Bishop's property $(\beta)$ and Riesz Idempotent for k-quasi-paranormal operators

Abstract

The study of operators satisfying Bishop's property $(\beta)$ is of significant interest and is currently being done by a number of mathematicians around the world. Recently Uchiyama and Tanahashi [Oper. Matrices 4 (2009), 517--524] showed that a paranormal operator has Bishop's property $(\beta)$. In this paper we introduce a new class of operators which we call the class of $k$-quasi-paranormal operators. An operator $T$ is said to be a $k$-quasi-paranormal operator if it satisfies $||T^{k+1}x||^{2}\leq||T^{k+2}x|||T^{k}x||$ for all $x\in H$ where k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi-class $A$ operators. We prove basic properties and give a structure theorem of $k$-quasi-paranormal operators. We also show that Bishop's property $(\beta)$ holds for this class of operators. Finally, we prove that if $E$ is the Riesz idempotent for a nonzero isolated point $\lambda_{0}$ of the spectrum of a $k$-quasi-paranormal operator $T$, then $E$ is self-adjoint if and only if the null space of $T-\lambda_{0},\, \ker(T-\lambda_{0})\subseteq \ker(T^{*}-\overline{\lambda_{0}})$.