Abstract
If F is an automorphism of Ω n , then 2-dimensional spectral unit ball, we show that, in a neighborhood of any cyclic matrix of Ωn, the mapF can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. Baribeau and T.J. Ransford, Nonlinear spectrumpreserving maps,Bull. London Math. Soc. 32 (2000), 8–14.
L. Baribeau and S. Roy, Caractérisation spectrale de la forme de Jordan,Linear Algebra Appl. 320 (2000), 183–191.
R.A. Horn and C.R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1985.
R.A. Horn and C.R. Johnson,Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
N. Nikolov, P.J. Thomas, and W. Zwonek, Discontinuity of the Lempert function and the Kobayashi-Royden metric of the spectral ball, (preprint, 2007), (arXiv:math.CV/0704.2470).
T.J. Ransford and M.C. White, Holomorphic selfmaps of the spectral unit ball,Bull. London Math. Soc. 23 (1991), 256–262.
J. Rostand, On the automorphisms of the spectral unit ball,Studia Math. 155 (2003), 207–230.
W. Zwonek, Proper holomorphic mappings of the spectral unit ball, (preprint, 2007), (arXiv:math.CV/0704.0614).
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper was made possible in part by a grant from the French Ministry of Foreign Affairs in the framework of the ECO Net programme, file 10291 SL, coordinated by Ahmed Zeriahi.
Rights and permissions
About this article
Cite this article
Thomas, P.J. A local form for the automorphisms of the spectral unit ball. Collect. Math. 59, 321–324 (2008). https://doi.org/10.1007/BF03191190
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03191190