Aluthge Transforms of $(\mathcal{C}_{p},\alpha)$-Hyponormal Operators

Abstract

Recently, the class of $(\mathcal{C}_{p},\alpha)$-hyponormal operators is introduced and the Aluthge transforms of such operators is discussed by some researchers. This paper is to give a further development of the Aluthge transforms of $(\mathcal{C}_{p},\alpha)$-hyponormal operators by using Loewner-Heinz inequality, Furuta inequality and Lauric's lemma. Especially, it is shown that, if $p\ge 1$, $\alpha\ge 1/2$ and $T$ is $(\mathcal{C}_{p},\alpha)$-hyponormal, then the Aluthge transform $T(1/2,1/2)$ is $(\mathcal{C}_{4p\alpha/\beta},\beta)-hyponormal$ where $0 \lt \beta \le 1$ and $T(1/2,1/2)=|T|^{1/2}U|T|^{1/2}$.