The generalized joint spectral radius. A geometric approach

The properties of the joint spectral radius with an arbitrary exponent p ∈ [1, +∞] are investigated for a set of finite-dimensional linear operators A1, . . . ,Ak, ρ̂p = limn→∞(1/kn∑∥A σ(1)mellipAσ(n)∥p) 1/pn, p < ∞, ρ̂∞ = limn→∞ maxσ∥Aσ(1)⋯A σ(n)∥1/n, where the summation and maximum extend over all maps σ: 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 = ∞. The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius ρ̂p. The problem of calculating ρ̂p 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. ©1997 RAS(DoM) and LMS.