The finiteness conjecture for the generalized spectral radius of a set of matrices

Abstract

The generalized spectral radius \ ̄g9(∑) of a set ∑ of n × n matrices is \ ̄g9(∑) = lim sup<inf>k→∞</inf> \ ̄g9<inf>k</inf>(∑)^{1 k}, where \ ̄g9<inf>k</inf>(∑) = sup{ρ{variant}(A<inf>1</inf>A<inf>2</inf>...A<inf>k</inf>): each A<inf>i</inf> ∈ ∑}. The joint spectral radius \ ̂g9(∑) is \ ̂g9(∑) = lim sup<inf>k→∞</inf> \ ̂g9<inf>k</inf>(∑)^{1 k}, where \ ̂g9<inf>k</inf>(∑) = sup{∥A<inf>1</inf> ... A<inf>k</inf>∥:each A<inf>i</inf> ∈ ∑}. It is known that \ ̂g9(∑) = \ ̄g9(∑) holds for any finite set ∑ of n × n matrices. The finiteness conjecture asserts that for any finite set ∑ of real n × n matrices there exists a finite k such that \ ̂g9(∑) = \ ̄g9(∑) = \ ̄g9<inf>k</inf>(∑)^{1 k}. The normed finiteness conjecture for a given operator norm asserts that for any finite set ∑ = {A<inf>1</inf>,..., A<inf>m</inf>} having all ∥A<inf>i</inf>∥<inf>op</inf> ≤ 1, either \ ̂g9(∑) < 1 or \ ̂g9(∑) = \ ̄g9(∑) = \ ̄g9<inf>k</inf>(∑)^{1 k} = 1 for some finite k. It is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms. The normed finiteness conjecture is proved for a large class of operator norms, extending results of Gurvits. In particular, for polytope norms and for the Euclidean norm, explicit upper bounds are given for the least k having \ ̄g9(∑) = \ ̄g9<inf>k</inf>(∑)^{1 k}. These results imply upper bounds for generalized critical exponents for these norms. © 1995.