Convergence of Aluthge iteration in semisimple Lie groups (Q482857)

Let \(0 < \lambda <1\). The \(\lambda\)-Aluthge transform of a complex matrix \(X\) with the polar decomposition \(X=UP\) is defined to be \(\Delta_\lambda(X):=P^\lambda UP^{1-\lambda}\) and the \(\lambda\)-Aluthge sequence \(\{\Delta_\lambda^m(X)\}_{n=1}^\infty\) is defined by \(\Delta_\lambda^m:=\Delta_\lambda(\Delta_\lambda^{m-1}(X))\) with \(\Delta_\lambda^1(X):=\Delta_\lambda(X)\). It is known that under certain conditions this sequence is convergent. \textit{J. Antezana} et al. [Adv. Math. 226, No. 2, 1591--1620 (2011; Zbl 1213.37047)] proved that the sequence converges to a normal matrix \(\Delta_\lambda^\infty(X)\) and the function \(\Delta_\lambda^\infty:\mathrm{GL}_n(\mathbb{C})\to\mathrm{GL}_n(\mathbb{C})\) defined by \(X\mapsto \Delta_\lambda^\infty(X)\) is continuous. In the paper under review, the authors extend this result in the context of real noncompact connected semisimple Lie groups.