On quasi $*$-paranormal operators

Abstract

‎An operator $T\in B(H)$ is called quasi $*$-paranormal if‎ ‎$||T^{*}Tx||^{2}\leq||T^{3}x|||Tx||$ for all $x\in H$‎. ‎If $\mu$ is‎ ‎an isolated point of the spectrum of $T$‎, ‎then the Riesz idempotent‎ ‎$E$ of $T$ with respect to $\mu$ is defined by‎ ‎$$E:= {1\over {2\pi i}}\int_{\partial D}(\mu I‎ - ‎T)^{-1}d\mu,$$‎ ‎where $D$ is a closed disk centered at $\mu$ which contains no other‎ ‎points of the spectrum of $T$‎. ‎Stampfli [Trans‎. ‎Amer‎. ‎Math‎. ‎Soc.‎, ‎117 (1965)‎, ‎469-476]‎, ‎showed that if $T$ satisfies the growth‎ ‎condition $G_{1}$‎, ‎then $E$ is self-adjoint and $E(H)=N(T-\mu)$‎. ‎Recently‎, ‎Uchiyama and Tanahashi [Integral Equations and Operator‎ ‎Theory‎, ‎55 (2006)‎, ‎145-151] obtained Stampfli's result for‎ ‎paranormal operators‎. ‎In general even though $T$ is a paranormal‎ ‎operator‎, ‎the Riesz idempotent $E$ of $T$ with respect to $\mu \in ‎{\rm iso\,}\sigma(T)$ is not necessary self-adjoint‎. ‎In this paper‎ ‎$2\times 2$ matrix representation of a quasi $*$-paranormal operator‎ ‎is given‎. ‎Using this representation we show that if $E$ is the Riesz‎ ‎idempotent for a nonzero isolated point $\lambda_{0}$ of the‎ ‎spectrum of a quasi $*$-paranormal operator $T$‎, ‎then $E$ is‎ ‎self-adjoint if and only if the null space of $T-\lambda_{0}$‎ ‎satisfies $N(T-\lambda_{0})\subseteq‎ ‎N(T^{*}-\overline{\lambda_{0}})$‎. ‎Other related results are also‎ ‎given‎.