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The normality of \(p\)-\(w\)-hyponormal operators. (Chinese. English summary) Zbl 1299.47040
Summary: Let \(T\in B(H)\). If \(|\widetilde{T}|^p\geqslant |T|^p\geqslant |\widetilde{T}^*|^p\) for some \(p>0\), then \(T\) is \(p\)-\(w\)-hyponormal. The quasinormal and subnormal relations of \(T\) and its Aluthge transform \(\widetilde{T}\) are studied. It is proved that \(\widetilde{T}\) is quasinormal if and only if \(T\) is quasinormal. An example is given to illustrate that there exists a non-subnormal \(p\)-\(w\)-hyponormal operator \(T\) so that \(\widetilde{T}\) is subnormal.
MSC:
47B20 Subnormal operators, hyponormal operators, etc.
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