Journal of the
Korean Mathematical Society JKMS

An operator $T\in {\mathcal L}({\mathcal H})$ is said to be $p$-$paranormal$ if
$$ |||T|^pU|T|^p x|| \,\, ||x|| \geq |||T|^p x||^2 $$ for all $x \in{\mathcal H}$ and $p>0$, where $T=U|T|$ is the polar decomposition of $T$. It is easy that every 1-paranormal operator is paranormal, and every $p$-paranormal operator is paranormal for $0< p <1$. In this note, we discuss some properties for $p$-paranormal operators.

Keywords: paranormal, $p$-paranormal, polar decomposition

MSC numbers: 47B20