A note on \(({\mathcal C}_p,\alpha )\)-hyponormal operators
Let \(T\) be a bounded linear operator on a complex Hilbert space \({\mathcal H}\) and \({\mathcal C}_p\) be the Schatten \(p\)-class. For \(\alpha>0\), let \(D_T^\alpha= (T^*T)^\alpha-(TT^*)^\alpha\). An operator \(T\) is said to be \(({\mathcal C}_p, \alpha)\)-hyponormal (or \(T\in {\mathcal H}_p^\alpha({\mathcal H})\)) if and only if \(D_T^\alpha=P+K\), where \(P\geq 0\) and \(K\in{\mathcal C}_p\). The authors obtain inclusion relations of the class of \(({\mathcal C}_p, \alpha)\)-hyponormal operators among some \(\alpha>0\). In fact, they obtain \({\mathcal H}_p^\alpha({\mathcal H})\subseteq {\mathcal H}_{\frac{\alpha p}{\beta}}^\beta({\mathcal H})\) for \(\alpha>0\), \(p\geq 1\) and \(0<\beta\leq \alpha\). This is an extension of the well-known result that if \(T\) is \(\alpha\)-hyponormal, then \(T\) is \(\beta\)-hyponormal for \(0<\beta\leq \alpha\). Moreover, the authors discuss properties of the Aluthge transform and the square of a \(({\mathcal C}_p, \alpha)\)-hyponormal operator. These results are extensions of well-known properties of \(\alpha\)-hyponormal operators.
\((\mathcal{C}_{p},\alpha)\)-hyponormal operators
\((\mathcal{C}_{p},\alpha)\)-normal operators
scientific article; zbMATH DE number 5823182
scientific article