Convergence of Aluthge iteration in semisimple Lie groups. (English) Zbl 1305.22012
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. 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.
Reviewer: Mohammad Sal Moslehian (Mashhad)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 47B20 | Subnormal operators, hyponormal operators, etc. |
Citations:
Zbl 1213.37047References:
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