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 × r complex matrix with polar decomposition T = U|T|. Then the λ-Aluthge transform is defined by

$$\Delta_{\lambda} (T) = |T|^{\lambda}U|T|^{1-\lambda}$$

. Let $\Delta^{n}_{\lambda} (T)$ denote the n-times iterated Aluthge transform of T, $n\, \in \, {\mathbb{N}}$. We prove that the sequence ${\{\Delta^{n}_{\lambda} (T)\}}_{n \in {\mathbb{N}}}$ converges for every r × r diagonalizable matrix T. We show regularity results for the two parameter map $(\lambda, T) \longmapsto \Delta^{\infty}_ {\lambda} (T)$, and we study for which matrices the map $(0, 1) \ni \lambda \longmapsto \Delta^{\infty}_ {\lambda} (T)$ is constant.