In this paper, we show that if T is a hyponormal operator on a non-separable Hilbert space H, then $Re;{omega}^0_{alpha}(T);{subset};{omega}^0_{alpha}(Re;T)$, where ${omega}^0_{alpha}(T)$ is the weighted Weyl spectrum of weight a with ${alpha};with;{aleph}_0{leq}{alpha}{leq}h:=dim;H$. We also give some conditions under which the product of two ${alpha}-Weyl$ operators is ${alpha}-Weyl$ 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.