Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Abstract

Given an r × r complex matrix T ,ifT = U|T | is the polar decomposition of T , then, the Aluthge transform is defined by Δ(T)=|T |1/2U|T |1/2. Let Δn(T ) denote the n-times iterated Aluthge transform of T ,i.e.Δ0 (T ) = T and Δn(T ) = Δ(Δn-1(T )), n ∈ N. We prove that the sequence {Δn(T )}n∈N converges for every r × r diagonalizable matrix T .We show that the limit Δ∞(·) is a map of class C ∞ on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r × r matrices with r different eigenvalues.