Abstract
Let A ∈ SL(n,ℝ). We show that for all n>2 there exist dimensional strictly positive constants C n such that
where ||A|| denotes the operator norm of A (which equals the largest singular value of A), ρ denotes the spectral radius, and the integral is with respect to the Haar measure on O n , normalized to be a probability measure. The same result (with essentially the same proof) holds for the unitary group U n in place of the orthogonal group. The result does not hold in dimension 2. This answers questions asked in [3, 5, 4]. We also discuss what happens when the integral above is taken with respect to measure other than the Haar measure.
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Avila, A., Bochi, J.: A formula with some applications to the theory of Lyapunov exponents. Israel J. Math. 131, 125–137 (2002)
Brualdi, R., Mellendorf, S.: Regions in the complex plane containing the eigenvalues of a matrix. Am. Math. Monthly 10, 975–985 (1994)
Burns, K., Pugh, C., Shub, M., Wilkinson, A.: Recent results about stable ergodicity. Proc. Symposia AMS 69, 327–366 (2001)
Dedieu, J-P., Shub, M.: On random and mean exponents for unitarily invariant and probability measures on Technical report, IBM, 2001. http://www.research.ibm.com/people/s/shub/Dedieu-Shub.pdf, 2001
Ledrappier, F., Simo, C., Shub, M., Wilkinson, A.: Random versus deterministic exponents in a rich family of diffeomorphisms. J. Stat. Phys. to appear
Hardy, H.G., Littlewood, J., Polya, G.: Inequalities. 2nd Edition, Cambridge: Cambridge University Press, 1988
Horn, R., Johnson, C.R.: Matrix Analysis. Corrected reprint, Cambridge: Cambridge University Press, England, 1990
Kato, T.: Perturbation Theory for Linear Operators. Reprint of the 1980 edition, Berlin Heidelberg-New York: Springer, 1995
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Communicated by P. Sarnak
The author is supported by the NSF DMS.
Acknowledgement The author would like to thank Amie Wilkinson for bringing the question to his attention.
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Rivin, I. On Some Mean Matrix Inequalites of Dynamical Interest. Commun. Math. Phys. 254, 651–658 (2005). https://doi.org/10.1007/s00220-004-1282-5
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DOI: https://doi.org/10.1007/s00220-004-1282-5