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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.

MSC:

22E46 Semisimple Lie groups and their representations
47B20 Subnormal operators, hyponormal operators, etc.

Citations:

Zbl 1213.37047

References:

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