Convergence of the iterated Aluthge transform sequence for diagonalizable matrices

Given an r×rr×r complex matrix T  , if T=U|T|T=U|T| is the polar decomposition of T, then, the Aluthge transform is defined byΔ(T)=|T|1/2U|T|1/2.Δ(T)=|T|1/2U|T|1/2. Let Δn(T)Δn(T) denote the n-times iterated Aluthge transform of T  , i.e. Δ0(T)=TΔ0(T)=T and Δn(T)=Δ(Δn−1(T))Δn(T)=Δ(Δn−1(T)), n∈Nn∈N. We prove that the sequence {Δn(T)}n∈N{Δn(T)}n∈N converges for every r×rr×rdiagonalizable matrix T  . We show that the limit Δ∞(⋅)Δ∞(⋅) is a map of class C∞C∞ on the similarity orbit of a diagonalizable matrix, and on the (open and dense) set of r×rr×r matrices with r different eigenvalues.