On log-hyponormal operators

Abstract

LetT∈B(H) be a bounded linear operator on a complex Hilbert spaceH.T∈B(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)≤log (T ^* T). Since log: (0, ∞)→(−∞,∞) is operator monotone, for 0<p≤1, every invertiblep-hyponormal operatorT, i.e., (TT ^*)^p≤(T ^* T)^p, is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform$\tilde T = |T|^s U|T|^t$ is$\frac{{\min (s,t)}}{{s + t}}$. Moreover, ifmeas (σ(T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.