The properties of the joint spectral radius with an arbitrary exponent p â [1, +â] are investigated for a set of finite-dimensional linear operators A1, . . . ,Ak, ÏÌp = limnââ(1/knââ¥A Ï(1)mellipAÏ(n)â¥p) 1/pn, p < â, ÏÌâ = limnââ maxÏâ¥AÏ(1)â¯A Ï(n)â¥1/n, where the summation and maximum extend over all maps Ï: 1, . . . ,n â 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 = â. The paper concludes with some assertions on the properties of invariant bodies and their relationship to the spectral radius ÏÌp. The problem of calculating ÏÌp 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. Â©1997 RAS(DoM) and LMS.
|Titolo:||The generalized joint spectral radius. A geometric approach|
|Data di pubblicazione:||1997|
|Appare nelle tipologie:||1.1 Articolo in rivista|