Simple proof of the \(p\)-hyponormality of the Aluthge transformation
Let \({\mathcal H}\) be a complex Hilbert space and \({\mathcal B}({\mathcal H})\) be the set of all bounded linear operators on \({\mathcal H}\). An operator \(T\in{\mathcal B}({\mathcal H})\) is said to be \(p\)-hyponormal if \((T^*T)^p\geq (TT^*)^p\) \((p>0)\). For \(T\in{\mathcal B}({\mathcal H})\) with polar decomposition \(T= U| T|\) and for any \(s\) and \(t\) such as \(s\geq 0\) and \(t\geq 0\), the operator \(T(s,t)=| T|^s U| T|^t\) is called Aluthge transform of \(T\). In this paper, the authors give a simple proof of the special case \((0< p\leq 1)\) of the result given by reviewer about what kind of \(p\)-hyponormality is preserved under the Aluthge transform.
scientific article; zbMATH DE number 1271377