices. Linear Alge-bra Appl. 1 (1968), 563–569.
H. Langer, Über Lancaster’s Zerlegung von Matritzen-Scharen. Archive Rat.Mech. Appl. 29 (1968), 75–80.
P. Lancaster and M. Tismenetsky, The Theory of Matrices, 2nd. ed. AcademicPress: New York, 1985.
A.S. Marcus, Introduction to the Spectral Theory of Polynomial Operator Pencils.Translation of Mathematical Monograph, vol. 71, Amer. Math. Soc., Providence, 1988.
R.T. Rau, On the Peripheral Spectrum of a Monic Operator Polynomial withPositive Coefficients. Integral Equations and Operator Theory 15 (1992), 479–495.
L. Rodman, An Introduction to Operator Polynomials, Operator Theory:Advances and Applications, vol. 38. Birkhäuser: Basel-Boston-Berlin, 1989.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Förster, KH., Nagy, B. (2001). Linear Independence of Jordan Chains. In: Bart, H., Ran, A.C.M., Gohberg, I. (eds) Operator Theory and Analysis. Operator Theory: Advances and Applications, vol 122. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8283-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8283-5_8
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9502-6
Online ISBN: 978-3-0348-8283-5
eBook Packages: Springer Book Archive