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 by Δ ( T ) = | T | 1 / 2 U | T | 1 / 2 . Let Δ n ( T ) denote the n -times iterated Aluthge transform of T , i.e. Δ 0 ( T ) = T and Δ n ( T ) = Δ ( Δ n − 1 ( T ) ) , n ∈ N . We prove that the sequence { Δ n ( 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.