The generalized joint spectral radius. A geometric approach

The properties of the joint spectral radius with an arbitrary exponent p is an element of [1, +infinity] are investigated for a set of finite-dimensional linear operators A(1),...,A(k), where the summation and maximum extend over all maps sigma:{1,...,n} --> {1,...,k}. Using the operation of generalized addition of convex sets, we extend the Dranishnikov-Konyagin theorem on invariant convex bodies, which has hitherto been established only for the case p = infinity. The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius <(rho)over cap>(p). The problem of calculating <(rho)over cap>(p) for even integers p is reduced to determining the usual spectral radius for an appropriate finite-dimensional operator. For other values of p, a geometric analogue of the method with a pre-assigned accuracy is constructed and its complexity is estimated.