Asymptotic behavior of Gelfand-Naimark decomposition

Abstract

Let X = LσU be the Gelfand-Naimark decomposition of X ∈ GLn(C) , where L is unit lower triangular, σ is a permutation matrix, and U is upper triangular. Call u(X) := diagU the u -component of X . We show that in a Zariski dense open subset of the ω -orbit of certain Bruhat decomposition, lim m→∞ |u(X m)|1/m = diag (|λω(1) |, · · · , |λω(n) |). The other situations where lim m→∞ |u(X m)|1/m converge to different limits or diverge are also discussed. Mathematics subject classification (2000): 15A23, 15A42.