Iterations of the inverse Aluthge transform

Abstract

We prove that for $\lambda \in R$ and $\lambda \ne \frac{1}{2}$, the λ-Aluthge transform is a $C^{\infty }$ diffeomorphism acting on the Lie group of invertible matrices ${\mathrm{GL}}_n$. In particular, this provides a one-parameter family in ${\text{Diff}}^{\infty }\left({\mathrm{GL}}_n\right)$. We also characterize the inverse. This characterization is expressed in terms of twisted polar decompositions defined in Bushell's equations and polar decompositions, Mathematische Nachrichten 282 (2009). This will allow us to study the dynamics of the Aluthge transforms for $\lambda \notin [0,1]$. In this range of values, we prove that the backward iterations of the Aluthge transform converge. This complements the results in The iterated Aluthge transforms of a matrix converge, Advances in Mathematics, 226 (2011), where the proof of the forward convergence was proved for $\lambda \in (0,1)$. Since neither the backward iterations for $\lambda \in (0,1)$ nor the forward iterations for $\lambda \notin (0,1)$ can converge for a non-normal matrix, this completes the study of the dynamics of the one-parameter family of λ-Aluthge transforms in ${\mathrm{GL}}_n$. Some open problems and possible future lines of research are mentioned throughout the paper.