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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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