Positive operator majorization and \(p\)-hyponormality (Q1423745)

The paper under review concerns \(p\)-hyponormal operators. Let \(\mathcal{H}\) be a separable complex Hilbert space and \(\mathcal{L(H)}\) be the algebra of bounded linear operators acting on \(\mathcal{H}\). An operator \(T\in\mathcal{L(H)}\) is called a \(p\)-hyponormal opeorator if \((T^*T)^p\geq (TT^*)^p\) for some \(p\in (0,\infty)\). It is well-known that if \(0<p\leq q\), then \(q\)-hyponormality implies \(p\)-hyponormality. The authors of this paper give a method for generating examples of \(p\)-hyponormal operators which are not \(q\)-hyponormal for any \(q>p\). The main result of this paper is as follows. There exist two operators \(A\) and \(B\) in \(\mathcal{L(H)}\) such that \(A\geq B\) and \(A^p\ngeq B^p\) for any \(p\in (0,\infty)\), but \(B=WAW^*\) for some unitary operator \(W\). As a corollary, they show that there exists a positive operator \(A\in\mathcal{L(H)}\) such that \(A\geq UAU^*\), but \(A^p\ngeq UA^pU^*\) for any \(p\in (0,\infty)\). They also apply this method to get implications for the class of Furuta type inequalities.