Abstract. In this paper, we show that the following spectral mapping theorem holds: Let $T = H + iK$ be hyponormal and $\varphi $ be a strictly monotone increasing continuous function on $\sigma(H)$. We define $\tilde{\varphi }(x+iy)=\varphi(x)+iy$ for $x \in\sigma (H), y \in{\msbm R}$ and $\tilde{\varphi }(T)=\varphi(H)+iK$. Then $$ \sigma_{na}(\tilde{\varphi }(T)) = \tilde{\varphi }(\sigma_{na}(T)), \sigma_{a}(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma_{a}(T)) \mbox{ and } \sigma(\tilde{\varphi }(T)) = \tilde{\varphi } (\sigma(T)). $$ We also show that Weyl's theorem holds for $\tilde{\varphi }(T)$ and study the G$_1$ property of the operator $\tilde{\varphi }(T)$.

AMS Subject Classification (1991): 47B20

Received March 11, 2002. (Registered under 2931/2009.)