THE FINITENESS CONJECTURE FOR THE GENERALIZED SPECTRAL-RADIUS OF A SET OF MATRICES

The generalized spectral radius ($) over bar rho(Sigma) of a set Sigma of n x n matrices is ($) over bar rho(Sigma) = lim sup(k-->infinity)($) over bar rho(k)(Sigma)(1/k), where ($) over bar rho(k)(Sigma) = sup{rho(A(1)A(2)...A(k)):each A(i) is an element of Sigma}. The joint spectral radius ($) over cap rho(Sigma) is ($) over cap rho(Sigma) = lim sup(k-->infinity)($) over cap rho(k)(Sigma)(1/k), where ($) over cap rho(k)(Sigma) = sup{\\A(1)... A(k)\\:each A(i) is an element of Sigma}. It is known that ($) over cap rho(Sigma) = ($) over bar rho(Sigma) holds for any finite set Sigma of n x n matrices. The finiteness conjecture asserts that for any finite set Sigma of real n x n matrices there exists a finite k such that ($) over cap rho(Sigma) = ($) over bar rho(Sigma) = ($) over bar rho(k)(Sigma)(1/k). The normed finiteness conjecture for a given operator norm asserts that for any finite set Sigma = {A(1),...,A(m)} having all \\A(i)\\(op) less than or equal to 1, either ($) over cap rho(Sigma) < 1 or ($) over cap rho(Sigma) = ($) over bar rho(Sigma) = ($) over bar rho(k)(Sigma)(1/k) = 1 for some finite k. It is shown that the finiteness conjecture is true if and only if the normed finiteness conjecture is true for all operator norms. The normed finiteness conjecture is proved for a large class of operator norms, extending results of Gurvits. In particular, for polytope norms and for the Euclidean norm, explicit upper bounds are given for the least k having ($) over bar rho(Sigma) = ($) over bar rho(k)(Sigma)(1/k). These results imply upper bounds for generalized critical exponents for these norms.