Convergence of the Iterated Aluthge Transform Sequence for Diagonalizable Matrices II: λ-Aluthge Transform

Abstract

Let λ∈ (0,1) and let T be a r x r complex matrix with polar decomposition T=U|T|. Then, the λ- Aluthge transform is defined by Δλ(T)= |T|λU |T |1-λ. Let Δnλ(T) denote the n-times iterated Aluthge transform of T, n ∈ N. We prove that the sequence {Δnλ(T)} n ∈ N converges for every r x r diagonalizable matrix T. We show regularity results for the two parameter map (λ , T) → Δ ∞ λ(T), and we study for which matrices the map (0,1) ∋ λ → Δ∞ λ(T) is constant.