On some mean matrix inequalites of dynamical interest

Abstract

Let A ∈ SL (n, ℝ). We show that for all n > 2 there exist dimensional strictly positive constants Cn such that ∫On log ρ(AX)dX ≥ Cn log∥A∥, 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 On, normalized to be a probability measure. The same result (with essentially the same proof) holds for the unitary group Un 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. © Springer-Verlag 2005.