On log-hyponormal operators (Q1301586)

The following inequalities are proved: For a log-hyponormal operator \(T\) (i.e., \(T\) is invertible and \(\log T^*T\geq \log TT^*)\) with its polar decomposition \(T= U|T|\), let \(T(s, t)= |T|^sU|T|^t\) with positive numbers \(0< s,t\). Then \[ \{T(s, t)T(s, t)^*\}^{{\min(s,t)\over s+t}}\leq|T|^{2\min(s,t)}\leq \{T(s, t)^* T(s,t)\}^{{\min(s,t)\over s+t}}. \] These inequalities are just same as the case where \(p= 0\) of the results given by the reviewer for a \(p\)-hyponormal operator \(T\) (i.e., \((T^*T)^p\geq (TT^*)^p\) for some \(p>0\)).