The generalized joint spectral radius. A geometric approach

Abstract

The properties of the joint spectral radius with an arbitrary exponent are investigated for a set of finite-dimensional linear operators where the summation and maximum extend over all maps 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 . The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius . The problem of calculating for even integers is reduced to determining the usual spectral radius for an appropriate finite-dimensional operator. For other values of , a geometric analogue of the method with a pre-assigned accuracy is constructed and its complexity is estimated.