2000]15a45, 14l24 Inequalities for Numerical Invariants of Sets of Matrices

The spectral radius of every d× d matrix A is bounded from below by c ‖A‖ ‖A‖, where c = c(d) > 0 is a constant and ‖·‖ is any operator norm. We prove an inequality that generalizes this elementary fact and involves an arbitrary number of matrices. In the proof we use geometric invariant theory. The generalized spectral radius theorem of Berger and Wang is an immediate consequence of our inequality. 1. Motivation and statement of the result Let M(d) be the space of d×d complex matrices. If A ∈ M(d), we indicate by ρ(A) the spectral radius of A, that is, the maximum absolute value of an eigenvalue of A. Given a norm ‖·‖ in Cd, we endow the space M(d) with the operator norm ‖A‖ = sup {‖Av‖; ‖v‖ = 1}. For every A ∈ M(d) and every norm ‖·‖ in Cd, we have ρ(A) ≤ ‖A‖. On the other hand, there is also a lower bound for ρ(A) in terms of norms: ‖A‖ ≤ Cρ(A)‖A‖, where C = 2 − 1. (1) In particular, if ρ(A) ≪ ‖A‖ then ‖Ad‖ ≪ ‖A‖d. Inequality (1) is a very simple consequence of the Cayley-Hamilton theorem. Indeed, let p(z) = zd − σ1z d−1 + · · · + (−1)σd be the characteristic polynomial of A. Since p(A) = 0,