中心算子的诱导 Aluthge 变换迭代的极限
Limit of iteration of the induced Aluthge transformations of centered operators
AI总结：
本文研究中心算子的诱导 Aluthge 变换迭代的收敛性，证明其极限为正规算子，并推广到紧算子 $C^*$-代数及 $II_1$-因子情形。
摘要：
Aluthge 变换是定义在有界线性算子上的一个著名映射。特别是，其迭代的收敛性质已被许多作者研究。本文讨论诱导 Aluthge 变换的问题，该变换是 2021 年定义的 Aluthge 变换的推广。我们给出中心算子的诱导 Aluthge 变换的极分解，并证明其迭代收敛到一个正规算子。特别地，如果 $T$ 是一个可逆中心矩阵，那么任何诱导 Aluthge 变换的迭代都收敛。利用矩阵代数的标准标准形，我们证明关于加权算术平均和幂平均的任何诱导 Aluthge 变换的迭代都收敛。这些观察被推广到无限维 Hilbert 空间上紧算子的 $C^*$-代数，并且作为一个应用，我们证明了在诱导 Aluthge 变换的迭代下 $\mathcal{AN}$ 和 $\mathcal{AM}$ 性质的稳定性。我们还给出中心矩阵的极限点的具体形式以及若干例子。此外，我们讨论了在单射 $II_1$-因子 $\mathcal{M}$ 中关于幂平均的诱导 Aluthge 变换的极限点，并确定了 $\mathcal{M}$ 中某些中心算子的极限形式。
英文摘要：
Aluthge transform is a well-known mapping defined on bounded linear operators. Especially, the convergence property of its iteration has been studied by many authors. In this paper, we discuss the problem for the induced Aluthge transforms which is a generalization of the Aluthge transform defined in 2021. We give the polar decomposition of the induced Aluthge transformations of centered operators and show its iteration converges to a normal operator. In particular, if $T$ is an invertible centered matrix, then iteration of any induced Aluthge transformations converges. Using the canonical standard form of matrix algebras we show that the iteration of any induced Aluthge transformations with respect to the weighted arithmetic mean and the power mean converge. Those observation are extended to the $C^*$-algebra of compact operators on an infinite dimensional Hilbert space, and as an application we show the stability of $\mathcal{AN}$ and $\mathcal{AM}$ properties under the iteration of the induced Aluthge transformations. We also provide concrete forms of their limit points for centered matrices and several examples. Moreover, we discuss the limit point of the induced Aluthge transformation with respect to the power mean in the injective $II_1$-factor $\mathcal{M}$ and determine the form of its limit for some centered operators in $\mathcal{M}$.
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