Putnam’s Inequality for log-Hyponormal Operators

Abstract.

Let T be a bounded linear operator on a complex Hilbert space H. T $/in$ B(H) is called a log-hyponormal operator if T is invertible and log (TT ^*) ≤ log (T ^* T). Since a function log : (0,∞) → (-∞,∞) is operator monotone, every invertible p-hyponormal operator T, i.e., (TT ^*) ^p ≤ (T ^* T ^p is log-hyponormal for 0 < p ≤ 1. Putnam‘s inequality for p-hyponormal operator T is the following:

$ \| (T^*T)^p-(TT^*)^p \|\leq\frac{p}{\pi}\int\int_{\sigma(T)}r^{2p-1}drd\theta $.

In this paper, we prove that if T is log-hyponormal, then

$ \| log(T^*T)-log(TT^*) \|\leq\frac{1}{\pi}\int\int_{\sigma(T)}r^{-1}drd\theta $.