Let $cal{K}$ be the extension Hilbert space of a Hilbert space $cal{H}$ and let $Phi$ be the faithful $ast$-representation of $cal{B}(cal{H})$ on $cal{k}$. In this paper, we show that if T is an irreducible ${omega}-hyponormal$ operators such that $ker(T);{subset};ker(T^{*})$ and $T^{*}T;-;TT^{ast}$ is compact, then $sigma_{e}(T);=;sigma_{e}(Phi(T))$.