Abstract
The notions of norm and spectral radius of a matrix are generalized to spaces with an indefinite metric. A spectral radius formula is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AI] T.Ya. Azizov and I.S. Iokhvidov,Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, 1989.
[BG] A. Ben-Artzi and I. Gohberg,s-Numbers of contractions in spaces with an indefinite metric, preprint.
[Bo] J. Bognár,Indefinite Inner Product Spaces, Springer-Verlag, 1974.
[EGL] R.L. Ellis, I. Gohberg and D.C. Lay, On two theorems of M.G. Krein concerning polynomials orthogonal on the unit circle,Integral Equations and Operator Theory 11 (1988), 87–104.
[F] H. Flanders, On the Norm and Spectral Radius,Linear and Multilinear Algebra, Vol. 2, 1974, pp. 239–240.
[G] I. Gohberg (ed.),Orthogonal Matrix-Valued Polynomials and Applications, Operator Theory: Advances and applications 34 (1988) Birkhäuser Verlag, Basel.
[GLR] I. Gohberg, P. Lancaster and L. Rodman,Matrices and Indefinite Scalar Products, Operator Theory: Advances and applications 8 (1983) Birkhäuser Verlag, Basel.
[GL] I. Gohberg and L. Lerer, Matrix generalizations of M.G. Krein theorems on orthogonal polynomials, inOrthogonal Matrix-Valued Polynomials and Applications (I. Gohberg, ed.),Operator Theory: Advances and applications 34 (1988), 137–202, Birkhäuser Verlag.
[IKL] I.S. Iohvidov, M.G. Krein and H. Langer,Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Akademie-Verlag, 1982.
[KoSV] J. Kovač-Striko and K. Veselić, Trace minimization and definiteness of symmetric pencils,Linear Algebra and Applications 216 (1995), 139–158.
[Kr1] M.G. Krein, On the location of the roots of polynomials which are orthogonal on the unit circle with respect to an indefinite weight,Teor. Funkcii, Funkcional. Anal. i Prilozen 2 (1966), 131–137 (Russian).
[LT] P. Lancaster and M. Tismenetsky,The Theory of Matrices, Academic Press, New York, 1985.
[PS] G. Pólya and G. Szegö,Problems and theorems in Analysis, Vol. 1, Springer Verlag, 1979.
[Pt1] V. Pták, Norms and the spectral radius of matrices,Czech. Math. J. 12 (1962), 555–557.
[Pt2] V. Pták, Spectral radius, norm of iterates, and the critical exponent,Linear Algebra and Its Applications 1 (1968), 245–260.
[V] K. Veselić, A Jacobi eigenreduction algorithm for definite matrix pairs,Numer. Math. 64 (1993), 241–269.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ben-Artzi, A., Gohberg, I. On contractions in spaces with an indefinite metric: G-norms and spectral radii. Integr equ oper theory 24, 422–469 (1996). https://doi.org/10.1007/BF01191620
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01191620