On the Aluthge Transformations of ∞-Hyponormal Operators
Shizuo Miyajima2005
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Abstract
A bounded linear operator T is called ∞-hyponormal if T is p-hyponormal for every p> 0. In this paper ∞-hyponormality of the Aluthge transformations of ∞-hyponormal operators is investigated. It is shown that the Aluthge transfor- mation of an ∞-hyponormal operator is not necessarily ∞-hyponormal. It is also shown that the (generalized) Aluthge transformation of an ∞-hyponormal operator T is ∞-hyponormal provided |T ||T ∗ | = |T ∗ ||T |. Moreover we give an example of an ∞-hyponormal operator T whose Aluthge transformation ˜ T is ∞-hyponormal but |T ||T ∗ | � |T ∗ ||T |.
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