Bull. Korean Math. Soc. 2006 Vol. 43, No. 4, 715-722 Published online December 1, 2006

Subhash Chander Arora and Preeti Dharmarha University of Delhi, University of Delhi

Abstract : In this paper, we show that if $T$ is a hyponormal operator on a non-separable Hilbert space $\mathcal H$, then $\Re\omega_\alpha^0(T)\subset \omega_\alpha^0(\Re T)$, where $\omega_\alpha^0(T)$ is the weighted Weyl spectrum of weight $\alpha$ with $\aleph_0 \le \alpha \le \ h:={\rm dim} \ {\mathcal H}$. We also give some conditions under which the product of two \mbox{$\alpha$-Weyl} operators is \mbox{$\alpha$-Weyl} and its converse implication holds, too. Finally, we show that the weighted Weyl spectrum of a hyponormal operator satisfies the spectral mapping theorem for analytic functions under certain conditions.