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On the polar decompositon of the Aluthge transformation and related results. (English) Zbl 1104.47004

Summary: Let \(T=U| T|\) be the polar decomposition of a bounded linear operator \(T\) on a Hilbert space. The transformation \(\tilde T=| T|^{\frac{1}{2}}U| T|^{\frac{1}{2}}\) is called the Aluthge transformation and its \(n\)-th iterate \(\tilde T_n\) the \(n\)-th Aluthge transformation. In this paper, firstly, we show that \(\tilde T=VU|\tilde T|\) is the polar decomposition of \(\tilde T\), where \(| T|^{\frac{1}{2}}| T^*|^{\frac{1}{2}}=V|| T|^{\frac{1}{2}}| T^*|^{\frac{1}{2}}|\) is the polar decomposition. Secondly, we show that \(\tilde T=U|\tilde T|\) if \(T\) is binormal, i.e., \([| T|,| T^*|]=0\), where \([A,B]=AB-BA\) for any operators \(A\) and \(B\). Lastly, we show that \(\tilde T_n\) is binormal for all non-negative integer \(n\) if and only if \(T\) is centered, i.e., \(T^n(T^n)^*\) and \((T^m)^*T^m\) commute for all \(n, m\in \mathbb N\).

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B20 Subnormal operators, hyponormal operators, etc.
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