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

The generalized spectral radiusimageg9(∑) of a set ∑ of n × n matrices is image, where image. The joint spectral radiusimageg9(∑) is image, where image. It is known that image 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 image. The normed finiteness conjecture for a given operator norm asserts that for any finite set ∑ = {A1,…, Am} having all short parallelAishort parallelop ≤ 1, either image or image 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 image. These results imply upper bounds for generalized critical exponents for these norms.