An operator \(T\) on a Hilbert space \({\mathcal H}\) is said to be \(p\)-hyponormal for \(p>0\) if \((T^*T)^p\geq (TT^*)^p\). By using Ando’s result, it is known that every \(p\)-hyponormal operator is paranormal (i.e., \(\|Tx\|^2\leq\|T^2x\|\) \(\|x\|\) for all \(x\in{\mathcal H}\)). Let \(T= U|T|\) be the polar decomposition of \(T\). Then it is also known that every \(p\)-hyponormal operator \(T\) satisfies the condition \[ \bigl||T|^{{1\over 2}} U|T|^{{1\over 2}}\bigr|\geq |T|\geq \bigl||T|^{{1\over 2}} U^*|T|^{{1\over 2}}\bigr|.\tag{\(\sharp\)} \] In this paper, the authors call the operator \(T\) which satisfies the condition \((\sharp)\) is \(w\)-hyponormal and prove that every \(w\)-hyponormal operator is also paranormal.
[For part I see ibid. 36, No. 1, 1-10 (2000; Zbl 0938.47021)].