Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Abstract

Given an r × r complex matrix T, if T = U | T | is the polar decomposition of T, then, the Aluthge transform is defined byΔ (T) = | T |1 / 2 U | 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∞ (ṡ) 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.